Rankine Passive Earth Pressure Coefficient

Also known as Kp · passive earth pressure

K_p = \tan^{2}\!\left(45^\circ + \frac{\phi}{2}\right)

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

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Rankine Passive Earth Pressure Coefficient explained

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°, $K_p$ = tan²(60°) = 3.0, exactly nine times the active coefficient. That factor of $K_p/K_a = 9$ 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 $K_p$ , 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

K_p = \tan^{2}\!\left(45^\circ + \frac{\phi}{2}\right)

Where

$K_p$ = Passive earth pressure coefficient
$\phi$ = Angle of internal friction (°)

Missing one of these? Work it out first, then come back

Passive earth pressure coefficient — Rankine Active Earth Pressure Coefficient, At-Rest Earth Pressure Coefficient (Jaky)
Angle of internal friction — Rankine Active Earth Pressure Coefficient, At-Rest Earth Pressure Coefficient (Jaky)

Part of this formula set

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.