RHCES Slope Stability — Bishop Simplified

Inputs

Slope Height H (m)

Slope Angle β (°)

Cohesion c' (kPa)

Friction φ' (°)

Unit Weight γ (kN/m³)

Pore pressure ru0 = dry, 0.4 typical wet

Number of slices

Trial radius factorR = factor × H

Method — Bishop's simplified

FoS = Σ[ (c'b + (W − ub)tanφ')/mα ] / Σ(W·sinα), mα = cosα + sinα·tanφ'/FoS.
Iterates from FoS=1.0; converges in 4–6 cycles.
Single trial circle — toe-circle approximation. Use trial radius × height.
For full stability search, use commercial slope-stability software.

Results

Failure circle R—

Centre (xc, yc)—

Driving Σ(W·sinα)—

Resisting (numerator)—

FoS Bishop—

Status—

About this calculator

This tool estimates the factor of safety (FoS) against slope failure along a single trial circular slip surface using Bishop's Simplified method for c'–φ' soils. It's built for quick, preliminary slope screening by geotechnical, highway and earthworks engineers — a single trial circle only, not an automated search for the critical (minimum-FoS) surface.

Method & Formulas

Bishop's Simplified method slices the soil mass above the trial circle into vertical slices and sums moment-equilibrium contributions from each:

FoS = Σ[(c'·b + (W − u·b)·tanφ') / m_α] / Σ(W·sinα)

m_α = cosα + sinα·tanφ'/FoS

where b = slice width, W = slice weight, α = inclination of the slice base, and u = pore pressure at the slice base (u = r_u·γ·h for a given pore-pressure ratio r_u).

Because m_α depends on the FoS being solved for, the equation is implicit and is solved iteratively: start from a trial FoS (commonly 1.0), recompute m_α and a new FoS for every slice, and repeat until the value converges — typically within 4–6 cycles (Bishop, 1955).

Worked Example

Consider one representative slice from a trial circle: base width b = 1.0 m, weight W = 150 kN, base angle α = 20°, dry slope so u = 0, with c' = 20 kPa and φ' = 25° (this tool's default inputs). This illustrates the iteration mechanics only — the real solution sums the ratio over every slice around the whole circle.

Trial FoS = 1.50: m_α = cos20° + sin20°·tan25°/1.50 = 0.940 + 0.106 = 1.046.

Numerator = (20×1.0 + 150×tan25°)/1.046 = (20 + 69.9)/1.046 ≈ 86.0 kN; denominator = W·sinα = 150×sin20° ≈ 51.3 kN → FoS = 86.0/51.3 ≈ 1.68.

Repeating with FoS = 1.68 gives m_α ≈ 1.035 and a new FoS ≈ 1.69; the value settles around FoS ≈ 1.70 for this single slice.

Assumptions & Limitations

Circular slip surface only, sized by a single trial radius (R = radius factor × H) — the tool does not search for the critical (minimum-FoS) circle.

Assumes a uniform, homogeneous soil (single c', φ', γ) for the whole slope; layered or anisotropic profiles are not modeled.

Pore pressure is simplified to one ratio r_u; it does not represent an actual piezometric surface or transient seepage.

Seismic (pseudo-static), rapid-drawdown and reinforcement (geogrid, soil nailing, piles) effects are not included.

Verify results against the governing code and a licensed engineer's judgment, and confirm with a full slope-stability program that searches multiple trial circles before design use.

FAQ

What FoS is considered acceptable?

This tool flags FoS ≥ 1.5 as stable and FoS between 1.0 and 1.5 as marginal, following common long-term static-slope screening practice — always confirm the minimum required FoS in your project's governing geotechnical guidelines.

Why does the calculator iterate instead of solving directly?

m_α depends on the very FoS being solved for, so Bishop's method is implicit. The tool feeds each new FoS estimate back into m_α and repeats until the value stops changing.

Does a larger trial-radius factor always give a lower (more critical) FoS?

Not necessarily — the critical circle depends on geometry and soil strength together. Try several radius factors to bracket the true minimum FoS, or use dedicated slope-stability software for a full search.

InputsReset