Nonlinear response of laterally loaded piles and pile groups

In spite of extensive studies on laterally loaded piles carried out over years, none of them offers an expedite approach as to gaining the nonlinear response and its associated depth of mobilization of limiting force along each pile in a group. To serve such a need, elastic–plastic solutions for free‐head, laterally loaded piles were developed recently by the author. They allow the response to be readily computed from elastic state right up to failure, by assigning a series of slip depths, and a limiting force profile. In this paper, equivalent solutions for fixed‐head (FixH) single piles were developed. They are subsequently extended to cater for response of pile groups by incorporating p‐multipliers. The newly established solutions were substantiated by existing numerical solutions for piles and pile groups. They offer satisfactory prediction of the nonlinear response of all the 6 single piles and 24 pile groups investigated so far after properly considering the impact of semi‐FixH restraints. They also offer the extent to ultimate state of pile groups via the evaluated slip depths. The study allows ad hoc guidelines to be established for determining input parameters for the solutions. The solutions are tailored for routine prediction of the nonlinear interaction of laterally loaded FixH piles and capped pile groups. Copyright © 2008 John Wiley & Sons, Ltd.     

Axial and lateral response of pile groups embedded in nonhomogeneous soils

A numerical method of analysis based on elasticity theory is presented for the analysis of axially and laterally loaded pile groups embedded in nonhomogeneous soils. The problem is decomposed into two systems, namely the group piles acted upon by external applied loads and pile–soil interaction forces, and a layered soil continuum acted upon by a system of pile–soil interaction forces at the imaginary positions of the piles. The group piles are discretized into discrete elements while the nonhomogeneous soil behaviour is determined from an economically viable finite element procedure. The load–deformation relationship of the pile group system is then determined by considering the equilibrium of the pile–soil interaction forces, and the compatibility of the pile and soil displacements. The influence of soil nonlinearity can be studied by limiting the soil forces at the pile–soil interface, and redistributing the ‘excess forces’ by an ‘initial stress’ process popular in elasto-plastic finite element analysis. The solutions from this approach are compared with some available published solutions for single piles and pile groups in homogeneous and nonhomogeneous soils. A limited number of field tests on pile groups are studied, and show that, in general, the computed response compares favourably with the field measurements.     

A modulus‐multiplier approach for non‐linear analysis of laterally loaded pile groups

A modulus‐multiplier approach, which applies a reduction factor to the modulus of single pile p–y curves to account for the group effect, is presented for analysing the response of each individual pile in a laterally loaded pile group with any geometric arrangement based on non‐linear pile–soil–pile interaction. The pile–soil–pile interaction is conducted using a 3D non‐linear finite element approach. The interaction effect between piles under various loading directions is investigated in this paper. Group effects can be neglected at a pile spacing of 9 times the pile diameter for piles along the direction of the lateral load and at a pile spacing of 6 times the pile diameter for piles normal to the direction of loading. The modulus multipliers for a pair of piles are developed as a function of pile spacing for departure angle of 0, 90, and 180sup>/sup> with respect to the loading direction. The procedure proposed for computing the response of any individual pile within a pile group is verified using two well‐documented full‐scale pile load tests. Copyright © 2006 John Wiley & Sons, Ltd.     

A simplified analysis method for piled raft foundations in non‐homogeneous soils

A simplified method of numerical analysis based on elasticity theory has been developed for the analysis of axially and laterally loaded piled raft foundations embedded in non‐homogeneous soils and incorporated into a computer program “PRAB”. In this method, a hybrid model is employed in which the flexible raft is modelled as thin plates and the piles as elastic beams and the soil is treated as springs. The interactions between structural members, pile–soil–pile, pile–soil–raft and raft–soil–raft interactions, are approximated based on Mindlin's solutions for both vertical and lateral forces with consideration of non‐homogeneous soils. The validity of the proposed method is verified through comparisons with some published solutions for single piles, pile groups and capped pile groups in non‐homogeneous soils. Thereafter, the solutions from this approach for the analysis of axially and laterally loaded 4‐pile pile groups and 4‐pile piled rafts embedded in finite homogeneous and non‐homogeneous soil layers are compared with those from three‐dimensional finite element analysis. Good agreement between the present approach and the more rigorous finite element approach is demonstrated. Copyright © 2002 John Wiley & Sons, Ltd.     

