Slope Stability: Method

Slope Stability

What the tool does

The tool performs the limit-equilibrium analysis of a two-dimensional slope the way the U.S. Army Corps of Engineers manual Slope Stability (EM 1110-2-1902, 2003) describes it: the soil above a trial slip surface is cut into vertical slices, the shear strength along the surface is divided by a factor of safety F so that the sliding mass is just in equilibrium, and F is found for the Ordinary Method of Slices, the Simplified Bishop method, the Simplified Janbu method (with Janbu's correction) and Spencer's method. A specified circle is analysed by all four; a search over a grid of centres and a set of tangent elevations finds the critical circle, which is then analysed by all four as well; a non-circular (polyline) surface is analysed by Janbu and Spencer; and the infinite-slope closed form covers shallow sliding parallel to the face. Every factor of safety is compared with the required value the analyst enters (EM 1110-2-1902 Table 3-1: 1.3 at the end of construction, 1.5 for long-term steady seepage, 1.4 for the maximum surcharge pool, 1.1 to 1.3 for rapid drawdown; FHWA NHI-06-088 Table 6-1: 1.25 to 1.5 for highway embankments) as a plain requirement. Civil practice puts the factor of safety on the strength, so the tool carries no load-level radio and no margin boxes.

Geometry, layers and loads

The ground surface is a polyline with x increasing (the slope may face either way; the tool finds the toe end of each slip surface from the elevations of its two ends and works internally with the toe at the left). The soil is up to five layers, each the material below its top boundary: a horizontal elevation or a polyline. A point belongs to the deepest layer whose boundary lies above it, the profile-line rule of the Corps' UTEXAS2 program, so a boundary that stops short lets the layer above fill the gap: a key trench is entered by stopping the sand boundary on either side of it. Unit weights are total unit weights everywhere, as the manual directs; an optional saturated unit weight applies below the piezometric line. Cohesion and friction angle are those of the soil at the bottom of each slice, which do not depend on the soils above it (EM F-2a(5)).

The piezometric line gives the pore pressure at the centre of each slice base, u = γ_w(h_p − y), zero above the line; where the line stands above the ground the water is an external hydrostatic load P on the slice top, applied normal to the top and entered in the equations and in the moment about the circle centre exactly as EM Figures F-3 and F-5 do, so a reservoir against the face is entered simply by raising the line. A uniform surcharge adds q times the covered width to the slice weights. A horizontal pseudo-static coefficient k_h puts a force k_hW toward the toe at the centroid of every slice: its moment about the circle centre enters the OMS and Bishop denominators, its component normal to the base enters the OMS normal force, and its horizontal component enters the Janbu and Spencer force equilibrium; the manual (paragraph C-3b) notes that the Simplified Bishop method is of doubtful suitability with such a force, so the Spencer value should be read beside it. A tension crack of the entered depth ends every slip surface where its depth below the ground first reaches that value walking down from the crest; the crack face carries no shear and no water (EM paragraph C-1; UTEXAS2 suggests 2c/(γ√K_a) as a guide to the depth).

Slices are cut at every vertex of the surface, of the layer boundaries and of the piezometric line, at every crossing of the slip surface with a boundary or the water line, and at the ends of the surcharge, and then subdivided to at least the entered count (30 by default; the manual's hand examples use 6 to 12). The base of a slice is the chord between the slip-surface points at its two edges, α its inclination (positive where the base rises toward the crest, the manual's Figure F-2 convention), Δℓ its length; the weight is Simpson's rule over the slice of the column weight from the slip surface to the ground, layer by layer, and the centroid height comes from the same integral. Units are plain numbers in the system of the radio (ft, lb/ft³, lb/ft² with γ_w = 62.4 lb/ft³; or m, kN/m³, kPa with 9.81 kN/m³); weights and forces are per unit thickness of slope.

The four methods

Ordinary Method of Slices (Fellenius 1936; EM eq. C-12 with the external water of C-14): side forces neglected, the normal force from the resolution of the effective weight perpendicular to the base, moments about the centre of the circle:

F = Σ[c Δℓ + (W cos α + P cos(α − β) − u Δℓ cos²α − k_hW sin α) tan φ] / [Σ W sin α + Σ k_hW (y_c − ȳ)/R − Σ M_P/R],

where β is the inclination of the slice top and M_P the moment of the external water load about the centre, positive when it resists. The manual keeps the OMS for reference: it can be as much as 20 % low with high pore pressures, it equals the rigorous methods when φ = 0, and it needs a circle (paragraph C-2).

Simplified Bishop (Bishop 1955; EM eqs C-15 and C-16): horizontal side forces, vertical force equilibrium of each slice and moment equilibrium about the centre:

F = Σ{[c Δx + (W + P cos β − u Δx) tan φ] / m_α} / [Σ W sin α + Σ k_hW (y_c − ȳ)/R − Σ M_P/R],   m_α = cos α + sin α tan φ / F,

solved by the fixed-point iteration of the manual's Figure F-1 from F = 1.5 until two successive values agree to 10⁻¹⁰ (the chart of F against iteration shows the sequence). The manual accepts it for circular surfaces (within about 5 % of complete equilibrium, Fredlund and Krahn 1977) and asks that major structures be checked by Spencer's method (paragraph C-3c).

