{"id":176,"date":"2025-01-08T23:48:35","date_gmt":"2025-01-08T23:48:35","guid":{"rendered":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/chapter\/3-3-stresses-in-the-soil-due-to-a-line-load\/"},"modified":"2026-03-16T13:57:04","modified_gmt":"2026-03-16T13:57:04","slug":"3-3-stresses-in-the-soil-due-to-a-line-load","status":"publish","type":"chapter","link":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/chapter\/3-3-stresses-in-the-soil-due-to-a-line-load\/","title":{"raw":"3.3 Stresses in the soil due to a line load","rendered":"3.3 Stresses in the soil due to a line load"},"content":{"raw":"Loading transferred to the soil from e.g., a rail track of limited width (Figure 3.6<\/a>) can be modelled as an \"infinitely long\" line load acting on the surface of a homogeneous elastic half-space. As plane-strain symmetry conditions apply, additional soil stresses due to the application of a line load can be calculated as:\n\n[caption id=\"attachment_175\" align=\"aligncenter\" width=\"500\"]\"Diagram Figure 3.6. Stresses due to line load acting on the surface of a homogeneous elastic half-space (right), and in the vicinity of a retaining wall (left). Line loads carry units of force per running meter.[\/caption]\n\n(3.5)<\/strong> [latex]\\Delta {\\sigma _z} = \\dfrac{{2{Q_{ext}}{z^3}}}{{\\pi {{\\left( {{x^2} + {z^2}} \\right)}^2}}}[\/latex]\n\n(3.6)<\/strong> [latex]\\Delta {\\sigma _x} = \\dfrac{{2{Q_{ext}}{x^2}z}}{{\\pi {{\\left( {{x^2} + {z^2}} \\right)}^2}}}[\/latex]\n\n(3.7)<\/strong> [latex]\\Delta {\\tau _{zx}} = \\dfrac{{2{Q_{ext}}x{z^2}}}{{\\pi {{\\left( {{x^2} + {z^2}} \\right)}^2}}}[\/latex]\n\nThe stress increment components and the vertical z<\/em> and horizontal x<\/em> distance from the line load considered in Eqs. 3.5 to 3.7 are defined in Figure 3.6<\/a>.\n\nIn the special case where the line load is acting near the crest of a retaining wall, the horizontal stress distribution along the wall height can be determined as function of the height of the wall Ho<\/sub><\/em> (Figure 3.6<\/a>):\n\n(3.8)<\/strong> [latex]\\Delta {\\sigma _x} = \\dfrac{{4{Q_{ext}}{a^2}b}}{{\\pi {H_o}{{\\left( {{a^2} + {b^2}} \\right)}^2}}}[\/latex]\n\nand the resultant horizontal force per running meter of the wall is equal to (Figure 3.6<\/a>):\n\n (3.9)<\/strong> [latex]\\Delta {Q_x} = \\dfrac{{2{Q_{ext}}}}{{\\pi \\left( {{a^2} + 1} \\right)}}[\/latex]\n\nNote that Eq. 3.8 yields identical results to Eq. 3.5, the only difference between these expressions is that in Eq. 3.8 the coordinates x<\/em> and z<\/em> as expressed as function of Ho<\/sub><\/em>.","rendered":"

Loading transferred to the soil from e.g., a rail track of limited width (Figure 3.6<\/a>) can be modelled as an “infinitely long” line load acting on the surface of a homogeneous elastic half-space. As plane-strain symmetry conditions apply, additional soil stresses due to the application of a line load can be calculated as:<\/p>\n

\"Diagram
Figure 3.6. Stresses due to line load acting on the surface of a homogeneous elastic half-space (right), and in the vicinity of a retaining wall (left). Line loads carry units of force per running meter.<\/figcaption><\/figure>\n

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

(3.6)<\/strong> [latex]\\Delta {\\sigma _x} = \\dfrac{{2{Q_{ext}}{x^2}z}}{{\\pi {{\\left( {{x^2} + {z^2}} \\right)}^2}}}[\/latex]<\/p>\n

(3.7)<\/strong> [latex]\\Delta {\\tau _{zx}} = \\dfrac{{2{Q_{ext}}x{z^2}}}{{\\pi {{\\left( {{x^2} + {z^2}} \\right)}^2}}}[\/latex]<\/p>\n

The stress increment components and the vertical z<\/em> and horizontal x<\/em> distance from the line load considered in Eqs. 3.5 to 3.7 are defined in Figure 3.6<\/a>.<\/p>\n

In the special case where the line load is acting near the crest of a retaining wall, the horizontal stress distribution along the wall height can be determined as function of the height of the wall Ho<\/sub><\/em> (Figure 3.6<\/a>):<\/p>\n

(3.8)<\/strong> [latex]\\Delta {\\sigma _x} = \\dfrac{{4{Q_{ext}}{a^2}b}}{{\\pi {H_o}{{\\left( {{a^2} + {b^2}} \\right)}^2}}}[\/latex]<\/p>\n

and the resultant horizontal force per running meter of the wall is equal to (Figure 3.6<\/a>):<\/p>\n

(3.9)<\/strong> [latex]\\Delta {Q_x} = \\dfrac{{2{Q_{ext}}}}{{\\pi \\left( {{a^2} + 1} \\right)}}[\/latex]<\/p>\n

Note that Eq. 3.8 yields identical results to Eq. 3.5, the only difference between these expressions is that in Eq. 3.8 the coordinates x<\/em> and z<\/em> as expressed as function of Ho<\/sub><\/em>.<\/p>\n","protected":false},"author":1,"menu_order":3,"template":"","meta":{"pb_show_title":"","pb_short_title":"3.3 Stresses in the soil due to a line load","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-176","chapter","type-chapter","status-publish","hentry"],"part":165,"_links":{"self":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapters\/176","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\/176\/revisions"}],"predecessor-version":[{"id":177,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapters\/176\/revisions\/177"}],"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\/176\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/media?parent=176"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapter-type?post=176"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/contributor?post=176"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/license?post=176"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}