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.     

Pile‐head kinematic bending in layered soil

Kinematic effects at the head of a flexible vertical pile embedded in a two‐layer soil deposit are investigated by means of rigorous three‐dimensional elastodynamic finite‐element analyses. Both pile and soil are idealized as linearly viscoelastic materials, modelled by solid elements, without the restrictions associated with the use of strength‐of‐materials approximations. The system is analyzed by a time‐Fourier approach in conjunction with a modal expansion in space. Constant viscous damping is considered for each natural mode, and an FFT algorithm is employed to switch from frequency to time domain and vice versa in natural or generalized coordinates. The scope of the paper is to: (a) elucidate the role of a number of key phenomena controlling the amplitude of kinematic bending moments at the pile head; (b) propose a simplified semi‐analytical formula for evaluating such moments; and (c) provide some remarks about the role of kinematic bending in the seismic design of pile foundations. The results of the study provide a new interpretation of the interplay between interface kinematic moments and corresponding head moments, as a function of layer thickness, pile‐to‐soil stiffness ratio, and stiffness contrast between the soil layers. In addition, the role of diameter in designing against kinematic action, with or without the presence of an inertial counterpart, is discussed. Copyright © 2012 John Wiley & Sons, Ltd.     

地震作用下预应力混凝土管桩运动响应三维数值分析

基于黏弹性人工边界,建立上部结构-桩-土的共同作用三维有限元模型,分析地震作用下预应力混凝土管桩的运动响应特性。分别针对预应力混凝土管桩的桩径、双层软硬土剪切波速比值、上覆土层厚度、上部结构荷载等影响因素进行数值计算。参数分析表明:在地震作用下,桩径的增大会导致桩身整体弯矩相应增加,特别是桩身土层分界面处增大明显;软硬土层剪切波速比及上覆土层厚度的增加,引起土层分界面处桩身峰值弯矩增加;固定桩头条件下,桩头与桩身软硬土层分界面处均会产生较大的运动弯矩;上部结构的惯性荷载对固定桩头的内力有着较大影响,对桩身深处段弯矩影响较小。本文研究结论可为预应力混凝土管桩抗震设计提供有益的理论参考。     

Transient kinematic pile bending in two-layer soil

The dynamic response of piles to seismic loading is explored by means of an extensive parametric study based on a properly calibrated Beam-on-Dynamic-Winkler-Foundation (BDWF) model. The investigated problem consists of a single vertical cylindrical pile, modelled as an Euler–Bernoulli beam, embedded in a subsoil consisting of two homogeneous viscoelastic layers of sharply different stiffness resting on a rigid stratum. The system is subjected to vertically propagating seismic S waves, in the form of a transient motion imposed on rock outcrop. Several accelerograms recorded in Italy are employed as input motions in the numerical analyses. The paper highlights the severity of kinematic pile bending in the vicinity of the interface separating the two soil layers. In addition to factors already investigated such as layer stiffness contrast, relative soil–pile stiffness, interface depth and intensity of ground excitation, the paper focuses on additional important factors, notably soil material damping, stiffness of Winkler springs and frequency content of earthquake excitation. Existing predictive equations for assessing kinematic pile bending at soil layer interfaces are revisited and new regression analyses are performed. A synthesis of findings in terms of a set of simple equations is provided. The use of these equations is discussed through examples.     

Kinematic internal forces in deep foundations with inclined piles

This paper presents a parametric study that looks into the influence of pile rake angle on the kinematic internal forces of deep foundations with inclined piles. Envelopes of maximum kinematic bending moments, shear forces and axial loads are presented along single inclined piles and 2 × 2 symmetrical square pile groups with inclined elements subjected to an earthquake generated by vertically incident shear waves. Inclination angles from 0° to 30° are considered, and three different pile–soil stiffness ratios are studied. These results are obtained through a frequency–domain analysis using a boundary element–finite element code in which the soil is modelled by the boundary element method as a homogeneous, viscoelastic, unbounded region, and the piles are modelled by finite elements as Euler–Bernoulli beams. The rotational kinematic response of the pile foundations is shown to be a key factor on the evolution of the kinematic internal forces along the foundations. Copyright © 2015 John Wiley & Sons, Ltd.     

