{"id":657,"date":"2025-03-31T05:11:25","date_gmt":"2025-03-31T05:11:25","guid":{"rendered":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/chapter\/example-6-16-estimation-of-the-p-y-curve-reduction-factor-for-a-laterally-loaded-pile-group\/"},"modified":"2026-03-16T14:16:08","modified_gmt":"2026-03-16T14:16:08","slug":"example-6-16-estimation-of-the-p-y-curve-reduction-factor-for-a-laterally-loaded-pile-group","status":"publish","type":"chapter","link":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/chapter\/example-6-16-estimation-of-the-p-y-curve-reduction-factor-for-a-laterally-loaded-pile-group\/","title":{"raw":"Example 6.16","rendered":"Example 6.16"},"content":{"raw":"Determine the reduction factor \u03b2<\/em>g<\/sub>5<\/sup> for pile #5 shown in the figure below, when the lateral load acts along the weak axis of the group (X-X), and along the strong axis of the group (Y-Y). The diameter of all piles is equal to D.<\/em>\n\n[caption id=\"attachment_656\" align=\"aligncenter\" width=\"350\"]\"Diagram Example 6.16. Problem description and input parameters.[\/caption]\n\n1. Load along the weak axis, HX-X<\/sub><\/em> :\n

Interaction of pile #5 with all the remaining piles in the group must be considered, according to Eq. 6.166 and Figure 6.130<\/a>, as:<\/p>\n[latex]\\beta _g^5 = \\begin{array}{*{20}{l}}{\\beta _{g,s}^{5 \\leftrightarrow 4}{\\rm{ }}\\left( {{\\rm{side \\: by \\:side}}} \\right)}\\\\{ \\times \\beta _{g,s}^{5 \\leftrightarrow 6}{\\rm{ (side \\:by \\:side)}}}\\\\{ \\times \\beta _{g,l}^{5 \\leftrightarrow 2}{\\rm{ (leading \\:in \\:line)}}}\\\\{ \\times \\beta _{g,sk}^{5 \\leftrightarrow 1}{\\rm{ (leading \\:skewed)}}}\\\\{ \\times \\beta _{g,sk}^{5 \\leftrightarrow 3}{\\rm{(leading \\:skewed)}}}\\end{array}[\/latex]\n

The individual interaction factors are determined from Figures 6.127<\/a>-6.129<\/a> as:<\/p>\n[latex]\\beta _{g,s}^{5 \\leftrightarrow 4} = \\beta _{g,s}^{5 \\leftrightarrow 6} = 0.64 \\times {3^{0.34}} = 0.929[\/latex]\n\n[latex]\\beta _{g,l}^{5 \\leftrightarrow 2} = 0.70 \\times {2.5^{0.26}} = 0.888[\/latex]\n\n[latex]\\left( \\begin{array}{l}\\beta _{g,a}^{5 \\leftrightarrow 1} = 1.0\\\\\\beta _{g,b}^{5 \\leftrightarrow 1} = 0.7{\\left( {\\sqrt {{3^2} + {{2.5}^2}} } \\right)^{0.26}} = 0.997\\\\\\omega = {\\tan ^{ - 1}}\\left( {\\dfrac{3}{{2.5}}} \\right) = 50.2{\\rm{ \\:deg}}\\end{array} \\right) \\to \\beta _{g,sk}^{5 \\leftrightarrow 1} = \\beta _{g,sk}^{5 \\leftrightarrow 3} = \\sqrt {{{0.997}^2}{{\\cos }^2}\\left( {50.2} \\right) + {1^2}{{\\sin }^2}\\left( {50.2} \\right)} = 0.998[\/latex]\n\nThe reduction factor for pile #5 is calculated from Eq. 6.166 as:\n\n[latex]\\beta _g^5 = \\beta _{g,s}^{5 \\leftrightarrow 4} \\times \\beta _{g,s}^{5 \\leftrightarrow 6} \\times \\beta _{g,l}^{5 \\leftrightarrow 2} \\times \\beta _{g,sk}^{5 \\leftrightarrow 1} \\times \\beta _{g,sk}^{5 \\leftrightarrow 3} = 0.929 \\times 0.929 \\times 0.888 \\times 0.998 \\times 0.998 = 0.763[\/latex]\n\n2. Load along the strong axis, HY-Y<\/sub><\/em> :\n

