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    Calculates locally in your browser · Bishop simplified slope stability
    Geotechnical / slopes

    Slope stability

    A circular slip surface is divided into vertical slices, and the factor of safety is the ratio of resisting to driving moments. Bishop’s simplification assumes the interslice shear forces cancel, which makes the equation implicit in F — so it is solved by iteration, and the reported value is the lowest over a search of centres rather than one arbitrary circle.

    VerdictStableTarget1.50ru0.20
    Factor of safety
    1.35
    Stable
    Critical circle
    2.2, 22.0
    centre relative to the toe, radius 22.1 m
    Driving moment
    0.5
    kN·m per metre of slope
    Resisting moment
    0.7
    kN·m per metre of slope
    The search, and what it found
    Bishop is solved by iteration for each trial circle; the reported value is the lowest over the grid.
    F = 1.35
    The search, and what it found
    QuantityValueNote
    Slope height10.0 mas entered
    Slope angle30°run of 17.3 m
    Unit weight18.0 kN/m³total unit weight
    Cohesion c’10.0 kPaeffective
    Friction φ’25°effective
    Pore pressure ratio ru0.20u = ru γ h
    Critical centre x2.2 mfrom the toe
    Critical centre y22.0 mabove the toe
    Radius22.1 mthrough the toe
    Driving moment0.5 kN·mΣ W sinα
    Resisting moment0.7 kN·mΣ (c b + (W − u b)tanφ)/m_α
    Factor of safety1.35Stable

    The critical circle is centred 2.2 m from the toe at 22.0 m height, with a radius of 22.1 m. A factor of safety of 1.35 is stable against the usual 1.5 target for a permanent slope.

    Inspect the model
    Choose a quick result view, a scaled comparison, or the complete diagnostic set.
    Factor of safety against slope angle
    Falls steeply — the 1.5 line is crossed around 30° for these soil parameters.
    Factor of safety against slope angle0.0000.5221.041.572.092.6115°15°20°20°25°25°30°30°35°35°40°40°50°50°FoS
    Cohesion against friction
    Either one can carry the slope; the pair is what the factor of safety is made of.
    Cohesion against friction0.0000.3190.6390.9581.281.600.434c’ onlyc’ only0.671φ’ onlyφ’ only1.35BothBothFoS
    Canonical equations
    The exact relationships implemented by this engine.

    F = Σ[(c b + (W − ub)tanφ)/m_α] / Σ W sinα · Simplified, interslice shear ignored.

    cosα + sinα tanφ / F · Implicit in F, hence the iteration.

    W = γ b h · Per metre of slope.

    u = ru γ h · The ru simplification.

    engineering methodology

    A clear path from inputs to results.

    A circular slip surface is divided into vertical slices, and the factor of safety is the ratio of resisting to driving moments. Bishop’s simplification assumes the interslice shear forces cancel, which makes the equation implicit in F — so it is solved by iteration, and the reported value is the lowest over a search of centres.

    Step-by-step methodWorked exampleAssumptions explained

    How to use

    1. Enter the slope height and angle.
    2. Set the unit weight, cohesion and friction angle.
    3. Add the pore pressure ratio.
    4. Read the minimum factor of safety and the critical circle.

    The formula

    F=∑[c′b+(W−ub)tan⁡ϕ′mα]∑Wsin⁡αF = \frac{\sum \left[\frac{c'b + (W - ub)\tan\phi'}{m_\alpha}\right]}{\sum W \sin\alpha}

    Reference: Bishop (1955), simplified method; the ru pore-pressure simplification is standard practice.

    Worked example

    A 10 m slope at 30°

    Inputs
    • 10 m high, 30° from horizontal.
    • Unit weight 18 kN/m³, c’ = 10 kPa, φ’ = 25°.
    • Pore pressure ratio 0.2.
    Calculation path
    1. 1

      The slope runs 17.3 m horizontally.

    2. 2

      A grid of trial circles through the toe is analysed, each with 40 slices.

    3. 3

      For each circle the factor of safety is iterated until it converges.

    4. 4

      The minimum over the grid is the reported value.

    5. 5

      Removing the cohesion drops the factor of safety sharply; raising ru does the same.

    Result

    A minimum factor of safety in the stable range for these parameters.

    Interpreting the result

    The pore pressure ratio is the variable that changes most in service: a slope that was stable when drained can fail after a wet season without any change in geometry. That is why the ru slider moves the answer more than the height slider does.

    Assumptions & limitations

    • A homogeneous soil in a single layer.
    • Circular slip surfaces through the toe.
    • Simplified Bishop, with interslice shear neglected.
    • No seismic loading, no external surcharge and no reinforcement.

    Frequently asked questions

    What factor of safety is acceptable?

    1.5 for a permanent slope and 1.3 for a temporary one in most codes. Below 1.0 the slope is already failing in the model.

    What does Bishop’s simplification assume?

    That the interslice shear forces cancel. It is accurate to within a few percent for circular surfaces and is the standard method for routine design.

    How do I get c’ and φ’?

    From triaxial or direct shear tests on undisturbed samples, or from correlations with SPT or CPT data. The factor of safety is far more sensitive to those than to the geometry.

    Why search over centres?

    Because the critical circle is not known in advance. A single assumed circle can overstate the factor of safety by 20% or more.

    When is a circular surface wrong?

    When there is a thin weak layer, a stiff crust over soft clay, or an existing slip plane. Those need a non-circular search such as Spencer or Morgenstern-Price.

    Methodology last reviewed 2026-10-22. Results are informational and not financial or engineering advice — see our disclaimer.