SlipSurface IQ Settle — theory
What SlipSurface IQ Settle computes, and where the method stops being valid. Depth z is measured from the ground surface; z_b = z − Df from the foundation base.
1. In-situ stresses
σv0(z) = Σ γ·Δz above the water table and Σ γsat·Δz below it; u0 = γw·(z − z_w) below the water table; σ'v0 = σv0 − u0. The last layer is taken as continuing below the profile for the stress calculation only. In a cohesive layer σ'p = OCR·σ'v0 (OCR constant per layer; OCR < 1 is not supported and is set to 1 with a warning).
2. Net pressure
q_net = q − σv0(Df) when the excavated overburden is deducted, q otherwise. Once the pore pressures have equilibrated the effective stress change equals the total stress change, so the same q_net drives both the elastic and the consolidation settlement. A zero or negative q_net is a compensated foundation: nothing settles.
3. Stress increase
Rectangle — Newmark's integration of Boussinesq under the corner of an a × b area, m = a/z, n = b/z:
I = 1/(4π) · [ 2mn√(m²+n²+1)/(m²+n²+1+m²n²) · (m²+n²+2)/(m²+n²+1)
+ atan2(2mn√(m²+n²+1), m²+n²+1−m²n²) ]
atan2 takes the branch that the textbook formula corrects by adding π. Any plan point (inside, on the edge or outside) is the signed sum of four corner rectangles.
Strip — Δσ/q = [θ₁ − θ₂ + sinθ₁cosθ₁ − sinθ₂cosθ₂]/π with θ = atan((x ± B/2)/z).
Circle — integrating Boussinesq over the radius leaves Δσ/q = (1/2π) ∮ [g(r₁) − g(r₂)] dθ, g(r) = z³/(z² + r²)^{3/2}, where r₁, r₂ are the distances at which the ray from the point in direction θ enters and leaves the circle. Both are explicit for a circle, so the integral is one-dimensional and exact up to the quadrature (720 steps); at the centre the closed form 1 − (z²/(z²+R²))^{3/2} is used.
2:1 — q·B·L/((B+z)(L+z)), q·B/(B+z), q·D²/(D+z)²: an average over the widened area, the same at every point, so under 2:1 the consolidation settlement does not vary in plan.
Embankment — a long fill on the ground surface: crest width b, height H, slope angles β_L, β_R, unit weight γ. The load is p(x) = γH under the crest, falling linearly to zero over the slope runs H/tan β. For a segment where p = A + Bξ, Flamant's line load 2p z³/(π((ξ−x)² + z²)²) integrates, with u = ξ − x, to
Δσ = (A + Bx)·[atan(u/z) + uz/(u² + z²)]/π − B·z³/(π(u² + z²))
between the ends of the segment; the embankment is the sum of its three segments, so the stress is exact at any point (plane strain). Under 2:1 the fill is replaced by the uniform strip of the same load, of width b + (run_L + run_R)/2. The evaluation points are the crest centre, the crest edge, the middle of the (right) slope and its toe. The elastic settlement superposes plane-strain Steinbrenner strips — the crest as one, each slope as 16 slices carrying the load at their middle (converged to 0.2 %). A fill is flexible; Schmertmann's footing diagram is not applied to it, and its angular distortion is not checked. The fill's own compression, undrained lateral spreading and stability are outside the program.
4. Influence depth
The profile below the base is cut into sublayers (0.25 m by default). Settlement is summed down to the bottom of the deepest sublayer where Δσ at the centre is still at least the chosen fraction of σ'v0 (0.1 by default; 0 sums the whole profile). The base of the profile is incompressible; the program warns when Δσ at the base is still ≥ 0.1·σ'v0.
