Lateral Earth Pressure and Thrust (Rankine)

The push of retained soil: the Rankine active coefficient Ka = tan^2(45 - phi/2) and passive Kp = 1/Ka, the triangular active thrust Pa = 0.5 x Ka x gamma x H^2 at H/3, the rectangular surcharge term Ka x q x H at H/2, the combined resultant and its height, and the passive resistance. A 10 ft wall of phi = 30 sand carries exactly 2,000 lb/ft active against 18,000 lb/ft passive -- the 9:1 ratio that is why passive is never counted on lightly; a 250 psf parking surcharge grows the thrust 42% and lifts its arm. Cohesionless, vertical, frictionless, dry Rankine case only. A design aid, not a substitute for a geotechnical engineer's report.

Ka = tan^2(45 - phi/2) = (1 - sin phi) / (1 + sin phi); Kp = 1/Ka; Pa = 0.5 Ka gamma H^2 at H/3, plus surcharge Ka q H at H/2; Pp = 0.5 Kp gamma H^2.

Rankine (1857) lateral earth 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 coefficient arithmetic is public and printed in every soil-mechanics text.

Estimate. AHJ and licensed professional govern.

Field names used by the API: phi, gamma, h_ft, ka, kp, pa_base, pa_tot, y_bar, pp

• Rankine conditions cohesionless backfill, vertical frictionless wall face, level dry backfill; Coulomb / cohesive / submerged / sloped cases need their own analysisRankine 1857 / Das / NAVFAC DM-7.02
• Limit states mobilized active and passive values require enough wall movement to mobilize them; a braced non-yielding wall sees the higher at-rest K0 = 1 - sin phiDas / NAVFAC DM-7.02
• Passive used cautiously the passive thrust is an upper bound needing large movement to develop; designs commonly reduce or neglect itNAVFAC DM-7.02