{"id":173,"date":"2025-01-08T22:48:59","date_gmt":"2025-01-08T22:48:59","guid":{"rendered":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/chapter\/3-2-stresses-in-the-soil-due-to-a-point-load\/"},"modified":"2026-03-16T13:56:55","modified_gmt":"2026-03-16T13:56:55","slug":"3-2-stresses-in-the-soil-due-to-a-point-load","status":"publish","type":"chapter","link":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/chapter\/3-2-stresses-in-the-soil-due-to-a-point-load\/","title":{"raw":"3.2 Stresses in the soil due to a point load","rendered":"3.2 Stresses in the soil due to a point load"},"content":{"raw":"Load transferred to the soil from e.g., the foundation of an electric power pole can be simulated as a point load Qext<\/sub><\/em> acting on the surface of a homogeneous elastic half-space (Figure 3.5<\/a>). As both load and geometry are axisymmetric, additional soil stresses due to the application of the load can be calculated as:\n\n[caption id=\"attachment_172\" align=\"aligncenter\" width=\"400\"]\"Diagram Figure 3.5. Stresses due to point load acting on the surface of a homogeneous elastic half-space.[\/caption]\n\n(3.1)<\/strong> [latex]\\Delta {\\sigma _z} = \\dfrac{{3{Q_{ext}}}}{{2\\pi {z^2}{{\\left[ {1 + {{\\left( {\\dfrac{r}{z}} \\right)}^2}} \\right]}^{\\tfrac{5}{2}}}}}[\/latex]\n\n(3.2) <\/strong>[latex]\\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) [\/latex]\n\n(3.3)<\/strong> [latex]\\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)[\/latex]\n\n(3.4)<\/strong> [latex]\\Delta {\\tau _{rz}} = \\dfrac{{3{Q_{ext}}}}{{2\\pi }}\\left( {\\dfrac{{r{z^2}}}{{{{\\left( {{r^2} + {z^2}} \\right)}^{\\tfrac{5}{2}}}}}} \\right)[\/latex]\n\nThe above expressions provide the stress increments <\/em>\u0394\u03c3<\/em> 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 <\/sub><\/em>is the Poisson ratio of the soil, while the corresponding stress components \u0394\u03c3<\/em>z<\/sub>, \u0394\u03c3<\/em>r<\/sub>, <\/em>\u0394\u03c3\u03b8<\/sub><\/em> and \u0394\u03c4<\/em>rz<\/sub> and the vertical z<\/em> and radial r<\/em> distance from the point load are defined in Figure 3.5<\/a>.","rendered":"

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

\"Diagram
Figure 3.5. Stresses due to point load acting on the surface of a homogeneous elastic half-space.<\/figcaption><\/figure>\n

(3.1)<\/strong> [latex]\\Delta {\\sigma _z} = \\dfrac{{3{Q_{ext}}}}{{2\\pi {z^2}{{\\left[ {1 + {{\\left( {\\dfrac{r}{z}} \\right)}^2}} \\right]}^{\\tfrac{5}{2}}}}}[\/latex]<\/p>\n

(3.2) <\/strong>[latex]\\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)[\/latex]<\/p>\n

(3.3)<\/strong> [latex]\\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)[\/latex]<\/p>\n

(3.4)<\/strong> [latex]\\Delta {\\tau _{rz}} = \\dfrac{{3{Q_{ext}}}}{{2\\pi }}\\left( {\\dfrac{{r{z^2}}}{{{{\\left( {{r^2} + {z^2}} \\right)}^{\\tfrac{5}{2}}}}}} \\right)[\/latex]<\/p>\n

The above expressions provide the stress increments <\/em>\u0394\u03c3<\/em> 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 <\/sub><\/em>is the Poisson ratio of the soil, while the corresponding stress components \u0394\u03c3<\/em>z<\/sub>, \u0394\u03c3<\/em>r<\/sub>, <\/em>\u0394\u03c3\u03b8<\/sub><\/em> and \u0394\u03c4<\/em>rz<\/sub> and the vertical z<\/em> and radial r<\/em> distance from the point load are defined in Figure 3.5<\/a>.<\/p>\n","protected":false},"author":1,"menu_order":2,"template":"","meta":{"pb_show_title":"","pb_short_title":"3.2 Stresses in the soil due to a point load","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-173","chapter","type-chapter","status-publish","hentry"],"part":165,"_links":{"self":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapters\/173","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/users\/1"}],"version-history":[{"count":1,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapters\/173\/revisions"}],"predecessor-version":[{"id":174,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapters\/173\/revisions\/174"}],"part":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/parts\/165"}],"metadata":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapters\/173\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/media?parent=173"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapter-type?post=173"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/contributor?post=173"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/license?post=173"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}