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Boussinesq Stress Distribution Calculator — Vertical Stress Below Footings

Compute the vertical stress increase Δσ_z at any depth below a point load, a line load, a strip footing, a rectangular footing or a uniformly loaded circle. Rectangles use the exact Newmark corner solution with superposition, so the point can be under the center, a corner, an edge or outside the footing. Results are compared with the 2:1 approximation (and Westergaard for point loads) and plotted against depth for settlement calculations.

Units

Surface load

Point load, Plb

Line load, qlb/ft

Applied pressure, qpsf (net, above existing)

Width, Bft

Length, Lft

Radius, Rft

Horizontal offset, xft — from load / center, across B

Horizontal offset, yft — from center, along L

Depth below load, zft

Chart depth (optional)ft

Vertical stress increase, Δσ_z—psf

Influence factor I = Δσz/q——

Ratio to surface pressure—

2:1 method (average)—psf

Westergaard (ν = 0, point load)—psf

Δσ_z versus depth at the chosen (x, y). Solid = Boussinesq; dashed = 2:1 approximation; dot = your depth z.

Point load (Boussinesq, 1885), r = horizontal distance:

$$ \Delta\sigma_z = \frac{3P}{2\pi}\,\frac{z^3}{(r^2+z^2)^{5/2}} $$

Line load q per unit length, and strip of width B integrated from it:

$$ \Delta\sigma_z = \frac{2q\,z^3}{\pi\,(x^2+z^2)^2}, \qquad \Delta\sigma_z = \frac{q}{\pi}\Big[\theta + \sin\theta\cos\theta\Big]_{\theta_1}^{\theta_2}, \quad \theta_{1,2} = \tan^{-1}\frac{x \mp B/2}{z} $$

Corner of a uniformly loaded rectangle (Newmark, 1935), m = B/z, n = L/z, V = m² + n² + 1:

$$ I = \frac{1}{4\pi}\left[ \frac{2mn\sqrt{V}}{V + m^2n^2}\cdot\frac{V+1}{V} + \tan^{-1}\frac{2mn\sqrt{V}}{V - m^2n^2} \right], \qquad \Delta\sigma_z = q\,I $$

Under the center of a uniformly loaded circle:

$$ \Delta\sigma_z = q\left[1 - \left(\frac{1}{1 + (R/z)^2}\right)^{3/2}\right] $$

Any point under or beside a rectangle is found by adding and subtracting corner rectangles (superposition). Off-center points under a circle are found by integrating the point-load solution over the loaded area. The arctangent is taken in the 0–π range, which fixes the well-known error in the tabulated form when V < m²n².

What the calculator assumes

Boussinesq's solution treats the ground as a semi-infinite, homogeneous, isotropic, linear-elastic mass loaded at its surface. Real soil is none of those things, yet the vertical stress Δσ_z turns out to be fairly insensitive to stiffness and Poisson's ratio, which is why the solution is still the standard way to estimate the stress increase used in consolidation and immediate-settlement calculations. The pressure q should be the net increase: the footing pressure minus the soil removed for the excavation.

The solution does not account for a stiff layer near the surface (which spreads load more widely) or a rigid base close below (which concentrates it). For layered sites with a stiff crust over soft clay, consider Westergaard or an elastic layered solution, and for very large or flexible loaded areas, check the edge as well as the center.

Newmark influence factors for a rectangle corner

I = Δσ_z/q below the corner of a uniformly loaded rectangle; m = B/z, n = L/z (symmetric in m and n)
m \ n	0.1	0.2	0.5	1	2	5	10
0.1	0.0047	0.0092	0.0198	0.0279	0.0311	0.0316	0.0316
0.2	0.0092	0.0179	0.0387	0.0547	0.0610	0.0620	0.0620
0.5	0.0198	0.0387	0.0840	0.1202	0.1350	0.1374	0.1374
1	0.0279	0.0547	0.1202	0.1752	0.1999	0.2044	0.2046
2	0.0311	0.0610	0.1350	0.1999	0.2325	0.2395	0.2398
5	0.0316	0.0620	0.1374	0.2044	0.2395	0.2486	0.2491
10	0.0316	0.0620	0.1374	0.2046	0.2398	0.2491	0.2498

As m and n grow, I approaches 0.25: a corner of an infinitely large loaded area carries a quarter of q. Some printed versions of Newmark's formula give the wrong value when m²n² > m² + n² + 1 because the arctangent is not shifted by π; this calculator uses the two-argument arctangent so the result is continuous.

Stress below the center vs depth

Δσ_z/q under the center (B = strip width, square side, or circle diameter)
z/B	Strip	Square	Circle (D = B)	2:1, strip	2:1, square
0.25	0.959	0.930	0.911	0.800	0.640
0.5	0.818	0.701	0.646	0.667	0.444
1	0.550	0.336	0.284	0.500	0.250
1.5	0.396	0.179	0.146	0.400	0.160
2	0.306	0.108	0.087	0.333	0.111
3	0.208	0.051	0.040	0.250	0.063
4	0.158	0.029	0.023	0.200	0.040
5	0.126	0.019	0.015	0.167	0.028

The table shows the familiar "pressure bulb" sizes: under a square footing the stress drops to about 10% of q at z ≈ 2B, while a long strip needs about 6B. That is why strip footings and embankments influence much deeper soil than isolated column footings of the same width.

Worked examples

Example 1 — Center and corner of a square footing

Given: 10 × 10 ft footing, net pressure q = 2,000 psf. Find Δσ_z at z = 10 ft below the center and below a corner.

