Kinematic Pile Bending Research Articles

Influence of phase difference between kinematic and inertial loads on seismic behaviour of pile foundations in layered soils

A series of dynamic centrifuge experiments was conducted on model pile foundations embedded in a two-layered soil profile consisting of soft-clay layer underlain by dense sand. These experiments were specifically designed to investigate the individual effect of kinematic and inertial loads on a single pile and a 3 × 1 row pile group during model earthquakes. It was observed that the ratio of free-field soil natural frequency to the natural frequency of structure might not govern the phase relationship between the kinematic and inertial loads for pile foundations as reported in some previous research. The phase relationship obtained in this study agrees well with the conventional phase variation between the force and displacement of a viscously damped simple oscillator subjected to a harmonic force. Further, as expected, the pile accelerations and bending moments can be smaller when the kinematic and inertial loads act against each other compared to the case when they act together on the pile foundations. This study also revealed that the peak kinematic pile bending moment will be at the interface of soil layers for both single pile and pile group. However, in the presence of both kinematic and inertial loads, the peak pile bending moment can occur either at the shallower depths or at the interface of soil layers depending on the pile cap rotational constraint.

Experimental investigation of kinematic pile bending in layered soils using dynamic centrifuge modelling

This research provides an insight into the previously unexplored aspects of kinematic pile bending, especially for large-intensity earthquakes where the soil behaviour is highly non-linear. In this study, a series of dynamic centrifuge experiments was conducted on pile foundations embedded in a two-layered soil profile to investigate the kinematic effects on pile foundations during model earthquakes. A single pile and a closely spaced 3 × 1 row pile group were used as model pile foundations, and the soil model consisted of a soft clay underlain by dense sand. It was observed that the peak kinematic pile bending moment occurs slightly beneath the interface of the soil layers and this depth is larger for the pile group compared to a single pile. Also, the piles in a group attract lower bending moments but carry larger residual kinematic pile bending moments compared to a single pile. Furthermore, the elastic solutions available in the literature for estimating the kinematic pile bending moments are shown to yield satisfactory results only for small-intensity earthquakes, but vastly underestimate for large-intensity earthquakes, if methods are applied injudiciously. The importance of considering soil non-linearity effects and accurate determination of shear strain at the interface of layered soils during large-intensity earthquakes for a reliable assessment of kinematic pile bending moment from methods in the literature is demonstrated using dynamic centrifuge test data.

Kinematic bending of single piles in layered soil

A continuum solution is proposed for the kinematic bending of single piles embedded in a layered soil deposit. A displacement model is proposed to represent the displacement of the model of interest incorporating soil–pile interaction. The displacement functions and the attenuation function are obtained in the coupled form by means of Hamilton’s principle. Then an iterative procedure is applied to solve the coupled functions. The contributions of the proposed model are threefold: (1) The continuity of the Winkler springs is considered. (2) The effects of the inertial force of the surrounding soil on the kinematic response of the piles are taken into account. (3) An improved formula is proposed for the kinematic pile bending strain at the interface of two-layer soil deposit. A parametric study is performed to investigate the influence of the soil inhomogeneity on the kinematic bending of single piles. The results highlight that the stiffness discontinuity significantly influences the kinematic bending of single piles. There is a critical depth considering the effect of the pile head estimated as a non-dimensional length of the pile.

A closed-form solution for kinematic bending of end-bearing piles

Kinematic soil-pile interaction is indispensable for seismic design of pile foundations and superstructures. The current understanding of kinematic pile bending mostly relies on either continuum approaches, requiring complex numerical calculation or iterative decoupling, or Winkler beam approaches, overlooking the continuity and shear stiffness of soils. In this study, a closed-form solution was proposed for the kinematic bending of end-bearing piles. In this solution, the conventional Beam-on-Dynamic-Winkler-Foundation (BDWF) was extended to incorporate the shear resistance of soils via Pasternak model. The proposed solution can provide an explicit analytical formulation for kinematic bending of end-bearing piles which is convenient for engineering practice. In addition, the proposed solution was compared with the conventional BDWF solution and the numerical solution, indicating that the proposed solution can maintain both accuracy and simplicity. At last, a series of parametric study was carried out and showed that the shear resistance of soils may considerably contribute to lowering the kinematic responses but increasing the maximum bending moment at the pile tip.