The procedure is similar, but notice that the relative position of the pile compared to its neighboring piles must be re-assessed according to Figure 6.130<\/a>, as:<\/p>\n[latex]\\beta _g^5 = \\begin{array}{*{20}{l}}{\\beta _{g,s}^{5 \\leftrightarrow 4}{\\rm{ }}\\left( {{\\rm{trailining\\: in \\:line}}} \\right)}\\\\{ \\times \\beta _{g,s}^{5 \\leftrightarrow 6}{\\rm{ (leading \\:in \\:line)}}}\\\\{ \\times \\beta _{g,l}^{5 \\leftrightarrow 2}{\\rm{ (side \\:by \\:side)}}}\\\\{ \\times \\beta _{g,sk}^{5 \\leftrightarrow 1}{\\rm{ (trailing \\:skewed)}}}\\\\{ \\times \\beta _{g,sk}^{5 \\leftrightarrow 3}{\\rm{(leading \\:skewed)}}}\\end{array}[\/latex]\n

The individual interaction factors are determined from Figures 6.127<\/a>-6.129<\/a> as:<\/p>\n[latex]\\beta _{g,s}^{5 \\leftrightarrow 4} = 0.48 \\times {3^{0.38}} = 0.728[\/latex]\n\n[latex]\\beta _{g,s}^{5 \\leftrightarrow 6} = 0.70 \\times {3^{0.26}} = 0.931[\/latex]\n\n[latex]\\beta _{g,l}^{5 \\leftrightarrow 2} = 0.64 \\times {2.5^{0.34}} = 0.874[\/latex]\n\n[latex]\\left( \\begin{array}{l}\\beta _{g,a}^{5 \\leftrightarrow 1} = 1.0\\\\\\beta _{g,b}^{5 \\leftrightarrow 1} = 0.48{\\left( {\\sqrt {{3^2} + {{2.5}^2}} } \\right)^{0.39}} = 0.805\\\\\\omega = {\\tan ^{ - 1}}\\left( {\\dfrac{{2.5}}{3}} \\right) = 39.8{\\rm{ \\:deg}}\\end{array} \\right) \\to \\beta _{g,sk}^{5 \\leftrightarrow 1} = \\sqrt {{{0.805}^2}{{\\cos }^2}\\left( {39.8} \\right) + {1^2}{{\\sin }^2}\\left( {39.8} \\right)} = 0.890[\/latex]\n\n[latex]\\left( \\begin{array}{l}\\beta _{g,a}^{5 \\leftrightarrow 3} = 1.0\\\\\\beta _{g,b}^{5 \\leftrightarrow 3} = 0.70{\\left( {\\sqrt {{3^2} + {{2.5}^2}} } \\right)^{0.26}} = 0.997\\\\\\omega = {\\tan ^{ - 1}}\\left( {\\dfrac{{2.5}}{3}} \\right) = 39.8{\\rm{ \\:deg}}\\end{array} \\right) \\to \\beta _{g,sk}^{5 \\leftrightarrow 1} = \\sqrt {{{0.997}^2}{{\\cos }^2}\\left( {39.8} \\right) + {1^2}{{\\sin }^2}\\left( {39.8} \\right)} = 0.998[\/latex]\n\nThe reduction factor for pile #5 is calculated from Eq. 6.166 as:\n\n[latex]\\beta _g^5 = \\beta _{g,s}^{5 \\leftrightarrow 4} \\times \\beta _{g,s}^{5 \\leftrightarrow 6} \\times \\beta _{g,l}^{5 \\leftrightarrow 2} \\times \\beta _{g,sk}^{5 \\leftrightarrow 1} \\times \\beta _{g,sk}^{5 \\leftrightarrow 3} = 0.728 \\times 0.931 \\times 0.874 \\times 0.890 \\times 0.998 = 0.526[\/latex]","rendered":"