Center: four 5 × 5 ft quarter-rectangles, m = n = 5/10 = 0.5 → I = 0.0840. Δσ_z = 4 × 0.0840 × 2,000 = 672 psf.

Corner: one 10 × 10 ft rectangle, m = n = 1.0 → I = 0.1752. Δσ_z = 0.1752 × 2,000 = 350 psf.

2:1 method: 2,000 × 10 × 10 / (20 × 20) = 500 psf — 26% low at the center, 43% high at the corner.

Under the center Δσ_z = 0.336q; under the corner 0.175q.

Example 2 — Point beside a footing (superposition)

Given: same footing, point 10 ft from the center (5 ft beyond the edge) on the centerline, z = 10 ft.

Make two 15 × 5 ft rectangles that reach from the point across the footing (m = 1.5, n = 0.5) and subtract two 5 × 5 ft rectangles that cover the unloaded strip between the point and the edge (m = n = 0.5).

I(1.5, 0.5) = 0.1313 and I(0.5, 0.5) = 0.0840, so Δσ_z = 2 × 2,000 × (0.1313 − 0.0840) = 189 psf.

Δσ_z = 189 psf (0.095q) — confirmed by direct numerical integration of the point-load solution.

Example 3 — Point load

Given: P = 10,000 lb at the surface. Find Δσ_z at z = 10 ft, r = 0 and r = 5 ft.

r = 0: Δσ_z = 3(10,000)/(2π × 10²) = 0.4775 × 100 = 47.7 psf. Westergaard (ν = 0): 10,000/(π × 100) = 31.8 psf.

r = 5 ft: R² = 125 ft², Δσ_z = 3(10,000)(1,000)/(2π × 125^2.5) = 27.3 psf.

47.7 psf directly below; 27.3 psf at 5 ft off-axis.

Limits of this tool

Flexible, uniformly loaded areas. A rigid footing redistributes contact pressure, but at depths beyond about B/2 the difference in Δσ_z is small.
Elastic half-space: no layering, no rigid base, no stiff crust. Results near the surface (z < 0.1B) under an edge are sensitive to the exact load shape.
Off-center points under a circle use numerical integration (about 38,000 cells); expect agreement with published Foster–Ahlvin charts to about 3 significant figures.
Only the vertical stress increase is reported — not horizontal or shear stresses — and the existing overburden stress must be added separately.

Frequently asked questions

What is the Boussinesq equation?

Boussinesq (1885) solved for the stresses in a homogeneous, isotropic, linear-elastic half-space under a vertical point load P at the surface. The vertical stress at depth z and horizontal distance r is Δσz = 3P·z³ / [2π(r² + z²)^(5/2)]. Directly below the load it reduces to 0.4775P/z². Solutions for line, strip, rectangular and circular loads are found by integrating it over the loaded area.

How do you find the stress below the center of a rectangular footing?

Split the footing into four equal rectangles that meet at the center. Each has a corner at the point of interest, with m = (B/2)/z and n = (L/2)/z. Look up or compute the Newmark corner influence factor I and multiply: Δσz = 4·q·I. For a 10 × 10 ft footing at z = 10 ft, m = n = 0.5, I = 0.0840, so Δσz = 0.336q.

How do you find the stress outside a loaded area?

Use superposition. Build a large rectangle that has a corner at the point and covers the footing, then subtract the rectangles that are not loaded and add back any that were subtracted twice. This calculator does it automatically with signed corner rectangles for any offset (x, y).

How accurate is the 2:1 method?

The 2:1 method spreads the load at 2 vertical to 1 horizontal, so the stress at depth z is q·B·L/[(B + z)(L + z)]. It is an average over the spread area: it underestimates the stress under the center at shallow and moderate depths (0.25q vs 0.336q for a square at z = B) and overestimates it under corners and edges. It is fine for quick estimates of average settlement, not for peak stress.

When should I use Westergaard instead of Boussinesq?

Westergaard assumed the soil is reinforced by thin, closely spaced rigid horizontal layers that prevent lateral strain, a model for stratified or varved clays. With Poisson's ratio taken as zero, it gives Δσz = P/(πz²)·[1 + 2(r/z)²]^(-3/2), about two-thirds of Boussinesq directly below the load. Boussinesq is the usual, more conservative choice.

What depth should I check stress to for settlement?

Commonly to the depth where the stress increase falls to about 10% of the applied pressure, or to where it is about 10% of the existing effective overburden stress. For a square footing that is about 2B, and for a strip about 4B to 6B. Use the chart to see where your profile crosses those limits.

References: Boussinesq, J. (1885), Application des potentiels à l'étude de l'équilibre et du mouvement des solides élastiques, Gauthier-Villars. Newmark, N.M. (1935), "Simplified computation of vertical pressures in elastic foundations," Univ. of Illinois Eng. Exp. Station Circular 24. Westergaard, H.M. (1938), "A problem of elasticity suggested by a problem in soil mechanics: soft material reinforced by numerous strong horizontal sheets." Das, B.M. and Sivakugan, N., Principles of Foundation Engineering, 9th ed., Cengage (2019), ch. 8. NAVFAC DM-7.01 (1986), ch. 4.

Related tools

How to calculate stress below a footing — step-by-step guide
Worked example: footing settlement on clay — Boussinesq vs 2:1 averaging
Consolidation settlement — use Δσz at mid-layer as the stress increase
Bearing capacity — shallow foundations
Lateral earth pressure
Soil classification (USCS + AASHTO)
Bearing capacity factors