Soil‐pile interaction in the pile vertical vibration considering true three‐dimensional wave effect of soil

The dynamic response of an end bearing pile embedded in a linear visco‐elastic soil layer with hysteretic type damping is theoretically investigated when the pile is subjected to a time‐harmonic vertical loading at the pile top. The soil is modeled as a three‐dimensional axisymmetric continuum in which both its radial and vertical displacements are taken into account. The pile is assumed to be vertical, elastic and of uniform circular cross section. By using two potential functions to decompose the displacements of the soil layer and utilizing the separation of variables technique, the dynamic equilibrium equation is uncoupled and solved. At the interface of soil‐pile system, the boundary conditions of displacement continuity and force equilibrium are invoked to derive a closed‐form solution of the vertical dynamic response of the pile in frequency domain. The corresponding inverted solutions in time domain for the velocity response of a pile subjected to a semi‐sine excitation force applied at the pile top are obtained by means of inverse Fourier transform and the convolution theorem. A comparison with two other simplified solutions has been performed to verify the more rigorous solutions presented in this paper. Using the developed solutions, a parametric study has also been conducted to investigate the influence of the major parameters of the soil‐pile system on the vertical vibration characteristics of the pile. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

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 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.     

Torsional dynamic response of a pile embedded in layered soil based on the fictitious soil pile model

The torsional dynamic response of a pile embedded in layered soil is investigated while considering the influence of the pile end soil. The finite soil layers under the end of the pile are modeled as a fictitious soil pile that has the same cross-sectional area as the pile and is in perfect contact with the pile end. To allow for variations of the modulus or cross-sectional area of the pile and soil, the soil surrounding and below the pile is vertically decomposed into finite layers. Using the Laplace transform and impedance function transfer method, the analytical solution for the dynamic response of the pile head in the frequency domain is then obtained, and the relevant semi-analytical solution in the time domain is derived using the inverse Fourier transform and convolution theorem. The rationality and accuracy of the solution is verified by comparing the torsional dynamic behavior of the pile calculated with the fictitious soil pile with those based on a rigid support model and a viscoelastic support model. Finally, a parametric study is conducted to investigate the influence of the properties and thickness of the pile end soil on the torsional dynamic response of the pile.     

Dynamic characteristics of a large‐diameter pile in saturated soil and its application

This study theoretically investigates the dynamic response of an end‐bearing pile embedded in saturated soil considering the transverse inertial effect of the pile. The saturated soil surrounding the pile is described by Biot poroelastic theory, and the pile is represented by a Rayleigh‐Love rod because both the vertical and radial displacements at the soil‐pile interface are considered. The potential function decomposition method and variable separation method are introduced to solve the governing equations of the soil, in which the vertical and radial displacement components are coupled. The governing equation of the pile is solved using the continuity conditions at the pile‐soil interface. Next, the velocity admittance in the frequency domain and the velocity response in the time domain at the pile top are presented based on the Laplace transform and inverse Fourier transform, respectively. Subsequently, the reduced solution is compared with a 1‐dimensional model solution to verify the validity, and the influences of the slenderness ratio of the pile on the transverse inertial effect of the pile are analyzed. Moreover, Poisson ratio, the slenderness ratio of the pile, and the pile‐soil modulus ratio are studied. Finally, the theoretical and measured curves in the engineering project are compared, and the results demonstrate the good application prospects of the solution presented in this article.     

An analytical solution for wave propagation in a square pile due to transient point load

This paper presents an analytical solution for wave propagation in a square pile due to transient point load. The differential equation of dynamic equilibrium is established considering propagation of waves in both vertical and transverse directions. The soil resistance is simulated by Voigt model. The three-dimensional analytical solution is deduced by using Fourier transform and the separation of variable method. The arithmetical results of the proposed solution show that the velocity responses along the radial direction at the pile top are highly non-uniform. In addition, Young’s modulus and the pile side length exert undisputable influences on the velocity responses.     

Development of a three-dimensional soil model for the dynamic analysis of end-bearing pile groups subjected to vertical loads

This paper presents a new analytical model for calculating the steady-state impedance of pile groups subjected to vertical dynamic loads. The derived solution allows considering effects of radially but also vertically propagating soil waves on the soil attenuation function, pile interaction factor, and pile group impedance. The proposed model provides accurate estimates of the soil stress field and of the response of the pile group in the low as well as in the high-frequency range, unlike earlier solutions based on the plane-strain model to describe the soil surrounding the piles, which ignores the vertical soil stress gradient. The latter assumption results in underestimating pile group impedance and overestimating radiation damping for frequencies lower than the cutoff frequencies of the system, which are explicitly captured with the proposed solution.     