Response evaluation for horizontally loaded fixed‐head pile groups using 3‐D non‐linear analysis

The response of laterally loaded pile foundations may be significantly important in the design of structures for such loads. A static horizontal pile load test is able to provide a load–deflection curve for a single free‐head pile, which significantly differs from that of a free‐ or fixed‐head pile group, depending on the particular group configuration. The aim of this paper is to evaluate the influence of the interaction between the piles of a group fixed in a rigid pile cap on both the lateral load capacity and the stiffness of the group. For this purpose, a parametric three‐dimensional non‐linear numerical analysis was carried out for different arrangements of pile groups. The response of the pile groups is compared to that of the single pile. The influence of the number of piles, the spacing and the deflection level to the group response is discussed. Furthermore, the contribution of the piles constituting the group to the total group resistance is examined. Finally, a relationship is proposed allowing a reasonable prediction of the response of fixed‐head pile groups at least for similar soil profile conditions. Copyright © 2005 John Wiley & Sons, Ltd.     

A method for the analysis of large vertically loaded pile groups

An iterative method is described for the analysis of vertically loaded pile groups with a large number of vertical piles. The individual pile response is modelled using load-transfer (t–z) curves while pile–soil–pile interaction is determined using Mindlin's solution. The present method not only keeps all the advantages of the so-called ‘hybrid method’, but also makes it possible for practising engineers to solve problems of large non-uniformly arranged pile groups in a time-saving way using a personal computer. Good agreement between the present method of analysis and the direct method is observed. A case history is analysed and the computed response of a large pile group compares favourably with the field measurement. Copyright © 1999 John Wiley & Sons, Ltd.     

Analysis of vertically loaded pile groups

An approach is presented for the analysis of linear and non-linear responses of vertically loaded pile groups. The soil behaviour of individual piles in modelled using load-transfer curves and the pile-soil-pile interaction is determined based on Mindlin's solution. Good agreement between the present method of analysis and the rigorous boundary integral method is observed for the computation of the response of pile groups embedded in a homogeneous, isotropic elastic half-space. The computed non-linear response of pile groups compares favourably with measured results from field load tests.     

Variation of limiting lateral soil pressure with depth for pile rows in clay

Pile group interaction effects on the lateral pile resistance are investigated for the case of a laterally loaded row of piles in clay. Both uniform undrained shear strength and linearly increasing with depth shear strength profiles are considered. Three-dimensional finite element analyses are presented, which are used to identify the predominant failure modes and to calculate the reduction in lateral resistance due to group effects. A limited number of two-dimensional analyses are also presented in order to examine the behaviour of very closely spaced piles. It is shown that, contrary to current practice, group effects vary with depth; they are insignificant close to the ground surface, increase to a maximum value at intermediate depths and finally reduce to a constant value at great depth. The effect of pile spacing and pile–soil adhesion are investigated and equations are developed for the calculation of a depth dependent reduction factor, which when multiplied by the limiting lateral pressure along a single pile, provides the corresponding variation of soil pressure along a pile in a pile row. This reduction factor is used to perform p–y analyses, which show that, due to this variation of group effects on the lateral soil pressures with depth, the overall group interaction effects depend on the pile length. Comparisons are also made with approaches used in practice that assume constant with depth reduction factors.     

Three‐dimensional nonlinear finite element analysis of pile groups in saturated porous media using a new transmitting boundary

The dynamic behaviour of pile groups subjected to an earthquake base shaking is analysed. An analysis is formulated in the time domain and the effects of material nonlinearity of soil, pile–soil–pile kinematic interaction and the superstructure–foundation inertial interaction on seismic response are investigated. Prediction of response of pile group–soil system during a large earthquake requires consideration of various aspects such as the nonlinear and elasto‐plastic behaviour of soil, pore water pressure generation in soil, radiation of energy away from the pile, etc. A fully explicit dynamic finite element scheme is developed for saturated porous media, based on the extension of the original formulation by Biot having solid displacement (u) and relative fluid displacement (w) as primary variables (u–w formulation). All linear relative fluid acceleration terms are included in this formulation. A new three‐dimensional transmitting boundary that was developed in cartesian co‐ordinate system for dynamic response analysis of fluid‐saturated porous media is implemented to avoid wave reflections towards the structure. In contrast to traditional methods, this boundary is able to absorb surface waves as well as body waves. The pile–soil interaction problem is analysed and it is shown that the results from the fully coupled procedure, using the advanced transmitting boundary, compare reasonably well with centrifuge data. Copyright © 2007 John Wiley & Sons, Ltd.     

