STRESS BULB · FOOTING ON SOIL

What this page computes

The calculator determines the stress bulb in the ground under a rectangular footing — how the vertical stress \(\sigma_z\) transmitted by the footing diffuses and dissipates with depth — plus the settlement under the footing centre. It can run on two engines: the analytical Boussinesq solution (the default, server-side) or a 3D finite-element model (CalculiX, through the desktop agent).

What the stress bulb is

A footing does not transmit its pressure straight down alone: the stress spreads laterally as it descends, like a "bulb" that widens and attenuates with depth. For a square footing, \(\sigma_z\) drops to 10% of the contact pressure at about \(2B\) below the base.

This diffusion is why the settlement of a footing depends on the layers at depth, not just the one directly beneath it: a compressible layer within the bulb, even a few metres below the footing, contributes to the total settlement. The bulb shows visually how far down the ground "feels" the load.

The Boussinesq solution

For a homogeneous elastic half-space loaded at the surface, Boussinesq (1885) gives the stresses in closed form. For a rectangular area under uniform load the primitive exists — the Newmark influence factor, which gives \(\sigma_z\) below the corner of the rectangle:

\(I = \frac{1}{4\pi}\left[\frac{2mn\sqrt{A}}{A + C}\cdot\frac{A+1}{A} + \arctan\frac{2mn\sqrt{A}}{A - C}\right]\)

where \(m = a/z\), \(n = b/z\), \(A = m^2+n^2+1\) and \(C = m^2n^2\). The stress below any point follows by superposing the four rectangles that have the point as a corner — with signs, which makes the formula valid outside the footprint too.

One consequence catches people out: \(\sigma_z\) depends on neither the modulus of elasticity nor Poisson's ratio. The vertical stress bulb is identical in clay and in sand. It follows directly from the equilibrium equations — the modulus only enters at the strain stage, that is, at settlement.

The two engines

The analytical engine is the default because it covers the common case and needs no installation. With layered soil, Boussinesq computes the stresses as if the ground were homogeneous, and the layering only enters through the moduli used for settlement. That is the usual engineering approximation and works well when the contrast between layers is moderate — but for a much stiffer layer close to the base, the FE model is the correct one.

The two should agree under the classical assumptions (homogeneous ground, deep model). When you switch to FEM, the page shows the Boussinesq value alongside at the probe point precisely as a cross-check.

Settlement

It follows from summing the strains over the layers:

\(s = \sum_i \frac{\Delta\sigma_{z,avg,i} \cdot h_i}{E_{oed,i}}\)

\(\Delta\sigma_{z,avg}\) is the average over the layer thickness, not the mid-layer value — \(\sigma_z\) decays strongly nonlinearly, and in the first layer the difference between the two reaches a few percent.

The oedometric modulus is used, not \(E\):

\(E_{oed} = \frac{E(1-\nu)}{(1+\nu)(1-2\nu)}\)

Under a wide load, lateral strain is restrained by the surrounding soil, so the soil column is stiffer than in a uniaxial test. As \(\nu \to 0.5\) the expression tends to infinity — the undrained, incompressible case.

Input data

• Footing dimensions and the contact pressure.
• Soil layers — thickness, modulus \(E\), Poisson's ratio \(\nu\). The thicknesses sum to the model depth.
• Solver — analytical or FEM.
• Mesh density — FEM mode only.

Assumptions and limitations

• Both engines assume linear-elastic soil; real behaviour, especially under high loads, is nonlinear.
• The footing is assumed flexible, with uniform contact pressure. A rigid footing redistributes pressure towards the edges, with peaks at the corners.
• In analytical mode the half-space is infinite: there is no rigid base at depth \(H\). The FE model has one, which raises its stresses near the base — which is why the two differ there.
• The settlement computed is the oedometric one, by layer summation. It excludes time-dependent consolidation and the distortion component under a narrow footing.
• Not covered: pore water pressure, eccentric or inclined loads, overlapping neighbouring footings.

Related calculations

• Winkler pressures — the contact pressures at the footing base, the input for the bulb
• Isolated footing (full design) — the structural design of the footing
• Slope stability — the other major geotechnical check