Pseudo-static slope stability analysis represents earthquake shaking as a constant horizontal force applied to the sliding mass, expressed as a fraction of its weight through a seismic coefficient. The limit equilibrium calculation is then carried out as in the static case, and the result is a seismic factor of safety.
The method replaces a transient, oscillating, direction-reversing load with a steady push in the least favourable direction. That substitution is severe, and it is made deliberately, because the alternative requires a time history analysis that most projects neither justify nor can parameterise.
Understanding what the resulting number means matters more here than in any other routine geotechnical calculation, because a seismic factor of safety below 1.0 does not mean the slope collapses.
A pseudo-static factor of safety is not a prediction of stability during an earthquake. It is a screening index calibrated so that slopes passing it deform acceptably.
What Is the Pseudo-static Method?
A static analysis with additional inertial forces.
Each slice in the method of slices carries its weight W. The pseudo-static method adds a horizontal force kh · W and, where it is included, a vertical force kv · W. These act at the centroid of the slice, in the direction that reduces stability: the horizontal force pointing out of the slope, the vertical force usually taken upward because reducing the weight reduces the normal force and therefore the frictional resistance.
Everything else is unchanged. The slip surface search, the strength model, the treatment of pore pressures and the interslice assumptions are the same as in the static analysis, which is why the seismic case is normally run with the same method and compared directly.
How Does the Seismic Force Enter the Slice Equations?
Through both the numerator and the denominator, which is why the effect is more than proportional.
For an effective stress analysis by the ordinary method of slices:
FS = Σ { c′ · l + [ ( 1 − kv ) · W · cos α − kh · W · sin α − u · l ] · tan φ′ } / Σ [ ( 1 − kv ) · W · sin α + kh · W · cos α ]
where kh and kv are the horizontal and vertical seismic coefficients, α is the slice base inclination and the other symbols are as in the static case.
The horizontal term appears twice with opposite effect. In the numerator it reduces the normal force on slices whose base dips towards the slope, lowering the available friction. In the denominator it adds directly to the driving force. A coefficient of 0.15 therefore does considerably more than reduce the factor of safety by fifteen per cent.
How Is the Seismic Coefficient Chosen?
From the design ground motion, reduced.
The starting point is the peak ground acceleration for the site and return period, taken from the seismic hazard map or the site-specific study that the governing code requires. The horizontal coefficient is then a fraction of that value, and the fraction is prescribed by the code rather than selected by the engineer. TBDY 2018 and Eurocode 8 Part 5 each set their own rule for that fraction, so the coefficient follows from whichever code governs the project.
The reduction is not arbitrary, and the reasoning behind it is the substance of the method. It is worth stating explicitly rather than applying as a rule.
Why Is kh Smaller Than the Peak Ground Acceleration?
Because peak acceleration acts for an instant, on part of the mass, in alternating directions.
Three effects justify the reduction. The peak occurs in a single cycle and lasts a fraction of a second, while the pseudo-static force is applied permanently. The acceleration is not uniform over the depth of the sliding mass, so the average acting on the whole body is lower than the peak at any point. And the motion reverses: a slope pushed outward for a fraction of a second is pushed back immediately afterwards, so a momentary exceedance produces a small displacement rather than a collapse.
Applying the full peak acceleration as a sustained force would therefore produce factors of safety that condemn slopes which have demonstrably survived similar events. The reduced coefficient is a calibration against observed performance rather than a mechanical derivation.
Does the Vertical Coefficient Matter?
Less than the horizontal one, and not always in the direction expected.
Upward vertical acceleration reduces the effective weight of the mass, which reduces both the driving component and the normal force. Because it works on both sides of the equation, its net effect is small for typical geometries and is often less than the uncertainty in the strength parameters.
Many codes permit it to be omitted for slope analysis or set it as a fraction of the horizontal coefficient. Where it is included, it should be applied in the direction that produces the lower factor of safety rather than assumed, because which direction that is depends on the geometry and on the friction angle.
Why Is a Seismic Factor of Safety Below 1.0 Not Failure?
Because the loading reverses before failure can develop.
A static factor of safety below 1.0 describes a slope that cannot be in equilibrium: it moves, and it keeps moving. A pseudo-static factor below 1.0 describes an instant during shaking in which the driving force momentarily exceeds the available resistance. The mass accelerates for that instant, then the ground motion reverses and the movement stops.
The consequence is permanent displacement rather than collapse. Whether that displacement matters depends on what sits on the slope. A few centimetres on a rural embankment is not a failure; the same movement beneath a structure with a rigid foundation may be.
This is the reasoning behind displacement-based methods, which integrate the accelerations exceeding the yield value to estimate the accumulated movement. They answer the question the pseudo-static factor cannot, and they require the design acceleration time history as an input, which is what keeps them outside routine practice.
What Factor of Safety Is Required in the Seismic Case?
Lower than in the static case, by design.
Because the load is transient and the criterion is tolerable displacement rather than equilibrium, codes set a lower required margin for the seismic combination, commonly close to unity rather than the value required for permanent static conditions. TBDY 2018 sets the values for Turkish practice, along with the coefficient used to derive kh from the design acceleration; Eurocode 8 Part 5 does the same for projects designed to the European codes.
The two numbers travel together. A seismic factor of safety compared against a static requirement is meaningless, and so is a static factor compared against a seismic one. Reporting which case each value belongs to, and under which code, is not a formality.
How Does Liquefaction Change the Problem?
It removes the strength the analysis assumed.
