{"id":1538,"date":"2026-10-20T17:25:09","date_gmt":"2026-10-20T14:25:09","guid":{"rendered":"https:\/\/setaf2018.com\/?p=1538"},"modified":"2026-10-20T17:25:09","modified_gmt":"2026-10-20T14:25:09","slug":"fellenius-vs-bishop-slope-stability","status":"publish","type":"post","link":"https:\/\/setaf2018.com\/en\/post\/fellenius-vs-bishop-slope-stability\/","title":{"rendered":"Fellenius vs Bishop: Limit Equilibrium Methods and the Factor of Safety"},"content":{"rendered":"\n

Limit equilibrium slope stability analysis divides a potential sliding mass into vertical slices, writes equilibrium for each, and reports the ratio of available shear strength to the shear stress required for equilibrium. That ratio is the factor of safety, and the two methods most often used to calculate it are the ordinary method of slices and the simplified Bishop method.<\/p>\n\n\n\n

\"Slope<\/figure>\n\n\n\n

They differ in one assumption, they differ in their answers by a margin that is usually between five and fifteen per cent, and the difference is systematic rather than random. Reporting a factor of safety without naming the method leaves the reader unable to interpret it.<\/p>\n\n\n\n

A factor of safety is not a property of a slope. It is a property of a slope, a slip surface, a strength model and a method of slices.<\/strong><\/p>\n\n\n\n

What Does the Factor of Safety Actually Mean?<\/h2>\n\n\n\n

The ratio between the shear strength available along a surface and the shear stress needed to keep the mass in equilibrium.<\/p>\n\n\n\n

At FS = 1.0 the two are equal and the slope is at the point of failure. At FS = 1.5 the soil has half again as much strength as equilibrium demands. What the number does not say is how much the slope will move, or whether the movement will be sudden or gradual. Limit equilibrium answers whether, not how much.<\/p>\n\n\n\n

It is also a global figure. A factor of safety of 1.4 does not mean every part of the surface has a margin of 1.4. Strength is mobilised unevenly, and in a strain-softening soil the parts that reach peak first may already be past it while others are still loading.<\/p>\n\n\n\n

How Does the Method of Slices Work?<\/h2>\n\n\n\n

By making a statically indeterminate problem solvable.<\/p>\n\n\n\n

The sliding mass is divided into vertical slices. Each slice has a weight, a base with a length and an inclination, pore water pressure at the base, and forces from the slices on either side. The number of unknowns exceeds the number of equilibrium equations, so every method in this family introduces assumptions about the interslice forces to close the system.<\/p>\n\n\n\n

What separates the methods is which assumption they make and which equilibrium conditions they satisfy. That is the whole of the difference between Fellenius and Bishop.<\/p>\n\n\n\n

The Ordinary Method of Slices<\/h2>\n\n\n\n

Fellenius (1936) makes the simplest assumption: the forces between slices are ignored entirely, or more precisely, the resultant interslice force is taken as parallel to the slice base so that it does not affect the normal force.<\/p>\n\n\n\n

For an effective stress analysis the factor of safety follows directly:<\/p>\n\n\n\n

FS = \u03a3 [ c\u2032 \u00b7 l + ( W \u00b7 cos \u03b1 \u2212 u \u00b7 l ) \u00b7 tan \u03c6\u2032 ] \/ \u03a3 ( W \u00b7 sin \u03b1 )<\/strong><\/p>\n\n\n\n

where c\u2032 is the effective cohesion, \u03c6\u2032 the effective friction angle, l the base length of the slice, \u03b1 the inclination of the slice base, W the slice weight and u the pore water pressure at the base.<\/p>\n\n\n\n

Because nothing on the right depends on the factor of safety, the calculation is direct. No iteration is required, and the result can be reproduced by hand for a small number of slices. That is why the method survives as a check even where it is not used for design.<\/p>\n\n\n\n

The Simplified Bishop Method<\/h2>\n\n\n\n

Bishop (1955) keeps vertical equilibrium of each slice and assumes the interslice forces are horizontal, which means they have no vertical component. The normal force at the slice base changes as a result, and so does the factor of safety:<\/p>\n\n\n\n

FS = \u03a3 [ ( c\u2032 \u00b7 b + ( W \u2212 u \u00b7 b ) \u00b7 tan \u03c6\u2032 ) \/ m\u03b1 ] \/ \u03a3 ( W \u00b7 sin \u03b1 )<\/strong><\/p>\n\n\n\n

with<\/p>\n\n\n\n

m\u03b1 = cos \u03b1 \u00b7 ( 1 + tan \u03b1 \u00b7 tan \u03c6\u2032 \/ FS )<\/strong><\/p>\n\n\n\n

where b is the horizontal width of the slice.<\/p>\n\n\n\n

The factor of safety appears on both sides. The solution is iterative: assume a value, compute the right-hand side, substitute, repeat until the two agree. Convergence is normally rapid, but it is not guaranteed, and non-convergence in a particular geometry is itself information rather than a software fault.<\/p>\n\n\n\n

