{"id":625,"date":"2025-03-14T04:54:37","date_gmt":"2025-03-14T04:54:37","guid":{"rendered":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/chapter\/6-26-rigorous-elasticity-method-for-estimating-pile-deformations\/"},"modified":"2026-03-16T14:15:15","modified_gmt":"2026-03-16T14:15:15","slug":"6-26-rigorous-elasticity-method-for-estimating-pile-deformations","status":"publish","type":"chapter","link":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/chapter\/6-26-rigorous-elasticity-method-for-estimating-pile-deformations\/","title":{"raw":"6.26 Rigorous elasticity method for estimating pile deformations","rendered":"6.26 Rigorous elasticity method for estimating pile deformations"},"content":{"raw":"As previously mentioned, Brom\u2019s method for estimating pile head deflection requires estimating the coefficient of subgrade reaction Kh<\/sub><\/em> by means of empirical formulas. As such, the method is inherently approximate. A rigorous method for estimating the deflection but also the rotation of the head of a single pile has recently been developed by Zheng et al.<\/em> (2024). Their formulation is based on modelling soil as elastic Tajimi-type medium, and ignoring possible slippage or separation at the soil-pile interface, therefore is applicable mainly to serviceability conditions. Zheng et al.<\/em>\u2019s method allows estimating the swaying Khh<\/sub><\/em>, rocking Krr<\/sub><\/em> and swaying-rocking Krh <\/sub><\/em>= Khr<\/sub><\/em> components of the head stiffness of a single floating pile embedded in elastic soil of finite thickness (Figure 6.109a<\/a>).\n\nIt is reminded that the swaying stiffness Khh<\/sub><\/em> is the lateral load Hw<\/sub><\/em> that, when applied to the head of a pile results in head deflection y<\/em>(0) = 1 (Figure 6.109b<\/a>). The rocking stiffness Krr<\/sub><\/em> is the bending moment Mw<\/sub><\/em> that, when applied to the head of a pile results in head rotation \u03c9<\/em>(0) = 1 (Figure 6.109b<\/a>). The swaying-rocking stiffness Krh<\/sub><\/em> is the bending moment that will develop to a pile with fixed head rotation \u03c9<\/em>(0) = 0 when the pile is subjected to unit displacement at its head y<\/em>(0) = 1, considering the sign convention in Figure 6.109b<\/a>. The swaying-rocking stiffness Khr<\/sub><\/em> is the lateral force that will develop to a pile with fixed head displacement y<\/em>(0) = 0 when the pile is subjected to unit rotation at its head \u03c9<\/em>(0) = 1, considering the sign convention in Figure 6.109b<\/a>. Moreover, it is Krh<\/sub><\/em> = Khr<\/sub><\/em>.\n\nUsing Zheng et al.<\/em>\u2019s method to calculate the stiffness components allows establishing the stiffness [K<\/strong>] and compliance [S<\/strong>] matrixes of the pile, as:\n\n(6.128a)<\/strong> [latex]\\left[ {\\begin{array}{*{20}{c}}{{H_w}}\\\\{{M_w}}\\end{array}} \\right] = \\left[ {\\bf{K}} \\right]\\left[ {\\begin{array}{*{20}{c}}{y(0)}\\\\{\\omega (0)}\\end{array}} \\right] = \\left[ {\\begin{array}{*{20}{c}}{{K_{hh}}}&{{K_{hr}}}\\\\{{K_{rh}}}&{{K_{rr}}}\\end{array}} \\right]\\left[ {\\begin{array}{*{20}{c}}{y(0)}\\\\{\\omega (0)}\\end{array}} \\right][\/latex]\n\n(6.128b) <\/strong>[latex]\\left[ {\\begin{array}{*{20}{c}}{y(0)}\\\\{\\omega (0)}\\end{array}} \\right] = \\left[ {\\bf{S}} \\right]\\left[ {\\begin{array}{*{20}{c}}{{H_w}}\\\\{{M_w}}\\end{array}} \\right] = {\\left[ {\\bf{K}} \\right]^{ - 1}}\\left[ {\\begin{array}{*{20}{c}}{{H_w}}\\\\{{M_w}}\\end{array}} \\right][\/latex]\n\nIt is reminded that the stiffness and compliance matrixes are independent of the load conditions, and of any restrains imposed to pile head deflection or rotation. The charts provided in Figure 6.110<\/a> allow calculating directly the components of the stiffness matrix for soil with Poisson\u2019s ratio vs<\/sub><\/em> = 0.3 and different values of the pile-soil moduli ratio Ep<\/sub><\/em>\/Es<\/sub><\/em>, the slenderness of the pile L<\/em>\/D<\/em> and the dimensionless thickness of the foundation soil layer H<\/em>\/L<\/em>. Having estimated these components one can calculate the compliance matrix by inverting the stiffness matrix, and subsequently the deflection y<\/em>(0) and the rotation \u03c9<\/em>(0) of the pile head for any load combination acting at the pile head, as well as head fixity conditions.\n\nIt is interesting to observe in Figure 6.110<\/a> that there exists a critical pile slenderness ratio L<\/em>\/D<\/em>, beyond which pile stiffness becomes independent of the length of the pile, and the bottom part of the pile no longer contributes to its response to lateral load. Therefore one can argue that, if the top part of the soil profile of dimensionless thickness equal to the critical pile slenderness comprises uniform soil conditions, this method (but also Broms method) will provide reasonably accurate results, even when the pile is installed in layered soil. This is not limited to serviceability conditions, but can be\u00a0extrapolated to the estimation of the collapse lateral load. However, the critical pile slenderness is not the same for all components of stiffness, and on top of that it varies with the pile-soil moduli ratio Ep<\/sub><\/em>\/Es<\/sub><\/em>.\n\n[caption id=\"attachment_623\" align=\"aligncenter\" width=\"800\"]\"Figure Figure 6.109. (a) Schematic of a floating pile embedded in a soil layer of finite thickness and subjected to lateral load and\/or bending moment at its head; (b) Definition of the components of pile stiffness (Zheng et al.<\/em> 2024).[\/caption]\n\n[caption id=\"attachment_623\" align=\"aligncenter\" width=\"936\"]\"Twelve Figure 6.110. Variation of pile head stiffness components with pile-soil moduli ratio Ep<\/sub>\/Es<\/sub><\/em>, pile slenderness L\/D<\/em> and dimensionless thickness of the foundation soil layer H\/L<\/em>. Results for soil Poisson\u2019s ratio vs<\/sub><\/em> = 0.3 (Results from Zheng et al.<\/em> 2024).[\/caption]","rendered":"

