Mononobe-Okabe Seismic Active Earth Pressure
The seismic increment retaining-wall-stability names as its own gap. Mononobe-Okabe is Coulomb's wedge with the earthquake body force folded in as an inertia angle psi = arctan(kh / (1 - kv)), so Kae = cos^2(phi - theta - psi) / [cos(psi) cos^2(theta) cos(delta + theta + psi) (1 + sqrt(sin(phi + delta) sin(phi - psi - alpha) / (cos(delta + theta + psi) cos(theta - alpha))))^2] and Pae = 0.5 Kae gamma H^2 (1 - kv). A 12 ft wall on a phi = 35 level backfill with two-thirds-phi wall friction goes from Ka 0.246 and 2,127 lb/ft static to Kae 0.341 and 2,942 lb/ft at kh = 0.15 -- an 816 lb/ft dynamic increment, against the 972 lb/ft Seed-Whitman envelope, acting higher on the wall (4.89 ft up, not H/3). Reduces exactly to Coulomb at kh = kv = 0, and reports when the seismic coefficient is too large for the friction angle to hold an active wedge at all. Cohesionless, dry, pseudo-static. A design aid, not a substitute for a geotechnical engineer's report.
Formula and source
psi = arctan(kh / (1 - kv)); Kae = cos^2(phi - theta - psi) / [cos(psi) cos^2(theta) cos(delta + theta + psi) (1 + sqrt(sin(phi + delta) sin(phi - psi - alpha) / (cos(delta + theta + psi) cos(theta - alpha))))^2]; Pae = 0.5 Kae gamma H^2 (1 - kv); dPae = Pae - Pa(Coulomb); Seed-Whitman dP = 0.375 kh gamma H^2 at 0.6H.
Mononobe-Okabe pseudo-static active earth pressure (Okabe 1926; Mononobe and Matsuo 1929) with the Seed and Whitman (1970) simplified increment, as compiled in NAVFAC DM-7.02, FHWA earth-retaining-structures guidance, and Kramer, Geotechnical Earthquake Engineering, by name.
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