Coulomb Active Earth Pressure (Wall Friction and Batter)
The active thrust Rankine leaves on the table: Coulomb 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 and source
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]; 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.
Coulomb (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.
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