考虑桩周土竖向作用大直径楔形桩纵向振动特性

将桩土系统划分为数量足够多的微元段,相邻微元桩段接触面处的环形凸面与土的相互作用采用单个Voigt体模拟,求得Voigt体的弹簧和黏壶系数。结合相邻微元桩段接触面上的应力平衡条件和位移连续条件,得到修正的阻抗函数递推法,桩身采用Rayleigh杆考虑桩身的横向惯性效应。结合桩底的边界条件,运用拉普拉斯变换和修正的阻抗函数递推法求得了平面应变条件下成层土中考虑桩周土竖向作用时大直径楔形桩桩顶复阻抗的解析解。通过与已有解对比,研究了桩周土竖向作用对桩顶复刚度和桩顶在瞬态激振下的速度响应的影响,并在低频域内详细分析了桩周土的竖向作用与桩土系统参数对桩顶复刚度的影响的耦合作用。     

Analysis of piled raft with nodular pile subjected to vertical load using a Winkler model approach

An investigation is made to present analytical solutions provided by a Winkler model approach for analysis of piled rafts with nodular pile subjected to vertical loads in nonhomogeneous soils. The vertical stiffness coefficient along a piled raft with the nodular pile in nonhomogeneous soils is derived from the displacement given by the Mindlin solution for elastic continuum analysis. The vertical stiffness coefficients for the bases of the raft and the nodular part in the nodular pile in a soil are expressed by the Muki solution for the 3‐D elastic analysis. The relationship between settlement and vertical load on the pile base is presented considering the Mindlin solution and the equivalent thickness in the equivalent elastic method. The interaction factor between the shaft of the nodular pile and the soil is expressed taking into account the Mindlin solution and the equivalent elastic modulus. The relationship between settlement and vertical load for a piled raft with the nodular pile in nonhomogeneous soils is obtained by using the recurrence equation of influence factors of the pile for each layer. The percentage of each load carried by both nodular pile and raft subjected to vertical load is represented through the vertical influence factors proposed here. Comparison of the results calculated by the present method for piled rafts with nodular piles in nonhomogeneous soils has shown good agreement with those obtained from the finite element method and a field test. Copyright © 2016 John Wiley & Sons, Ltd.     

Improving dynamic soil parameters and advancing the pile signal matching technique

The pile signal matching technique widely used for estimating vertical resistances of piles during construction is highly influenced by the assumed dynamic soil parameters. Due to the lack of understanding and supporting data, constant soil parameters for the entire pile length have been routinely used. This practice is unrealistic and compromises the signal match quality. Using recently completed field tests, this paper develops empirical equations for dynamic soil parameters in terms of measureable soil properties and proposes an improved signal matching technique, thereby allowing for better match quality.     

饱和黏弹性地基土中管桩纵向振动研究

用解析方法在频率域内研究考虑质量耦合效应的饱和黏弹性地基土中管桩的纵向振动特性。基于Biot理论,采用薄层法,推导得到饱和黏弹性地基土的位移、应力等的表达式。将管桩等效为一维弹性杆件处理。根据界面连续性条件,给出饱和黏弹性地基土中管桩的纵向振动一般分析方法和桩顶动力复刚度的表达式。在该基础上,对比分析饱和地基土中实心桩和管桩纵向振动特性。通过算例分析,考察桩周土和桩芯土的力学参数对桩顶刚度因子和等效阻尼的影响。研究表明,饱和黏弹性地基土中实心桩和管桩的纵向振动有明显的差异。     

Analysis of pile‐raft foundations under complex loads in layered soils

Considering there is hardly any concerted effort to analyze the pile‐raft foundations under complex loads (combined with vertical loads, horizontal loads and moments), an analysis method is proposed in this paper to estimate the responses of pile‐raft foundations which are subjected to vertical loads, horizontal loads and moments in layered soils based on solutions for stresses and displacements in layered elastic half space. Pile to pile, pile to soil surface, soil surface to pile and soil surface to soil surface interactions are key ingredients for calculating the responses of pile‐raft foundations accurately. Those interactions are fully taken into account to estimate the responses of pile‐raft foundations subject to vertical loads, horizontal loads and moments in layered soils. The constraints of the raft on vertical movements, horizontal movements and rotations of the piles as well as the constraints of the raft on vertical movements and horizontal movements of the soils are considered to reflect the coupled effect on the raft. The method is verified through comparisons with the published methods and FEM. Then, the method is adopted to investigate the influence of soil stratigraphy on pile responses. The study shows that it is necessary to consider the soil non‐homogeneity when estimating the responses of pile‐raft foundations in layered soils, especially when estimating the horizontal responses of pile‐raft foundations. The horizontal loads and the moments have a significant impact on vertical responses of piles in pile‐raft foundations, while vertical loads have little influence on horizontal responses of piles in pile‐raft foundations in the cases of small deformations. The proposed method can provide a simple and useful tool for engineering design. Copyright © 2013 John Wiley & Sons, Ltd.     

