SlipSurface IQ LE — theory

This is what SlipSurface IQ LE actually computes. It is written so that a reviewer can check the numbers by hand.

1. Coordinates and sign conventions

x increases to the right and y is elevation. Internally the solver works in a canonical frame in which the crest is on the right and the sliding mass moves towards −x. A model drawn the other way round is mirrored (x → −x) on input and every result is mirrored back, so the user never sees the internal frame. "direction" in the analysis options forces the choice for geometries that are ambiguous, such as a two-sided embankment.

For each slice:

Symbol	Meaning
b, l	slice width and base length, l = b / cos α
α	base inclination from the +x axis, measured on the chord between the two base corners. Positive under the crest, negative near the toe
W	total slice weight, including surcharge and any ponded water above the ground surface
u	pore water pressure at the mid-point of the base
c, φ	strength parameters of the material immediately below the base
N	total normal force on the base
S_m	mobilised shear on the base, S_m = (c l + (N − u l) tan φ) / F
E, X	horizontal and vertical interslice forces, both counted as the force exerted by the material on the left of a boundary on the material to its right, E positive in compression and X positive upwards

Driving quantities are counted positive. For a circular surface the driving moment of the weights about the centre of rotation is Σ W (x_base − x_centre) and the driving horizontal force is Σ N sin α + Σ k_h W.

2. Slice equilibrium

Vertical equilibrium of a slice, with the vertical seismic coefficient k_v acting downwards:

N cos α + S_m sin α = W (1 + k_v) + (X_right − X_left)

Substituting the Mohr-Coulomb shear and solving for N:

N = [ W (1 + k_v) + (X_right − X_left) − (c l − u l tan φ) sin α / F ] / m_α
m_α = cos α + sin α tan φ / F

m_α below 0.2 is reported as poorly conditioned; below 0.02 it is clamped so a single bad slice cannot produce a meaningless factor of safety. The effective normal force N − u l is clipped at zero, which is the usual treatment of tensile slices.

Horizontal equilibrium of the same slice gives the interslice normal force by recursion from E = 0 at the toe end:

E_{i+1} = E_i − N sin α + S_m cos α − k_h W + Q_x,i

where Q_x,i is any external horizontal point force applied within that slice (reinforcement, water thrust in a tension crack). The interslice shear follows the assumed force function,

X_i = λ f(x_i) E_i ,   X = 0 at both ends

with f(x) = 1 for Spencer and a half sine, constant or trapezoid for Morgenstern-Price. f(x) is fixed at the chord inclination for Corps of Engineers #1 and at the average of ground and base inclination for Lowe-Karafiath, both with λ = 1.

3. Factor of safety

Moment equilibrium about the axis (x_O, y_O):

F_m = Σ S_avail ρ / [ Σ M_driving + M_external − Σ N γ ]

ρ = (x_b − x_O) sin α − (y_b − y_O) cos α          ( = R for a circle )
γ = (x_b − x_O) cos α + (y_b − y_O) sin α          ( = 0 for a circle )
M_driving = W (1 + k_v)(x_b − x_O) + k_h W (y_O − y_cg)
S_avail = c l + (N − u l) tan φ

For a circular surface the base shear acts at radius R and the normal force passes through the centre, so this reduces to the familiar F = Σ S_avail R / Σ W (x_b − x_c). For a non-circular surface the axis is the centre of the circle fitted through the first, middle and last points of the surface, and the result depends on that choice unless force equilibrium is also satisfied — which is why Spencer or Morgenstern-Price should be used there.

Horizontal force equilibrium of the whole mass:

F_f = Σ S_avail cos α / [ Σ N sin α + Σ k_h W − Q_x ]

Both are solved by fixed-point iteration with under-relaxation, starting from the factor of safety found during the search.

Spencer and Morgenstern-Price solve both at once. For a trial λ the interslice forces are iterated to convergence and F_m(λ) and F_f(λ) are computed; λ is then found by bisection on F_m − F_f, seeded by a scan over λ ∈ [−0.6, 1.4]. Repeated evaluations (during surface optimisation) use a secant iteration warm-started from the previous λ. If the two curves never cross, there is no Spencer solution and SlipSurface IQ LE says so instead of reporting a number.

Ordinary / Fellenius ignores interslice forces completely and resolves the weight (and seismic force) perpendicular to the base:

N = W (1 + k_v) cos α − k_h W sin α

Janbu corrected multiplies the simplified force solution by

f0 = 1 + b1 [ d/L − 1.4 (d/L)² ]

with d the maximum depth of the surface below its chord, L the chord length, and b1 = 0.69 for φ = 0, 0.31 for c = 0 and 0.50 for c-φ soils.

