GEO-10

Boussinesq point load

Vertical stress under a point load on an elastic half-space.

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Stress distributionBoussinesq 1885

Governing equation

\sigma_z = \dfrac{3Q}{2\pi}\dfrac{z^3}{R^5},\quad R=\sqrt{r^2+z^2}

where

Q: Point load (kN)
z: Depth (m)
r: Radial offset (m)
\sigma_z: Vertical stress (kPa)

Lecture brief

Historical brief

Soil mechanics became a quantitative laboratory after Karl von Terzaghi’s 1925–1943 work on effective stress, consolidation and bearing. These sheets still size shallow foundations, retaining walls, piles and drainage in SI. This sheet (GEO-10 — Boussinesq point load) is the form associated with Boussinesq 1885. Working symbols: $Q$ , $z$ , $r$ $\rightarrow$ $\sigma_z$ . Boussinesq integrated the Kelvin solution for a vertical force on a homogeneous isotropic elastic half-space, giving σz = (3Q/2π) z³/R⁵.

Purpose

Purpose: compute

\sigma_z

from

Q

z

r

in Geotechnical engineering via

\sigma_z = \dfrac{3Q}{2\pi}\dfrac{z^3}{R^5},\quad R=\sqrt{r^2+z^2}

Vertical stress under a point load on an elastic half-space. Use it when a real geotechnical engineering question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

Q = 1000.000\,\mathrm{kN}

z = 4.000\,\mathrm{m}

r = 2.000\,\mathrm{m}

, the governing relation

\sigma_z = \dfrac{3Q}{2\pi}\dfrac{z^3}{R^5},\quad R=\sqrt{r^2+z^2}

yields

\sigma_z = 17.082\,\mathrm{kPa}

. Infinite half-space, no groundwater, superposition for groups. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Point load Q1000 kNDepth z4 mRadial offset r2 m

Outputs

Vertical stress \sigma_z17.082 kPa

Reading speed

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Narration of this film

Infinite half-space, no groundwater, superposition for groups.

Boussinesq integrated the Kelvin solution for a vertical force on a homogeneous isotropic elastic half-space, giving σz = (3Q/2π) z³/R⁵.

Reading speed

Watch on YouTube