Seismic earth pressure is the change in lateral pressure on a retaining structure caused by ground acceleration during an earthquake. Inertia acting on the soil mass behind the wall increases the force driving it toward the excavation, while inertia acting on the soil in front reduces the resistance available to hold it back.
The effect is not an additional load applied to a wall that otherwise behaves as it did before.
An earthquake does not add a force to the wall. It changes the state the soil is in.
Two approaches are in general use. The Mononobe–Okabe method modifies the earth pressure coefficients themselves, so that the analysis proceeds with altered active and passive limits. The equivalent static approach calculates a single inertial force from the weight of a soil wedge and applies it to the wall as a static action. They produce different pressure distributions and are not interchangeable.
How Seismic Acceleration Enters the Calculation
Ground acceleration is represented by dimensionless seismic coefficients, defined as fractions of gravitational acceleration:
kh = ah / g and kv = av / g
The horizontal coefficient kh is always taken as positive, and its effect is always adverse: horizontal inertia pushes the soil wedge toward the excavation regardless of the direction of shaking, because the analysis considers the critical direction.
The vertical coefficient kv can be positive or negative. When the equivalent acceleration acts downward, the inertia force on the wedge acts upward, reducing its effective weight. When it acts upward, the inertia force presses the wedge down. Both cases must be considered, because which one is critical depends on the geometry and the relative magnitudes of the coefficients.
In excavation support design the vertical component is frequently neglected by taking kv = 0. This is a defensible simplification for temporary works, where the duration of exposure is limited and the vertical effect is small relative to the uncertainty in the horizontal coefficient.
The Seismic Inertia Angle
Both coefficients combine into a single angle that represents the resultant of gravity and inertia acting on the soil wedge:
ψ = arctan ( kh / ( 1 ± kv ) )
This angle is the mechanism by which seismic loading enters the earth pressure equations. Geometrically, it is equivalent to rotating the whole problem — wall, ground surface, and gravity vector — by ψ. A slope that was stable at its static angle becomes, in the rotated frame, steeper by ψ.
That equivalence explains why seismic loading is so much more onerous for slopes and for walls retaining sloping ground than for walls retaining level ground.
Determining the Coefficients
The horizontal coefficient is derived from the design spectral acceleration for the site, reduced by a factor that reflects the type of support system and the displacement it can tolerate.
The reduction is not arbitrary. A structure that can accommodate small permanent displacement without loss of function does not need to be designed for the peak acceleration, because the peak acts for a fraction of a second and produces a limited displacement rather than a collapse. A rigid structure that must not move at all is designed for a higher coefficient than a flexible embedded wall that can accept some permanent movement.
Turkish practice under TBDY derives kh from the short-period design spectral acceleration coefficient SDS with a reduction factor selected according to the support type. Eurocode 8 Part 5 follows a comparable logic with different notation.
The Mononobe–Okabe Method
Mononobe–Okabe extends Coulomb’s wedge analysis by adding the inertia forces to the equilibrium of the failure wedge. The result is a modified pair of coefficients, Kae for the active case and Kpe for the passive case, that replace the static Ka and Kp.
The expressions include the same four angles as the static Coulomb solution — the effective friction angle φ′, the wall back inclination α, the ground slope β, and the wall friction angle δ — with the seismic inertia angle ψ added.
Kae = cos²( φ′ − ψ − α ) / { cos ψ · cos²α · cos( δ + α + ψ ) · ( 1 + √( sin( φ′ + δ ) · sin( φ′ − ψ − β ) / ( cos( δ + α + ψ ) · cos( α − β ) ) ) )² }
The passive coefficient follows the same construction with the signs of δ and ψ reversed in the appropriate terms.
Setting ψ = 0 recovers the static Coulomb coefficients exactly. Mononobe–Okabe contains the static solution as a special case, which is what makes it usable within an analysis that also handles static stages.
The Constraint That Limits the Method
The term under the square root in the active expression contains sin(φ′ − ψ − β). When ψ + β approaches φ′, that term approaches zero and the coefficient increases without limit.
The physical meaning is clear enough. When the seismic inertia angle plus the ground slope reaches the friction angle of the soil, the slope behind the wall is at failure under its own weight in the rotated frame, and no finite wall force can retain it. The mathematics does not break down; the situation does.
Practical implementations enforce the constraint:
ψ ≤ φ′ − β for the active case, and ψ ≤ φ′ + β for the passive case
Where the calculated ψ exceeds the limit, it is capped at that value. Reaching the cap is a signal, not a formality: it indicates that the retained slope is marginal under the design earthquake and that the design requires reconsideration rather than a larger wall.
Where the Resultant Acts
Under static conditions, the active thrust from a triangular pressure distribution acts at one third of the wall height above the base. Under seismic conditions the additional dynamic component acts higher, because the inertia force is distributed over the wedge rather than concentrated near its base.
The consequence is that the seismic increment produces disproportionately more overturning moment than its magnitude alone suggests, and a wall checked only for total force can be under-designed for moment.
The Equivalent Static Method
The second approach treats the seismic action as a single force rather than as a modification of the pressure coefficients.
The weight of the soil wedge within an assumed failure surface behind the wall is calculated, multiplied by the horizontal seismic coefficient, and applied to the wall as a static horizontal force. The point of application is commonly taken at about two thirds of the wall height above the base, reflecting the higher line of action of the dynamic component discussed above. The force is then distributed among the nodes of the wall model.
Static active and passive pressures, determined by Rankine or Coulomb theory, remain in place, and the dependent pressures framework continues to apply. The seismic action is superimposed on that system rather than replacing its limits.
