Pile–soil–pile interaction in pile groups with batter piles under dynamic loads

A simple analytical solution is developed for computing the dynamic interaction tensor for floating pile groups with batter piles. For this purpose, the governing differential equations are derived for an unloaded batter floating pile closely spaced to another loaded pile with the same properties. The reaction of soil against pile deformation is modeled by the springs and dashpots along the length of the pile. The soil is assumed linear viscous elastic and the pile behavior is linear elastic. The closed form solutions of governing equations are obtained using appropriate boundary conditions. The results are verified and compared with rigorous and approximate analytical solutions for vertical piles. The proposed method can be readily applied by engineers in the design of pile groups with batter piles.     

Cone models for nearly incompressible soil

Cones can be used to model soil in a unified strength-of-materials approach. For the vertical and rocking motions involving predominantly compressional-extensional deformation, the corresponding dilatational wave velocity tends to infinity for Poisson's ratio approaching 1/2. Based on the rigorous solution for the dynamic stiffness of a rigid disk for all frequencies, whereby the partition of the power among P-, S- and Rayleigh waves is also discussed, two special features are necessary for the vertical and rocking motions for nearly incompressible soil with Poisson's ratio between 1/3 and 1/2: (1) The appropriate wave velocity is selected as twice the shear wave velocity and not as the dilatational wave velocity; (2) A trapped mass which increases linearly with Poisson's ratio is introduced. The trapped mass can be assigned to the base mat, allowing the cone model to be constructed in the same way for all Poisson's ratios. The realization of cone models for surface foundations on a homogeneous half-space and on a layer on a flexible half-space and for embedded and pile foundations is addressed.     

单排弹性实心桩对SH波的散射求解问题

采用波函数展开法结合格拉夫加法定理分析单排实心弹性桩的SH波散射问题,通过改变单一变量,分析桩土剪切模量之比、桩体个数等因素对排桩隔振的影响效果,得到如下结论：当无量纲频率为低频和中频时,排桩后无量纲位移曲线在无限远处趋于0,随着桩体根数的增加隔振效果逐渐提高;在排桩后0～70a(a为桩径)范围内无量纲位移之比变化幅度较大,且不稳定,隔振效果较差;当桩土剪切模量之比小于500时,在250a～380a范围内减隔振效果较好;当桩土剪切模量之比大于或等于500时,隔振效果并不会随着其增加而大幅增加,即此时可将桩体认为是刚性桩体。当无量纲频率为中频时,在100a～120a范围内隔振效果较好,之后此范围内无量纲位移之比先骤升后逐渐下降,但当无量纲频率上升为高频时,各桩体根数下排桩隔振效果相差不大,且会在距离排桩更近处就已经达到很好的隔振效果。     

Soil–pile interaction during the passage of rayleigh waves: An analytical solution

A simple analytical solution is presented to calculate the single-pile response when excited by the passage of Rayleigh seismic waves. Closed-form expressions for the horizontal and vertical displacement distributions are presented for piles with finite or infinite length. The analytical results for both free-head and fixed-head piles are obtained through a dynamic Winkler model, with realistic frequency-dependent ‘springs’ and ‘dashpots’. The results of the presented method are in excellent agreement with results of a rigorous solution. It is shown that in vertical motion, the differences between pile and soil displacements are far more significant than in horizontal motion, and therefore, further work is needed to investigate the importance of pile-soil-pile interaction (group effects), because of the vertical component of Rayleigh seismic waves.     

Impedance of flexible suction caissons

The dynamic response of offshore wind turbines is affected by the properties of the foundation and the subsoil. The aim of this paper is to evaluate the dynamic soil–structure interaction of suction caissons for offshore wind turbines. The investigations include evaluation of the vertical and coupled sliding–rocking vibrations, influence of the foundation geometry and examination on the properties of the surrounding soil. The soil is simplified as a homogenous linear viscoelastic material and the dynamic stiffness of the suction caisson is expressed in terms of dimensionless frequency‐dependent coefficients corresponding to different degrees of freedom. The dynamic stiffness coefficients for the skirted foundation are evaluated using a three‐dimensional coupled boundary element/finite element model. Comparisons with known analytical and numerical solutions indicate that the static and dynamic behaviours of the foundation are predicted accurately using the applied model. The analysis has been carried out for different combinations of the skirt length, Poisson's ratio of the subsoil and the ratio of the soil stiffness to the skirt stiffness. Copyright © 2007 John Wiley & Sons, Ltd.     

