About Mononobe-Okabe Seismic Earth Pressure Calculator
This Mononobe-Okabe calculator returns the pseudo-static seismic active earth pressure on a gravity or cantilever retaining wall. It extends Coulomb wedge theory by adding horizontal and vertical seismic inertia forces to the failure wedge, giving the seismic active coefficient KAE and the total seismic thrust PAE.
Enter the soil friction angle, the wall friction angle, the wall batter and backfill slope, the horizontal and vertical seismic coefficients, the backfill unit weight, and the wall height. The tool reports the seismic inertia angle, KAE and the static coefficient KA, the seismic and static thrusts, the dynamic increment from the Seed-Whitman method, and the height at which the resultant acts. When the seismic coefficients and all geometry angles are zero, KAE reduces exactly to the Rankine active coefficient Ka = tan^2(45 - phi/2).
How It Works
- Compute the seismic inertia angle theta = arctan(kh / (1 - kv)) from the horizontal and vertical seismic coefficients.
- Evaluate the Mononobe-Okabe seismic active coefficient KAE; a negative term under the square root means the active wedge cannot form (an unstable combination) and is flagged as an error rather than returning a number.
- Evaluate the static Coulomb coefficient KA (the same expression with no seismic action) and the corresponding thrusts PAE and PA.
- Report the dynamic increment dPAE = PAE - PA and locate the resultant: the static part acts at H/3 and the dynamic increment at about 0.6 H above the base (Seed-Whitman).
Worked Example
A vertical wall (beta = 0) retains a level cohesionless backfill (i = 0) with phi = 35 degrees, wall friction delta = 17.5 degrees, unit weight gamma = 18 kN/m^3 and height H = 6 m. For horizontal seismic coefficient kh = 0.2 and vertical kv = 0, the inertia angle is theta = arctan(0.2/1) = 11.31 degrees. The Mononobe-Okabe coefficient is KAE = 0.380 and the static Coulomb coefficient is KA = 0.246. The total seismic thrust is PAE = 0.5 × 18 × 6^2 × (1 - 0) × 0.380 = 123.0 kN/m and the static thrust is PA = 0.5 × 18 × 6^2 × 0.246 = 79.7 kN/m, giving a dynamic increment of 43.3 kN/m. With the static part at H/3 = 2.0 m and the increment at 0.6 H = 3.6 m, the resultant acts 2.56 m above the base. As a separate check of the static limit, a vertical wall with phi = 30 degrees and kh = kv = delta = beta = i = 0 returns KAE = KA = 0.333, exactly the Rankine value tan^2(45 - 30/2) = tan^2(30) = 0.333.
Formulas
- Seismic inertia angle
theta = arctan( kh / (1 - kv) )- Mononobe-Okabe seismic active coefficient
KAE = cos^2(phi - theta - beta) / [ cos(theta) cos^2(beta) cos(delta + beta + theta) (1 + sqrt( sin(phi+delta) sin(phi-theta-i) / (cos(delta+beta+theta) cos(i-beta)) ))^2 ]- Seismic and static thrust
PAE = 0.5 gamma H^2 (1 - kv) KAE; PA = 0.5 gamma H^2 KA; dPAE = PAE - PA- Point of application (Seed-Whitman)
h = [ PA (H/3) + dPAE (0.6 H) ] / PAE
Standards & References
- Mononobe & Matsuo (1929); Okabe (1926)
- Seed & Whitman (1970), Design of Earth Retaining Structures for Dynamic Loads
- Kramer, Geotechnical Earthquake Engineering
- Eurocode 8 Part 5 / AASHTO seismic design of retaining walls
Frequently Asked Questions
Does the Mononobe-Okabe coefficient reduce to the static case?
Yes. With horizontal and vertical seismic coefficients both zero the inertia angle theta becomes zero and the formula collapses to the static Coulomb active coefficient. If in addition the wall friction, wall batter, and backfill slope are all zero, KAE equals the Rankine active coefficient Ka = tan^2(45 - phi/2). This calculator is verified against that limit.
Why does the calculator sometimes report that no solution exists?
The Mononobe-Okabe expression contains a square root of sin(phi - theta - i) divided by cosine terms. When the seismic inertia angle plus the backfill slope approaches the friction angle (phi - theta - i becomes negative), the active wedge can no longer form and the soil mass is unstable. The tool flags this as an error instead of returning a meaningless number.
Where does the total seismic thrust act on the wall?
Following Seed and Whitman, the static component of the thrust is taken to act at one third of the wall height, while the dynamic increment acts higher, at about 0.6 of the height, because the inertia force is distributed more towards the top of the backfill. The reported point of application is the thrust-weighted average of these two.
How do I choose the seismic coefficients kh and kv?
The horizontal coefficient kh is commonly taken as a fraction of the peak ground acceleration (often kh = 0.5 to 1.0 times amax/g depending on allowable wall displacement), and the vertical coefficient kv is frequently taken as zero or as one half to two thirds of kh. Follow the governing seismic code (for example Eurocode 8 or AASHTO) for the appropriate values.