Nonlinear lateral interaction in pile dynamics

A model for pile lateral response to transient dynamic loading and to harmonic loading is presented allowing for nonlinear soil behaviour, discontinuity conditions at the pile-soil interface and energy dissipation through different types of damping. The approach is used to establish equivalent linear stiffness and damping parameters of single piles as well as dynamic interaction factors for approximate nonlinear analysis of pile groups. The applicability of these parameters to the pile-group analysis was examined, and a reasonable agreement with the direct analysis was found. The superposition technique may be used to analyze the response of small pile groups. Also, the dynamic stiffness of pile groups is greatly affected by both the nonlinear behavior of the soil and the slippage and gapping between the pile and soil. For a basic range of soil and pile parameters, equivalent linear stiffness and damping parameters of single piles and interaction factors for approximate nonlinear analysis are provided.     

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.     

A method for assessing kinematic bending moments at the pile head

The conventional design methods for seismically loaded piles still concentrate in providing adequate resistance from the pile to withstand only the inertial bending moments generated from the oscillation of the superstructure, thus neglecting the effect of kinematic interaction between pile and soil. By contrast there has been extensive research on kinematic effects induced by earthquakes and a number of simplified methods are available for a preliminary evaluation of kinematic bending moments at the interface between two soil layers. Less attention has been paid to the effects of kinematic interaction at the pile‐head. The paper summarizes recent research work on kinematic response analysis of fixed‐head piles aimed at the performance evaluation of a piled foundation. Results from an extensive parametric study, undertaken by means of three‐dimensional FE analyses, suggest a new criterion to predict kinematic bending effects at the pile head, where the combination of kinematic and inertial effect may be critical. Copyright © 2010 John Wiley & Sons, Ltd.     

Shear modulus and damping ratio of grouted sand

An experimental comparative study of three different grouted sands in terms of their effects on the values of two dynamic properties is presented. The dynamic properties studied are the shear modulus and the damping ratio which are determined with resonant column tests and cyclic triaxial tests. The behaviour of a pure Fontainebleau sand is compared with the behaviour of a Fontainebleau sand grouted with a silicate grout, a micro-fine cement grout and a mineral grout. The effects of the grouting treatment, the type of grout, the confining pressure, and the strains, on the shear modulus and the damping ratio are studied. The test results have shown that grouting improves the stiffness of the sand especially for small strains. Whatever the type of material, confining stress improves the shear modulus whereas it has a negligible effect on the damping ratio. When strain increases, the shear modulus decreases and the damping ratio increases.     

桩基—非线性框剪结构相互作用体系（上）——群桩阻抗函数的求取

为研究桩基－非线性框剪结构相互作用体系的地震反应,需要求解群桩基础的动力阻抗函数。本文利用单桩阻抗和群桩动力相互作用因子计算群桩动力阻抗函数,计算并讨论了不同构形群桩阻抗函数在软、中和硬土地基中的变化规律和特点。研究表明：群桩阻抗表现出很强的频率相关性,随土体剪切波速的增大,群桩阻抗有较大幅度的增加,但土体剪切波速的变化对群桩效率（规格化阻抗函数）影响不大。     

Soil-pile interaction in vertical vibration

The interaction between a soil layer and an end bearing pile is theoretically investigated. The pile is assumed to be vertical and elastic, the soil is considered as a linear visco-elastic layer with hysteretic type damping. The layer alone is solved first and the wave modes of the layer are used in the analysis of the pile response. The pile response to a harmonic load is obtained in a closed form and used to define stiffness and damping at the level of the pile head. The dimensionless parameters of the problem are identified. A parametric study is conducted to determine the main features of the response and of the equivalent stiffness and damping. The validity of equivalent viscous damping is examined. A comparison is made with the simpler plane strain theory used previously and its accuracy is assessed.     

