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Engineering & geotechnicsFree

Slope Stability

Factor of safety of cuts, embankments and natural slopes by the method of slices, with water and earthquakes.

Searches hundreds of circular slip surfaces through a slope and finds the one with the lowest factor of safety by the simplified Bishop method, checked against the ordinary method of slices. The slope can carry a surcharge on its crest, sit on a firm layer, hold a water table or a pore-pressure ratio, and be shaken by an earthquake. A second search over slope angles gives the steepest face that meets the required factor of safety.

Open the tool
Slope Stability
στFitted envelopec = 18 kPaφ = 32°

Overview

Slope failures in road cuts, quarry faces, canal banks and embankments usually happen along a curved surface after heavy rain or an earthquake. The tool reproduces the classical design check of geotechnical engineering: the slope is cut into slices, the forces on each slice are balanced, and the ratio of available shear strength to the strength needed for equilibrium is the factor of safety.

The section is drawn to true scale with the critical circle, the slices, the sliding mass, the water table and the coloured grid of centres, so that the student sees why one circle is critical and the engineer sees how far the result depends on the search.

What you enter · What you get

What you enter

  • Height, slope angle, crest surcharge and depth of a firm layer
  • Effective cohesion, friction angle and unit weights
  • Water table or pore-pressure ratio ru, seismic coefficient kh and the required factor of safety

What you get

  • Lowest factor of safety (Bishop and Fellenius) and the critical circle
  • Section to scale with slices, sliding mass and search grid
  • Factor of safety against slope angle and the steepest safe angle
  • Slice table (CSV)

How it works in the field

Enter the slope
Enter the soil and water
Read the factor of safety and the critical circle
Find the safe angle

Method and equations

Simplified Bishop method

Moment equilibrium about the circle centre with interslice normal forces; the factor of safety appears on both sides and is found by iteration (Bishop 1955). Pore pressure acts on the base of each slice; the seismic force kh·W acts at the slice centroid.

FS = Σ[(c′b + (W − ub) tan φ′) / mα] / Σ[W sin α + kh W ȳ / R] ; mα = cos α + sin α tan φ′ / FS

Ordinary method

Fellenius (1936) ignores interslice forces and resolves normal to each base; it is shown as a conservative check.

FS = Σ[c′l + (W cos α − kh W sin α − ul) tan φ′] / Σ[W sin α + kh W ȳ / R]

Search

A grid of 22 × 22 centres above the slope is tried with circles through the toe, through three points of the face, beyond the toe and tangent to six levels down to the firm layer; the best centre is then refined with 11 × 11 sub-centres and continuous radii. The slope-angle curve repeats the search at 5° steps.

Accuracy and error limits

  • For a given geometry and strength the simplified Bishop factor is within about 5 % of rigorous limit-equilibrium methods (Spencer, Morgenstern–Price) for circular surfaces; the search can miss the true minimum by 1–3 %. Non-circular failures along a weak layer, a joint set or a fault are not covered and can be 10–30 % lower.
  • The largest error comes from the inputs: a change of 2° in φ′ or of 5 kPa in c′ typically changes FS by 5–15 %, and the water table can change it by 20–40 %. Pseudo-static seismic analysis does not give displacement.
  • To reduce the error: measure c′ and φ′ on undisturbed samples at the expected stress, use residual strength where old slips exist, record the highest seasonal water level in piezometers, and check plane and wedge failures in rock with the stereonet tool.

Who uses it

Geotechnical and highway engineersQuarry and mine plannersEngineering geology students

Standards and references

  1. Bishop, A.W. (1955). The use of the slip circle in the stability analysis of slopes. Géotechnique, 5, 7–17.
  2. Fellenius, W. (1936). Calculation of the stability of earth dams. Transactions, 2nd Congress on Large Dams, Washington, 4, 445–462.
  3. Duncan, J.M., Wright, S.G. & Brandon, T.L. (2014). Soil Strength and Slope Stability, 2nd ed. Wiley.
  4. Abramson, L.W., Lee, T.S., Sharma, S. & Boyce, G.M. (2002). Slope Stability and Stabilization Methods, 2nd ed. Wiley.
  5. Kramer, S.L. (1996). Geotechnical Earthquake Engineering. Prentice Hall.
Results support professional judgement. They do not replace a licensed engineer’s or geologist’s sign-off.

At a glance

Category
Engineering & geotechnics
Runs on
Phone, tablet and desktop browser
Export
Section and slope-angle chart as PNG, JPG and PDF; slices as CSV; full PDF report
Access
Free

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