考虑挤土效应时楔形桩纵向振动阻抗研究

基于复刚度传递多圈层平面应变模型,研究考虑桩周土挤土效应时成层地基中楔形桩的纵向振动问题。首先根据桩周土体的纵向成层情况并考虑楔形桩的变截面特性,将桩土系统沿纵向划分为有限个微元段,对每个微元段的桩周土体建立复刚度传递多圈层平面应变模型,并通过剪切复刚度递推方法求得桩周土作用在桩身的剪切复刚度;然后将求得的剪切复刚度代入桩身纵向振动控制方程,运用Laplace变换技术和阻抗函数递推方法,推导得到考虑桩周土挤土效应时成层地基中楔形桩纵向振动时桩顶复阻抗的解析解;最后,采用参数研究方法在低频范围内分析挤土效应对桩顶复阻抗的影响及其规律。     

The effects of liquefaction-induced lateral spreading on pile foundations

Observations of pile foundation performance during previous earthquakes have shown that pile failure has been caused by lateral ground movements resulting from soil liquefaction. The recognition that lateral ground movements may play a critical role in pile performance during an earthquake has important implications for design and risk assessment, and requires that analytical models be devised to evaluate these potential problems.In this paper, parametric studies were conducted to estimate the maximum bending moments induced in piles subjected to lateral ground displacement. The results are summarized in charts using dimensionless parameters.The analyses reveal that the existence of a nonliquefiable layer at the ground surface can affect significantly the maximum bending moment of the pile. When a relatively thick nonliquefiable layer exists above a liquefiable layer, neither the material nonlinearity of the soil nor loss of soil stiffness within the liquefiable layer significantly affect the maximum bending moment. When the thickness of the liquefiable soils is greater than about three times that of an overlying intact layer, soil stiffness in the liquefiable layer must be chosen carefully when evaluating the maximum bending moment.     

Kinematic response of single piles to vertically incident P‐waves

Dynamic response of single piles to seismic waves is fundamentally different from the free‐field motion because of the interaction between the pile and the surrounding soil. Considering soil–pile interaction, this paper presents a new displacement model for the steady‐state kinematic response of single piles to vertically incident P‐waves on the basis of a continuum model. The governing equations and boundary conditions of the two undetermined functions in the model are obtained to be coupled by using Hamilton's principle. Then, the two unknown functions are decoupled and solved by an iterative algorithm numerically. A parametric study is performed to investigate the effects of the properties of the soil–pile system on the kinematic response of single piles. It is shown that the effects of the pile–soil modulus ratio, the slenderness ratio of the pile, and the frequency of the incident excitations are very significant. By contrast, the influence of soil damping on the kinematics of the system is slight and can be neglected. Copyright © 2013 John Wiley & Sons, Ltd.     

A numerical and experimental study of hollow steel pile in layered soil subjected to vertical dynamic loading

This paper presents an investigation of the nonlinear behaviour of single piles subjected to varying levels of vertical dynamic load. A good number of tests are performed for the understanding of the dynamic behaviour of single hollow steel piles embedded in layered soil. Experimental results are validated with results obtained from a nonlinear numerical analysis using commercially available Finite Element Method (FEM) based software. The results of numerical analysis and experimental investigations showed that the length of pile has significant influence on resonant frequency and amplitude of the pile foundation. It has also been found that the slippage of pile from the surrounding soil considerably affects the resonance frequency and amplitude of the soil–pile foundation system.     

Dynamic stiffness and kinematic response of single piles in inhomogeneous soil

The effect of soil inhomogeneity on dynamic stiffness and kinematic response of single flexural elastic piles to vertically-propagating seismic SH waves is explored. A generalized parabolic function is employed to describe the variable shear wave propagation velocity in the inhomogeneous stratum. A layered soil with piece-wise homogeneous properties is introduced to approximate the continuous inhomogeneity in the realm of a Beam-on-Dynamic-Winkler-Foundation model. The problem is treated numerically by means of a layer transfer-matrix (Haskell–Thompson) formulation, and validated using available theoretical solutions and finite-element analyses. The role of salient model parameters such as pile-head fixity conditions, pile-to-soil stiffness ratio, surface-to-base shear wave velocity ratio and rate of inhomogeneity is elucidated. A new normalization scheme for inertial and kinematic response of such systems is presented based on an average Winkler wavenumber. With reference to long piles in moderately inhomogeneous soils, results indicate that: (a) kinematic pile response is essentially governed by a single dimensionless frequency parameter accounting for pile-to-soil stiffness ratio, pile slenderness and soil inhomogeneity and (b) definition of a characteristic pile wavelength allows an approximate estimation of pile elastodynamic response for preliminary design or analysis. Issues related to domain discretization and Winkler moduli are discussed.     

