Boussinesq Stress Bulb — A Complete Worked Example

Why the shape of the bulb matters, not just the surface value

Every foundation transfers a contact pressure to the soil beneath it — but that stress does not simply stop at some depth. It fades gradually, in a characteristic shape engineers call the stress bulb: wide right under the footing, narrowing steadily with depth, like an inverted onion.

That shape matters for two practical reasons: settlement (calculated by summing the deformation of the layers that actually sit inside the bulb — a layer below it barely "feels" the foundation) and checking the effect on neighbouring structures (a new foundation next to an existing one can add meaningful stress to the soil under it — exactly the requirement in EN 1997-1 §2.4.2, checking the effect on neighbouring construction and assets).

Below is the theory, a complete worked example, and at the end a free online calculator that draws the bulb for any rectangular footing.

The Boussinesq solution — and why it surprises people

In 1885, Boussinesq solved the problem of a point load applied to the surface of an elastic half-space — homogeneous, isotropic, of infinite extent. The vertical stress at any point in the soil, at depth \(z\) and horizontal distance \(r\) from the load \(P\):

\(\sigma_z = \frac{3P}{2\pi} \cdot \frac{z^3}{R^5}, \qquad R = \sqrt{r^2+z^2}\)

The detail that catches most people off guard the first time: \(\sigma_z\) does not depend on the soil's modulus of elasticity \(E\). A loose sand and a stiff clay, with completely different stiffness, receive exactly the same vertical stress distribution under the same load — because the formula comes straight from equilibrium equations, not from a constitutive law. Stiffness governs how much the soil settles under that stress, not how large the stress is; Poisson's ratio \(\nu\) only enters the horizontal stresses, never \(\sigma_z\).

For a uniformly loaded rectangular area, integrating the point solution over the full footprint gives Newmark's closed-form expression — an exact formula, not a numerical integration. Under the corner of a rectangle with sides \(a\) and \(b\), at depth \(z\), with \(m=a/z\) and \(n=b/z\):

\(I_{corner}(m,n) = \frac{1}{4\pi}\left[\frac{2mn\sqrt{A}}{A+C}\cdot\frac{A+1}{A} + \arctan\left(\frac{2mn\sqrt{A}}{A-C}\right)\right], \quad A=m^2+n^2+1, \quad C=m^2n^2\)

Under the center of a \(B\times L\) footing, four such identical corners (each with \(a=B/2\), \(b=L/2\)) superpose:

\(\sigma_z = 4 \cdot q \cdot I_{corner}\left(\frac{B}{2z}, \frac{L}{2z}\right)\)

The same corner expression, applied with sign to four rectangles cut at any arbitrary point, gives \(\sigma_z\) anywhere — including outside the loaded footprint, where stress does not vanish abruptly but fades off laterally. That superposition principle is exactly what makes the formula useful for adjacent foundations: the stress under a neighbouring footing is found by summing each foundation's contribution, computed separately, at the point of interest.

Worked example — a 2 × 3 m isolated footing

Take an isolated rectangular footing with:

• width \(B = 2\) m, length \(L = 3\) m
• net contact pressure \(q = 200\) kN/m²

The calculation below is the one produced by JointLib's analytic solver (the Newmark formula above, evaluated exactly — not read off a chart or interpolated from a table), for the stress under the footing's center, at six depths:

At a depth equal to the footing width (\(z=B=2\) m), stress has already dropped to 43% of the contact pressure. At \(z=3B=6\) m, only 7.4% remains. The 10% threshold — the usual convention for the "active depth," below which settlement is neglected — is reached a little past \(z=5\) m, roughly \(2.5B\) for this geometry.

The 2:1 approximate method — how far off it runs

A quick hand-calculation method, common where no chart is at hand, assumes the load spreads uniformly on a cone with a 2-vertical-to-1-horizontal slope, giving a very simple formula:

\(\sigma_z \approx \frac{qBL}{(B+z)(L+z)}\)

Comparing it against the exact Boussinesq solution, for the same 2×3 m footing:

The pattern is systematic, not random: the 2:1 method underestimates stress close to the footing (where the real bulb is more concentrated than a straight 2:1 cone assumes) and overestimates it at depth (where the real bulb has already narrowed more than the geometric cone). The two curves cross around \(z \approx 4.5\) m for this geometry — there is no "safe" depth at which the approximate method is right everywhere; the error just changes sign.

