Boussinesq Vertical Stress Distribution

Computes vertical stress increase Δσ_z due to different surface loads using Boussinesq's solution: point load, circular UDL, rectangular UDL, and 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

Uniform pressure q kPa

Width B m

Length L m

Uniform pressure q kPa

Width B of strip m

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 basic stats here.

Profile, table & exports

Boussinesq vertical stress profile

Depth z	r/z	I_B	Δσ_z
No results yet.

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

Theory & formulas

Boussinesq point load
Vertical stress at depth z and radial distance r:

Δσz = Q / z² · I_B

I_B = (3 / 2π) · [1 + (r/z)²]^(-5/2)

Circular uniformly distributed load (on axis):

Δσz = q · [ 1 - (1 + (R/z)²)^(-3/2) ]

Rectangular UDL beneath centre is obtained by numerically integrating the point‑load solution over the loaded area (flexible foundation assumption).

Strip load is treated with a simple approximate expression for a long strip. All formulas assume a homogeneous, isotropic, semi‑infinite elastic half‑space and neglect soil self‑weight (unless accounted separately).