\"Slope<\/figure>\n\n\n\n

Why Do the Two Methods Not Give the Same Answer?<\/h2>\n\n\n\n

Because ignoring the interslice forces is not a neutral simplification.<\/p>\n\n\n\n

In the ordinary method the normal force at the slice base is W \u00b7 cos \u03b1, which underestimates the normal force where the base is steeply inclined. A lower normal force gives a lower frictional resistance, so the computed factor of safety is lower. Bishop, by retaining vertical equilibrium, produces a larger normal force on those slices and a higher factor of safety.<\/p>\n\n\n\n

The gap widens in three situations. Deep circles with steeply inclined slices at the ends, where the cos \u03b1 term is furthest from the truth. High pore water pressures, where the ( W \u00b7 cos \u03b1 \u2212 u \u00b7 l ) term in the ordinary method can become small or even negative in the steep slices. And soils with a high friction angle, where the normal force matters most.<\/p>\n\n\n\n

Fellenius is therefore conservative, and the conservatism is not uniform across geometries. The practical consequence is that a design economy achieved by switching methods is real but has to be declared, because the two numbers are not interchangeable.<\/p>\n\n\n\n

Which Slip Surface Is the Critical One?<\/h2>\n\n\n\n

The one with the lowest factor of safety, which has to be found rather than assumed.<\/p>\n\n\n\n

For a circular surface the search runs across centre coordinates and radii, and the result is a grid of factors of safety from which the minimum is taken. Automatic optimisation refines the circle around that minimum. The search has to be wide enough that the minimum is genuinely interior to it, and a critical circle sitting at the edge of the search grid means the search was too small.<\/p>\n\n\n\n

Non-circular surfaces matter where the geology imposes them: a soft layer, a bedding plane, an old failure surface or the interface between fill and natural ground. A circular search will not find a surface that follows a thin weak seam, and where such a seam exists the polygonal analysis is not an optional refinement.<\/p>\n\n\n\n

How Are Pore Pressures Handled?<\/h2>\n\n\n\n

As the input that most often decides the answer.<\/p>\n\n\n\n

Effective stress analysis needs pore water pressure at the base of each slice. It can come from a defined water table, from a pressure distribution, or from a pore pressure ratio applied to the overburden. The choice matters: a water table and an equivalent ratio are not the same thing on a slope where seepage is not parallel to the surface.<\/p>\n\n\n\n

Total stress analysis with undrained strength avoids the pore pressure question and answers a different one. It represents the short-term condition in a cohesive soil before drainage. For a cut slope the long-term drained case usually governs; for an embankment on soft clay, the short-term undrained case usually does. Analysing only one of them is a decision, and it belongs in the report.<\/p>\n\n\n\n

Parameter selection sits beneath all of this. Whether the friction angle<\/a> came from a direct measurement or a correlation changes what the factor of safety is worth.<\/p>\n\n\n\n

How Do Reinforcement and Surcharge Enter?<\/h2>\n\n\n\n

As additional forces on the sliding mass.<\/p>\n\n\n\n

A soil nail<\/a> or ground anchor<\/a> crossing the slip surface contributes a force whose magnitude is limited by the lesser of its tensile capacity and its pullout resistance beyond the surface. That force has components along and normal to the base, and both matter: the component along the surface reduces the driving moment, and the normal component increases the frictional resistance.<\/p>\n\n\n\n

Surcharge is applied as additional weight on the slices it covers. The worst position is not always the crest, and a surcharge that increases the driving moment on one surface may increase the resisting moment on another, which is why it is analysed in position rather than as an average.<\/p>\n\n\n\n

What Factor of Safety Is Enough?<\/h2>\n\n\n\n

It depends on the case being analysed and on what the number is being asked to protect.<\/p>\n\n\n\n

Typical practice requires a higher margin for permanent slopes than for temporary ones, and a lower margin for the seismic case than for the static case, because the seismic loading is itself a short-duration event applied through a pseudo-static coefficient rather than a sustained condition. The relevant code sets the values, and the seismic coefficients<\/a> used in the pseudo-static case have to be consistent with the framework that sets the required margin.<\/p>\n\n\n\n

What is not defensible is comparing a factor of safety against a required value without stating the method, the strength model and the drainage condition it was computed with.<\/p>\n\n\n\n

When Is Limit Equilibrium Not Appropriate?<\/h2>\n\n\n\n

When the question is deformation rather than stability.<\/p>\n\n\n\n

Limit equilibrium says nothing about how much a slope moves before failure, and a slope with an adequate factor of safety can still displace enough to damage what sits on it. Where movement is the design criterion, a finite element analysis<\/a> that computes displacement is the appropriate tool.<\/p>\n\n\n\n

Two other cases fall outside it. Progressive failure in strain-softening soils, where peak strength is not mobilised simultaneously along the surface, is not represented by a single global factor. And flow failures in liquefiable<\/a> material are a different mechanism altogether, since the strength that limit equilibrium assumes is the strength the event removes.<\/p>\n\n\n\n