The stress bulb — a visual

Suggested figure: isobars (lines of equal \(\sigma_z/q\)) drawn under the footing, at ratios 0.9 · 0.7 · 0.5 · 0.3 · 0.1 — the characteristic inverted-onion shape, widest right under the footing and narrowing with depth:

Common mistake: "the active depth is 1.5·B"

A rule of thumb repeated often in courses and technical notes says a foundation's influence effectively stops at \(z \approx 1.5B\). Calculated exactly, for a square footing, at \(z=1.5B\) the remaining stress is still 17.9% of \(q\) — nearly double the usual 10% threshold for the "active depth." The depth where 10% is actually reached is \(z \approx 2.09B\), not \(1.5B\) — verified directly from the calculator's analytic solver, by bisection on the \(\sigma_z(z)\) function.

The difference is not cosmetic: for a \(B=2\) m footing, ignoring the extra \(\approx0.6B\) means leaving out of the settlement calculation more than a meter of soil — exactly the layer where, if a soft clay sits there, the additional settlement can be significant.

Practical note: how the bulb changes in layered soil

The Boussinesq formula assumes a homogeneous half-space. JointLib's calculator also offers an alternative engine — a 3D FEM model solved live through CalculiX — for layered soil, with different stiffness per layer. Comparing the two on the same case (layered soil vs. an equivalent homogeneous soil) shows a pattern that is not random: a layer stiffer than its neighbours attracts extra stress, much like a stiffer column in a static frame carries proportionally more force. Boussinesq, with its homogeneity assumption, distributes stress "correctly" only when all layers have similar stiffness — in a profile with a rock or stiff clay layer sitting shallow under a soft layer, the real stress concentrated in the stiff layer can exceed what the classical formula predicts.

Where Eurocode 7 fits in

EN 1997-1 does not itself contain stress-distribution formulas — much like slab moment coefficients, the code leaves the calculation method to the engineer's judgement and only requires the result (a reasonable settlement estimate, a check on the effect on neighbouring structures). Boussinesq remains the de facto method for:

• Settlement calculation (informative Annex F of EN 1997-1) — the stress increase in each layer, combined with that layer's oedometric modulus \(E_{oed}\).
• §2.4.2 — checking the effect of new construction on neighbouring foundations and structures, by superposing the stresses the new foundation produces at points of interest under the neighbour.
• Geotechnical category selection — a bulb that reaches a problematic layer (sensitive clay, uncompacted fill) signals a need for further investigation, not just a more careful calculation.

Assumptions and limitations

• Homogeneous, isotropic, linearly elastic half-space of infinite extent — real layered, anisotropic, or nonlinear soil deviates from the exact solution (see the practical note above).
• \(\sigma_z\) is a total stress increase, not effective — for consolidation settlement, pore water pressure needs to be subtracted separately where relevant.
• The corner/center formula assumes uniform contact pressure; for eccentric or moment-loaded footings, the real contact pressure is not uniform (see Winkler footing pressures).
• It does not account for the footing's own stiffness or soil-structure interaction — this is a "free field" soil solution, not a coupled analysis.

Calculate it online — no charts, no table interpolation

JointLib's stress bulb calculator evaluates the exact Newmark formula above (or, on request, a 3D FEM model for layered soil) for any rectangular footing: enter \(B\), \(L\), \(q\) and the soil layers, and get the bulb drawn out, the \(\sigma_z\) profile along any vertical, and the depth at which stress drops below whatever threshold you choose.

Related calculations

• Winkler footing pressures — the real contact pressure at the base, including eccentricity
• Active earth pressure calculator (Rankine) — the thrust on a retaining wall
• Isolated footing — full design — sizing, reinforcement, and punching shear check