Which Method Should Be Used?
| Mononobe–Okabe | Equivalent static | |
|---|---|---|
| Mechanism | Modifies Ka and Kp through ψ | Applies a separate horizontal force |
| Pressure distribution | Altered throughout the wall | Static distribution plus a point force |
| Passive resistance | Explicitly reduced | Unchanged unless separately factored |
| Behaviour at high ψ | Constrained by ψ ≤ φ′ − β | No equivalent constraint |
| Transparency | Coefficients change; effect is distributed | Force is explicit and easy to check |
| Suits | Walls where the pressure state itself changes | Cases where a resultant force is the required output |
Mononobe–Okabe represents the physics more completely, particularly on the passive side. Reducing passive resistance during shaking is a real effect, and a method that leaves passive resistance at its static value while increasing active thrust is optimistic in a way that is not always recognised.
The equivalent static method is more transparent and easier to verify by hand, which has value where an independent check is required or where the seismic action is small relative to the static case.
Both remain pseudo-static: they replace a dynamic problem with an equivalent static one and say nothing about how long the acceleration acts, how many cycles occur, or how much permanent displacement accumulates. For most retaining structures that abstraction is accepted, because design is based on limiting permanent displacement rather than on preventing any movement at all.
What Seismic Analysis of a Wall Does Not Cover
Two mechanisms are frequently more damaging than the increase in earth pressure, and neither is addressed by the methods above.
Liquefaction eliminates the effective stress on which the earth pressure coefficients depend. A wall retaining liquefied soil is subject to something closer to fluid pressure, at a magnitude no earth pressure coefficient produces, and passive resistance in front of it may disappear entirely. Where the ground is susceptible, liquefaction assessment governs the design and the earth pressure calculation becomes secondary.
Overall slope instability is a soil mass failure that the wall analysis does not represent. It is checked by limit equilibrium with the seismic coefficient applied to the slices, and the slope stability result can govern where the wall section checks all pass.
Common Errors
| Error | Consequence |
|---|---|
| Increasing active pressure without reducing passive resistance | Wall stability overestimated on the resisting side |
| Applying the seismic increment at one third of the wall height | Overturning moment underestimated |
| Ignoring the ψ ≤ φ′ − β constraint | Unbounded coefficients, or a marginal slope treated as a wall problem |
| Combining full seismic action with full surcharge and full water pressure | Load combination that no code requires and no event produces |
| Using peak ground acceleration without the code reduction factor | Excessive design action for a structure that tolerates small displacement |
| Ignoring liquefaction because the wall checks pass | The governing mechanism not examined |
| Using kv = 0 without confirming it is the less critical case | The more onerous vertical case not identified |
Where SETAF2018 Fits
SETAF2018 offers both approaches within the staged excavation analysis, applied to whichever construction stages the seismic case is defined for.
Seismic parameters are entered as the short-period design spectral acceleration coefficient SDS with a reduction factor selected by support type, from which the horizontal coefficient kh is derived, with kv entered separately and available to be set to zero. The seismic inertia angle is calculated from both coefficients, and the ψ ≤ φ′ − β constraint is enforced automatically with the capped value applied where the limit is reached.
Under the Mononobe–Okabe option, Kae and Kpe replace the static coefficients, and the analysis proceeds through the same dependent pressures iteration described in the guide to beam-on-spring analysis, with the seismic coefficients defining the new limits. Under the equivalent static option, the wedge force is calculated, applied at the appropriate height, and distributed among the wall nodes, with static earth pressure coefficients retained as the bounds.
Because the seismic case is analysed as a construction stage rather than as a separate model, the anchor and strut forces, wall section design, and displacement checks are all produced for the seismic condition alongside the static ones. The wall section can be transferred into the slope stability module for the overall stability check under the same seismic coefficient, and liquefaction potential is evaluated separately from SPT data using the code methodology. Equations and code references appear in the calculation reports.
Frequently Asked Questions
What is seismic earth pressure?
It is the change in lateral pressure on a retaining structure caused by ground acceleration during an earthquake. Inertia on the retained soil increases active thrust, while inertia on the soil in front of the wall reduces the passive resistance available.
What is the Mononobe–Okabe method?
It is an extension of Coulomb’s wedge analysis that includes seismic inertia forces in the equilibrium of the failure wedge, producing modified active and passive coefficients Kae and Kpe. Setting the seismic inertia angle to zero recovers the static Coulomb coefficients.
What are kh and kv?
They are the horizontal and vertical seismic coefficients, expressed as fractions of gravitational acceleration. kh is always taken as positive and always adverse. kv can be positive or negative and is often taken as zero in temporary excavation support design.
Why does the Mononobe–Okabe coefficient become unbounded?
Because when the seismic inertia angle plus the ground slope reaches the effective friction angle of the soil, the retained slope is at failure under its own weight in the rotated frame. No finite wall force retains it, and the equation reflects that. The angle is capped at the limit, and reaching the cap indicates the design requires reconsideration.
Where does the seismic thrust act on the wall?
Higher than the static thrust. The static component acts at about one third of the wall height, while the dynamic increment acts higher because inertia is distributed over the wedge. The seismic increment therefore produces more overturning moment than its magnitude alone suggests.
Should passive resistance be reduced during an earthquake?
Yes. Inertia acts on the soil in front of the wall as well as behind it, reducing available passive resistance. A method that increases active thrust while leaving passive resistance at its static value is optimistic on the resisting side.
Is pseudo-static analysis sufficient?
For most retaining structures it is the accepted approach and is what design codes require. It replaces a dynamic problem with an equivalent static one and does not predict permanent displacement, which is acceptable because design is based on limiting displacement rather than preventing all movement.
What if the soil can liquefy?
Then liquefaction assessment governs. Earth pressure coefficients depend on effective stress, and liquefaction removes it. A liquefaction check should precede the seismic earth pressure calculation rather than follow it.
