Mononobe-Okabe Seismic Active Earth Pressure

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.

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.

NAVFAC DM-7.02 and the FHWA earth-retaining-structures manuals are free public US government design documents; the Mononobe-Okabe coefficient is public and printed in every geotechnical-earthquake text.

Estimate. AHJ and licensed professional govern.

Field names used by the API: phi, delta, gamma, h_ft, kh, kae, ka_static, psi_deg, pae, pa_static, d_pae, sw_increment

• Reduces to Coulomb at kh = kv = 0 the inertia angle psi is 0 and Kae equals the static Coulomb Ka exactly; the tile reports both and the bounds fuzzer pins the identityMononobe-Okabe / Coulomb 1776
• Seed-Whitman is an envelope the simplified 0.375 kh gamma H^2 increment at 0.6H is conservative by design and normally exceeds the exact M-O increment; it is shown for comparison, not substitutedSeed and Whitman 1970
• Active wedge must exist phi - psi - alpha < 0 means the seismic coefficient exceeds what the friction angle and backfill slope can hold in an active state; no finite thrust exists and the tile errorsstandard Mononobe-Okabe limitation
• Dry cohesionless backfill, pseudo-static no water, no cohesion, one equivalent inertia rather than a time history; a submerged backfill needs buoyant unit weight plus a hydrodynamic termscope of this tile