Determine the reduction factor \u03b2<\/em>g<\/sub>5<\/sup> for pile #5 shown in the figure below, when the lateral load acts along the weak axis of the group (X-X), and along the strong axis of the group (Y-Y). The diameter of all piles is equal to D.<\/em><\/p>\n

\"Diagram
Example 6.16. Problem description and input parameters.<\/figcaption><\/figure>\n

1. Load along the weak axis, HX-X<\/sub><\/em> :<\/p>\n

Interaction of pile #5 with all the remaining piles in the group must be considered, according to Eq. 6.166 and Figure 6.130<\/a>, as:<\/p>\n

[latex]\\beta _g^5 = \\begin{array}{*{20}{l}}{\\beta _{g,s}^{5 \\leftrightarrow 4}{\\rm{ }}\\left( {{\\rm{side \\: by \\:side}}} \\right)}\\\\{ \\times \\beta _{g,s}^{5 \\leftrightarrow 6}{\\rm{ (side \\:by \\:side)}}}\\\\{ \\times \\beta _{g,l}^{5 \\leftrightarrow 2}{\\rm{ (leading \\:in \\:line)}}}\\\\{ \\times \\beta _{g,sk}^{5 \\leftrightarrow 1}{\\rm{ (leading \\:skewed)}}}\\\\{ \\times \\beta _{g,sk}^{5 \\leftrightarrow 3}{\\rm{(leading \\:skewed)}}}\\end{array}[\/latex]<\/p>\n

The individual interaction factors are determined from Figures 6.127<\/a>–6.129<\/a> as:<\/p>\n

[latex]\\beta _{g,s}^{5 \\leftrightarrow 4} = \\beta _{g,s}^{5 \\leftrightarrow 6} = 0.64 \\times {3^{0.34}} = 0.929[\/latex]<\/p>\n

[latex]\\beta _{g,l}^{5 \\leftrightarrow 2} = 0.70 \\times {2.5^{0.26}} = 0.888[\/latex]<\/p>\n

[latex]\\left( \\begin{array}{l}\\beta _{g,a}^{5 \\leftrightarrow 1} = 1.0\\\\\\beta _{g,b}^{5 \\leftrightarrow 1} = 0.7{\\left( {\\sqrt {{3^2} + {{2.5}^2}} } \\right)^{0.26}} = 0.997\\\\\\omega = {\\tan ^{ - 1}}\\left( {\\dfrac{3}{{2.5}}} \\right) = 50.2{\\rm{ \\:deg}}\\end{array} \\right) \\to \\beta _{g,sk}^{5 \\leftrightarrow 1} = \\beta _{g,sk}^{5 \\leftrightarrow 3} = \\sqrt {{{0.997}^2}{{\\cos }^2}\\left( {50.2} \\right) + {1^2}{{\\sin }^2}\\left( {50.2} \\right)} = 0.998[\/latex]<\/p>\n

The reduction factor for pile #5 is calculated from Eq. 6.166 as:<\/p>\n

[latex]\\beta _g^5 = \\beta _{g,s}^{5 \\leftrightarrow 4} \\times \\beta _{g,s}^{5 \\leftrightarrow 6} \\times \\beta _{g,l}^{5 \\leftrightarrow 2} \\times \\beta _{g,sk}^{5 \\leftrightarrow 1} \\times \\beta _{g,sk}^{5 \\leftrightarrow 3} = 0.929 \\times 0.929 \\times 0.888 \\times 0.998 \\times 0.998 = 0.763[\/latex]<\/p>\n

2. Load along the strong axis, HY-Y<\/sub><\/em> :<\/p>\n

The procedure is similar, but notice that the relative position of the pile compared to its neighboring piles must be re-assessed according to Figure 6.130<\/a>, as:<\/p>\n

