Footing Settlement on Clay — Boussinesq Stress to Consolidation Settlement
A square footing bears on sand above a 10-ft layer of normally consolidated clay. The settlement depends on the stress increase the footing causes in the clay, and that stress drops by a factor of four from the top of the layer to the bottom. We compute it with Newmark's solution, compare the mid-layer shortcut, Simpson's rule, five sublayers and the 2:1 method, and see how much each choice moves the answer.
The site
| Footing | 8 × 8 ft square, base 3 ft below grade, net bearing pressure q = 3,000 psf |
| 0–10 ft | Medium dense sand, γ = 120 pcf (treated as incompressible) |
| Groundwater | At 10 ft, the top of the clay |
| 10–20 ft | Normally consolidated clay, γsat = 115 pcf, Cc = 0.30, e0 = 0.85 |
| Below 20 ft | Dense sand (incompressible) |
Boussinesq tool: Rectangular, q 3000, B 8, L 8, x = y = 0, z = 7, 12 and 17 ft (depths below the footing base).
Open the Boussinesq calculator →Measure z from the base, not the ground surface
Four corner rectangles, Newmark's factor
z = 12 ft: m = n = 0.333 → Icorner = 0.0447 → Δσ = 4 × 0.0447 × 3,000 = 537 psf
| Point in clay | z (ft) | I = Δσ/q | Δσ Boussinesq (psf) | Δσ 2:1 (psf) |
|---|---|---|---|---|
| Top | 7 | 0.4021 | 1,206 | 853 |
| Middle | 12 | 0.1789 | 537 | 480 |
| Bottom | 17 | 0.0968 | 290 | 307 |
At the middle of the clay (15 ft below grade)
Sc = CcH/(1 + e0) · log[(σ′0 + Δσ)/σ′0]
Mid-layer stress: 19.46 × log(1,999.8/1,463) = 19.46 × 0.1357 = 2.64 in
Simpson average: 19.46 × log(2,070.3/1,463) = 19.46 × 0.1508 = 2.93 in
| Sublayer (depth below grade) | z (ft) | Δσ (psf) | σ′0 (psf) | S (in) |
|---|---|---|---|---|
| 10–12 ft | 8 | 1,008 | 1,253 | 0.998 |
| 12–14 ft | 10 | 723 | 1,358 | 0.721 |
| 14–16 ft | 12 | 537 | 1,463 | 0.528 |
| 16–18 ft | 14 | 412 | 1,568 | 0.394 |
| 18–20 ft | 16 | 324 | 1,673 | 0.299 |
| Total, five 2-ft sublayers | 2.94 | |||
Which shortcut is good enough?
| Stress method | Settlement (in) | vs sublayers |
|---|---|---|
| 2:1 at mid-layer (480 psf) | 2.40 | −18% |
| Boussinesq at mid-layer (537 psf) | 2.64 | −10% |
| Boussinesq, Simpson average (607 psf) | 2.93 | −0.2% |
| Boussinesq, 5 sublayers | 2.94 | — |
Tools used in this example
Boussinesq stress distribution and consolidation settlement. Guide: How to calculate stress below a footing.
Frequently asked questions
Which stress increase should I use for consolidation settlement?
The average over the compressible layer. Using only the mid-layer value underestimates the average when the layer is thick compared with the footing, because the stress curve is concave. Simpson's rule, Δσav = (Δσt + 4Δσm + Δσb)/6, or splitting the layer into sublayers gets close to the exact answer. Here mid-layer gave 2.64 in, Simpson 2.93 in and five sublayers 2.94 in.
Should I use gross or net footing pressure?
Net: the gross bearing pressure minus the weight of soil removed for the footing. Only the increase over the existing overburden causes consolidation. In this example the net pressure is 3,000 psf.
Is the 2:1 method conservative for settlement?
Not under the center of a footing over a deep layer. Here it gave a mid-clay stress of 480 psf against 537 psf by Boussinesq, and a settlement of 2.40 in against 2.94 in — 18% low. It is better treated as an average-stress estimate.
What is the consolidation settlement equation for normally consolidated clay?
Sc = Cc·H/(1 + e0)·log10[(σ′0 + Δσ)/σ′0], where Cc is the compression index, H the layer thickness, e0 the initial void ratio and σ′0 the initial effective stress at the middle of the layer (or sublayer). For overconsolidated clay use the recompression index Cr up to the preconsolidation stress.