Operations, Logistics & Engineering
Mononobe-Okabe Seismic Active Earth Pressure Coefficient (K_AE) Formula: Retaining Wall Thrust
Calculate earthquake pseudo-static retaining-wall thrust P_AE = 0.5 * gamma * H^2 * (1 - k_v) * K_AE and dynamic increment Delta_K_AE = K_AE - K_A using N. Mononobe & S. Okabe’s (1926/1929) AASHTO/Eurocode 8 wedge formula.
Born from the 1923 Great Kanto Earthquake: how horizontal acceleration k_h tilts Coulomb’s soil wedge!
Following widespread retaining-wall collapses in the 1923 Great Kanto Earthquake, Japanese civil engineers Nagaho Mononobe and Saburo Okabe (1926/1929) extended Coulomb’s active soil wedge by rotating gravity through the seismic inertia angle theta = DEGREES(ATAN(k_h / (1 - k_v))), where k_h and k_v are horizontal and vertical peak ground acceleration coefficients! Today, Mononobe-Okabe is the AASHTO LRFD Bridge Design and Eurocode 8 standard for seismic retaining walls and bridge abutments.
For a vertical wall backface (beta = 0°) and level dry granular backfill (i = 0°) with soil friction angle phi (deg), wall-soil interface friction delta (deg), and seismic inertia angle theta < phi, the Seismic Active Earth Pressure Coefficient is K_AE = (COS(phi - theta))^2 / (COS(theta) * COS(delta + theta) * (1 + SQRT(SIN(phi + delta) * SIN(phi - theta) / COS(delta + theta)))^2), and total seismic thrust per unit wall length is P_AE = 0.5 * gamma * H^2 * (1 - k_v) * K_AE! Notice that when k_h = 0 and k_v = 0 (theta = 0°), K_AE collapses identically to static Coulomb K_A!
| Seismic Acceleration Regime (phi=34°, delta=17°, gamma=18.5 kN/m^3, H=6 m) | Seismic Angle theta (deg) | K_AE & Total Wall Thrust P_AE (kN/m) |
|---|---|---|
| Static Design (k_h = 0.00, k_v = 0.00) | 0.00° | K_A = 0.2564 | P_A = 85.39 kN/m |
| Moderate Seismic Zone (k_h = 0.20, k_v = 0.00) | 11.31° | K_AE = 0.4021 (+56.8%!) | P_AE = 133.90 kN/m |
| Extreme Shaking (theta >= phi -> Fluidized Slope!) | theta >= 34° | Wedge Instability Guard Triggered! |
Mononobe-Okabe Seismic Active Coefficient K_AE (Vertical Wall, Level Granular Backfill)
Computes seismic inertia angle D2 = DEGREES(ATAN(k_h / (1 - k_v))) and evaluates K_AE when theta < phi.
=IF(OR(B2 <= 0, B2 >= 50, B3 < 0, B4 < 0, B5 >= 1, D2 >= B2), "Invalid Seismic Wedge (Requires theta < phi & kv < 1)", ROUND((COS(RADIANS(B2 - D2)) ^ 2) / (COS(RADIANS(D2)) * COS(RADIANS(B3 + D2)) * ((1 + SQRT(SIN(RADIANS(B2 + B3)) * SIN(RADIANS(B2 - D2)) / COS(RADIANS(B3 + D2)))) ^ 2)), 4))Total Seismic Active Thrust P_AE = 0.5 * gamma * H^2 * (1 - k_v) * K_AE (kN/m)
Computes total static + dynamic earth thrust per meter of retaining wall.
=IF(OR(B2 <= 0, B2 >= 50, B3 < 0, B4 < 0, B5 >= 1, B6 <= 0, B7 <= 0, D2 >= B2), "Invalid Seismic Wedge Inputs", ROUND(0.5 * B6 * (B7 ^ 2) * (1 - B5) * ((COS(RADIANS(B2 - D2)) ^ 2) / (COS(RADIANS(D2)) * COS(RADIANS(B3 + D2)) * ((1 + SQRT(SIN(RADIANS(B2 + B3)) * SIN(RADIANS(B2 - D2)) / COS(RADIANS(B3 + D2)))) ^ 2))), 2))Try the example
Use the original values or change them to test your understanding.