---
title: "Limitations of Boussinesq and Westergaard"
subject: "Soil Mechanics"
chapter: 8
key: "semester_4/soil_mechanics:limitations-boussinesq-westergaard"
source: https://www.bctengineer.com/q/semester_4/soil_mechanics/limitations-boussinesq-westergaard
site: BCT Engineering
---

# Limitations of Boussinesq and Westergaard

Soil Mechanics · Chapter 8: Vertical Stresses Below Applied Loads

- Asked in: 2078 Chaitra, 3 marks; 2078 Baisakh, 2 marks; 2075 Baisakh, 2 marks; 2075 Bhadra, 2 marks; 2076 Baisakh, 2 marks (asked 5 times)
- Video: [Soil Mechanics-8.3 | Comparison Between Westergaard and Boussinesq Theory | Shubham Sarathe](https://www.youtube.com/watch?v=_VKG76pkErA) — EIQ
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/soil_mechanics/limitations-boussinesq-westergaard
- Whole chapter: https://www.bctengineer.com/md/semester_4/soil_mechanics/chapter-8

## Question

Describe the limitations (conditions for use) of Boussinesq's and Westergaard's theory / analysis.

## Short answer

Both theories assume ideal elastic soil. Boussinesq applies to a homogeneous, isotropic half-space and ignores foundation depth; it overestimates stress in layered soils. Westergaard is for stratified deposits with rigid horizontal layers preventing lateral strain (ν=0), giving lower stresses, but its assumptions are idealised.

## Answer

Both theories assume an ideal elastic soil; real soil only approximates this.

#### Boussinesq's theory (point load on a half-space)

- The soil is a semi-infinite, homogeneous, isotropic, linearly elastic mass (stress is proportional to strain).
- The soil is weightless and continuous; the stress-strain relation does not depend on the stress level.
- The load acts at the surface of a horizontal half-space as a point load (loaded areas are handled by integration).
- It is valid only while the stresses are well below failure and the soil is fairly uniform. It gives results close to observed values for homogeneous deposits, and somewhat high stresses for stratified soils.

Limitations: real soils are neither perfectly elastic nor homogeneous; layered soils, a stiff layer below, or a very soft layer limit its use. It also ignores the effect of foundation depth (the load is taken at the surface).

#### Westergaard's theory

- The soil is an elastic medium reinforced by thin, perfectly rigid horizontal sheets (e.g. sand or silt layers within clay) which allow no lateral strain.
- Poisson's ratio is taken as zero ($\nu = 0$).
- It suits stratified, anisotropic deposits (alternating sand and clay) and soft clay with thin sand seams. It gives a lower stress than Boussinesq (about 2/3 on the load axis).

Limitations: the assumptions of perfectly rigid layers and $\nu = 0$ are idealised, and the actual stress usually lies between the two theories.

| Condition | Boussinesq | Westergaard |
|---|---|---|
| Soil | Homogeneous, isotropic | Stratified (rigid sheets) |
| Poisson's ratio | Any (not in the formula) | 0 |
| Stress at the axis (point load) | $0.4775\,Q/z^2$ | $0.3183\,Q/z^2$ |

## Check yourself

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

1. What does Boussinesq assume about soil?
   A. Semi-infinite, homogeneous, isotropic, linearly elastic
   B. Layered with rigid sheets
   C. Plastic
   D. Poisson's ratio zero
   Answer: A. These are Boussinesq's conditions. (Key point: Boussinesq assumption)

2. What does Westergaard assume?
   A. Homogeneous isotropic half-space
   B. Elastic medium with thin rigid sheets and nu = 0
   C. Rigid soil
   D. Soil with a free surface
   Answer: B. Rigid horizontal sheets prevent lateral strain and Poisson's ratio is zero. (Key point: Westergaard assumption)

3. Point load stress coefficients on the axis are:
   A. Boussinesq 0.4775, Westergaard 0.3183
   B. Boussinesq 0.3183, Westergaard 0.4775
   C. Both 0.4775
   D. Both 0.3183
   Answer: A. Westergaard gives lower stress, about 2/3 of Boussinesq. (Key point: Axis coefficient)

4. Westergaard suits which deposit?
   A. Homogeneous sand
   B. Stratified deposits like clay with sand seams
   C. Rock
   D. Dry gravel
   Answer: B. Thin rigid layers represent stratified soils. (Key point: Suitable soil)

## Related in Soil Mechanics

- [Boussinesq vs Westergaard solutions](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Aboussinesq-westergaard-difference) — Soil Mechanics, chapter 8
- [Limitations of Mohr-Coulomb theory](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Amohr-coulomb-limitations) — Soil Mechanics, chapter 10
- [Boussinesq stress below circle center](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Aboussinesq-circle-center-stress) — Soil Mechanics, chapter 8

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