Longitudinal vibration of a pile embedded in layered soil considering the transverse inertia effect of pile

The dynamic response of a viscoelastic bearing pile embedded in multilayered soil is theoretically investigated considering the transverse inertia effect of the pile. The soil layers surrounding the pile are modeled as a set of viscoelastic continuous media in three-dimensional axisymmetric space, and a simplified model, i.e., the distributed Voigt model, is proposed to simulate the dynamic interactions of the adjacent soil layers. Meanwhile, the pile is assumed to be a Rayleigh–Love rod with material damping and can be divided into several pile segments allowing for soil layers and pile defects. Both the vertical and radial displacement continuity conditions at the soil–pile interface are taken into account. The potential function decomposition method and the variable separation method are introduced to solve the governing equations of soil vibration in which the vertical and radial displacement components are coupled. On this basis, the impedance function at the top of the pile segment is derived by invoking the force and displacement continuity conditions at the soil–pile interface as well as the bottom of pile segment. The impedance function at the pile head is then obtained by means of the impedance function transfer method. By means of the inverse Fourier transform and convolution theorem, the velocity response in the time domain can also be obtained. The reasonableness of the assumptions of the soil-layer interactions have been verified by comparing the present solutions with two published solutions and a set of well-documented measured pile test data. A parametric analysis is then conducted using the present solutions to investigate the influence of the transverse inertia effect on the dynamic response of an intact pile and a defective pile for different design parameters of the soil–pile system.     

非均质土中桩端扩散虚土桩法的桩基纵向振动研究

根据桩端土应力扩散的规律,建立了桩端扩散虚土桩模型。基于该模型对非均质土中桩-土纵向耦合振动进行研究。利用复刚度传递多圈层平面应变模型,得到桩与虚土桩桩侧土的剪切复刚度。结合边界条件、初始条件和连续条件,对扩散虚土桩和实体桩动力方程从底层往顶层逐层进行求解,得到桩顶动力响应的频域解析解和时域半解析解。通过对桩端扩散虚土桩扩散角、扩散层厚度、桩侧土非均质性和桩长的影响进行计算分析,得到基于扩散虚土桩法桩-土纵向振动响应特性。研究结论可为桩基础动力设计和动态检测提供理论依据。     

Vertical dynamic response of a pipe pile in saturated soil layer

An analytical solution is developed in this paper to investigate the vertical time-harmonic response of a pipe pile embedded in a viscoelastic saturated soil layer. The wave propagation in the saturated soil is simulated by Biot’s 3D poroelastic theory and that in the pipe pile is simulated by 1D elastodynamic theory. Potential functions are applied to decouple the governing equations of the soil. The analytical solutions of the outer and inner soil in frequency domain are obtained by the method of separation of variables. The vertical response of the pipe pile is then obtained based on the continuity assumption of the displacement and stress between the pipe pile and both the outer and inner soil. The solution is compared with existing solutions to verify the validity. Numerical examples are presented to analyze the vibration characteristics of the pile.     

Method of initial parameters for laterally loaded piles embedded in layered soils

The method of initial parameters (MIP) was originally developed to solve the problem of a beam on an elastic foundation with applied concentrated forces along the span, which introduce discontinuities in the mathematical formulation of the problem. MIP is modified in this paper so that it can be used for solving the problem of a laterally loaded pile with discontinuities due to soil layering along the length of the pile. In this paper, the basis of MIP is outlined, and its use to find the deflection, slope, bending moment and shear force of laterally loaded piles is illustrated. Example problems of laterally loaded piles embedded in multi-layered soil media are provided.     

横观各向同性地基中管桩扭转振动响应解析解

考虑地基沉积过程中产生的竖向和水平向力学性质的差异,对横观各向同性地基中管桩扭转振动频域响应进行了理论研究。基于横观各向同性材料的本构关系以及桩-土耦合扭转振动,建立了桩土系统定解问题,通过Laplace变换和分离变量法求得了桩周土和桩芯土扭转振动位移形式解。通过桩-土接触面的连续条件,求得了管桩扭转频域响应解析解,并得到了桩顶复动刚度和速度导纳的表达式。将所得解退化到横观各向同性地基中实心桩解以及均匀地基中管桩解,并与已有文献进行了对比,验证了解的合理性。通过数值算例,分析了桩周土和桩芯土的横观各向同性力学参数对桩顶扭转复刚度及速度导纳的影响。     

Torsional dynamic response of a large‐diameter pipe pile in viscoelastic saturated soil

