---
title: "Boussinesq vs Westergaard solutions"
subject: "Soil Mechanics"
chapter: 8
key: "semester_4/soil_mechanics:boussinesq-westergaard-difference"
source: https://www.bctengineer.com/q/semester_4/soil_mechanics/boussinesq-westergaard-difference
site: BCT Engineering
---

# Boussinesq vs Westergaard solutions

Soil Mechanics · Chapter 8: Vertical Stresses Below Applied Loads

- Asked in: 2074 Bhadra, 2 marks
- Video: [Westergaard's Theory - Stress Distribution in Soils - Foundation Engineering GATE](https://www.youtube.com/watch?v=Fa8qiYApnLU) — Ekeeda GATE & ESE
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/soil_mechanics/boussinesq-westergaard-difference
- Whole chapter: https://www.bctengineer.com/md/semester_4/soil_mechanics/chapter-8

## Question

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?

## 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.

## Answer

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.

| | Boussinesq | Westergaard |
|---|---|---|
| Medium | Homogeneous, isotropic | Stratified, laterally restrained |
| Point load formula | $\Delta\sigma_z = \dfrac{3Q}{2\pi z^2}\left[1+(r/z)^2\right]^{-5/2}$ | $\Delta\sigma_z = \dfrac{Q}{\pi z^2}\left[1+2(r/z)^2\right]^{-3/2}$ |
| Coefficient at $r=0$ | 0.4775 | 0.3183 |

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

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The basic difference is in:
   A. Assumed soil behaviour
   B. Load magnitude
   C. Footing size
   D. Water table
   Answer: A. Boussinesq assumes a homogeneous isotropic half-space; Westergaard assumes rigid sheets preventing lateral strain. (Key point: Basic difference)

2. Westergaard takes Poisson's ratio as:
   A. 0.5
   B. 0
   C. 0.3
   D. 1
   Answer: B. nu = 0. (Key point: Poisson's ratio)

3. Directly under the load, which gives smaller stress?
   A. Boussinesq
   B. Westergaard
   C. Equal
   D. Depends on k
   Answer: B. 0.3183 vs 0.4775 coefficient. (Key point: Axis stress)

## Related in Soil Mechanics

- [Limitations of Boussinesq and Westergaard](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Alimitations-boussinesq-westergaard) — Soil Mechanics, chapter 8, asked 5×
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