On the response prediction of horizontally loaded fixed-head pile groups in sands

The aim of this paper is to investigate the behavior of laterally loaded pile groups in sands with a rigid head and correlate the response of a pile group it to that of a single pile. For this purpose, a computationally intensive study using 3-D nonlinear numerical analysis was carried out for different pile group arrangements in sandy soils. The responses of the pile groups were compared to that of the single pile and the variation of the displacement amplification factor R_a was then quantified. The influence of the number of piles, the spacing, and the deflection level on the group response is discussed. A relationship for predicting the response of a pile group, based on its configuration and the response of a single pile, has been formulated allowing also for soil shear strength which was found to affect the group response. The relationship provides a reasonable prediction for various group configurations in sandy soils.     

Study on laterally loaded piles with rectangular and circular cross sections

The conventional approach in the design of laterally loaded piles with rectangular cross section involves the simplification of converting the rectangular cross section of the pile to an equivalent circular cross section. An analysis to determine the response of laterally loaded rectangular or circular piles in elastic soil is presented in which this simplification is not required. The analysis is based on the solution of differential equations governing the displacements of the pile–soil system derived using energy principles. The pile geometry and the elastic constants of the soil and pile are the input parameters to the analysis. Using this analysis, comparisons are made between the response of rectangular and circular piles in elastic soil. Based on the proposed solution scheme, a user-friendly spreadsheet program (LATPAXL) was developed that can be used to perform the analysis. In addition, simple equations obtained by regression analysis of the pile head deflection and bending moment profiles are proposed. Examples illustrate the use of the analysis.     

软土地层高承台桥梁群桩基础横向承载特性研究

高承台群桩基础是高速铁路桥梁基础的一种常用形式,受到风、地震等荷载作用影响,常常需要承受较大的横向荷载。采用室内物理模型试验和三维有限元程序ABAQUS对软土地层中单桩、群桩的横向承载特性进行了研究,软土采用修正剑桥黏土本构模型,试验结果与有限元计算结果吻合较好。群桩研究方案包括了桩数的变化以及桩间距的变化。结果表明,群桩基础的基桩平均横向承载力（总承载力/桩数）较单桩基础显著增加,且水平荷载方向桩间距越大,其横向承载力越大;群桩基础基桩受力存在三维空间效应,不同位置基桩受力大小排序为角桩最大,其次为边桩,最小为中间桩,弯矩极值差异可达20%,群桩基础桩周土影响范围距外围基桩边缘净距离约为16D (D为桩径)。桩与桩相互影响效应对群桩水平承载不利,承台约束效应对水平承载有利。探讨了考虑上述两种效应的群桩效应系数计算方法,通过计算验证了该方法在软土地区高承台群桩基础横向承载力计算中的适用性。     

Calculation of cyclic response of laterally loaded piles

A numerical model based on discrete elements has been developed which calculates the cyclic response of laterally loaded foundation piles. The soil behaviour is modeled with the so-called HYGADE-element. This element models the gap formation around the pile, the degradation of the soil strength and the backsliding of the soil into the gap. The friction between the pile and the gap walls and the plastic soil behaviour at larger depths can also be taken into account.

Numerical verification of quasi-static and cyclic experiments confirm the validity of the model.

The resulting program is of interest to designers of foundations and researchers on structure-foundation interaction.     

Numerical study of group effects for pile groups in sands

The OpenSees finite element framework was used to simulate the response of 3×3 and 4×3 pile groups founded in loose and medium dense sands. Several numerical static pushover tests were conducted to investigate the interaction effects for pile groups. The results were then compared with those from centrifuge study. It is shown that our simulations can predict the behaviour of pile groups with good accuracy. Special attention was given to the three dimensional distribution of bending moment. It was found that bending moment develops in the plane perpendicular to the loading direction. In addition, bending moment data from simulations was used to derive p–y curves for individual piles, which were used to illustrate different behaviour of individual piles in the same group. Copyright © 2003 John Wiley & Sons, Ltd.     

