Boussinesq Vertical Stress Distribution

Computes vertical stress increase Δσ_z due to different surface loads using Boussinesq's solution. Available modes: **Point Load**, **Circular UDL**, **Rectangular UDL**, **Strip UDL**.

Inputs

Units

Current units: SI — Q in kN, q in kPa, distances in m, stresses in kPa.

Load case

Point load Q at surface, vertical stress at depth z and radial distance r.

Point load Q kN

Radial distance r m (0 = directly under load)

Uniform pressure q kPa

Radius R of loaded circle m

Stress is computed on the axis below the centre of a flexible circular area: Δσ_z = q [1 − (1 + (R/z)²)^−3/2].

Uniform pressure q kPa

Width B m

Length L m

Stress is computed beneath the centre of a flexible rectangle B×L by numerically integrating the point‑load Boussinesq solution over the loaded area.

Uniform pressure q kPa

Width B of strip m

Stress is computed using the Boussinesq solution for a strip load, assuming the strip is sufficiently long.

Start depth z_min m

End depth z_max m

Step Δz m

Summary

Enter load, geometry, and depth range, then click Compute profile to see vertical stress vs depth, and download the table or PDF.

Profile, table & exports

Boussinesq vertical stress profile

Depth z	r/z	I_B	Δσ_z
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Representative diagram: surface point load Q and stress Δσ_z at depth z and radial distance r.

Theory and Formulas

Boussinesq's Method for Vertical Stress Distribution:
The stress at a depth z and radial distance r below a point load is given by Boussinesq's solution for a semi‑infinite, homogeneous, isotropic elastic mass. The general formula is:

σ_z = Q / z² × I_B

where I_B = (3 / 2π) × 1 / [1 + (r/z)²]^5/2.

For Circular UDL:
The vertical stress at a depth z below a uniformly loaded circular area of radius R is calculated as:

σ_z = q × (1 - 1 / [1 + (R/z)²]^3/2)

For Rectangular UDL (Centre):
The vertical stress beneath the centre of a uniformly loaded rectangular area B × L is calculated using the numerical integration of point load solutions:

σ_z = q × I₂(B/z, L/z)

For Strip UDL:
The vertical stress beneath a sufficiently long strip is approximated using a Boussinesq solution for a strip load.

These methods assume homogeneous, isotropic, and weightless soil conditions. Stresses due to self‑weight of the soil are typically neglected unless explicitly accounted for.