Time-domain analysis of velocity waves in a pipe pile due to a transient point load

The propagation of stress waves in a pipe pile subjected to a transient point load cannot be expressed using traditional one-dimensional (1D) wave theory. This paper presents an analytical solution used to investigate the wave propagation in a pipe pile under an axial point load. The soil resistance is simulated using the Winkler model, and the excitation force is simulated with a semi-sinusoidal impulse. A time-domain analytical solution for the three-dimensional wave equation is derived using the separation of variables and variation of constants methods. The solution is verified with a frequency domain analytical solution in which the time-domain response is calculated by numerical Fourier inverse transformation. Furthermore, the solution proposed in this paper is compared with the results of model testing and 3D FEM analysis. The comparisons show that the analytical solution proposed in this study agrees well with the results of previous studies. The proposed solution is subsequently applied in case studies. The vertical velocity responses in the circumferential and axial directions are analyzed to reveal the propagation characteristics of transient waves in the pipe pile. Moreover, the effects of the location and period of the excitation force, the side and tip resistances and high-order modes are studied in detail.     

Analytical solution for distributed torsional low strain integrity test for pipe pile

Low strain integrity tests (LSITs) are the most popular non-destructive methods for pile testing. However, traditional LSITs have encountered unprecedented challenges as the need for long pile and existing pile testing keeps multiplying. Compared to traditional longitudinal excitations, the torsional wave is less influenced by the velocity attenuation effect and can be subjected at the pile shaft for existing piles. Distributed torsional LSIT is proposed in this article with the presentation of the corresponding analytical solutions that exhibiting the velocity responses along the pile shaft. The solution is verified with previous simplified theoretical and rigorous finite element method (FEM) answers. At the end, the application of this method is exhibited through the identification of necking and concrete segregation defects on pipe piles, which shows the advantage of this method on long pile testing.     

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.     

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.     

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。     

A novel analytical approach for predicting the noncylindrical pile penetration‐induced soil displacement in undrained soil by combining use of cavity expansion and strain path methods

Since development of cavity expansion theory and strain path method, almost all the conventional analyses of pile penetration problem have been based on circular cross section penetrometer. However, noncylindrical pile (with noncircular cross section) is also required in geotechnical engineering such as rectangular cross‐sectional pile, X‐sectional cast‐in‐place concrete pile, H‐shaped steel pile, prefabricated vertical drains, and flat dilatometer. This paper presents a novel and general analytical approach for capturing the soil deformation mechanism around the pile with arbitrary cross section. The penetration problem is simulated by a new 2‐dimensional (radial and circumferential) cavity expansion model. Based on the theoretical framework of strain path method, the kinematics (velocity field) of the noncylindrical cavity expansion is reduced to solve the Laplace equation with arbitrary velocity boundary conditions by using the conformal mapping technique. Then, solutions for the strain and displacement, which could consider the large deformation effect, are obtained by the integration of the strain rate and velocity along the streamline. The analytical solution is validated by comparing the degenerate solution of this study with conventional circular (cylindrical) cavity expansion theory. Subsequently, typical numerical examples for the deformation mechanism of elliptical and rectangular cavity expansion are presented to prove the advantage of the proposed new solution particularly in capturing the noncylindrical symmetric displacement field. A brief application of the proposed new analytical solution to the interpretation of the smear effect of prefabricated vertical drain installation confirms its useful in geotechnical engineering.     

基于径向不变假定的现浇大直径管桩纵向振动响应频域解

假定各物理量沿径向不变化,建立了低应变瞬态集中荷载作用下现浇大直径管桩振动响应的计算模型和波动方程。采用Laplace变换法,求得了波动方程的频域解析解,采用Fourier逆变换求得了时域响应。将文中解计算结果与三维频域解析解进行了对比分析,文中解的入射峰-反射峰时间差与三维频域解差别很小,说明对于PCC桩这种大直径薄壁管桩,采用径向不变假定对计算结果几乎没有影响。因此,在PCC桩低应变检测波形分析时,采用基于径向不变假定的二维解是完全合理的。将文中二维频域解的计算结果与二维时域解的结果进行了对比分析,结果表明,2种解在入射波、第一个桩底反射波、第二个桩底反射波峰值大小和到达时间非常吻合,这说明考虑环向位移与否对计算结果没有太大的影响,研究PCC桩低应变动力响应的问题时忽略水平方向的位移是完全可以的。     