Closure to “Kinematic Bending of Fixed-Head Piles in Nonhomogeneous Soil” by Raffaele Di Laora and Emmanouil Rovithis

The writers would like to thank the Discussers for their comments and solutions, which contribute to the state of the art on the topic. The writers believe that the closed-form solution for active pile length derived by the Discussers in Eq. (3) is helpful in computations, especially when non-typical values of problem parameters are examined (e.g., very high pile–soil stiffness ratios). In the latter conditions, if no iterations are performed to assess the active pile length via the approximate procedure suggested in the original paper, the errors in the computed bendingmoment could reach 5–10%. The writers also agree with the Discussers that the absolute value of pile-head bending moment fluctuates with frequency, yet this is true if soil amplification is taken into account (e.g., when acceleration is specified at rock level). In this case, the absolute value of bending moment fluctuates with frequency due to the fluctuation in ground surface acceleration. In the original paper, the dynamic modification factor employs a pre-specified soil surface acceleration to allow (1) a straightforward physical interpretation, and (2) a direct implementation in design codes, which often prescribe a value for surface acceleration that encompasses (frequency-dependent) site effects. With reference to the decrease of the dynamic reduction factor with frequency, while the writers agree as to the validity of the formula derived by the Discussers [Eq. (13)], they believe that it should not be correlated with an “increased dynamic resistance of the pile, mainly due to inertia,” as mentioned by the Discussers. This is evident in Eq. (13) in the discussion, which does not involve pile mass in the right-hand side. The decreasing trend is rather explainable by Fig. 11 in the original paper: the excitation frequency modifies the distribution of soil shear strain with depth, indicating that soil deformations loading the pile are the only source of frequency effects. Pile inertia is proved to be negligible in the kinematic interaction problem, as also recognized by Anoyatis et al. (2013). In other words, although the development of pile bending moments is a frequency-related phenomenon, the dynamic effect is limited to the dynamic response of soil, while pile–soil interaction is essentially a static phenomenon. In a Winkler framework, applying statically, at any time step, the free-field soil displacements (derived in the dynamic regime) to the pile-supporting springs will produce accurate predictions of kinematic pile bending. Finally, the writers correctly highlight that, due to a clerical error in Eq. (22), the value of surface acceleration is missing. The correct solution for shear strain reads

1-g Experimental investigation of bi-layer soil response and kinematic pile bending

The effect of soil inhomogeneity and material nonlinearity on kinematic soil–pile interaction and ensuing bending under the passage of vertically propagating seismic shear waves in layered soil, is investigated by means of 1-g shaking table tests and nonlinear numerical simulations. To this end, a suite of scale model tests on a group of five piles embedded in two-layers of sand in a laminar container at the shaking table facility in BLADE Laboratory at University of Bristol, are reported. Results from white noise and sine dwell tests were obtained and interpreted by means of one-dimensional lumped parameter models, suitable for inhomogeneous soil, encompassing material nonlinearity. A frequency range from 0.1Hz to 100Hz and 5Hz to 35Hz for white noise and sine dwell tests, respectively, and an input acceleration range from 0.015g to 0.1g, were employed. The paper elucidates that soil nonlinearity and inhomogeneity strongly affect both site response and kinematic pile bending, so that accurate nonlinear analyses are often necessary to predict the dynamic response of pile foundations.

Strain effects on kinematic pile bending in layered soil

The kinematic bending of single piles in two-layer soil is explored to account for soil stiffness degradation and associated damping increase with increasing levels of shear strain, a fundamental aspect of soil behaviour which is not incorporated in current simplified seismic design methodologies for pile foundations.A parametric study of a vertical cylindrical pile embedded in a two-layer soil profile to vertically-propagating S waves, carried out in the time domain by a pertinent beam-on-dynamic-Winkler-foundation (BDWF) model, is reported. Strain effects are treated by means of the equivalent-linear procedure which provides soil stiffness and damping ratio as function of shear strain level. Whereas the approach still represents a crude representation of the actual soil behaviour to dynamic loading, it is more realistic than elementary solutions based on linear visco-elasticity adopted in earlier studies.The paper highlights that soil nonlinearity may have either a detrimental or a beneficial effect on kinematic pile bending depending on the circumstances. The predictive equations for kinematic pile bending in visco-elastic soil recently developed by the Authors are extended to encompass strain effects. Numerical examples and comparisons against experimental data from case histories and shaking table tests are presented.