As previously mentioned, Brom\u2019s method for estimating pile head deflection requires estimating the coefficient of subgrade reaction Kh<\/sub><\/em> by means of empirical formulas. As such, the method is inherently approximate. A rigorous method for estimating the deflection but also the rotation of the head of a single pile has recently been developed by Zheng et al.<\/em> (2024). Their formulation is based on modelling soil as elastic Tajimi-type medium, and ignoring possible slippage or separation at the soil-pile interface, therefore is applicable mainly to serviceability conditions. Zheng et al.<\/em>\u2019s method allows estimating the swaying Khh<\/sub><\/em>, rocking Krr<\/sub><\/em> and swaying-rocking Krh <\/sub><\/em>= Khr<\/sub><\/em> components of the head stiffness of a single floating pile embedded in elastic soil of finite thickness (Figure 6.109a<\/a>).<\/p>\n

It is reminded that the swaying stiffness Khh<\/sub><\/em> is the lateral load Hw<\/sub><\/em> that, when applied to the head of a pile results in head deflection y<\/em>(0) = 1 (Figure 6.109b<\/a>). The rocking stiffness Krr<\/sub><\/em> is the bending moment Mw<\/sub><\/em> that, when applied to the head of a pile results in head rotation \u03c9<\/em>(0) = 1 (Figure 6.109b<\/a>). The swaying-rocking stiffness Krh<\/sub><\/em> is the bending moment that will develop to a pile with fixed head rotation \u03c9<\/em>(0) = 0 when the pile is subjected to unit displacement at its head y<\/em>(0) = 1, considering the sign convention in Figure 6.109b<\/a>. The swaying-rocking stiffness Khr<\/sub><\/em> is the lateral force that will develop to a pile with fixed head displacement y<\/em>(0) = 0 when the pile is subjected to unit rotation at its head \u03c9<\/em>(0) = 1, considering the sign convention in Figure 6.109b<\/a>. Moreover, it is Krh<\/sub><\/em> = Khr<\/sub><\/em>.<\/p>\n

