Theory

What is behind the analyses

A short account of the methods, with the assumptions Soilicon makes. For how well they match published results, see Verification.

1. Conventions

• Units are SI throughout: m, kN, kPa, kN/m³, degrees. Nothing is converted internally.
• x points right and y is elevation (up). Angles are counter-clockwise from +x.
• Stress and strain are tension positive. Pore pressure is positive as a pressure.
• Coordinates are on a 1 mm grid, so the same model always gives the same result.
• Every result value carries its unit and a note on its sign.

2. Limit equilibrium with slices

The soil above a trial slip surface is divided into vertical slices. Soilicon adds a slice boundary at every vertex of the ground, the layer boundaries and the water line, so each slice base lies in one material. The factor of safety F is the factor by which the shear strength has to be reduced to bring the mass into limit equilibrium. The strength on the base of slice i is Mohr–Coulomb in effective stress:

$$\tau_f = c' + (\sigma_n - u)\tan\varphi', \qquad S_{m,i} = \frac{\tau_f\, l_i}{F}$$

Here l_i is the base length and u is the pore pressure at the base. u comes from a phreatic line, from r_u, or from a seepage result (§7). Undrained soil uses total stress with τ_f = c_u.

Each slice carries its weight W_i, which uses γ above the phreatic line and γ_sat below it. It can also carry external loads (strip and line loads), reinforcement forces, and the pseudo-static forces k_hW (horizontal, at the slice's centre of gravity) and k_vW (vertical). The methods differ in which equilibrium conditions they satisfy and in what they assume about the forces between slices.

3. Simplified Bishop (slope.bishop)

Bishop (1955) works on circles. It satisfies moment equilibrium about the circle's centre and the vertical force equilibrium of each slice, and it neglects the interslice shear (X = 0). The base normal force then follows from vertical equilibrium, and moment equilibrium gives F:

$$F = \frac{\displaystyle\sum_i \frac{c' b_i + (W_i - u_i b_i)\tan\varphi'}{m_{\alpha,i}}}{\displaystyle\sum_i W_i \sin\alpha_i}, \qquad m_{\alpha,i} = \cos\alpha_i + \frac{\sin\alpha_i \tan\varphi'}{F}$$

This is the dry form without loads. Soilicon's implementation also includes ponded water, external loads (each with its own moment arm), reinforcement and seismic forces. In the seismic case the weight becomes W(1 − k_v) and the moment k_hW(y_c − y_g) is added. F appears on both sides, so it is found by iteration. The result also reports the Fellenius (ordinary method of slices) value for the same surface.

• The m_α guard. At steep exit angles near the toe, m_α tends to zero and the base normal force blows up. Soilicon rejects surfaces with m_α ≤ 0 rather than reporting a meaningless F, and reports the smallest m_α of the critical surface. The non-circular search rejects trial surfaces with m_α < 0.2.
• Pieces of the slip mass. A circle may cut the ground more than once. Each piece of soil above it is assessed on its own, and the lowest F is reported.
• Rigid regions. Walls and other rigid regions can be made impenetrable, so that no searched circle crosses them.

4. General limit equilibrium (slope.gle)

The general limit equilibrium formulation (Fredlund & Krahn 1977) satisfies both force and moment equilibrium. The interslice shear is tied to the normal force by an assumed shape f(x) and a scale λ:

$$X = \lambda\, f(x)\, E$$

For each λ, two factors of safety are found: F_f from overall horizontal force equilibrium and F_m from moment equilibrium. The solution is the λ at which they are equal. Soilicon finds it with Brent's method and solves each slice in turn, as in Zhu et al. (2005), which avoids the inner iteration of the original formulation.

• Spencer (1967): f(x) = 1, so all interslice forces are parallel.
• Morgenstern–Price (1965): f(x) is a half-sine over the slip mass.

GLE works on circles and on arbitrary polylines, which is what thin weak layers need. At λ = 0, F_m on a circle is exactly simplified Bishop. Soilicon checks this identity to 10⁻⁹. The result reports λ, the force and moment residuals, and the interslice forces E and X of every slice, so you can judge whether a solution is physically acceptable (for example, no tension between slices).

