{"id":566,"date":"2025-03-31T04:39:08","date_gmt":"2025-03-31T04:39:08","guid":{"rendered":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/chapter\/example-6-9-calculation-of-settlement-of-an-end-bearing-pile-in-layered-soil\/"},"modified":"2026-03-16T14:13:12","modified_gmt":"2026-03-16T14:13:12","slug":"example-6-9-calculation-of-settlement-of-an-end-bearing-pile-in-layered-soil","status":"publish","type":"chapter","link":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/chapter\/example-6-9-calculation-of-settlement-of-an-end-bearing-pile-in-layered-soil\/","title":{"raw":"Example 6.9","rendered":"Example 6.9"},"content":{"raw":"Calculate the immediate settlement of the driven solid concrete pile shown in the Figure below, using i) the method of Poulos and Davis (1991) for end-bearing piles in uniform soil underlaid by rigid bedrock (claystone) and ii) the method of Zheng et al.<\/em> (2023) for end-bearing piles in two-layered soil underlaid by rigid bedrock. Repeat the calculation, this time considering that instead of rigid claystone, the bedrock consists of hard alluvial clay with Eb<\/sub><\/em> = 55 MPa.\n\n[caption id=\"attachment_565\" align=\"aligncenter\" width=\"400\"]\"Schematic Example 6.9. Problem description and input parameters.[\/caption]\n\n1. Calculation of settlement according to Poulos and Davis (1991) for Eb<\/sub><\/em> = inf<\/em>:\n\nWe estimate the pile head stiffness factor Ib<\/sub><\/em> from Figure 6.71<\/a> using a weighted average soil modulus from Eq. 6.94 equal to Es<\/sub><\/em> = 5.5 MPa. For Kp<\/sub><\/em> = Ep<\/sub><\/em>\/Es<\/sub><\/em> (solid pile) = 25,000\/5.5 = 4,545 and L<\/em>\/D<\/em> = 20 it is Ib<\/sub><\/em> \u2248 0.98 (see Figure below).\n\n[caption id=\"attachment_565\" align=\"aligncenter\" width=\"400\"]\"A Example 6.9. Calculation of Ib<\/sub>.<\/em>[\/caption]\n\nTherefore settlement of the end-bearing pile is equal to:\n\n[latex]\\dfrac{{{E_p}{A_p}{\\rho _e}}}{{{Q_w}L}} = {I_b} = 0.98[\/latex]\n\n[latex]{\\rho _e} = 0.98 \\times \\dfrac{{1000 \\times 10}}{{25000000 \\times \\left( {\\dfrac{{\\pi {{0.5}^2}}}{4}} \\right)}} = 1.99{\\rm{\\: mm}}[\/latex]\n\n2.\u00a0Calculation of settlement according to Zheng et al.<\/em> (2024) for Eb<\/sub><\/em> = inf<\/em>:\n\nIn lack of charts for soil Poisson\u2019s ratio vs<\/sub><\/em> = 0.5, we will use the chart of Figure 6.74<\/a> to estimate the pile head stiffness Qw<\/sub><\/em>\/Ep<\/sub>D<\/em>\u03c1<\/em>e<\/sub>. Since here Ep<\/sub><\/em>\/Es2<\/sub><\/em> = 25,000 we will use the chart for the nearest Kp<\/sub><\/em> value of K<\/em>p<\/sub>= Ep<\/sub><\/em>\/Es2<\/sub><\/em> (solid pile) = 5,000 and Es<\/sub><\/em>1<\/sub>\/Es<\/sub><\/em>2<\/sub> = 10, L<\/em>\/D<\/em> = 20. As shown below, we obtain Qw<\/sub><\/em>\/Ep<\/sub>D<\/em>\u03c1<\/em>e<\/sub> \u2248 0.048.\n\n[caption id=\"attachment_565\" align=\"aligncenter\" width=\"400\"]\"ChartEs2. The chart contains multiple curves, each one for different Es1\/Es2 values. For L\/D = 20 and Es1\/Es2 = 10 its is found that the pile head stiffness is approximately 0.048.\" width=\"400\" height=\"379\"> Example 6.9. Calculation of pile head stiffness.[\/caption]\n\nTherefore pile settlement is estimated to be:\n\n[latex]{\\rho _e} = \\dfrac{{{Q_w}}}{{{E_p}D \\times 0.048}} = \\dfrac{{1000}}{{25000000 \\times 0.5 \\times 0.048}} = 1.66{\\rm{ \\:mm}}[\/latex]\n\nThis settlement value is lower to that computed with the method of Poulos and Davis, which requires using a weighted average soil modulus if the pile is driven in layered soil.\n\n3. Calculation of settlement for hard alluvial clay bedrock (Eb<\/sub><\/em> = 55 MPa) :\n\nWe first consider the pile as friction pile, and obtain the factor Is<\/sub><\/em> from Figure 6.66<\/a>. For Kp<\/sub><\/em> = Ep<\/sub><\/em>\/Es<\/sub><\/em> (solid pile) = 25,000\/5.5 = 4,545 and L<\/em>\/D<\/em> = 20 it is Is<\/sub><\/em> \u2248 0.09.\n\nThus, from Eq. 6.95 it is:\n\n[latex]{\\rho _{e,friction}} = \\dfrac{{{Q_w}}}{{{E_s}D}}{I_s} = \\dfrac{{1000}}{{5500 \\times 0.5}}0.09 = 32.7{\\rm{ \\:mm}}[\/latex]\n\nWe can now refer to Figure 6.72<\/a> and estimate the settlement of the end-bearing pile, as function of \u03c1<\/em>e,friction<\/sub> while considering pile slenderness L<\/em>\/D<\/em> = 20 and Eb<\/sub><\/em>\/Es<\/sub><\/em> = 55\/5.5 = 10. Note that the particular figure presents results for Kp<\/sub><\/em> = 1000, therefore will provide an upper bound of settlement as here Kp<\/sub><\/em> = 4,545.\n\n[caption id=\"attachment_565\" align=\"aligncenter\" width=\"400\"]\"Graph Example 6.9. Calculation of \u03c1e,end bearing<\/sub><\/em>\/\u03c1e,friction<\/sub>.<\/em>[\/caption]\n\nThe above figure suggests that \u03c1<\/em>e,friction<\/sub>\/\u03c1<\/em>e,end bearing <\/sub>= 0.6, therefore\u00a0 \u03c1<\/em>e<\/sub> = 0.6 \u00d7 32.7 = 19.6 mm.\n\nIt is clear from the above that: i) Considering a weighted average modulus may result in overestimating settlement when the surficial layer (crust) is stiffer than the bottom layer (soft soil), and ii) Assuming that the bedrock is incompressible can result in significantly underestimating settlement of relatively short piles (low L<\/em>\/D<\/em> values), when the ratio of Young\u2019s modulus of the bedrock over the Young\u2019s modulus of the soil layer is Eb<\/sub><\/em>\/Es<\/sub><\/em> = 10, or less.","rendered":"