列车运行时由轨道不平顺引起的地基振动研究

采用半解析法研究了列车荷载作用下列车-轨道-饱和地基系统的耦合振动问题。研究模型共分为3部分：车体简化为一个多刚体系统,在车轮与钢轨之间引入线性Hertizian弹簧接触模型模拟轮轨动力相互作用、采用离散轨枕支撑的弹性Euler梁来模拟轨道系统、下卧土体采用多孔饱和半空间模型。列车荷载分为轴重和由轨道不平顺引起的轮轨动力相互作用力。采用Fourier变换分别求解各子系统的控制方程,并通过动力子结构法对各子系统进行耦合。土体在时域内的动力响应通过快速Fourier变换求得。在分析了轮轨动力相互作用力的基础上,研究了轮轨动力作用力和列车轴重作用下饱和地基的动力响应,并分析了轨枕间距和土体渗透系数对饱和地基振动响应的影响。研究表明,轮轨动力作用力对地基远场振动有重要贡献,同时枕木间距对轨道与地基振动响应有较大影响。     

移动荷载作用下饱和土中单桩的动力响应

根据Biot动力理论,采用Fourier和Hankel变换方法得到了半空间饱和土受移动载荷及土体内受垂直简谐载荷作用下频域内基本解。根据虚拟桩法,得到了移动载荷作用下桩基的第2类Fredholm积分方程,并应用IFFT方法得到时间、空间域内单桩的动力响应。数值结果表明,移动荷载会引起桩身的负摩擦力;桩身最大轴力、孔压随移动荷载速度增加而增大;此外,在桩上端部会出现孔压集中现象。     

变截面桩竖向承载特性的数值模拟研究

以变截面桩为研究对象,采用FLAC3D对其竖向承载特性进行了数值分析,在分析中考虑了变截面桩的几何尺寸和土层分布的影响。研究结果表明:变截面桩中扩径部分的直径与未扩径部分的直径的比值δ越大,变截面桩的竖向极限承载力逐渐增大;变截面处接近桩顶时的竖向极限承载力比接近桩底时小;当扩径部分的长度逐渐增加时,变截面桩的竖向极限承载力逐渐增大;当变截面部分所处的土性为粘质土、变截面部分以下桩体所处的土性为砂土时,变截面桩的竖向极限承载能力明显高于相反的情况。     

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。     

Dynamic response of a pile considering the interaction of pile variable cross section with the surrounding layered soil

Assuming that the pile variable cross section interacts with the surrounding soil in the same way as the pile toe does with the bearing stratus, the interaction of pile variable cross section with the surrounding soil is represented by a Voigt model, which consists of a spring and a damper connected in parallel, and the spring constant and damper coefficient are derived. Thus, a more rigid pile–soil interaction model is proposed. The surrounding soil layers are modeled as axisymmetric continuum in which its vertical displacements are taken into account and the pile is assumed to be a Rayleigh–Love rod with material damping. Allowing for soil properties and pile defects, the pile–soil system is divided into several layers. By means of Laplace transform, the governing equations of soil layers are solved in frequency domain, and a new relationship linking the impedance functions at the variable‐section interface between the adjacent pile segments is derived using a Heaviside step function, which is called amended impedance function transfer method. On this basis, the impedance function at pile top is derived by amended impedance function transfer method proposed in this paper. Then, the velocity response at pile top can be obtained by means of inverse Fourier transform and convolution theorem. The effects of pile–soil system parameters are studied, and some conclusions are proposed. Then, an engineering example is given to confirm the rationality of the solution proposed in this paper. Copyright © 2017 John Wiley & Sons, Ltd.     

基于薄层法和移动坐标系法的地铁竖向振动在成层土层中传播

针对地铁竖向振动在成层地基中的传播,提出了移动荷载作用下层状地基的分析模型。基于该模型,利用狄拉克函数及三重傅里叶变换将时空域内单个移动简谐荷载转换为频率-波数域内的荷载,结合薄层法和移动坐标系法推导了单个移动简谐荷载和移动简谐线荷载作用下三维层状地基动力响应的解析解,并给出了移动简谐线荷载动力响应解析解中参数n的经验取值范围;分析了荷载的移动速度对层状地基动力响应的影响以及弹性模量、泊松比、阻尼比和荷载频率对土体临界速度的影响。结果表明：荷载的移动速度对不同频带的动力响应的影响范围不同;移动速度对低频响应的影响程度大于对高频响应的影响程度;相比于泊松比和阻尼比,弹性模量对土体临界速度的影响最大;频率越接近荷载振动频率的振动响应,其幅值越大,临界速度越小。