液化侧扩流场地桥梁群桩效应分析

基于u-p有限元公式模拟饱和砂土中水和土颗粒完全耦合效应,建立液化侧向流场地群桩动力反应分析的三维数值模型。模型中,砂土采用多屈服面弹塑性本构模型模拟、黏土采用多屈服面运动塑性模型模拟,群桩在计算过程中保持线弹性状态;采用20节点的六面体单元和考虑孔压效应的20-8节点分别划分黏土层和饱和砂层;选用剪切梁边界处理计算域的人工边界,模拟地震过程中土层的剪切效应;应用瑞利阻尼考虑体系的阻尼效应。随后对比分析2×2群桩中各单桩的地震反应规律,结果表明,各单桩的弯矩、位移时程规律基本一致,峰值弯矩及峰值位移出现时刻滞后于输入加速度峰值时刻,上坡向桩的弯矩和位移峰值大于下坡向的桩的反应值。接着通过改变桩间距研究群桩效应,随着桩间距增加,群桩中各单桩的弯矩最大值均出现在土层分界处,且各单桩的弯矩、桩顶位移逐渐增大。最后给出液化侧向流场地群桩效应的基本原因,得出该类场地群桩抗震设计的基本认识。     

Identification of dynamic soil properties through shaking table tests on a large saturated sand specimen in a laminar shear box

Many laminar shear boxes have recently been developed into sliding-frame containers that can reproduce 1D ground-response boundary conditions. The measured responses of such large specimens can be utilized to back-calculate soil properties. This study investigates how the boundary effect in large specimens affects the identified soil properties through shaking table tests on a soil-filled large laminar box conducted at the National Center for Research on Earthquake Engineering in Taiwan. The tested soil-box system is unique because only 80% of the container is filled with soil. This system can be regarded as a two-layer system: an empty top and soil-filled bottom. The dynamic properties of this two-layer system are identified through various approaches, including theoretical solutions of wave propagation, free vibration, and nonparametric stress–strain analyzes. Therefore, the coupling effect of the box and soil can be evaluated. Results show that, compared with the two-layer system considering the influence of the box, the conventional approach with a single-layer system slightly underestimates shear wave velocity but obtains the same damping ratio of the soil layer. In addition, the identified modulus reduction and damping curves in the two-layer system are consistent with those obtained in a laboratory test on a small specimen. Furthermore, based on detailed acceleration measurements along different depths of soil, a piecewise profile of shear wave velocity is built. The identified shear wave velocity increases with depth, which is not uniform and differs from the constant velocity typically assumed for the specimen.     

单桩-土-结构振动台试验模型数值计算分析

为研究地震荷载作用下桩基-土-核电结构的抗震性能及土结动力反应规律,对拟开展的地震模拟振动试验模型进行数值计算分析。核电工程结构上部质量大和刚度大,试验模型不同于一般的工程结构,为检验振动台试验模型设计、传感器布设方案,对试验模型进行了数值模拟。数值模拟以单端承桩为研究对象,计算了上部结构质量和刚度变化时,在脉冲荷载及基于RG1.60谱人工合成地震动作用下桩身的地震反应规律。数值模拟表明:在水平地震动作用下,桩身剪力和弯矩包络线呈"X"状分布,桩底和顶处剪力弯矩较大;上部结构质量越大,桩身的剪力与弯矩越大;上部结构的刚度越大,桩身的剪力与弯矩越小;随着上部结构质量的增大和刚度的减小,反弯点逐渐向桩顶移动。桩顶发生最大位移时所对应的桩身挠度随着上部结构质量的增加而增大并且随着上部结构刚度的增大而减小。土层分界面处,桩身内力发生突变。此外,在脉冲荷载输入下,桩身反弯点位置与输入荷载的周期有关。计算结果为振动台试验模型设计提供了理论依据。     