Calculate the immediate settlement of the driven solid concrete pile shown in the Figure below, using i) the method of Poulos and Davis (1991) for end-bearing piles in uniform soil underlaid by rigid bedrock (claystone) and ii) the method of Zheng et al.<\/em> (2023) for end-bearing piles in two-layered soil underlaid by rigid bedrock. Repeat the calculation, this time considering that instead of rigid claystone, the bedrock consists of hard alluvial clay with Eb<\/sub><\/em> = 55 MPa.<\/p>\n

\"Schematic
Example 6.9. Problem description and input parameters.<\/figcaption><\/figure>\n

1. Calculation of settlement according to Poulos and Davis (1991) for Eb<\/sub><\/em> = inf<\/em>:<\/p>\n

We estimate the pile head stiffness factor Ib<\/sub><\/em> from Figure 6.71<\/a> using a weighted average soil modulus from Eq. 6.94 equal to Es<\/sub><\/em> = 5.5 MPa. For Kp<\/sub><\/em> = Ep<\/sub><\/em>\/Es<\/sub><\/em> (solid pile) = 25,000\/5.5 = 4,545 and L<\/em>\/D<\/em> = 20 it is Ib<\/sub><\/em> \u2248 0.98 (see Figure below).<\/p>\n

\"A
Example 6.9. Calculation of Ib<\/sub>.<\/em><\/figcaption><\/figure>\n

Therefore settlement of the end-bearing pile is equal to:<\/p>\n

[latex]\\dfrac{{{E_p}{A_p}{\\rho _e}}}{{{Q_w}L}} = {I_b} = 0.98[\/latex]<\/p>\n

[latex]{\\rho _e} = 0.98 \\times \\dfrac{{1000 \\times 10}}{{25000000 \\times \\left( {\\dfrac{{\\pi {{0.5}^2}}}{4}} \\right)}} = 1.99{\\rm{\\: mm}}[\/latex]<\/p>\n

2.\u00a0Calculation of settlement according to Zheng et al.<\/em> (2024) for Eb<\/sub><\/em> = inf<\/em>:<\/p>\n

