---
title: "Boussinesq stress below circle center"
subject: "Soil Mechanics"
chapter: 8
key: "semester_4/soil_mechanics:boussinesq-circle-center-stress"
source: https://www.bctengineer.com/q/semester_4/soil_mechanics/boussinesq-circle-center-stress
site: BCT Engineering
---

# Boussinesq stress below circle center

Soil Mechanics · Chapter 8: Vertical Stresses Below Applied Loads

- Asked in: 2077 Chaitra, 3 marks
- Video: [Finding Vertical Stress in a Soil Below a Uniformly Loaded Circular Area| Solved Problem](https://www.youtube.com/watch?v=bsNiwi84Uvc) — Master Civil Engineering
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/soil_mechanics/boussinesq-circle-center-stress
- Whole chapter: https://www.bctengineer.com/md/semester_4/soil_mechanics/chapter-8

## Question

Using Boussinesq's equation for point load, determine the increment in vertical stress below the center of a uniformly loaded circle. Assume all necessary conditions.

## Short answer

Integrating the Boussinesq point load over a circular area gives Δσz = q[1 − 1/(1+(R/z)2)3/2]. For R = 2 m, q = 100 kPa, stresses at z = 1, 2, 4 m are 91.1, 64.6, 28.4 kPa.

## Answer

#### Assumptions

Uniform pressure $q$ on a flexible circular area of radius $R$ on the ground surface; the soil is homogeneous, isotropic and elastic (Boussinesq). The stress is required at depth $z$ below the centre.

#### Derivation

Boussinesq's equation for a point load $dQ$ at horizontal distance $r$:

$$
d\sigma_z = \frac{3\,dQ}{2\pi z^2}\left[\frac{1}{1+(r/z)^2}\right]^{5/2}
$$

An elemental ring of radius $r$ and width $dr$ carries $dQ = q\,(2\pi r\,dr)$. All points of the ring are at the same distance $r$ from the axis, so:

$$
\Delta\sigma_z = \int_0^R \frac{3q\,(2\pi r\,dr)}{2\pi z^2}\left[\frac{1}{1+(r/z)^2}\right]^{5/2} = 3q\int_0^R \frac{r\,dr}{z^2}\left[1+\frac{r^2}{z^2}\right]^{-5/2}
$$

Let $t = 1 + r^2/z^2$, so $dt = 2r\,dr/z^2$:

$$
\Delta\sigma_z = \frac{3q}{2}\int_1^{1+R^2/z^2} t^{-5/2}\,dt = q\left[1 - \frac{1}{\left(1+(R/z)^2\right)^{3/2}}\right]
$$

$$
\boxed{\Delta\sigma_z = q\left[1 - \frac{1}{\left(1+(R/z)^2\right)^{3/2}}\right]}
$$

#### Example

Take $q = 100$ kPa and $R = 2$ m.

| $z$ (m) | $R/z$ | $\Delta\sigma_z$ (kPa) |
|---|---|---|
| 1 | 2.0 | 91.1 |
| 2 | 1.0 | 64.6 |
| 4 | 0.5 | 28.4 |

The stress decreases with depth, and is nearly $q$ close to the surface.

## Check yourself

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

1. How is the stress below the centre of a uniformly loaded circle derived from the point-load equation?
   A. By integrating over rings of load dQ = q(2 pi r dr)
   B. By averaging the stress at the edge
   C. By using the 2:1 spread
   D. By multiplying the point load stress by the area
   Answer: A. The area is divided into elemental rings, each carrying dQ = q 2 pi r dr, and the Boussinesq point-load stress is integrated from 0 to R. (Key point: Method of derivation)

2. The stress at depth z below the centre of a circle of radius R and pressure q is
   A. q (R/z)^2
   B. q [1 - 1/(1+(R/z)^2)^(3/2)]
   C. q / (1+(R/z)^2)^(5/2)
   D. q [1 - 1/(1+(R/z)^2)^(5/2)]
   Answer: B. The integration gives delta_sigma_z = q[1 - (1+(R/z)^2)^(-3/2)]. (Key point: Final formula)

3. For q = 100 kPa and R = 2 m, how does the stress on the axis vary?
   A. Increases from 91 kPa at 1 m to 28 kPa at 4 m
   B. Decreases with depth: 91.1, 64.6, 28.4 kPa at 1, 2, 4 m
   C. Stays constant at q
   D. Is zero at the surface
   Answer: B. It is nearly q near the surface and decreases with depth, giving 91.1, 64.6 and 28.4 kPa. (Key point: Variation with depth)

## Related in Soil Mechanics

- [Methods for vertical stress increment](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Amethods-vertical-stress-increment) — Soil Mechanics, chapter 8
- [Rectangular foundation, stress at point P](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Arectangular-foundation-point-p-stress) — Soil Mechanics, chapter 8
- [Assumptions of Boussinesq's equation](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Aboussinesq-assumptions) — Soil Mechanics, chapter 8
- [Excavation effect on vertical stress](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Aexcavation-effect-vertical-stress) — Soil Mechanics, chapter 8
- [Boussinesq vs Westergaard solutions](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3Aboussinesq-westergaard-difference) — Soil Mechanics, chapter 8
- [T-shaped foundation, stress at P](https://www.bctengineer.com/question.md?key=semester_4%2Fsoil_mechanics%3At-shaped-foundation-stress-p) — Soil Mechanics, chapter 8

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