SOIL–PILE–BRIDGE SEISMIC INTERACTION: KINEMATIC AND INERTIAL EFFECTS. PART I: SOFT SOIL

A substructuring method has been implemented for the seismic analysis of bridge piers founded on vertical piles and pile groups in multi-layered soil. The method reproduces semi-analytically both the kinematic and inertial soil–structure interaction, in a simple realistic way. Vertical S-wave propagation and the pile-to-pile interplay are treated with sufficient rigor, within the realm of equivalent-linear soil behaviour, while a variety of support conditions of the bridge deck on the pier can be studied with the method. Analyses are performed in both frequency and time domains, with the excitation specified at the surface of the outcropping (‘elastic’) rock. A parameter study explores the role of soil–structure interaction by elucidating, for typical bridge piers founded on soft soil, the key phenomena and parameters associated with the interplay between seismic excitation, soil profile, pile–foundation, and superstructure. Results illustrate the potential errors from ignoring: (i) the radiation damping generated from the oscillating piles, and (ii) the rotational component of motion at the head of the single pile or the pile-group cap. Results are obtained for accelerations of bridge deck and foundation points, as well as for bending moments along the piles. © 1997 by John Wiley & Sons, Ltd.     

Dynamic soil–structure interaction of monopile supported wind turbines in cohesive soil

Offshore wind turbines supported on monopile foundations are dynamically sensitive because the overall natural frequencies of these structures are close to the different forcing frequencies imposed upon them. The structures are designed for an intended life of 25 to 30 years, but little is known about their long term behaviour. To study their long term behaviour, a series of laboratory tests were conducted in which a scaled model wind turbine supported on a monopile in kaolin clay was subjected to between 32,000 and 172,000 cycles of horizontal loading and the changes in natural frequency and damping of the model were monitored. The experimental results are presented using a non-dimensional framework based on an interpretation of the governing mechanics. The change in natural frequency was found to be strongly dependent on the shear strain level in the soil next to the pile. Practical guidance for choosing the diameter of monopile is suggested based on element test results using the concept of volumetric threshold shear strain.     

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.     

Static equivalent method for the kinematic interaction analysis of single piles

This paper presents a static equivalent approach to estimate the maximum kinematic interaction effects on piles subjected to lateral seismic excitation. Closed-form expressions are reported for the evaluation of the maximum free-field soil movements and for the computation of maximum pile shear force and bending moments. Firstly, modal analysis, combined with a suitable damped response spectrum, is used to evaluate the maximum free-field response. Secondly, the pile is schematised as a Winkler's beam subjected to equivalent static forces defined according to soil vibration modal shapes and amplitude. The method may be applied by using response spectra suggested by National Standards or those obtained with accelerograms. The procedure proposed may be conveniently implemented in simple spreadsheets or in commercial finite element programs and easily used by practicing engineers. Method accuracy is demonstrated by comparing the results with those obtained with a more rigorous model. Good results may be achieved by considering only the first soil vibration mode making the procedure straightforward for practical design purposes.     

Effects of foundation embedment during building–soil interaction

A numerical solution for evaluating the effects of foundation embedment on the effective period and damping and the response of soil–structure systems is presented. A simple system similar to that used in practice to account for inertial interaction effects is investigated, with the inclusion of kinematic interaction effects for the important special case of vertically incident shear waves. The effective period and damping are obtained by establishing an equivalence between the interacting system excited by the foundation input motion and a replacement oscillator excited by the free-field ground motion. In this way, the use of standard free-field response spectra applicable to the effective period and damping of the system is permitted. Also, an approximate solution for total soil–structure interaction is presented, which indicates that the system period is insensitive to kinematic interaction and the system damping may be expressed as that for inertial interaction but modified by a factor due to kinematic interaction. Results involving both kinematic and inertial effects are compared with those obtained for no soil–structure interaction and inertial interaction only. The more important parameters involved are identified and their influences are examined over practical ranges of interest. © 1998 John Wiley & Sons, Ltd.