Coulomb Active Earth Pressure (Wall Friction and Batter)
The active thrust Rankine leaves on the table.
Example
You enter
- Soil friction angle phi (deg) 30
- Wall friction delta (deg, ~2/3 phi, 0 smooth) 20
- Soil unit weight (pcf) 120
- Retained height H (ft) 10
You get
- Active earth-pressure coefficient Ka 0.2973
- Active thrust Pa 1,784 lb/ft
- Horizontal component Pa_h (overturning) 1,676 lb/ft
- Vertical component Pa_v (down on the face) 610 lb/ft
Checked against published worked examples from: PDHonline (Weber).
Details, formula, and sources
ScopeCoulomb credits wall friction delta, a battered face theta, and a sloped backfill alpha, Ka = cos^2(phi - theta) / [cos^2(theta) cos(delta + theta) (1 + sqrt(sin(phi + delta) sin(phi - alpha) / (cos(delta + theta) cos(theta - alpha))))^2]. A rough vertical wall on a phi = 30 level backfill carries only 1,676 lb/ft of horizontal thrust once two-thirds-phi wall friction is credited, versus 2,000 lb/ft by Rankine -- a 16% cut in the overturning force plus a downward drag that resists sliding. Reduces exactly to Rankine when the wall is smooth, vertical, and the fill level. Cohesionless, active limit. A design aid, not a substitute for a geotechnical engineer's report.
Formula or tableKa = cos^2(phi - theta) / [cos^2(theta) cos(delta + theta) (1 + sqrt(sin(phi + delta) sin(phi - alpha) / (cos(delta + theta) cos(theta - alpha))))^2]; Pa = 0.5 Ka gamma H^2 at (delta + theta) from horizontal; Pa_h = Pa cos(delta + theta), Pa_v = Pa sin(delta + theta); reduces to Rankine (1 - sin phi)/(1 + sin phi) at delta = theta = alpha = 0.
Source and editionCoulomb (1776) active earth pressure with wall friction, wall batter, and sloped backfill, as compiled in Das, Principles of Foundation Engineering, and NAVFAC DM-7.02 (Foundations and Earth Structures), by name.
Free accessNAVFAC DM-7.02 is a free public US Navy design manual; the Coulomb active coefficient is public and printed in every soil-mechanics text.
What governsEstimate. AHJ and licensed professional govern.
API field namesphi, delta, gamma, h_ft, ka, pa, pa_h, pa_v
- Wall friction lowers the active thrust a rough wall (delta > 0) carries part of the backfill in shear, cutting the horizontal thrust below Rankine and adding a downward face drag that resists slidingCoulomb 1776 / Das / NAVFAC DM-7.02
- Angle ranges keep it well-posed delta in [0, phi], theta in [0, 40), alpha in [0, phi) keep the Coulomb radical non-negative and the denominator positive; the active case uses a planar failure surfaceDas / NAVFAC DM-7.02