How to Calculate Stress Below a Footing — Boussinesq, Newmark & 2:1
Settlement calculations need the increase in vertical stress the footing causes at depth. This guide covers the inputs that are easy to get wrong (net pressure, depth below the base), which elastic solution to use for each load shape, how Newmark's corner factor gives the stress at any point by superposition, and when the 2:1 method is good enough. Examples use a 10 × 10 ft footing at 2,000 psf, as in the free Boussinesq calculator.
Use the net pressure
Only the increase above the existing overburden compresses the soil. A footing 3 ft down in 120-pcf soil already "removes" 360 psf; using gross pressure overstates the stress increase and the settlement.
Measure depth from the footing base
Boussinesq's solution is for a load on the surface of a half-space, so z is measured from the bearing level, not from grade. The initial effective stress σ′0, on the other hand, is measured from the ground surface and uses buoyant unit weight below the water table. Mixing the two is a classic exam trap.
Pick the elastic solution for the load shape
| Load | Solution | Example |
|---|---|---|
| Point (column on a large mat, wheel) | Δσ = 3Pz³/[2π(r² + z²)5/2] | P = 10 kip, z = 10 ft below: 47.7 psf |
| Line (wall, rail) | Δσ = 2qz³/[π(x² + z²)²] | — |
| Strip footing | (q/π)[θ + sinθcosθ] between the edges | z = B below centerline: 0.55q |
| Rectangular footing | Newmark corner factor × superposition | see Step 4 |
| Circular (tank) | center: q[1 − (1 + (R/z)²)−3/2] | z = R: 0.646q |
Rectangles: corner factor and superposition
Corner: one 10 × 10 rectangle, m = n = 1.0 → I = 0.1752 → Δσ = 350 psf
Outside, 10 ft from center on the centerline: two 15 × 5 rectangles (m = 1.5, n = 0.5, I = 0.1314) minus two 5 × 5 (I = 0.0840) → Δσ = 2(2,000)(0.1314 − 0.0840) = 189 psf
Compare with the 2:1 method
The 2:1 value is an average over a spreading area, so it is reasonable for average settlement of a flexible footing but not for the maximum stress under the center.
Average over the compressible layer
Stress drops quickly with depth, so one mid-layer value underestimates the average in a thick layer. Use Simpson's rule, Δσav = (Δσtop + 4Δσmid + Δσbot)/6, or split the layer into sublayers and add their settlements.
Feed the settlement calculation
For normally consolidated clay, Sc = CcH/(1 + e0)·log[(σ′0 + Δσ)/σ′0]. The footing settlement worked example shows how the averaging choice moves the answer from 2.40 in (2:1) to 2.94 in (sublayers).
Point, line, strip, rectangle at any (x, y), or circle; Boussinesq vs 2:1 and Westergaard, with a stress-versus-depth plot.
Open the Boussinesq calculator →Worked examples
Frequently asked questions
What is Newmark's influence factor?
The ratio Δσz/q directly below the corner of a uniformly loaded rectangle, as a function of m = B/z and n = L/z. It comes from integrating Boussinesq's point-load solution over the rectangle. Because it is for a corner, any other point is handled by adding and subtracting rectangles that have a corner at the point.
How do I get the stress under the center of a rectangular footing?
Divide the footing into four equal quarters meeting at the center. Each quarter has a corner at the point with m = (B/2)/z and n = (L/2)/z. The center stress is 4 × q × I(m, n).
How deep does a footing influence the soil?
Under the center of a square footing the stress increase drops to about 10% of the applied pressure at roughly 2B below the base; under a long strip it takes about 6B. That depth range is where settlement mostly comes from.
Is the 2:1 method acceptable?
For quick estimates of average stress, yes. It underestimates the stress below the center at shallow to moderate depths (0.25q vs 0.34q at z = B for a square) and overestimates it below corners and edges.