Common Errors in Slope Stability Analysis<\/h2>\n\n\n\n
Error<\/th>Consequence<\/th><\/tr><\/thead>
Reporting a factor of safety without naming the method<\/td>Number cannot be compared with a required value<\/td><\/tr>
Search grid too small<\/td>Critical circle sits at the grid edge and the true minimum is never found<\/td><\/tr>
Circular search on a slope with a weak seam<\/td>The governing non-circular surface is not examined<\/td><\/tr>
Pore pressure ratio used where seepage is not parallel to the slope<\/td>Effective stresses wrong along the surface<\/td><\/tr>
Only the long-term or only the short-term case analysed<\/td>The governing condition may be the one omitted<\/td><\/tr>
Reinforcement force taken as tensile capacity without pullout check<\/td>Available force overestimated beyond the slip surface<\/td><\/tr>
Peak strength used in a strain-softening soil<\/td>Progressive failure not represented by a global factor<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

Where SETAF2018 Fits<\/h2>\n\n\n\n

SETAF2018 performs limit equilibrium slope analysis with both methods in the same model that carries the rest of the geotechnical design.<\/p>\n\n\n\n

Circular and polygonal slip surfaces are analysed by the ordinary method of slices and by the simplified Bishop method, with automatic optimisation of circular surfaces to locate the critical circle. The ground comes from the same layered profile used for foundations and excavation support, so a slope section and a wall section read the same stratigraphy, unit weights<\/a> and strength parameters rather than separately entered copies.<\/p>\n\n\n\n

Ground anchors, soil nails and surcharge loads are included as forces on the sliding mass, and the factor of safety is reported for the static case and for the earthquake case. A section from an excavation support model can be transferred into the slope analysis through the external stability check, so the same geometry is examined for wall behaviour and for global stability without being rebuilt.<\/p>\n\n\n\n

Geometry is exchanged with drawings through DWG import and export, and results are issued through calculation reports<\/a> that print the equations and the slice data rather than a single number, which is what allows a hand check of a governing circle. The wider slope stability<\/a> workflow covers the analysis; what it does not include is finite element deformation analysis or probabilistic assessment, and where those are the question a dedicated package remains the right tool.<\/p>\n\n\n\n

Frequently Asked Questions<\/h2>\n\n\n\n

What is the difference between the Fellenius and Bishop methods?<\/h3>\n\n\n\n

Fellenius ignores the interslice forces, so the normal force at each slice base is W \u00b7 cos \u03b1 and the calculation is direct. Simplified Bishop keeps vertical equilibrium of each slice and assumes horizontal interslice forces, which changes the normal force and makes the solution iterative.<\/p>\n\n\n\n

Which method gives a higher factor of safety?<\/h3>\n\n\n\n

Bishop, almost always. The ordinary method underestimates the normal force on steeply inclined slices, which reduces the frictional resistance and lowers the computed factor of safety, typically by five to fifteen per cent and more where circles are deep or pore pressures are high.<\/p>\n\n\n\n

Why does the simplified Bishop method need iteration?<\/h3>\n\n\n\n

Because the factor of safety appears on both sides of the equation, inside the m\u03b1 term. A value is assumed, the right-hand side is evaluated, the result is substituted and the process repeats until the two agree. Convergence is usually fast but is not guaranteed for every geometry.<\/p>\n\n\n\n

Is the ordinary method of slices still useful?<\/h3>\n\n\n\n

Yes, as a conservative check and as a calculation that can be reproduced by hand. Because it requires no iteration, a governing circle can be verified independently with a spreadsheet, which is difficult with methods that solve implicitly.<\/p>\n\n\n\n

How is the critical slip surface found?<\/h3>\n\n\n\n

By searching across centre coordinates and radii for circular surfaces and taking the minimum factor of safety, then refining around it. The search must be wide enough that the minimum lies inside it; a critical circle at the edge of the grid means the search was too small.<\/p>\n\n\n\n

When is a non-circular slip surface needed?<\/h3>\n\n\n\n

Where the geology imposes the failure path: a soft layer, a bedding plane, a pre-existing failure surface or a fill-to-natural-ground interface. A circular search cannot find a surface that follows a thin weak seam, so a polygonal analysis is required rather than optional.<\/p>\n\n\n\n

Should the short-term or long-term case be analysed?<\/h3>\n\n\n\n

Both, unless there is a reason not to. For a cut slope in clay the long-term drained condition usually governs as pore pressures equalise; for an embankment on soft clay the short-term undrained condition usually does. Which case was analysed belongs in the report.<\/p>\n\n\n\n

What does limit equilibrium not tell the designer?<\/h3>\n\n\n\n

How much the slope will move. It compares available strength with required strength along an assumed surface and reports a ratio. Deformation, progressive failure in strain-softening soils and flow failure in liquefiable material all fall outside what the method represents.<\/p>\n\n\n\n\n