Modelling of raked pile foundations in liquefiable ground

Raked piles are believed to behave better than vertical piles in a laterally flowing liquefied ground. This paper aims at numerically simulating the response of raked pile foundations in liquefying ground through nonlinear finite element analysis. For this purpose, the OpenSees computer package was used. A range of sources have been adopted in the definition of model components whose validity is assessed against case studies presented in literature. Experimental and analytical data confirmed that the backbone force density–displacement (p–y) curve simulating lateral pile response is of acceptable credibility for both vertical and raked piles. A parametric investigation on fixed-head piles subject to lateral spreading concluded that piles exhibiting positive inclination impart lower moment demands at the head while those inclined negatively perform better at liquefaction boundaries (relative to vertical piles). Further studies reveal substantial axial demand imposed upon negatively inclined members due to the transfer of gravity and ground-induced lateral forces axially down the pile. Extra care must be taken in the design of such members in soils susceptible to lateral spreading such that compressive failure (i.e. pile buckling) is avoided.     

Kinematic response of single piles for different boundary conditions: Analytical solutions and normalization schemes

Kinematic pile–soil interaction is investigated analytically through a Beam-on-Dynamic-Winkler-Foundation model. A cylindrical vertical pile in a homogeneous stratum, excited by vertically-propagating harmonic shear waves, is examined in the realm of linear viscoelastic material behaviour. New closed-form solutions for bending, displacements and rotations atop the pile, are derived for different boundary conditions at the head (free, fixed) and tip (free, hinged, fixed). Contrary to classical elastodynamic theory where pile response is governed by six dimensionless ratios, in the realm of the proposed Winkler analysis three dimensionless parameters suffice for describing pile–soil interaction: (1) a mechanical slenderness accounting for geometry and pile–soil stiffness contrast, (2) a dimensionless frequency (which is different from the classical elastodynamic parameter a₀=ω d/V_s), and (3) soil material damping. With reference to kinematic pile bending, insight into the physics of the problem is gained through a rigorous superposition scheme involving an infinitely-long pile excited kinematically, and a pile of finite length excited by a concentrated force and a moment at the tip. It is shown that for long piles kinematic response is governed by a single dimensionless frequency parameter, leading to a unique master curve pertaining to all pile lengths and pile–soil stiffness ratios.     

Model tests and numerical analyses on horizontal impedance functions of inclined single piles embedded in cohesionless soil

Horizontal impedance functions of inclined single piles are measured experimentally for model soil-pile systems with both the effects of local soil nonlinearity and resonant characteristics.Two practical pile inclinations of 5° and 10° in addition to a vertical pile embedded in cohesionless soil and subjected to lateral harmonic pile head loadings for a wide range of frequencies are considered.Results obtained with low-to-high amplitude of lateral loadings on model soil-pile systems encased in a laminar shear box show that the local nonlinearities have a profound impact on the horizontal impedance functions of piles.Horizontal impedance functions of inclined piles are found to be smaller than the vertical pile and the values decrease as the angle of pile inclination increases.Distinct values of horizontal impedance functions are obtained for the ’positive’ and ’negative’ cycles of harmonic loadings,leading to asymmetric force-displacement relationships for the inclined piles.Validation of these experimental results is carried out through three-dimensional nonlinear finite element analyses,and the results from the numerical models are in good agreement with the experimental data.Sensitivity analyses conducted on the numerical models suggest that the consideration of local nonlinearity at the vicinity of the soil-pile interface influence the response of the soil-pile systems.     

基于Pasternak地基模型和Adomian分解方法的扩底桩水平动力响应分析

工程上广泛采用基于Winkler模型的层状地基反力系数法对桩土水平动力响应进行分析,该方法忽略了地基土剪切作用的影响,与工程实际有一定偏差。另外,对桩土的非线性相互作用和如扩底桩、楔形桩等变截面桩问题常用的传递矩阵法或中心差分法,计算过程较为繁琐。基于Pasternak地基模型和Adomian分解方法,提出一种考虑地基土剪切作用的桩土水平动力相互作用近似计算方法,该方法计算简便且结果精度较高,对变截面桩问题有很好的适用性;并基于该方法,对扩底桩水平动力响应问题和影响因素进行分析。结果指出,扩底半径和上部桩周土弹性模量对扩底桩水平动力响应影响较大,随着扩底半径的增加和桩周土弹性模量的增大,扩底桩水平振动位移幅值逐渐减小。另外,在较低频率的荷载激励下,应考虑土层对桩的剪切作用。     

Response of piles and pile groups to travelling SH-waves

The response of single piles and pile groups under vertically and obliquely incident seismic waves is obtained using the hybrid boundary element (BEM) formulation. The piles are represented by compressible beam-column elements and the soil as a hysteretic viscoelastic half-space. A recently developed Green function corresponding to the dynamic Mindlin problem is implemented in the numerical formulation. Exact analytical solutions for the differential equations for the piles under distributed harmonic excitations are used. Treating the half-space as a three-dimensional elastic continuum, the interaction problem is formulated by satisfying equilibrium and displacement compatibility along the pile-soil interface. Solutions adopted for the seismic waves are obtained by direct integration of the differential equations in terms of amplitudes. Salient features of the seismic response are identified in several non-dimensional plots. Results of the analyses compare favourably with the limited data available in the literature.     