4. Weights, water and loads

The weight of a slice is integrated as a column at the slice mid-point: the column is split at every material boundary and at the water table, and each piece uses the saturated unit weight when it lies below the water table and the bulk unit weight above it. Surcharge pressure over the slice width is added to the weight, and so is any water standing above the ground surface.

Pore pressure at the base is u = γ_w (y_water − y_base) from the piezometric line, or u = r_u σ_v when the material carries a pore pressure ratio, with σ_v the total vertical stress of the column above the point. r_u takes precedence over the water table for that material.

Seismic loading is pseudo-static: a horizontal inertia force k_h W acting down-slope and a vertical force k_v W acting downwards. Surcharges only contribute inertia when "include_in_seismic" is set.

A tension crack truncates the crest end of the surface at the depth given, or at z = (2c/γ) tan(45° + φ/2) when "auto" is set. Water in the crack applies a horizontal thrust ½ γ_w h_w² at one third of the water depth above the base of the crack, in the driving direction.

Reinforcement applies its capacity as a force at the point where the slip surface crosses the element, directed along the element towards whichever end lies outside the sliding mass. The force enters the slice it acts in, the global force balance and the moment balance.

5. Searching for the critical surface

The default search is grid-and-tangent: circle centres are taken from a rectangular grid above the slope, and each centre is combined with a set of tangent elevations to give the radii. A trial circle is rejected when it does not daylight at both ends, when the arc rises above the ground surface between its ends, when it enters an impenetrable material, when it is shallower than the minimum depth, or when the chosen method does not converge on it. Where a circle cuts the ground more than twice — a benched slope — the largest valid arc is used.

The box adapts: if the best trial of a pass sits on the edge of the grid or at the extreme tangent elevation, that side is extended and the pass repeats, up to three times. This is what finds the deep foundation failures of an embankment on soft clay without the user setting anything. The grid is then refined around the best centre and tangent for a few passes, each pass covering one cell of the previous one.

Searching is done with a single method (Bishop by default, 25 slices) and every requested method is then evaluated on the resulting critical surface; the report says so. "search_each_method": true repeats the whole search for each method instead.

Non-circular optimisation starts from the critical circle, resamples it to 20 vertices, and moves each vertex in turn — interior vertices vertically, the two ends along the ground surface — keeping any move that lowers the factor of safety. The step halves when a pass makes no progress. Trials are ranked with Spencer, warm-started from the previous λ, because ranking non-circular surfaces with a moment-only method just walks the surface towards whatever moment axis flatters it.

A trial surface must stay below the ground and above any impenetrable material, its segments must be flatter than 80°, and its base inclination must increase monotonically from toe to crest (concave upwards). Without that last condition the optimiser invents zig-zag surfaces with a meaningless low factor of safety. The optimised surface is reported only if it beats the circular one on the same basis.

6. Numerical defaults

Slices for the final analysis	50 (boundaries are placed at every profile vertex, material boundary, water-table vertex and surcharge edge first, then the remaining width is divided evenly)
Slices during the search	25
Iteration tolerance on F	1e-6 relative, 80 iterations, relaxation 0.6
λ scan	15 points over [−0.6, 1.4], then bisection to 1e-5
Search grid	12 × 12 centres × 12 tangents, 3 refinement passes

References

Bishop, A. W. (1955). The use of the slip circle in the stability analysis of slopes. Géotechnique 5(1), 7-17.

Fellenius, W. (1936). Calculation of the stability of earth dams. Transactions, 2nd Congress on Large Dams, Washington DC.

Fredlund, D. G. and Krahn, J. (1977). Comparison of slope stability methods of analysis. Canadian Geotechnical Journal 14(3), 429-439.

Janbu, N. (1973). Slope stability computations. In Embankment Dam Engineering, Casagrande Volume, Wiley.

Krahn, J. (2003). The 2001 R. M. Hardy Lecture: The limits of limit equilibrium analyses. Canadian Geotechnical Journal 40(3), 643-660.

Morgenstern, N. R. and Price, V. E. (1965). The analysis of the stability of general slip surfaces. Géotechnique 15(1), 79-93.

Spencer, E. (1967). A method of analysis of the stability of embankments assuming parallel inter-slice forces. Géotechnique 17(1), 11-26.

Taylor, D. W. (1937). Stability of earth slopes. Journal of the Boston Society of Civil Engineers 24, 197-246.