Calculate vertical stress increase beneath a surface point load using the Boussinesq elastic half-space solution.
Boussinesq Vertical Stress Calculation
The Boussinesq point-load solution estimates the increase in vertical stress at a point within an idealised soil mass. It is commonly used as a building block for settlement calculations and for integrated solutions covering loaded areas.
Note: This is draft calculation guidance. The model assumes a homogeneous, isotropic, linear-elastic, semi-infinite medium and does not represent every layered or nonlinear ground profile.
Point-load equation
For a vertical surface point load \(Q\), depth \(z\) and radial offset \(r\):
$$ \Delta\sigma_z=\frac{3Q}{2\pi z^2}\frac{1}{\left[1+\left(\frac{r}{z}\right)^2\right]^{5/2}} $$An equivalent form using \(R=\sqrt{r^2+z^2}\) is:
$$ \Delta\sigma_z=\frac{3Qz^3}{2\pi R^5} $$With \(Q\) in kN and distances in metres, the stress increase is in kN/m² or kPa.
Directly below the load
At \(r=0\):
$$ \Delta\sigma_z=\frac{3Q}{2\pi z^2} $$The point-load solution is singular at \(z=0\), so it must not be used at the load application point.
Worked example
Assume:
- \(Q=500\ \text{kN}\)
- \(z=4.0\ \text{m}\)
- \(r=2.0\ \text{m}\)
Then:
$$ \Delta\sigma_z=\frac{3(500)}{2\pi(4)^2}\frac{1}{[1+(2/4)^2]^{5/2}} $$ $$ \Delta\sigma_z\approx8.54\ \text{kPa} $$Superposition
For multiple independent point loads in a linear-elastic model:
$$ \Delta\sigma_{z,total}=\sum_{j=1}^{n}\Delta\sigma_{z,j} $$Distributed loads may be represented by an approved integrated solution or a sufficiently refined load grid. Grid convergence and the treatment of foundation contact pressure should be checked.
Validation controls
- Require \(Q\ge0\), \(z>0\) and \(r\ge0\) for the standard compression workflow.
- Use consistent coordinates and units.
- Record whether the input is a point load or a discretised area load.
- Flag evaluation points very close to a discrete load where grid sensitivity is high.
- Do not add initial overburden stress twice.
- Separate total and effective stress increments where pore-pressure response matters.
Limitations
Layering, anisotropy, nonlinear stiffness, finite boundaries, excavation, rigid-foundation interaction and complex loading can require a different model. The equation estimates stress change, not settlement or bearing resistance by itself.