Sloped-Backfill Earth Pressure (Rankine Inclined Surface)
The push of a backfill that rises behind the wall at a slope beta, which the level Ka under-predicts: 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.
Formula and source
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.
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