{"id":183,"date":"2025-01-28T05:08:02","date_gmt":"2025-01-28T05:08:02","guid":{"rendered":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/chapter\/3-5-stresses-in-the-soil-due-to-a-circular-pressure\/"},"modified":"2026-03-16T13:57:23","modified_gmt":"2026-03-16T13:57:23","slug":"3-5-stresses-in-the-soil-due-to-a-circular-pressure","status":"publish","type":"chapter","link":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/chapter\/3-5-stresses-in-the-soil-due-to-a-circular-pressure\/","title":{"raw":"3.5 Stresses in the soil due to a circular pressure","rendered":"3.5 Stresses in the soil due to a circular pressure"},"content":{"raw":"Stresses transferred to the soil from a circular footing such as e.g., the foundation of a tank or a silo, can be modelled as a circular pressure acting on the surface of a homogeneous elastic half-space (Figure 3.8<\/a>). Considering a uniform vertical pressure, where both loading and geometry are axisymmetric, stress increments due to the application of the pressure qext<\/sub><\/em> below the axis of the applied pressure<\/em> can be calculated as:\n\n[caption id=\"attachment_182\" align=\"aligncenter\" width=\"400\"]\"Diagram Figure 3.8. Stresses under the centre of circular pressure acting on the surface of a homogeneous elastic half-space.[\/caption]\n\n(3.13) <\/strong>[latex]\\Delta {\\sigma _z} = {q_{ext}}\\left[ {1 - {{\\left( {\\dfrac{1}{{1 + {{\\left( {\\dfrac{{{r_o}}}{z}} \\right)}^2}}}} \\right)}^{\\tfrac{3}{2}}}} \\right][\/latex]\n\n(3.14)<\/strong> [latex]\\Delta {\\sigma _r} = \\Delta {\\sigma _\\theta } = \\dfrac{{{q_{ext}}}}{2}\\left[ {\\left( {1 + 2{\\nu _s}} \\right) - \\dfrac{{4\\left( {1 + \\nu } \\right)}}{{{{\\left[ {1 + {{\\left( {\\dfrac{{{r_o}}}{z}} \\right)}^2}} \\right]}^{\\tfrac{1}{2}}}}} + \\dfrac{1}{{{{\\left[ {1 + {{\\left( {\\dfrac{{{r_o}}}{z}} \\right)}^2}} \\right]}^{\\tfrac{3}{2}}}}}} \\right][\/latex]\n\nIn Eqs. 3.13 and 3.14 vs <\/sub><\/em>is the soil\u2019s Poisson ratio, ro<\/sub><\/em> is the radius of the area where pressure is applied, while the corresponding stress increment components \u0394\u03c3<\/em>z<\/sub>, \u0394\u03c3<\/em>r<\/sub> and \u0394\u03c3\u03b8<\/sub><\/em> are defined in Figure 3.8<\/a>.","rendered":"

Stresses transferred to the soil from a circular footing such as e.g., the foundation of a tank or a silo, can be modelled as a circular pressure acting on the surface of a homogeneous elastic half-space (Figure 3.8<\/a>). Considering a uniform vertical pressure, where both loading and geometry are axisymmetric, stress increments due to the application of the pressure qext<\/sub><\/em> below the axis of the applied pressure<\/em> can be calculated as:<\/p>\n

\"Diagram
Figure 3.8. Stresses under the centre of circular pressure acting on the surface of a homogeneous elastic half-space.<\/figcaption><\/figure>\n

(3.13) <\/strong>[latex]\\Delta {\\sigma _z} = {q_{ext}}\\left[ {1 - {{\\left( {\\dfrac{1}{{1 + {{\\left( {\\dfrac{{{r_o}}}{z}} \\right)}^2}}}} \\right)}^{\\tfrac{3}{2}}}} \\right][\/latex]<\/p>\n

(3.14)<\/strong> [latex]\\Delta {\\sigma _r} = \\Delta {\\sigma _\\theta } = \\dfrac{{{q_{ext}}}}{2}\\left[ {\\left( {1 + 2{\\nu _s}} \\right) - \\dfrac{{4\\left( {1 + \\nu } \\right)}}{{{{\\left[ {1 + {{\\left( {\\dfrac{{{r_o}}}{z}} \\right)}^2}} \\right]}^{\\tfrac{1}{2}}}}} + \\dfrac{1}{{{{\\left[ {1 + {{\\left( {\\dfrac{{{r_o}}}{z}} \\right)}^2}} \\right]}^{\\tfrac{3}{2}}}}}} \\right][\/latex]<\/p>\n

In Eqs. 3.13 and 3.14 vs <\/sub><\/em>is the soil\u2019s Poisson ratio, ro<\/sub><\/em> is the radius of the area where pressure is applied, while the corresponding stress increment components \u0394\u03c3<\/em>z<\/sub>, \u0394\u03c3<\/em>r<\/sub> and \u0394\u03c3\u03b8<\/sub><\/em> are defined in Figure 3.8<\/a>.<\/p>\n","protected":false},"author":1,"menu_order":5,"template":"","meta":{"pb_show_title":"","pb_short_title":"3.5 Stresses in the soil due to a circular pressure","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-183","chapter","type-chapter","status-publish","hentry"],"part":165,"_links":{"self":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapters\/183","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\/183\/revisions"}],"predecessor-version":[{"id":184,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapters\/183\/revisions\/184"}],"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\/183\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/media?parent=183"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapter-type?post=183"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/contributor?post=183"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/license?post=183"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}