Simplified Boundary Element Method for Kinematic Response of Single Piles in Two-Layer Soil

A simple approach is formulated to predict the elastic, kinematic pile bending during harmonic or transient excitation for a circular pile (rather than a simplified thin strip). The kinematic response of a pile embedded in two-layer soil is resolved in the frequency domain caused by the upward propagation of shear waves from the underlying bedrock. The simplified approach is generally valid to nonhomogeneous soil profiles, in light of the good comparison with the dynamic FE method and BDWF solution. It employs the soil-displacement-influence coefficientsIsto consider the pile-soil interaction (resembling the spring constantkxin the BDWF) and provides conservative estimations of maximum kinematic bending moments at the soil-layer interface (with a sharper stiffness contrast). The accuracy of the approach may be improved by incorporating the interaction of soil into the soil-displacement-influence coefficientsIsfor such cases withVb/Va<3.

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/Vs), 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.

Insight on kinematic bending of flexible piles in layered soil

The behaviour of a kinematically stressed pile in layered soil under the passage of vertically-propagating seismic S waves is investigated by means of rigorous three-dimensional Finite-Element (FE) analyses. Both pile and soil are idealized as linearly viscoelastic materials, modelled by solid elements and pertinent interpolation functions in the realm of classical elastodynamic theory. 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 a 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) provide some rigorous elastodynamic results in both frequency and time domains that can be used as reference cases; (b) elucidate the role of a number of key phenomena and salient dimensionless parameters for the amplitude of pile bending at an interface separating two soil layers of different stiffness; (c) propose a simplified semi-analytical formula for evaluating such moments; (d) provide some remarks about the role of kinematic bending in seismic design of pile foundations, with emphasis on the long-standing issue of establishing an optimal pile diameter to resist such bending. The results of the study offer a new interpretation of kinematic pile bending in terms of the interplay between pile and soil, expressed through dimensionless layer thickness, pile-to-soil stiffness ratio and impedance contrast at the layer interface. A case study from Japan is presented.

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.

Simplified Model for Seismic Pile Bending at Soil Layer Interfaces

ABSTRACT The curvatures and subsequent bending imposed to piles by the surrounding soil during the passage of seismic waves is studied. This type of bending develops even in the absence of a superstructure and is referred to as “kinematic” bending, to distinguish it from pile bending generated from inertia forces in the superstructure (termed “inertial” bending). Although kinematic bending may be severe in the presence of sharp stiffness discontinuities in the profile and may lead to damage, it has received little attention by engineers. The scope of this paper is threefold: (1) to review some existing design methods; (2) to present an improved analytical model for estimating kinematic pile bending moments at an interface between two thick soil layers under dynamic SH-wave excitation; and (3) to propose a simplified analysis procedure to be used for designing piles against kinematic loading. To this end, a dimensionless bending strain parameter (instead of the commonly-used bending moment) and a strain transmissibility function relating pile bending strain and corresponding soil shear strain are introduced. The two indices provide insight into the physics of the problem which is often obscured by the use of bending moments. Results from the model are in good agreement with more rigorous solutions. Numerical examples are presented.

Kinematic pile bending during earthquakes: analysis and field measurements

The passage of seismic waves through the soil surrounding a pile imposes lateral displacements and curvatures on the pile, thereby generating ‘kinematic’ bending moments even in the absence of a superstructure. These moments are concentrated in the vicinity of interfaces of alternating soft and stiff soil layers and, in the case of restrained-head piles, at the pile head. The scope of this paper is threefold: (a) to critically review some existing design methods for kinematic pile loading; (b) to develop new analytical results for piles in homogeneous and layered soils; (c) to present a case study in which theoretical predictions are tested against field measurements. To this end, an approximate beam-on-dynamic-Winkler-foundation (BDWF) model is implemented, specifically developed for the seismic response of piles in layered soil. Both fixed- and free-head piles, and different boundary conditions at the pile toe, are considered. It is shown that the magnitude of kinematic moments depends mainly on the stiffness contrast between the soil layers, the pile–soil stiffness contrast, the excitation frequency, and the number of excitation cycles. A unique case history involving the instrumented pile foundation of a multistorey building in Japan is presented. Time histories of bending and axial strains recorded at six locations along two piles are successfully compared with results computed from simple formulae and methods presented in the paper.

Kinematic pile bending during earthquakes: analysis and field measurements

Kinematic pile bending during earthquakes: analysis and field measurements