An analytical solution is developed in this paper to investigate the dynamic response of a large‐diameter end‐bearing pipe pile subjected to torsional loading in viscoelastic saturated soil. The wave propagation in saturated soil and pile are simulated by Biot's two‐phased linear theory and one‐dimensional elastic theory, respectively. The dynamic equilibrium equations of the outer soil, inner soil, and pile are established. The solutions for the outer and inner soils in frequency domain are obtained by Laplace transform technique and the separation of variables method. Then, the dynamic response of the pile is obtained on the basis of the perfect contacts between the pile and the outer soil as well as the inner soil. The results in this paper are compared with that of a solid pile in elastic saturated soil to verify the validity of the solution. Furthermore, the solution in this paper is compared with the classic plane strain solution to verify the solution further and check the accuracy of the plane strain solution. Numerical results are presented to analyze the vibration characteristics and illustrate the effect of the soil parameters and the geometry size of the pile on the complex impedance and velocity admittance of the pile head. Finally, the displacement of the soil at different depth and frequency is analyzed. Copyright © 2014 John Wiley & Sons, Ltd.     

Dynamic analysis of an axially loaded pile embedded in elastic-poroelasitc layered soil of finite thickness

This paper presents an analytical solution for determining the dynamic characteristics of axially loaded piles embedded in elastic-poroelastic layered soil of finite thickness. The interface between the elastic and poroelastic soil coincides with the groundwater table level, which is explicitly taken into account in the solution. The pile is modelled as elastic one-dimensional rod to account for the effect of its dynamic characteristics on the response of the soil-pile system. The solution is based on Biot's poroelastodynamic theory and the classical elastodynamic theory, which we use to establish the governing equations of the soil and pile. Accordingly, the pile base resistance, shaft reaction, and the complex impedance of soil-pile system are obtained using the method of Hankel integral transformation. Following the validation of the derived solution, we identify the main parameters affecting the vertical dynamic impedance of the pile via a parametric study. The presented method poses as an efficient alternative for quickly estimating the dynamic characteristics of axially loaded piles, without having to resort to complex numerical analyses.     

分层土中群桩水平动力阻抗的改进计算

采用动力Winkler地基梁模型,建立了主动桩和被动桩的简化解析运动方程,考虑地基土的分层特性、桩身剪切变形和转动惯量,提出了计算分层地基中单桩和群桩动力阻抗的改进计算方法,并与已有文献的计算结果进行了对比,结果表明：改进方法的计算结果与现场实测更接近。在低频段,改进方法的计算结果与已有文献结果较一致,但随着桩身剪切模量的减小,两者差异增大。差异主要是由考虑桩身剪切变形和转动惯量引起的,且前者影响较大,在低频段差异亦十分明显,而后者影响较小,在群桩阻抗峰值区域和高频段差异明显,需考虑桩身转动惯量的影响。     

Lateral dynamic response of a pipe pile in saturated soil layer

This paper presents an analytical solution for the lateral dynamic response of a pipe pile in a saturated soil layer. The wave propagations in the saturated soil and the pipe pile are simulated by Biot's three‐dimensional poroelastic theory and one‐dimensional elastic theory, respectively. The governing equations of soil are solved directly without introducing potential functions. The displacement response and dynamic impedances of the pipe pile are obtained based on the continuous conditions between the pipe pile and both the outer and inner soil. A comparison with an existing solution is performed to verify the proposed solution. Selected numerical results for the lateral dynamic responses and impedances of the pipe pile are presented to reveal the lateral vibration characteristics of the pile‐soil system. Copyright © 2015 John Wiley & Sons, Ltd.     

传递矩阵法分析层状地基中桩的扭转变形

研究了扭矩作用下单桩的扭转变形.采用积分变换和传递矩阵方法,求解了成层土在内部环形荷载作用下的基本解;利用此基本解并考虑桩土位移协调条件,提出了层状地基中单桩扭转变形分析的解析方法;并按此理论方法对匀质地基模型进行了数值计算,其结果与已有经典解答相当吻合.     

层状地基中群桩水平振动

在考虑地基土分层的基础上,采用动力Winkler地基模型模拟桩土相互作用并运用传递矩阵,求解层状地基中的单桩和群桩的阻抗函数.在计算动力相互作用因子时考虑了被动桩与土的相互作用.最后将相互作用因子和群桩阻抗的本文解与精确解进行对比,验证了本文方法的有效性.     