In lack of charts for soil Poisson\u2019s ratio vs<\/sub><\/em> = 0.5, we will use the chart of Figure 6.74<\/a> to estimate the pile head stiffness Qw<\/sub><\/em>\/Ep<\/sub>D<\/em>\u03c1<\/em>e<\/sub>. Since here Ep<\/sub><\/em>\/Es2<\/sub><\/em> = 25,000 we will use the chart for the nearest Kp<\/sub><\/em> value of K<\/em>p<\/sub>= Ep<\/sub><\/em>\/Es2<\/sub><\/em> (solid pile) = 5,000 and Es<\/sub><\/em>1<\/sub>\/Es<\/sub><\/em>2<\/sub> = 10, L<\/em>\/D<\/em> = 20. As shown below, we obtain Qw<\/sub><\/em>\/Ep<\/sub>D<\/em>\u03c1<\/em>e<\/sub> \u2248 0.048.<\/p>\n

\"image\"
Es2. The chart contains multiple curves, each one for different Es1\/Es2 values. For L\/D = 20 and Es1\/Es2 = 10 its is found that the pile head stiffness is approximately 0.048.” width=”400″ height=”379″> Example 6.9. Calculation of pile head stiffness.<\/figcaption><\/figure>\n

Therefore pile settlement is estimated to be:<\/p>\n

[latex]{\\rho _e} = \\dfrac{{{Q_w}}}{{{E_p}D \\times 0.048}} = \\dfrac{{1000}}{{25000000 \\times 0.5 \\times 0.048}} = 1.66{\\rm{ \\:mm}}[\/latex]<\/p>\n

This settlement value is lower to that computed with the method of Poulos and Davis, which requires using a weighted average soil modulus if the pile is driven in layered soil.<\/p>\n

3. Calculation of settlement for hard alluvial clay bedrock (Eb<\/sub><\/em> = 55 MPa) :<\/p>\n

We first consider the pile as friction pile, and obtain the factor Is<\/sub><\/em> from Figure 6.66<\/a>. For Kp<\/sub><\/em> = Ep<\/sub><\/em>\/Es<\/sub><\/em> (solid pile) = 25,000\/5.5 = 4,545 and L<\/em>\/D<\/em> = 20 it is Is<\/sub><\/em> \u2248 0.09.<\/p>\n

Thus, from Eq. 6.95 it is:<\/p>\n

[latex]{\\rho _{e,friction}} = \\dfrac{{{Q_w}}}{{{E_s}D}}{I_s} = \\dfrac{{1000}}{{5500 \\times 0.5}}0.09 = 32.7{\\rm{ \\:mm}}[\/latex]<\/p>\n

We can now refer to Figure 6.72<\/a> and estimate the settlement of the end-bearing pile, as function of \u03c1<\/em>e,friction<\/sub> while considering pile slenderness L<\/em>\/D<\/em> = 20 and Eb<\/sub><\/em>\/Es<\/sub><\/em> = 55\/5.5 = 10. Note that the particular figure presents results for Kp<\/sub><\/em> = 1000, therefore will provide an upper bound of settlement as here Kp<\/sub><\/em> = 4,545.<\/p>\n

\"Graph
Example 6.9. Calculation of \u03c1e,end bearing<\/sub><\/em>\/\u03c1e,friction<\/sub>.<\/em><\/figcaption><\/figure>\n

The above figure suggests that \u03c1<\/em>e,friction<\/sub>\/\u03c1<\/em>e,end bearing <\/sub>= 0.6, therefore\u00a0 \u03c1<\/em>e<\/sub> = 0.6 \u00d7 32.7 = 19.6 mm.<\/p>\n

It is clear from the above that: i) Considering a weighted average modulus may result in overestimating settlement when the surficial layer (crust) is stiffer than the bottom layer (soft soil), and ii) Assuming that the bedrock is incompressible can result in significantly underestimating settlement of relatively short piles (low L<\/em>\/D<\/em> values), when the ratio of Young\u2019s modulus of the bedrock over the Young\u2019s modulus of the soil layer is Eb<\/sub><\/em>\/Es<\/sub><\/em> = 10, or less.<\/p>\n","protected":false},"author":1,"menu_order":28,"template":"","meta":{"pb_show_title":"","pb_short_title":"Example 6.9","pb_subtitle":"Calculation of settlement of an end-bearing pile in layered soil","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-566","chapter","type-chapter","status-publish","hentry"],"part":421,"_links":{"self":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapters\/566","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\/566\/revisions"}],"predecessor-version":[{"id":567,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapters\/566\/revisions\/567"}],"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\/566\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/media?parent=566"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapter-type?post=566"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/contributor?post=566"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/license?post=566"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}