5. Immediate settlement
Elastic (Steinbrenner). Under the corner of a flexible a × b rectangle on an elastic layer of thickness H over a rigid base, with M = L/B, N = H/B (B the shorter side):
s = q·B/E · [ (1 − ν²)·F1 + (1 − ν − 2ν²)·F2 ]
F1 = (A0 + A1)/π, F2 = N/(2π)·atan(A2)
A0 = M·ln[(1+√(M²+1))·√(M²+N²) / (M·(1+√(M²+N²+1)))]
A1 = ln[(M+√(M²+1))·√(1+N²) / (M+√(M²+N²+1))]
A2 = M / (N·√(M²+N²+1))
The settlement of a layer between z₁ and z₂ below the base is q/E times the difference of the bracket at the two depths (the layered method), each layer with its own E and ν, and any point is superposed from four corners. A strip is a rectangle 200 B long; a circle is the square of the same area, its points placed at the same fraction of the half-width. Clay layers use their undrained E and ν (≈ 0.5), which makes this the undrained distortion settlement that precedes consolidation.
Schmertmann et al. (1978), granular layers only:
s = C1·C2·q_net·Σ (Iz/E)·Δz
C1 = 1 − 0.5·σ'v0(Df)/q_net ≥ 0.5, C2 = 1 + 0.2·log10(t/0.1 yr)
Izp = 0.5 + 0.1·√(q_net/σ'vp), σ'vp at the depth of the peak
The influence diagram runs from Iz = 0.1 at the base to Izp at B/2 and zero at 2B for L/B = 1, from 0.2 to Izp at B and zero at 4B for L/B ≥ 10, and is interpolated linearly in L/B between. The result is the footing's settlement; at the other points it is scaled by the ratio of the elastic settlement of the same granular sublayers at that point to the one at the centre. E is entered directly (Schmertmann suggests E = 2.5·qc for axisymmetric and 3.5·qc for plane-strain loading).
6. Primary consolidation
For each clay sublayer, at each evaluation point, with σ'f = σ'v0 + Δσ:
Δe = Cr·log10(min(σ'f, σ'p)/σ'v0) + Cc·log10(max(σ'f, σ'p)/σ'p)
s = Σ Δe/(1 + e0) · Δz
No Skempton–Bjerrum correction is applied; the one-dimensional value is conservative for over-consolidated clay under a narrow footing.
7. Time
Terzaghi's average degree of consolidation for a uniform initial excess pore pressure, U = 1 − Σ 2/M²·exp(−M²Tv), M = π(2m+1)/2, with √(4Tv/π) for Tv < 10⁻³; Tv = cv·t/H_dr², H_dr = H/2 for double and H for single drainage, H the thickness of the layer below the base. Tv(U) is found by bisection, so U(Tv(U)) = U exactly. Each clay layer consolidates independently; interaction between layers through a common drainage boundary is ignored. A clay without cv is taken as consolidating at once.
8. Secondary compression
From the end of primary consolidation, taken as U = 95 % (t_p), to the design life: s = Cα/(1 + e0)·Δz·log10(t/t_p). Since Cα/Cc is a soil constant (Mesri), where σ'f stays below σ'p the clay creeps along the recompression line and Cα·Cr/Cc is used instead. e0 is used for (1 + e_p).
9. Rigid foundation, distortion, checks
A rigid foundation settles uniformly by the settlement of its characteristic point (Grasshoff: 0.74 of the half-widths from the centre in a rectangle or strip, 0.845·R in a circle). The angular distortion of a flexible foundation is β = |s_centre − s_edge| / (B/2) (R for a circle); for a rigid one it is not checked. The total settlement of the governing point (centre, or characteristic point) and β are checked against their allowable values.
10. Studies
Inputs vary as a range (uniform in LHS / Monte Carlo, n evenly spaced points in a one-at-a-time sweep) or a distribution (normal; lognormal with the given mean and CoV; uniform over mean ± √3·σ). Exceedance probabilities P = k/n carry a Wilson 95 % interval; β = −Φ⁻¹(P), reported as "> −Φ⁻¹(3/n)" when nothing exceeded. Sensitivities are Spearman rank correlations.