Slope Stability Analysis

Theory: Slope Stability

How the method of slices finds the factor of safety of a slope on a circular slip surface, how Fellenius's and Bishop's methods differ, and how the critical circle is searched for.

Factor of safety

A slope fails when the soil along some slip surface can no longer carry the shear the weight above it puts there. The factor of safety is the ratio of the shear strength available on the surface to the shear needed for equilibrium; the slope's factor is the least over all surfaces, found on the critical one.

\mathrm{FS} = \frac{\tau_f}{\tau_d}

τ_f: shear strength · τ_d: shear mobilized · σ′: effective normal stress on the surface

Method of slices

For a circular surface, the sliding mass is cut into vertical slices. Taking moments about the circle's center, the weight of each slice drives rotation through W sin α, α being the inclination of its base, and the shear strength along its base resists it. The forces between slices make the problem statically indeterminate, and each method makes an assumption about them.

Fellenius

The ordinary method (Fellenius, 1936) leaves the forces between slices out, so the normal force on each base is just the weight resolved normal to it, less the water's push.

\mathrm{FS} = \frac{\sum \left[c' l + (W \cos\alpha - u l) \tan\phi'\right]}{\sum W \sin\alpha}

l = b/cos α: length of the base · u: pore pressure at the base

It needs no iteration but errs on the safe side, often by 5–20%, and more for deep circles with high pore pressure.

Bishop's simplified method

Bishop (1955) took the forces between slices as horizontal and resolved each slice's forces vertically. FS then appears on both sides of the equation, and is found by iteration.

\mathrm{FS} = \frac{\sum \left[c' b + (W - u b) \tan\phi'\right] / m_\alpha}{\sum W \sin\alpha}

b: width of the slice. Converges in a few iterations from Fellenius's value

Its results lie close to the rigorous methods' on circular surfaces. Where m_α becomes small — steep bases near the toe with a low FS — the method is unreliable; values below 0.2 are a warning sign (Whitman and Bailey, 1967). For φ′ = 0 both methods give the same result.

Pore pressure

r_u = \frac{u}{\gamma h}

h: height of the slice above its base. Bishop and Morgenstern (1960) used r_u to chart the stability of homogeneous slopes

The critical circle

Centers are tried over a grid above the slope, and about each one circles of several depths, tangent to levels from the firm layer up.
The least FS at each center, contoured, shows where the critical center lies; a finer grid about the best one refines it.
A firm layer limits how deep the circles go. In soft clay (φ′ = 0) the critical circle runs as deep as it's allowed.

References

Fellenius, W. Calculation of the stability of earth dams. Transactions of the 2nd Congress on Large Dams, Washington, v. 4, p. 445–462, 1936.
Bishop, A. W. The use of the slip circle in the stability analysis of slopes. Géotechnique, v. 5, n. 1, p. 7–17, 1955.
Bishop, A. W.; Morgenstern, N. Stability coefficients for earth slopes. Géotechnique, v. 10, n. 4, p. 129–150, 1960.
Whitman, R. V.; Bailey, W. A. Use of computers for slope stability analysis. Journal of the Soil Mechanics and Foundations Division, ASCE, v. 93, n. SM4, p. 475–498, 1967.
Das, B. M.; Sobhan, K. Principles of Geotechnical Engineering. 9th ed. Boston: Cengage, 2018.

Enter the slope — height, face angle, a firm layer below and any surcharge on the crest — and the soil's c′, φ′, unit weight and pore pressure ratio to find the critical slip circle and its factor of safety by Fellenius's and Bishop's methods of slices.

Slope

Height, HmFace angle, β27°Firm layer, DmSurcharge, qkPa

The face is 1V : 1.96H. D is the firm layer's depth below the toe; leave it blank for none. q acts on the crest.

Soil

c′kPaφ′25°γkN/m³r_u

r_u is the pore pressure over the vertical stress, u/γh: 0 for a dry slope, often 0.2–0.5 for a wet one.

Design

Required FS

Factors below it are marked. Usually 1.3–1.5 for permanent slopes.

A homogeneous slope with level ground at the toe and the crest; circular slip surfaces cut into 30 slices, split at the toe and the crest.
Each method's critical circle is found by trying circles of 18 depths about each of 24 × 22 centers over the slope, then a finer grid about the best. Without a firm layer, circles go no deeper than H below the toe.
Pore pressure is taken through r_u = u/γh (Bishop and Morgenstern). The surcharge adds to the weight of the slices under the crest.
Fellenius leaves the forces between slices out; Bishop's simplified method takes them horizontal and is solved by iteration. Circles where Bishop's m_α falls below 0.2 are left out (Whitman and Bailey).
Forces per metre of slope; x is measured from the toe, into the slope, and y up from the toe.
The 3D is an illustration: the slope is analysed per metre, so Bishop's critical slip surface is drawn running its whole length, its arc and the mass over it on the ends, and the lines where it comes out of the ground along it.