5. Finding the critical surface

• Circles (Bishop):
  1. A grid of centres is set up, with several radii per centre, between the shallowest cut and tangency with the model base. Each face of a benched slope is searched separately.
  2. The best local minima of the coarse grid are refined, then polished with Nelder–Mead.
  3. When the critical centre lies on the edge of the search box, the box is extended automatically.
• Non-circular (GLE):
  1. The search starts from the Bishop critical circle, and from seeds along weak layers.
  2. The vertices of a polyline are moved along the ground and up and down.
  3. Every trial goes through the same slip-mass code as a fixed surface, so the reported critical polyline reproduces exactly when you run it as a fixed surface.

No search is guaranteed to find the global minimum. Check the critical surface visually, and try fixed surfaces where you expect a mechanism.

6. Steady seepage (seepage.steady)

Darcy's law with mass conservation gives, for steady flow in the cross-section:

$$\nabla \cdot \big(k_{rel}(\psi)\, \mathbf{K}\, \nabla h\big) = 0, \qquad h = y + \frac{u}{\gamma_w}, \qquad \mathbf{K} = \operatorname{diag}(k_x, k_y)$$

The domain is meshed into quadratic triangles (T6) by gmsh, and the unknown is the pore pressure u. Head h and pressure head ψ = u/γ_w follow from it.

• Confined flow: k_rel = 1, so the equation is linear.
• Unconfined flow (a free surface): the soil above the phreatic line still conducts a little, with k_rel = max(10⁻⁴, 10^4ψ/ψ_k) for ψ < 0. The problem is solved by Picard iteration with relaxation. The phreatic line is the contour ψ = 0 of the result. A smaller ψ_k gives a sharper free surface at a higher cost.
• Boundary conditions are placed by position on the surface, left, right or base:
  • prescribed total head
  • no flow (the default)
  • seepage face: u = 0 where water leaves the domain, and no flow elsewhere, iterated inside the Picard loop
  A water condition (a reservoir level) can generate these boundaries for you.
• Outputs:
  • discharge in and out
  • flow through user-defined sections
  • the phreatic line
  • active seepage faces
  • the maximum hydraulic gradient
  • pore pressure and head fields
  The mass balance is checked on every run.

7. From seepage to stability

A water condition can take its pore pressures from a seepage result instead of a drawn phreatic line. Bishop and GLE then interpolate u from the FEM field at each slice base. The slope analysis's input includes the seepage input, so any change to the seepage model marks the slope result as out of date. A slope analysis refuses to run while the seepage result it needs is missing.

8. References

1. Bishop, A.W. (1955). The use of the slip circle in the stability analysis of slopes. Géotechnique 5(1), 7–17.
2. Fredlund, D.G. & Krahn, J. (1977). Comparison of slope stability methods of analysis. Canadian Geotechnical Journal 14(3), 429–439.
3. Morgenstern, N.R. & Price, V.E. (1965). The analysis of the stability of general slip surfaces. Géotechnique 15(1), 79–93.
4. Spencer, E. (1967). A method of analysis of the stability of embankments assuming parallel inter-slice forces. Géotechnique 17(1), 11–26.
5. Zhu, D.Y., Lee, C.F., Qian, Q.H. & Chen, G.R. (2005). A concise algorithm for computing the factor of safety using the Morgenstern–Price method. Canadian Geotechnical Journal 42(1), 272–278.
6. Abramson, L.W., Lee, T.S., Sharma, S. & Boyce, G.M. (1996). Slope Stability and Stabilization Methods. Wiley.
7. Harr, M.E. (1991). Groundwater and Seepage. Dover.
8. Polubarinova-Kochina, P.Ya. (1962). Theory of Ground Water Movement. Princeton University Press.
9. Casagrande, A. (1937). Seepage through dams. Journal of the New England Water Works Association 51(2).