Vertical impedance of an end‐bearing pile in viscoelastic soil

This paper presents a new method to derive the analytical solution for the vertical impedance of an end‐bearing pile in viscoelastic soil. The soil is assumed as a homogeneous and isotropic layer, and the pile is considered as a one‐dimensional Euler rod. Considering both the vertical and radial displacements of soil and soil–pile coupled vibration, the governing equations of the soil and pile are established. The volumetric strain of soil is obtained by transformation on the equations of soil and variable separation method. Then the vertical and radial displacements of soil are obtained accordingly. The displacement response and impedance function of pile are derived based on the continuity assumption of the displacement and stress between the pile and soil. The solution is verified by being compared with an existing solution obtained by introducing potential functions. Furthermore, a comparison with two other simplified solutions is conducted. Numerical examples are presented to analyze the vibration characteristics of the pile. Copyright © 2014 John Wiley & Sons, Ltd.     

基于传递矩阵法的层状土中管桩水平动力阻抗分析

在考虑土体分层特性的基础上,分别建立了管桩桩周土体和桩芯土体的水平振动控制方程。通过引入势函数并考虑桩周土和桩芯土径向位移和环向位移的边界条件及其奇偶性,求得了管桩-土动力相互作用的刚度系数和阻尼系数。将土体模拟为连续分布的弹簧-阻尼器,并考虑桩芯土和桩周土的作用,建立了层状土中管桩的水平振动方程。借助初参数法和传递矩阵法求解了管桩的水平振动,得到了管桩桩顶的水平动力阻抗。通过数值分析,得到了土层剪切模量、管桩壁厚、桩周土和桩芯土剪切模量比、土层厚度等对管桩桩顶动力阻抗的影响规律。土层剪切模量、管桩壁厚、桩周土和桩芯土剪切模量比对层状土中管桩水平振动的影响主要在低频处,土层厚度在较宽的频率范围内对管桩水平振动有影响;管桩壁越厚,桩周土的剪切模量越大时,管桩水平动力阻抗的绝对值越大。     

基于模型试验的静压群桩引起的土体变形分析

以往对沉桩挤土效应的研究主要集中于单桩或双桩,对群桩挤土效应分析较少。基于室内模型试验,分析了群桩压入软土后土体的变形情况,得到了沉桩过程中及压桩后土体侧向位移、竖向变形和土层表面隆起变化的规律。认识到单桩侧向挤土位移随着距桩边距离的增加而以近似对数规律衰减,群桩压入后土体水平侧移和地表隆起是不断累积的,存在着已压入桩的遮帘作用,而且桩施工顺序对土体位移起到了关键作用,应该在实际工程中得到重视。     

Vertical dynamic response of an inhomogeneous viscoelastic pile

The vertical dynamic response of an inhomogeneous viscoelastic pile embedded in layered soil subjected to axial loading has been investigated. The interaction between pile and soil is simulated by a general Voigt model, one that has been demonstrated by earlier investigators to be capable of representing the plane strain case of soil adequately. The analytical solutions of pile responses in the frequency domain are obtained by using the (two-sided) Laplace transform. The corresponding semi-analytical solutions in the time domain for the case of a pile subjected to an instantaneous half-sine exciting force applied at the pile top are obtained via Fourier transform inversion. Using these solutions, a parametric study of the influence of the pile and soil properties on the vertical dynamic responses has been undertaken. It is shown that an abrupt variation of the soil properties with depth cannot yield evident reflection signal that may lead geotechnical engineers to assess the pile integrity wrongly from the velocity curve of the pile top, and the influence of viscosity of the pile material on the response is different from that of the damping of the soil surrounding the pile. The theoretical model developed in the present paper has also been validated in field studies, where it is shown by means of three examples that the solution developed in this study has been adequately verified by comparison of the theoretical pile model and field measurements of the dynamic responses.     

Group interaction on vertically loaded piles in saturated poroelastic soil

The pile-to-pile interaction was obtained for vertically loaded piles embedded in homogeneous poroelastic saturated soil. Deduced from Biot’s theory, the fundamental functions of the quasi-static development for the force, displacement and pore pressure were acquired in cylindrical coordinates. The pile–soil system was decomposed into extended soil and fictitious piles, and the compatibility condition was set up between the axial strain of the fictitious piles and the corresponding average strain over the extended soil. This approach results in the governing equations, which consist of the Fredholm integral equations of the second kind and the basic unknowns of the axial forces along the fictitious pile shaft. The axial force and settlement along the pile shaft were calculated based on the axial forces of the fictitious piles. The interaction between the piles was investigated under different consolidation conditions through a two-pile model, and two pile interaction factors were obtained. Stemming from the two-pile analysis, numerical analyses on the settlement of the pile groups were conducted to probe pile interaction with consolidation. The conventional solutions for the single-phase soil-pile problem seem to underestimate the interaction factor if the consolidation effect is taken into account as pile settlement continues. The pile-to-pile interaction can also aggravate the percentage of consolidation settlement (PCS), and as the pile number increases, the value of the PCS will also increase. Several key factors, such as the pile stiffness, pile slenderness ratio and pile spacing, are investigated to better understand the impact of consolidation on pile analysis.     

