Rankine Active Earth Pressure on a Cohesive (Clay) Backfill

How cohesion cuts active earth pressure: it subtracts a constant 2 c sqrt(Ka) 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.

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.

NAVFAC DM-7.02 is a free public US Navy design manual; the Rankine c-phi active pressure, tension-crack depth, and critical-height results are public and printed in every soil-mechanics text.

Estimate. AHJ and licensed professional govern.

Field names used by the API: phi, c_psf, gamma, h_ft, ka, sigma_top, sigma_base, z_c_ft, pa_cracked, pa_uncracked, pw_crack, h_crit_ft

• Tension is not credited the negative upper pressure is discarded and the wall is loaded only below the crack; the cracked thrust exceeds the uncracked trapezoid, so using the textbook uncracked value understates the loadRankine / Das / NAVFAC DM-7.02
• Water-filled crack usually governs a tension crack open to the surface fills; full hydrostatic 0.5 gamma_w zc^2 is added over the crack depth and typically exceeds the relief the crack gaveDas / NAVFAC DM-7.02
• Hc is theory, not a safety allowance the critical unsupported height is the depth at which the uncracked active thrust is zero; OSHA 29 CFR 1926 Subpart P governs any unbraced excavation faceOSHA 29 CFR 1926 Subpart P
• phi = 0 allowed the undrained total-stress clay case is in scope (Ka = 1, zc = 2c/gamma, Hc = 4c/gamma), unlike the cohesionless siblings that require phi > 0scope of this tile