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Boussinesq vs Westergaard solutions

Short answer

Boussinesq assumes a homogeneous isotropic elastic half-space allowing lateral deformation, while Westergaard assumes a stratified medium with rigid horizontal layers that prevent lateral strain (ν=0). As a result, Westergaard’s vertical stress under a point load is about two-thirds of Boussinesq’s on the axis.

Vertical stress due to a point load can be calculated based on Boussinesq's and Westergaard's solutions. What is the basic difference between these two solutions?

The basic difference is in the assumed behaviour of the soil mass.

  • Boussinesq: the soil is a homogeneous, isotropic, elastic half-space. It can deform laterally freely; Poisson's ratio does not enter the vertical stress formula.
  • Westergaard: the soil is an elastic medium reinforced by thin, rigid, horizontal layers (like alternating sand/clay sheets) which prevent lateral strain; Poisson's ratio is taken as zero.
BoussinesqWestergaard
MediumHomogeneous, isotropicStratified, laterally restrained
Point load formulaΔσz=3Q2πz2[1+(r/z)2]−5/2\Delta\sigma_z = \dfrac{3Q}{2\pi z^2}\left[1+(r/z)^2\right]^{-5/2}Δσz=Qπz2[1+2(r/z)2]−3/2\Delta\sigma_z = \dfrac{Q}{\pi z^2}\left[1+2(r/z)^2\right]^{-3/2}
Coefficient at r=0r=00.47750.3183

Westergaard's solution gives smaller stress directly under the load.

✓ Answer checked by Claude Opus 5.5 (high effort)

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