A new analytical model to study the influence of weld on the vertical dynamic response of prestressed pipe pile

This investigation is concerned with the mathematical analysis of a viscoelastic prestressed pipe pile embedded in multilayered soil under vertical dynamic excitation. The pile surrounding soil is governed by the plane strain model, and the soil plug is assumed to be an additional mass connected to the pipe pile shaft by applying the distributed Voigt model. Meanwhile, the prestressed pipe pile is assumed to be a vertical, viscoelastic, and hollow cylinder governed by the one‐dimensional wave equation. Then, analytical solutions of the dynamic response of the pipe pile in the frequency domain are derived by means of the Laplace transform and impedance function transfer method. Subsequently, the corresponding quasi‐analytical solution in the time domain for the case of the prestressed pipe pile undergoing a vertical semi‐sinusoidal exciting force applied at the pile top is obtained by employing the inverse Fourier transform. Utilizing these solutions, selected results for the velocity admittance curve and the reflected wave curve are presented for different heights of the soil plug to examine the influence of weld properties on the vertical dynamic response of prestressed pipe pile. The reasonableness of the theoretical model is verified by comparing the calculated results based on the presented solutions with measured results. Copyright © 2017 John Wiley & Sons, Ltd.     

Vertical dynamic response of pile in a radially heterogeneous soil layer

An analysis of a pile vertical response considering soil inhomogeneity in the radial direction under dynamic loads is presented. The solution technique is based on a three‐dimensional axisymmetric model, which includes the consideration of the vertical displacement of the soil. The soil domain is subdivided into a number of annular vertical zones, and the continuity of the displacements and stresses are imposed at both the interface of pile–soil and the interfaces of adjacent soil zones to establish the dynamic equilibrium equations of the pile–soil interaction. Then, the equations of each soil zone and of the pile are solved one by one to obtain the analytical and semi‐analytical dynamic responses at the top of the pile in the frequency domain and time domain. Parametric studies have been performed to examine the influence of soil parameters' variations in the radial direction caused by the construction effect on the dynamic responses of pile. The results of the studies have been summarized and presented in figures to illustrate the influences of the soil parameters as they change radially. The effect of the radius of the disturbed soil zone caused by construction is also studied in this paper. Copyright © 2008 John Wiley & Sons, Ltd.     

黏弹性地基中PCC桩扭转振动响应解析方法研究

考虑土体材料的黏性阻尼和桩-土扭转耦合振动,把桩看作一维杆,将土体视作三维轴对称黏弹性介质,对黏弹性地基中现浇混凝土大直径管桩（简称PCC桩）扭转振动频域特性进行了理论研究,采用Laplace变换和分离变量的方法求得了桩顶扭转频域响应解析解。将所得解完全退化到实心桩的解,并与经典平面应变解进行对比,验证了解析解的合理性。分析了桩长以及土体黏性阻尼系数对桩顶速度导纳和复动刚度的影响,得到了各参数对桩扭转振动特性影响的规律。分析表明：桩周土黏性阻尼系数增大可以显著提高桩顶扭转复动刚度和减小速度导纳振荡幅值,而桩芯土黏性阻尼系数的影响不明显;桩长越长,桩顶复动刚度越大,速度导纳振荡幅值越小,但当桩长增大到一定程度时,再继续增加桩长,桩顶复动刚度基本没有改变。     

层状土中单桩竖向简谐动力响应的简化解析方法

运用土动力学和结构动力学原理,基于改进的Winkler地基梁模型,同时考虑桩周土的弱化效应和地基的成层非均质性,采用数理方程方法分别求解土与桩的振动微分方程,建立了竖向荷载作用下单桩简谐动力响应分析的计算力学模型和方法,并将所得结果与有限元计算结果进行了对比分析,验证了建议计算方法的合理性,通过对影响单桩竖向简谐动力响应的各相关参数进行变动参数比较分析,得到了各影响参数对单桩竖向简谐动力响应特性的一般影响规律。