Retaining & slopes · Slope stability
Slope stability: Spencer & Morgenstern–Price with reinforcement and earthquake calculator
Layered slope with a phreatic line, crest surcharge and a horizontal seismic coefficient. Finds the critical circle by Bishop's method, then reports Fellenius, Bishop, Janbu, Spencer and Morgenstern–Price factors of safety. Soil nails, ground anchors and geogrids add the lesser of their tensile and pull-out capacity beyond the slip surface.
Standard: GEO5 Slope Stability style · Spencer (1967) · Morgenstern–Price (1965) · Bishop · Janbu · AS 4678 · AS 1170.4 pseudo-static
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Inputs in the worked example
The live calculator starts with these values; change any of them and every result, check and diagram updates instantly.
| Input | Value | Unit |
|---|---|---|
| Slope height H | 8 | m |
| Slope angle β | 35 | ° |
| Layers (top elevation m, γ kN/m³, c′ kPa, φ′ °), top first | 8, 19, 5, 30 3, 20, 10, 27 | |
| Water table elevation behind the crest | 3 | m |
| Water table elevation at and beyond the toe | 0 | m |
| Crest surcharge | 10 | kPa |
| Horizontal seismic coefficient kh (0 = static) | 0 | |
| Reinforcement (face elevation m, inclination ° below horizontal, length m, tensile kN, pull-out kN/m, spacing m) | 1.5, 15, 9, 150, 25, 1.5 3.5, 15, 9, 150, 25, 1.5 5.5, 15, 9, 150, 25, 1.5 | |
| Slip circle | Search for the critical circle (Bishop) | |
| Manual circle centre x (crest edge = 0, positive toward toe) | 6 | m |
| Manual circle centre elevation | 16 | m |
| Manual circle radius | 16 | m |
| Lowest elevation the slip may reach (firm stratum) | -4 | m |
| Required FS, static | 1.5 | |
| Required FS, with earthquake | 1.1 |
Worked example — results
| Result | Value | Unit |
|---|---|---|
| Spencer (moment & force) | 1.554 | |
| Morgenstern–Price (half-sine) | 1.555 | |
| Bishop simplified | 1.552 | |
| Janbu corrected (f0 1.082) | 1.498 | |
| Fellenius / ordinary method | 1.341 | |
| Spencer interslice inclination θ | 15.5 | ° |
| M–P scaling factor λ | 0.359 | |
| Critical circle centre (x, y) | 7.46, 13.35 | m |
| Radius | 15.77 | m |
| Slip entry / exit x | -7.38 / 15.86 | m |
| Reinforcement crossing the slip | 1 of 3 | |
| Mobilised reinforcement force | 7.5 | kN/m |
- ✓ Governing FS (Spencer, M–P) ≥ 1.5 — 1.554
- ✓ Bishop and Spencer agree within 5 % — 1.552 vs 1.554
- ✓ M–P solution converged (Fm = Ff)
Method, formulas and assumptions
- Slices: 30 across the slip; W from each layer's thickness; u = γw × head above the base from the phreatic line; surcharge added to crest slices; kh acts on soil weight at slice mid-height.
- Bishop: F = Σ[c′b + (W − ub) tanφ′]/mα ÷ [ΣW sinα + kh ΣWs (yo − yc)/R − ΣT cos(α + δ)], mα = cosα + sinα tanφ′/F.
- Spencer: resultant interslice force Q at a constant inclination θ; F and θ solved so that ΣQ = 0 and ΣQ cos(α + θ) = 0 (Newton iteration).
- Morgenstern–Price: GLE with X = λ f(x) E, f(x) = half-sine; λ found where the moment and force factors of safety are equal.
- Reinforcement force per metre run = min(tensile capacity, pull-out × bond length beyond the slip) / spacing, acting at the slip at its inclination (active: reduces the driving moment).
- Critical circle by Bishop over a grid of centres and tangent depths, refined three times; M–P and Spencer are reported for that circle.
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For preliminary design and education. Check every result against the current standard and your own engineering judgement before use.