Force equilibrium with a constant interslice inclination (EM eqs C-19 to C-21): the interslice force on the downslope side of each slice follows from horizontal and vertical equilibrium of the slice for a trial F, slice by slice from the crest, with the base shear S = (c Δℓ + N′ tan φ)/F, the pore force uΔℓ, the surface load P, the weight and the seismic force; F is the value for which the force on the downslope side of the last slice vanishes (found by bracketing from the Bishop value and bisection). With θ = 0 this is the Simplified Janbu method, which underestimates F because horizontal side forces ignore moment equilibrium; the tool multiplies it by Janbu's (1973) correction factor f₀ = 1 + b₁[d/L − 1.4 (d/L)²], the common polynomial fit of his chart, with b₁ = 0.69 for c = 0, 0.31 for φ = 0 and 0.50 for a c-φ soil, d the greatest depth of the slip surface below its chord and L the chord length; an entered f₀ replaces the fit, 1.0 giving the uncorrected value. With θ equal to the average embankment slope the same recursion is the Corps' Modified Swedish method of EM Figures F-4, F-5, F-12 and F-18, and the tool reproduces those hand calculations (see the test).

Spencer (Spencer 1967, extended to any surface by Wright 1969; EM paragraph C-5): all side forces parallel at an inclination θ that is solved for, with the normal force at the centre of each base. For a trial θ the force-equilibrium factor F_f is the root of the recursion above and the moment-equilibrium factor F_m the root of the moment of all external forces (weights, seismic forces, surface loads, base normal and shear forces at the base midpoints) about the circle centre, or about a reference point above a non-circular mass, with the base forces from the same recursion; θ is adjusted by bisection, scanning outward from zero to ±60°, until F_f = F_m. Both residuals vanish at the solution, so it does not depend on the reference point; the chart shows F_f and F_m at each bisection step. θ is reported positive when the side force rises toward the crest, the manual's convention (UTEXAS2 prints the opposite sign for a slope facing right).

Circle search

The search follows the Corps' practice of centres on a grid combined with circles tangent to a set of elevations (R = y_c − y_t). The bounds are automatic from the geometry (centres from 0.3 of the slope's horizontal length beyond the toe and the crest, 0.2 H to 2.5 H above the crest; tangent elevations 0.6 H below to 0.6 H above the lowest surface point, H the slope height) or entered; the tangent set always includes every layer-boundary elevation and the lowest surface point inside the range, because the critical circle of a layered slope usually follows the weaker layer. Every circle is ranked by the search method (Simplified Bishop by default; OMS or uncorrected Janbu may be chosen), the three lowest distinct coarse minima are each refined by three rounds of a 5 × 5 × 5 grid at half the spacing inside the bounds, and the critical circle is analysed by all four methods. The table of the ten lowest lists the refined minimum and then the lowest coarse-grid circles with distinct centres, so that a second valley of nearly equal F (a toe circle beside a deep circle) is visible; the figure colours every centre by its lowest F and draws the ten circles. A minimum that sits at a bound is reported, since the true minimum may lie outside the grid: widen the bound, or enter the rock elevation as the lowest elevation if deeper circles are not physically possible. An optional floor rejects circles that reach below an entered elevation. The manual warns (paragraph C-1g) that a shallow infinite-slope mechanism in a cohesionless face may give the lowest F of all while deeper surfaces matter more; the tool's infinite-slope check covers it.

Non-circular surfaces and the infinite slope

A polyline slip surface is extended in the direction of its end segments to the ground on both sides, sliced the same way, and analysed by the force-equilibrium (Janbu) and Spencer methods, which admit any shape; OMS and Bishop need a circle. For a thin layer over firm material or a cohesionless face the plane parallel to the face at depth z gives F = [c + (σ − u) tan φ]/τ with σ = W_vcos²β − k_hW_v sin β cos β, τ = W_v sin β cos β + k_hW_vcos²β, u = m z γ_wcos²β and W_v = z[(1 − m)γ + m γ_sat], the water table standing m z above the plane with seepage parallel to the face (FHWA NHI-06-088 eq. 6-11; EM eqs C-25 to C-29 are its c = 0 and r_u forms; the USGS TRIGRS program uses the same expression cell by cell). The soil is that of layer 1.

Checks

The test script reproduces EM 1110-2-1902's Appendix F hand calculations from their geometry and from their printed slice values (Figure F-11: F = 1.33 by Bishop with the manual's ten slices; Figure F-12: a force imbalance of 75 kips at F = 1.6 and F = 1.36 by the Modified Swedish method at θ = 19.7°; Figures F-17 and F-18: F = 2.01 and 2.07 with external water and effective side forces, to 1 % because the manual prints the base inclinations to whole degrees), the UTEXAS2 user's manual's Example 1 (Spencer F = 2.899 at θ = 20.53° with a 6.3 ft tension crack, reproduced to 0.1 % and 0.01°) and Example 3 (six critical circles by Spencer and Bishop to 0.3 %, and the search recovering the manual's critical tangent circle), the FHWA NHI-06-089 16-slice OMS table (FS 1.36), the EM Figure E-13 infinite-slope chart example (1.63 against the closed form's 1.65) and the closed-form identities, and ten seeded random slopes against an independent numpy implementation to 10⁻⁶. The manual's Figures F-11 and F-12 print their width, height and base-length columns in reverse row order; the test records it.

Not covered

Three-dimensional effects, anisotropic or nonlinear (curved-envelope) strengths, strength increasing with depth as a profile (enter layers), rapid-drawdown staged strengths (EM Appendix G), reinforcement and anchors, water in the tension crack, Morgenstern–Price interslice functions (Spencer is the constant-ratio case), probabilistic analysis (ETL 1110-2-556), seepage analysis (the piezometric line is entered), and the "Lowe and Karafiath" and "Corps' Modified Swedish" interslice assumptions as separate results (the tool's Janbu value is the θ = 0 member of that family and Spencer solves θ).