PCC桩低应变反射波法检测时速度波形成机制探讨

现浇混凝土大直径管桩（PCC桩）低应变检测时,桩头附近存在着明显的三维效应,表现在与激振点夹角不同的测点的速度响应存在着明显的差别。基于所建立的解析公式,研究了桩顶各振动模式的动力响应特征,分析了桩顶总体速度响应沿环向和径向的变化规律。研究结果表明：90º点的总体速度响应主要来源于轴对称模式,45º点、135º点和180º点的桩底反射峰大小和到达时间与轴对称模式接近,但入射波峰大小和到达时间与轴对称模式差别较大;桩顶速度响应主要来源于径向第1阶的模式,第2阶以上的模式贡献较小;各点速度响应主要是前几阶模式的叠加结果,环向第10阶以上的振动模式对总体速度响应的贡献较小;各模式对入射波峰的形成都有较大贡献,但反射波峰主要来自轴对称模式的贡献;高频干扰峰主要来自第1阶非轴对称模式,各点干扰波峰值、相位不一,90º点干扰波峰值最小,0º～90º点与90º～180º点相位相反;环向各点的速度响应差别较大,但沿径向的变化却不很明显。     

均质土中PCC桩与实心桩动力响应对比分析

现浇混凝土大直径管桩(简称PCC桩)由于具有较大的内、外壁摩阻力,在竖向荷载作用下表现出与一般实心桩不同的静、动力特性。采用建立的解析公式,对PCC桩、等截面积实心桩和等外直径实心桩3种桩的动力响应进行了对比分析,研究结果表明,当桩端阻力相对较小时,3种模型位移幅值差别不大;当桩端阻力较大时,3种模型幅频曲线和相频曲线存在较大差别;等面积实心桩位移幅值比PCC桩稍大,而等外直径实心桩的位移幅值比PCC桩稍小;位移相频曲线呈现等间距的震荡,实心桩的震荡峰值比PCC桩要高,且震荡峰值对应的频率各不相同,PCC桩出现震荡峰值的频率比实心桩小;3种计算模型动刚度震荡幅值差别较大,等直径实心桩的动刚度最大,等面积实心桩次之,PCC桩最小;等直径实心桩的虚部最大,PCC桩次之,等面积实心桩最小,振动过程中等直径实心桩桩顶阻尼最大;等面积实心桩的速度导纳最大,PCC桩次之,等直径实心桩最小。PCC桩充分发挥了单方混凝土的性能,比等截面实心桩具有更小的动力响应,比等外径实心桩更节省混凝土。     

低应变下变阻抗薄壁管桩动力响应频域解析解

管桩低应变完整性检测时,桩顶一点受到瞬态集中荷载的作用,其动力响应问题是一个三维波动问题。基于三维波动理论,建立了任意段变阻抗薄壁管桩动力响应的计算模型和波动方程,结合初边值条件,采用Laplace变换法求得了该波动方程的频域解析解,采用Fourier逆变换求得了时域响应。将计算结果与三维有限元结果进行了对比分析,解析解的计算结果和三维有限元结果较为接近,两者在入射波峰、缺陷反射峰和桩底反射峰处都较为吻合,两种方法位移响应曲线基本相同。给出了桩顶不同点的动力响应,并探讨了高频干扰问题。对变截面桩和变模量桩的动力响应特性进行了分析。     

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.     

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.     

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.     

采用Laplace变换分析桩基应力波传播问题

将土体中的混凝土桩基简化为黏滞介质中的一维黏弹性杆,桩顶部受锤头冲击产生内部应力波。根据杆内微元应力平衡条件建立杆中一维黏弹性应力波传播的控制方程,结合桩顶锤头冲击条件和桩底弹-黏性约束条件给出桩基两端的耦合边界条件。对控制方程和定解条件作Laplace变换并求解变换后的常微分方程,得到变换域的应力像函数解析解。采用数值反变换技术将像函数转变为时间域的应力波形。应用此方法可以较方便地分析桩基中应力波的产生、传播、反射及相互作用过程。     

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.     

周期分布桩体对平面SH波隔振效应的解析求解

将排桩对平面SH波的隔振简化为弹性波散射的二维平面问题,基于全空间中无限周期结构的周期特性,给出了一种求解无限周期分布桩体对平面SH波隔振效应的解析方法。该方法采用波函数展开法并结合Graf加法定理,利用全空间中相邻周期单元的散射波场在频域内相差一个相位的特性,仅选取一个周期单元,将入射波场和所有散射波场的贡献叠加后,根据边界条件求解待定系数,从而求得整个散射波场。该解析解能够精确求解无限周期分布桩体的散射问题,分析周期分布桩体数量较多时的隔振规律,弥补了以往理论分析中桩体个数较多时难以求解的不足。重点讨论了桩体个数、桩体刚度、桩体间距和桩体类型等因素对隔振效果的影响,结果表明：（1）该方法显著降低了求解大量桩体问题时的存储量和计算量,有限周期模型计算结果随桩体个数增多收敛于无限周期模型,反映了该方法的正确性;（2）整体上桩体刚度增大有利于提高隔振效果,但桩体刚度对隔振效果的提升有限,桩体剪切波速为土体5倍时已具有足够的隔振效果;（3）桩间距对隔振效果有着直接影响,间距越小则低频禁带宽度越大;（4）桩体类型对隔振效果有着显著影响,实心桩有着良好的隔振效果,而具有柔性内填充的管桩在低频段有着更佳的隔振效果。     

