Rankine Active Earth Pressure on a Cohesive (Clay) Backfill
The reduction the lateral-earth-pressure tile leaves out: cohesion subtracts a constant 2 c sqrt(Ka) from the active pressure at every depth, so sigma_a(z) = Ka (gamma z + q) - 2 c sqrt(Ka). The top of a clay backfill goes into tension and cracks to depth zc = 2c / (gamma sqrt(Ka)), the tension is not credited (soil does not pull), and the design thrust acts on the cracked wall only. A 20 ft wall on a phi = 20, c = 300 psf, 115 pcf clay cracks 7.45 ft down and carries 4,439 lb/ft -- but fill that crack with water and it jumps to 6,172 lb/ft, which is why clay backfills get drained rather than trusted. Also reports the passive case (cohesion adds there) and the critical unsupported height Hc = 4c / (gamma sqrt(Ka)) = 2 zc. phi = 0 is the undrained clay case. Hc is theory, never an excavation-safety allowance -- OSHA Subpart P governs any unbraced face, and a licensed geotechnical engineer's report governs the design.
Formula and source
Ka = (1 - sin phi)/(1 + sin phi); sigma_a(z) = Ka (gamma z + q) - 2 c sqrt(Ka); zc = (2 c sqrt(Ka) - Ka q)/(Ka gamma); Pa_cracked = 0.5 sigma_a(H) (H - zc) at (H - zc)/3; Pw = 0.5 gamma_w zc^2; sigma_p(z) = Kp gamma z + 2 c sqrt(Kp), Pp = 0.5 Kp gamma H^2 + 2 c sqrt(Kp) H; Hc = 4 c / (gamma sqrt(Ka)) = 2 zc.
Rankine active and passive earth pressure on a c-phi soil, with the tension-crack depth and critical unsupported height, as compiled in Das, Principles of Foundation Engineering, and NAVFAC DM-7.02 (Foundations and Earth Structures), by name.
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