[latex]\\beta _g^5 = \\begin{array}{*{20}{l}}{\\beta _{g,s}^{5 \\leftrightarrow 4}{\\rm{ }}\\left( {{\\rm{trailining\\: in \\:line}}} \\right)}\\\\{ \\times \\beta _{g,s}^{5 \\leftrightarrow 6}{\\rm{ (leading \\:in \\:line)}}}\\\\{ \\times \\beta _{g,l}^{5 \\leftrightarrow 2}{\\rm{ (side \\:by \\:side)}}}\\\\{ \\times \\beta _{g,sk}^{5 \\leftrightarrow 1}{\\rm{ (trailing \\:skewed)}}}\\\\{ \\times \\beta _{g,sk}^{5 \\leftrightarrow 3}{\\rm{(leading \\:skewed)}}}\\end{array}[\/latex]<\/p>\n

The individual interaction factors are determined from Figures 6.127<\/a>–6.129<\/a> as:<\/p>\n

[latex]\\beta _{g,s}^{5 \\leftrightarrow 4} = 0.48 \\times {3^{0.38}} = 0.728[\/latex]<\/p>\n

[latex]\\beta _{g,s}^{5 \\leftrightarrow 6} = 0.70 \\times {3^{0.26}} = 0.931[\/latex]<\/p>\n

[latex]\\beta _{g,l}^{5 \\leftrightarrow 2} = 0.64 \\times {2.5^{0.34}} = 0.874[\/latex]<\/p>\n

[latex]\\left( \\begin{array}{l}\\beta _{g,a}^{5 \\leftrightarrow 1} = 1.0\\\\\\beta _{g,b}^{5 \\leftrightarrow 1} = 0.48{\\left( {\\sqrt {{3^2} + {{2.5}^2}} } \\right)^{0.39}} = 0.805\\\\\\omega = {\\tan ^{ - 1}}\\left( {\\dfrac{{2.5}}{3}} \\right) = 39.8{\\rm{ \\:deg}}\\end{array} \\right) \\to \\beta _{g,sk}^{5 \\leftrightarrow 1} = \\sqrt {{{0.805}^2}{{\\cos }^2}\\left( {39.8} \\right) + {1^2}{{\\sin }^2}\\left( {39.8} \\right)} = 0.890[\/latex]<\/p>\n

[latex]\\left( \\begin{array}{l}\\beta _{g,a}^{5 \\leftrightarrow 3} = 1.0\\\\\\beta _{g,b}^{5 \\leftrightarrow 3} = 0.70{\\left( {\\sqrt {{3^2} + {{2.5}^2}} } \\right)^{0.26}} = 0.997\\\\\\omega = {\\tan ^{ - 1}}\\left( {\\dfrac{{2.5}}{3}} \\right) = 39.8{\\rm{ \\:deg}}\\end{array} \\right) \\to \\beta _{g,sk}^{5 \\leftrightarrow 1} = \\sqrt {{{0.997}^2}{{\\cos }^2}\\left( {39.8} \\right) + {1^2}{{\\sin }^2}\\left( {39.8} \\right)} = 0.998[\/latex]<\/p>\n

The reduction factor for pile #5 is calculated from Eq. 6.166 as:<\/p>\n

[latex]\\beta _g^5 = \\beta _{g,s}^{5 \\leftrightarrow 4} \\times \\beta _{g,s}^{5 \\leftrightarrow 6} \\times \\beta _{g,l}^{5 \\leftrightarrow 2} \\times \\beta _{g,sk}^{5 \\leftrightarrow 1} \\times \\beta _{g,sk}^{5 \\leftrightarrow 3} = 0.728 \\times 0.931 \\times 0.874 \\times 0.890 \\times 0.998 = 0.526[\/latex]<\/p>\n","protected":false},"author":1,"menu_order":44,"template":"","meta":{"pb_show_title":"","pb_short_title":"Example 6.16","pb_subtitle":"Estimation of the p-y curve reduction factor for a laterally loaded pile group","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-657","chapter","type-chapter","status-publish","hentry"],"part":421,"_links":{"self":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapters\/657","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\/657\/revisions"}],"predecessor-version":[{"id":658,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapters\/657\/revisions\/658"}],"part":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/parts\/421"}],"metadata":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapters\/657\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/media?parent=657"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapter-type?post=657"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/contributor?post=657"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/license?post=657"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}