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3.2 Stresses in the soil due to a point load

Load transferred to the soil from e.g., the foundation of an electric power pole can be simulated as a point load Qext acting on the surface of a homogeneous elastic half-space (Figure 3.5). As both load and geometry are axisymmetric, additional soil stresses due to the application of the load can be calculated as:

Diagram depicting a point load acting on the surface of a half space, resulting in radial (Δσ_r), axial (Δσ_z), and tangential (Δσ_θ) stresses at depth z and radial distance r from the point load.
Figure 3.5. Stresses due to point load acting on the surface of a homogeneous elastic half-space.

(3.1) \Delta {\sigma _z} = \dfrac{{3{Q_{ext}}}}{{2\pi {z^2}{{\left[ {1 + {{\left( {\dfrac{r}{z}} \right)}^2}} \right]}^{\tfrac{5}{2}}}}}

(3.2) \Delta {\sigma _r} = \dfrac{{{Q_{ext}}}}{{2\pi }}\left( {\dfrac{{3{r^2}z}}{{{{\left( {{r^2} + {z^2}} \right)}^{\tfrac{5}{2}}}}} - \dfrac{{\left( {1 - 2{\nu _s}} \right)}}{{{r^2} + {z^2} + z{{\left( {{r^2} + {z^2}} \right)}^{\tfrac{1}{2}}}}}} \right)

(3.3) \Delta {\sigma _\theta } = \dfrac{{{Q_{ext}}}}{{2\pi }}\left( {1 - 2{\nu _s}} \right)\left( {\dfrac{z}{{{{\left( {{r^2} + {z^2}} \right)}^{\tfrac{3}{2}}}}} - \dfrac{1}{{{r^2} + {z^2} + z{{\left( {{r^2} + {z^2}} \right)}^{\tfrac{1}{2}}}}}} \right)

(3.4) \Delta {\tau _{rz}} = \dfrac{{3{Q_{ext}}}}{{2\pi }}\left( {\dfrac{{r{z^2}}}{{{{\left( {{r^2} + {z^2}} \right)}^{\tfrac{5}{2}}}}}} \right)

The above expressions provide the stress increments Δσ due to the application of the point load, which should be added to existing geostatic stresses to find the total stress in the soil. In Eqs. 3.1 to 3.4 vs is the Poisson ratio of the soil, while the corresponding stress components Δσz, Δσr, Δσθ and Δτrz and the vertical z and radial r distance from the point load are defined in Figure 3.5.

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Fundamentals of foundation engineering and their applications Copyright © 2025 by University of Newcastle & G. Kouretzis is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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