Load transfer approach for laterally loaded piles

A two‐parameter model has been proposed previously for predicting the response of laterally loaded single piles in homogenous soil. A disadvantage of the model is that at high Poisson's ratio, unreliable results may be obtained. In this paper, a new load transfer approach is developed to simulate the response of laterally loaded single piles embedded in a homogeneous medium, by introducing a rational stress field. The approach can overcome the inherent disadvantage of the two‐parameter model, although developed in a similar way. Generalized solutions for a single pile and the surrounding soil under various pile‐head and base conditions were established and presented in compact forms. With the solutions, a load transfer factor, correlating the displacements of the pile and the soil, was estimated and expressed as a simple equation. Expressions were developed for the modulus of subgrade reaction for a Winkler model as a unique function of the load transfer factor. Simple expressions were developed for estimating critical pile length, maximum bending moment, and the depth at which the maximum moment occurs. All the newly established solutions and/or expressions, using the load transfer factor, offer satisfactory predictions in comparison with the available, more rigorous numerical approaches. The current solutions are applicable to various boundary conditions, and any pile–soil relative stiffness. Copyright © 2001 John Wiley & Sons, Ltd.     

Predicted ultimate capacity of laterally loaded piles in clay using support vector machine

The determination of ultimate capacity of laterally loaded pile in clay is a key parameter for designing the laterally loaded pile. The available methods for determination of ultimate resistance of pile in clay are not reliable. This study investigates the potential of a support vector machine (SVM)-based approach to predict the ultimate capacity of laterally loaded pile in clay. The SVM, which is firmly based on statistical learning theory, uses a regression technique by introducing an ?-insensitive loss function. A sensitivity analysis has been carried out to determine the relative importance of the factors affecting ultimate capacity. The results show that SVM has the potential to be a practical tool for prediction of the ultimate capacity of pile in clay.     

Effect of Edge Distance from the Slope Crest on the Response of a Laterally Loaded Pile in Sloping Ground

Compared to the field tests, the numerical modelling is an economical way to analyze the response of laterally loaded piles in sloping grounds. This paper presents a three-dimensional finite element analysis to investigate the effect of edge distance from the slope crest of a laterally loaded pile embedded in the sloping ground for different slope angles and pile lengths. The results show that the pile top displacement and the bending moment in the pile decrease with an increase in the edge distance, whereas they increase as the slope angle is increased. The response of the pile in sloping ground is compared with its response in the level ground. The comparison is used to develop a simple methodology for estimating the pile top displacement and the maximum bending moment for any edge distance from the slope crest considering their values for level ground.     

Analysis of laterally loaded piles with rectangular cross sections embedded in layered soil

An analysis is developed to determine the response of laterally loaded rectangular piles in layered elastic media. The differential equations governing the displacements of the pile–soil system are derived using variational principles. Closed‐form solutions of pile deflection, the slope of the deflected curve, the bending moment and the shear force profiles can be obtained by this method for the entire pile length. The input parameters needed for the analysis are the pile geometry and the elastic constants of the soil and pile. The new analysis allows insights into the lateral load response of square, rectangular and circular piles and how they compare. Copyright © 2007 John Wiley & Sons, Ltd.     

Analysis of pile behaviour in liquefying sloping ground

In this paper, a numerical procedure based on the finite element method is outlined to investigate pile behaviour in sloping ground, which involves two main steps. First a free-field ground response analysis is carried out using an effective stress based stress path model to obtain the ground displacements, and the degraded soil stiffness and strength over the depth of the soil deposit. Next a dynamic analysis is carried out for the pile. The interaction coefficients and ultimate lateral pressure of soil at the pile–soil interface are calculated using degraded soil stiffness and strength due to build-up of pore pressures, and the soil in the far field is represented by the displacements calculated from the free-field ground response analysis. Pore pressure generation and liquefaction strength of the soil predicted by the stress path model used in the free-field ground response analysis are compared with a series of simple shear tests performed on loose sand with and without an initial static shear stress simulating sloping and level ground conditions, respectively. Also the numerical procedure utilised for the analysis of pile behaviour has been verified using centrifuge data, where soil liquefaction has been observed in laterally spreading sloping ground. It is demonstrated that the new method gives good estimate of pile behaviour, despite its relative simplicity.