Pseudo-static analysis adds a force while keeping the soil strength constant. In a liquefiable layer the shaking does something different: pore pressures rise, effective stress falls, and the shear strength drops towards a residual value that may be a small fraction of the original.
A slope containing a potentially liquefiable layer therefore needs two separate assessments. The pseudo-static analysis addresses inertial instability with intact strength. A post-liquefaction stability analysis uses residual strength in the affected layer, usually without the inertial force, because the strength loss persists after the shaking stops. Flow failure is governed by the second, and the second is the one that produces the large displacements.
What the Method Does Not Represent
Duration, frequency content and the history of the shaking.
Two earthquakes with the same peak acceleration and different durations impose very different demands on a slope, and the pseudo-static coefficient carries no information about which. Nor does it represent the progressive degradation of strength under cyclic loading, the build-up of pore pressure in partially drained material, or amplification of motion through the slope itself.
The topographic effect is worth noting separately: shaking is amplified near the crest of a slope relative to the free field, and a single coefficient applied uniformly to the sliding mass does not capture that.
Common Errors in Seismic Slope Stability
| Error | Consequence |
|---|---|
| Applying the peak ground acceleration directly as kh | Slopes condemned that have survived comparable events |
| Comparing a seismic factor of safety against the static requirement | A passing design judged as failing, or the reverse |
| Mixing the coefficient rule of one code with the required factor of another | A margin that belongs to neither code is reported |
| Treating a pseudo-static value below 1.0 as collapse | Mitigation specified where displacement would have been tolerable |
| Omitting post-liquefaction stability where a susceptible layer exists | The governing mechanism is never analysed |
| Using peak strength for a cyclically degrading soil | Available resistance overstated for the seismic case |
| Applying kv in the assumed rather than the worse direction | Less critical case reported |
| Reporting the seismic factor without stating kh | Result cannot be reproduced or compared |
Where SETAF2018 Fits
SETAF2018 reports the static and the earthquake factor of safety from the same slope model, so the two cases are produced together rather than as separate analyses. The earthquake case can be run to TBDY 2018 or to Eurocode 8, so one model serves a project under either code.
Circular and polygonal slip surfaces are analysed by the ordinary method of slices and the simplified Bishop method, with automatic optimisation of circular surfaces, and each surface carries both a static and a seismic result. Ground anchors, soil nails and surcharge loads are included as forces on the sliding mass in both cases, so a reinforced slope is checked under seismic loading with the reinforcement contributing as it does statically.
The ground model is the layered profile shared with the foundation and excavation support work, which means the unit weights and strength parameters used in the seismic slope check are the ones used everywhere else in the project. Where the profile contains a susceptible layer, liquefaction is assessed separately to TBDY 2018 Annex 16B with post-liquefaction settlement, so the two mechanisms are examined with their own methods rather than merged into one number.
Excavation support carries its own seismic treatment through Mononobe–Okabe or an equivalent static approach for seismic earth pressure, and a wall section can be transferred into the slope stability analysis through the external stability check. Calculation reports print the coefficients used and the slice data behind each factor of safety.
What the software does not perform is displacement-based seismic analysis or finite element dynamic analysis. Where permanent displacement is the design criterion rather than a screening factor of safety, those remain the appropriate tools.
Frequently Asked Questions
What is pseudo-static slope stability analysis?
It is a limit equilibrium analysis in which earthquake shaking is represented by a constant horizontal force equal to a seismic coefficient multiplied by the weight of the sliding mass, with an optional vertical component. The calculation otherwise proceeds exactly as in the static case.
How is the seismic coefficient determined?
From the design peak ground acceleration for the site and return period, reduced by a factor set in the governing code, whether that is TBDY 2018 or Eurocode 8 Part 5. The reduction accounts for the transient nature of the peak, the non-uniform acceleration over the sliding mass, and the reversing direction of the motion.
Why is the seismic coefficient lower than the peak ground acceleration?
Because the peak acts for a fraction of a second in one direction and then reverses, while the pseudo-static force is applied permanently. Using the full peak as a sustained force would condemn slopes that have demonstrably survived comparable earthquakes.
Does a seismic factor of safety below 1.0 mean the slope will fail?
No. It means the driving force momentarily exceeds the available resistance during shaking. The mass accelerates briefly and stops when the motion reverses, producing permanent displacement rather than collapse. Whether that displacement is acceptable depends on what the slope supports.
Should the vertical seismic coefficient be included?
It is often omitted for slopes, because it reduces both the driving force and the normal force and its net effect is usually smaller than the uncertainty in the strength parameters. Where it is included, it should be applied in whichever direction gives the lower factor of safety.
What required factor of safety applies to the seismic case?
A lower one than the static case, commonly close to unity, because the loading is transient and the real criterion is tolerable displacement. The governing code — TBDY 2018 in Turkish practice, Eurocode 8 Part 5 under the European codes — sets the value together with the rule for deriving the coefficient, and the two have to be used as a pair.
How is liquefaction handled in seismic slope stability?
Separately. Pseudo-static analysis adds a force while keeping strength constant, whereas liquefaction removes strength. A susceptible slope needs a post-liquefaction stability check using residual strength, usually without the inertial force, since the strength loss persists after shaking ends.
What does the pseudo-static method not account for?
Duration and frequency content of the shaking, cyclic degradation of strength, pore pressure build-up during the event, amplification of motion through the slope, and topographic amplification near the crest. It also gives no estimate of displacement.