Using Zheng et al.<\/em>\u2019s method to calculate the stiffness components allows establishing the stiffness [K<\/strong>] and compliance [S<\/strong>] matrixes of the pile, as:<\/p>\n

(6.128a)<\/strong> [latex]\\left[ {\\begin{array}{*{20}{c}}{{H_w}}\\\\{{M_w}}\\end{array}} \\right] = \\left[ {\\bf{K}} \\right]\\left[ {\\begin{array}{*{20}{c}}{y(0)}\\\\{\\omega (0)}\\end{array}} \\right] = \\left[ {\\begin{array}{*{20}{c}}{{K_{hh}}}&{{K_{hr}}}\\\\{{K_{rh}}}&{{K_{rr}}}\\end{array}} \\right]\\left[ {\\begin{array}{*{20}{c}}{y(0)}\\\\{\\omega (0)}\\end{array}} \\right][\/latex]<\/p>\n

(6.128b) <\/strong>[latex]\\left[ {\\begin{array}{*{20}{c}}{y(0)}\\\\{\\omega (0)}\\end{array}} \\right] = \\left[ {\\bf{S}} \\right]\\left[ {\\begin{array}{*{20}{c}}{{H_w}}\\\\{{M_w}}\\end{array}} \\right] = {\\left[ {\\bf{K}} \\right]^{ - 1}}\\left[ {\\begin{array}{*{20}{c}}{{H_w}}\\\\{{M_w}}\\end{array}} \\right][\/latex]<\/p>\n

It is reminded that the stiffness and compliance matrixes are independent of the load conditions, and of any restrains imposed to pile head deflection or rotation. The charts provided in Figure 6.110<\/a> allow calculating directly the components of the stiffness matrix for soil with Poisson\u2019s ratio vs<\/sub><\/em> = 0.3 and different values of the pile-soil moduli ratio Ep<\/sub><\/em>\/Es<\/sub><\/em>, the slenderness of the pile L<\/em>\/D<\/em> and the dimensionless thickness of the foundation soil layer H<\/em>\/L<\/em>. Having estimated these components one can calculate the compliance matrix by inverting the stiffness matrix, and subsequently the deflection y<\/em>(0) and the rotation \u03c9<\/em>(0) of the pile head for any load combination acting at the pile head, as well as head fixity conditions.<\/p>\n

It is interesting to observe in Figure 6.110<\/a> that there exists a critical pile slenderness ratio L<\/em>\/D<\/em>, beyond which pile stiffness becomes independent of the length of the pile, and the bottom part of the pile no longer contributes to its response to lateral load. Therefore one can argue that, if the top part of the soil profile of dimensionless thickness equal to the critical pile slenderness comprises uniform soil conditions, this method (but also Broms method) will provide reasonably accurate results, even when the pile is installed in layered soil. This is not limited to serviceability conditions, but can be\u00a0extrapolated to the estimation of the collapse lateral load. However, the critical pile slenderness is not the same for all components of stiffness, and on top of that it varies with the pile-soil moduli ratio Ep<\/sub><\/em>\/Es<\/sub><\/em>.<\/p>\n

\"Figure
Figure 6.109. (a) Schematic of a floating pile embedded in a soil layer of finite thickness and subjected to lateral load and\/or bending moment at its head; (b) Definition of the components of pile stiffness (Zheng et al.<\/em> 2024).<\/figcaption><\/figure>\n
\"Twelve
Figure 6.110. Variation of pile head stiffness components with pile-soil moduli ratio Ep<\/sub>\/Es<\/sub><\/em>, pile slenderness L\/D<\/em> and dimensionless thickness of the foundation soil layer H\/L<\/em>. Results for soil Poisson\u2019s ratio vs<\/sub><\/em> = 0.3 (Results from Zheng et al.<\/em> 2024).<\/figcaption><\/figure>\n","protected":false},"author":1,"menu_order":39,"template":"","meta":{"pb_show_title":"","pb_short_title":"6.26 Rigorous elasticity method for estimating pile deformations","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-625","chapter","type-chapter","status-publish","hentry"],"part":421,"_links":{"self":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapters\/625","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\/625\/revisions"}],"predecessor-version":[{"id":626,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapters\/625\/revisions\/626"}],"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\/625\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/media?parent=625"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/pressbooks\/v2\/chapter-type?post=625"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/contributor?post=625"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/libraryresources.nse.org.ng\/fundamentalsoffoundationengineering\/wp-json\/wp\/v2\/license?post=625"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}