饱和土体中排桩对移动荷载的被动隔振效果分析

利用Muki和Sternberg的虚拟桩法,研究了饱和土体地基中排桩对移动荷载引起振动的被动隔振效果。隔振桩作为一维杆,饱和土体满足Biot理论。利用已有的移动荷载作用下的饱和土体的自由波场解及饱和土体内部受竖向圆形分布荷载作用下的基本解,建立了频域内土-桩的第2类Fredholm积分方程。通过Fourier逆变换得到时间域内评价隔振效果的振幅比。与已知文献结果相比较,验证了方法的正确性。数值结果表明,荷载速度对排桩的隔振效果有一定影响,即在相同隔振系统情况下,单排桩对低速荷载引起振动的隔振效果比高速移动荷载效果好。同时,较高速时的最佳桩长比低速时要短。     

Load-displacement and upper-bound solutions of a loaded laterally pile in clay based on a total-displacement-loading EMSD method

A total-displacement-loading approach based on the extended mobilisable strength design (EMSD) method is developed for predicting nonlinear foundation responses and upper-bound capacities with continuous velocity fields. Different from EMSD, the new approach turns optimizing a minimum energy solution into solving an elastic problem with strain-dependent equivalent elastic modulus. The strain-dependent modulus is determined by a nonlinear iterative method. This transform guarantees that the reaction force is consistent with the elastic solution in early stages of loading and gradually approaches an upper-bound solution at a large value of displacement. The approach is demonstrated on a problem of the laterally loaded pile.     

http://www.sciencedirect.com/science/article/pii/S1674987114001364

Piles are long, slender structural elements used to transfer the loads from the superstructure through weak strata onto stiffer soils or rocks. For driven piles, the impact of the piling hammer induces compression and tension stresses in the piles. Hence, an important design consideration is to check that the strength of the pile is sufficient to resist the stresses caused by the impact of the pile hammer. Due to its complexity, pile drivability lacks a precise analytical solution with regard to the phenomena involved.In situations where measured data or numerical hypothetical results are available, neural networks stand out in mapping the nonlinear interactions and relationships between the system's predictors and dependent responses. In addition, unlike most computational tools, no mathematical relationship assumption between the dependent and independent variables has to be made. Nevertheless, neural networks have been criticized for their long trial-and-error training process since the optimal configuration is not known a priori. This paper investigates the use of a fairly simple nonparametric regression algorithm known as multivariate adaptive regression splines(MARS), as an alternative to neural networks, to approximate the relationship between the inputs and dependent response, and to mathematically interpret the relationship between the various parameters. In this paper, the Back propagation neural network(BPNN) and MARS models are developed for assessing pile drivability in relation to the prediction of the Maximum compressive stresses(MCS), Maximum tensile stresses(MTS), and Blow per foot(BPF). A database of more than four thousand piles is utilized for model development and comparative performance between BPNN and MARS predictions.     

Numerical modelling of the driving response of thin‐walled open‐ended piles

The driving response of thin‐walled open‐ended piles is studied using numerical simulation of the wave propagation inside the soil plug and the pile. An elastic finite element analysis is carried out to identify the stress wave propagation in the vicinity of the pile toe. It is found that the shear stress wave has the highest magnitude above the bottom of the soil plug. Below the bottom of the soil plug, the vertical stress wave has the highest magnitude. Although the shear stress wave propagating in the radial direction is similar in magnitude to the vertical stress wave at the bottom of the soil plug, it decays rapidly while travelling downwards. The highest vertical stress at the bottom of the soil plug appears after the vertical stress wave interacts with the shear stress wave travelling in the radial direction. Initially, the vertical stress wave propagates with the dilation wave velocity in both the radial and vertical directions. After it interacts with the shear stress wave, the vertical stress wave starts to propagate with the shear wave velocity in the radial direction and with the axial wave velocity downwards. It is concluded that at the bottom of the soil plug, the interaction between the waves travelling in radial and vertical directions is important. The capabilities of several one‐dimensional pile‐in‐pile models to reproduce the driving response given by a two‐dimensional axisymmetric finite element model is studied. It is seen that when the base of the soil plug fails, a one‐dimensional pile‐in‐pile model can be used to achieve results in agreement with the finite element model. However, when the pile is unplugged, where the base of the soil plug does not fail, a reduced finite element mesh that permits the radial wave propagation inside the soil plug must be used. Copyright © 2001 John Wiley & Sons, Ltd.