Sloped-Backfill Earth Pressure (Rankine Inclined Surface)

The Rankine inclined-surface coefficient Ka = cos b (cos b - sqrt(cos^2 b - cos^2 phi)) / (cos b + sqrt(cos^2 b - cos^2 phi)), the thrust Pa = 0.5 x Ka x gamma x H^2 acting parallel to the slope, and its horizontal (overturning) and vertical (heel) components. A 15 deg backfill on a phi = 30 sand raises Ka from 0.333 to 0.373 -- a 12% heavier thrust that also tilts and adds an uplift the level analysis never sees; beta must stay below phi or the slope cannot be retained on Rankine terms. Cohesionless, vertical wall face. A design aid, not a substitute for a geotechnical engineer's report.

Ka = cos b (cos b - sqrt(cos^2 b - cos^2 phi)) / (cos b + sqrt(cos^2 b - cos^2 phi)); Pa = 0.5 Ka gamma H^2 parallel to the slope at H/3; Pa_h = Pa cos b, Pa_v = Pa sin b; Ka0 = (1 - sin phi)/(1 + sin phi) for contrast.

Rankine sloped-backfill active pressure, as compiled in Das, Principles of Foundation Engineering, and NAVFAC DM-7.02 (Foundations and Earth Structures), by name.

NAVFAC DM-7.02 is a free public US Navy design manual; the Rankine inclined-surface coefficient is public and printed in every soil-mechanics text.

Estimate. AHJ and licensed professional govern.

Field names used by the API: phi, beta, gamma, h_ft, ka, pa, pa_h, pa_v

• Resultant parallel to the slope the sloped-backfill Rankine thrust acts at the backfill inclination beta, not horizontally; only the Pa cos beta component overturns the wall while Pa sin beta bears on the heelRankine / Das / NAVFAC DM-7.02
• beta below phi a backfill steeper than the soil's own friction angle is at or past its angle of repose; the Rankine active state breaks down and a gravity/cantilever wall cannot retain it on these termsDas / NAVFAC DM-7.02