成层粘弹性土中桩土耦合纵向振动时域响应研究

从三维轴对称角度出发,采用粘性阻尼粘弹性连续土介质模型,考虑桩土相互作用效应,对成层土中桩土纵向耦合振动时的桩顶时域响应进行了解析研究。求解时,首先建立定解问题,然后利用拉氏变换先对底部土层进行求解得到其振动位移形式解,然后利用桩土接触界面连续条件来考虑桩土耦合作用,分析底层土中桩段的动力反应,然后利用桩段阻抗函数的传递性,进行逐层递推求解,最终得到桩顶时域和频域响应的半解析解。通过参数影响分析和与工程实测曲线的对比,讨论分析了成层土中桩土耦合振动的响应特性,验证了本文解。基于本文研究可为桩基抗震、防震设计、桩基动力检测提供新的理论支持。     

Flexible support of a pile embedded in unsaturated soil under Rayleigh waves

Under the action of Rayleigh waves, pile head is easy to rotate with a concrete pile cap, and pure fixed-head condition is rarely achieved, which is a common phenomenon for it usually occurs on the precast piles with insufficient anchorage. In addition, the propagation characteristics of Rayleigh wave have been changed significantly due to the existence of capillary pressure and the coupling between phases in unsaturated soil, which significantly affects the pile-soil interaction. In order to study the above problems, a coupled vibration model of unsaturated soil–pile system subjected to Rayleigh waves is established on the basis that the pile cap is equivalent to a rigid mass block. Meanwhile, the soil constitution is simplified to linear-elastic and small deformations are assumed to occur during the vibration phase of soil–pile system. Then, the horizontal dynamic response of a homogeneous free-field unsaturated soil caused by propagating Rayleigh waves is obtained by using operator decomposition theory and variable separation method. The dynamic equilibrium equation of a pile is established by using the dynamic Winkler model and the Timoshenko beam theory, and the analytical solutions of the horizontal displacement, rotation angle, bending moment and shear force of pile body are derived according to the boundary conditions of flexible constraint of pile top. Based on the present solutions, the rationality of the proposed model is verified by comparing with the previous research results. Through parametric study, the influence of rotational stiffness and yield bending moment of pile top on the horizontal dynamic characteristics of Rayleigh waves induced pile is investigated in detailed. The analysis results can be utilized for the seismic design of pile foundation under Rayleigh waves.     

地震作用下液化场地变截面桩与等截面桩的动力响应对比分析

为研究液化场地变截面桩的动力响应,依托翔安大桥实体工程,采用有限元软件,建立变截面桩－土和等截面桩－土相互作用模型,模拟液化场地变截面桩及等截面桩在地震作用下的振动反应,分析在地震作用下变截面位置不同的变截面桩及等截面桩的动力响应特征。结果表明:地震作用下,液化土层不同深度处的孔压比变化规律基本相同,均从0逐渐增大最后趋于稳定;变截面桩的桩身加速度和桩身位移均大于等截面桩,且桩顶加速度峰值出现的时刻均滞后于桩底;在饱和砂土层处,桩身位移变化趋势均较陡;变截面桩的桩身弯矩峰值和桩身剪力峰值均大于等截面桩,且其峰值出现的位置较等截面桩深;地震作用下,变截面桩及等截面桩的弯矩与剪力均在安全范围之内;液化场地变截面梁桥桩基础抗震设计时,应着重分析液化土层与非液化土层分界面以下的抗弯能力设计及液化土层中抗剪能力设计。     

Dynamic nonlinear response of pile foundations under vertical vibration—Theory versus experiment

The influence of nonlinearity on the dynamic response of cast-in-situ reinforced concrete piles subjected to strong vertical excitation was studied. Forced vibration test of single piles (L/d=10, 15, 20) and 2×2 pile groups (s/d=2, 3, 4 for each L/d) were conducted in the field for two different embedded conditions of pile cap. From the measured nonlinear response curves, the effective pile–soil system mass, stiffness and damping were determined and the nonlinear response curves were back-calculated using the theory of nonlinear vibration. The test results were compared with the continuum approach of Novak with dynamic interaction factor approach using both linear and linear-equivalent numerical methods. Reasonable match between the measured and predicted response was found for linear-equivalent methods by introducing a weak boundary-zone around the pile to approximately account for the nonlinear behaviour of pile–soil system. The test data were used to establish the empirical relationship in order to estimate the extent of soil separation around the pile with soil under vertical vibration.