Boussinesq Vertical Stress Distribution

Computes vertical stress increase Δσ_z due to surface loading using classical elastic solutions. Choose a load case, enter inputs, and compute stress profile vs depth. Use Load sample to see a ready example.

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 QkN

Radial distance rm (0 = under load)

Uniform pressure qkPa

Radius Rm

Axis stress formula used: Δσz = q [1 − (1 + (R/z)²)^(-3/2)].

Uniform pressure qkPa

Width Bm

Length Lm

Centre stress computed by numerical integration of point-load Boussinesq solution.

Uniform pressure qkPa

Width Bm

Under centre (infinite strip): Δσz = (2q/π) · arctan(B/(2z)).

Start depth zminm

End depth zmaxm

Step Δzm

Summary

Enter inputs and click Compute. Use Load sample to see a complete example.

Results

Boussinesq vertical stress profile

Depth z	Ratio	Factor	Δσz
No results yet.

Theory & References

Open formulas and references

This tool uses classical elastic half-space solutions (Boussinesq-type). Results are idealized and mainly useful for preliminary checks and understanding stress distribution.

Point load (Boussinesq)

Δσz = (Q / z²) · IB IB = (3 / 2π) · 1 / (1 + (r/z)²)^(5/2)

Circular UDL (axis)

Δσz = q · [ 1 − (1 + (R/z)²)^(-3/2) ]

Rectangular UDL (centre)

Δσz is computed numerically by integrating point-load Boussinesq solution over the loaded area (flexible load assumption).

Strip UDL (centre, infinite length)

Δσz = (2q/π) · arctan( B / (2z) )

References

Boussinesq, J. (1885). Applications des potentiels...
Poulos & Davis (1974). Elastic Solutions for Soil and Rock Mechanics.
Das, B.M. Principles of Foundation Engineering (stress distribution chapter).