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.     

Torsional dynamic response of a pile embedded in layered soil based on the fictitious soil pile model

The torsional dynamic response of a pile embedded in layered soil is investigated while considering the influence of the pile end soil. The finite soil layers under the end of the pile are modeled as a fictitious soil pile that has the same cross-sectional area as the pile and is in perfect contact with the pile end. To allow for variations of the modulus or cross-sectional area of the pile and soil, the soil surrounding and below the pile is vertically decomposed into finite layers. Using the Laplace transform and impedance function transfer method, the analytical solution for the dynamic response of the pile head in the frequency domain is then obtained, and the relevant semi-analytical solution in the time domain is derived using the inverse Fourier transform and convolution theorem. The rationality and accuracy of the solution is verified by comparing the torsional dynamic behavior of the pile calculated with the fictitious soil pile with those based on a rigid support model and a viscoelastic support model. Finally, a parametric study is conducted to investigate the influence of the properties and thickness of the pile end soil on the torsional dynamic response of the pile.     

Elastic analysis of laterally loaded pile in multi-layered soil

An analysis is developed to determine the response of laterally loaded piles in layered elastic media. The differential equations governing pile deflections in different layers due to a concentrated static force and/or moment acting at the pile head are obtained using the principle of minimum potential energy and calculus of variations. The differential equations are solved analytically using the method of initial parameters. Pile deflection, slope of the deformed axis of the pile, bending moment and shear force can be reliably 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. It is observed that soil layering has a definite impact on pile response and must be taken into account for proper analysis and design. The analysis forms the basis for future formulations that can consider stress–strain nonlinearity.     

Nonlinear analysis of laterally loaded rigid piles in cohesionless soil

In this paper, a method is developed for nonlinear analysis of laterally loaded rigid piles in cohesionless soil. The method assumes that both the ultimate soil resistance and the modulus of horizontal subgrade reaction increase linearly with depth. By considering the force and moment equilibrium, the system equations are derived for a rigid pile under a lateral eccentric load. An iteration scheme containing three main steps is then proposed to solve the system equations to obtain the response of the pile. To determine the ultimate soil resistance and the modulus of horizontal subgrade reaction required in the analysis, related expressions are selected by reviewing and assessing the existing methods. The degradation of the modulus of horizontal subgrade reaction with pile displacement at ground surface is also considered. The developed method is validated by comparing its results with those of centrifugal tests and three-dimensional finite element analysis. Applications of the developed method to laboratory model and field test piles also show good agreement between the predictions and the experimental results.     

Soil resistances for laterally loaded rigid piles in multi-layered elastic soil

ABSTRACT

Short stubby piles like monopiles and large diameter drilled shafts undergo rigid body translation and rotation when subjected to a lateral force and/or a moment at the head. A method of analysis for these piles embedded in multi-layered elastic soil is developed using the variational principles of mechanics. Using this analysis, the soil resistance against pile movement can be rigorously related to the soil elastic constants, and the pile head displacement and rotation can be quickly calculated. The equilibrium equations for pile and soil displacements are obtained using the principle of virtual work and solved using an iterative algorithm. Pile responses obtained from the analysis match well with those obtained from three-dimensional finite element analyses in which the same inputs of loads, geometry, and material properties are given. Based on the new analysis, fitted equations for soil resistance parameters are developed, which can be used to directly calculate the pile head displacement and rotation without the use of the iterative algorithm. Numerical examples are provided that demonstrate how the method can be used to analyse practical problems.     

Boundary element analysis of axially loaded piles embedded in a multi-layered soil

In a field, piles are likely installed in a multi-layered soil. Analysis of axially loaded piles in a multi-layered soil is complicated and deserves more attention. A boundary element method is used in this study to analyze an axially loaded single pile in a multi-layered soil using the solution for vertical and horizontal axisymmetric ring loads in a multi-layered elastic medium. Good and reasonable agreement is obtained between the proposed and published solutions for a single pile in a homogenous soil, a finite soil, and a Gibson soil. The proposed solution is also used to evaluate an axially loaded single pile in a multi-layered (8 layers) soil.     

Nonlinear analysis of laterally loaded rigid piles in cohesive soil

This article presents a method for the nonlinear analysis of laterally loaded rigid piles in cohesive soil. The method considers the force and the moment equilibrium to derive the system equations for a rigid pile under a lateral eccentric load. The system equations are then solved using an iteration scheme to obtain the response of the pile. The method considers the nonlinear variation of the ultimate lateral soil resistance with depth and uses a new closed‐form expression proposed in this article to determine the lateral bearing factor. The method also considers the horizontal shear resistance at the pile base, and a bilinear relationship between the shear resistance and the displacement is used. For simplicity, the modulus of horizontal subgrade reaction is assumed to be constant with depth, which is applicable to piles in overconsolidated clay. The nonlinearity of the modulus of horizontal subgrade reaction with pile displacement at ground surface is also considered. The validity of the developed method is demonstrated by comparing its results with those of 3D finite element analysis. The applications of the developed method to analyze five field test piles also show good agreement between the predictions and the experimental results. The developed method offers an alternative approach for simple and effective analysis of laterally loaded rigid piles in cohesive soil. Copyright © 2011 John Wiley & Sons, Ltd.     

分布荷载推力桩计算的p-y曲线法研究

为提高现有水平梯形分布荷载推力桩设计计算水平,提出了地基系数按非线性土抗力.水平位移(p-y)曲线抗力模式表达的水平梯形分布荷载推力桩位移和内力计算的有限差分数值分析方法,并详细推导了桩身位移的差分格式.基于这些公式,编制了全桩位移、内力及侧土抗力的计算和图形处理程序,可适用于滑坡抗滑桩和深基坑悬臂支护桩的设计计算.结合新型桩型现浇混凝土薄壁管桩(PCC桩)作为滑坡抗滑桩进行对比分析,算例表明:该方法方便可靠.当有限差分段划分得足够小时,可使数值解接近于真实解.     

Longitudinal vibration of a pile embedded in layered soil considering the transverse inertia effect of pile

The dynamic response of a viscoelastic bearing pile embedded in multilayered soil is theoretically investigated considering the transverse inertia effect of the pile. The soil layers surrounding the pile are modeled as a set of viscoelastic continuous media in three-dimensional axisymmetric space, and a simplified model, i.e., the distributed Voigt model, is proposed to simulate the dynamic interactions of the adjacent soil layers. Meanwhile, the pile is assumed to be a Rayleigh–Love rod with material damping and can be divided into several pile segments allowing for soil layers and pile defects. Both the vertical and radial displacement continuity conditions at the soil–pile interface are taken into account. The potential function decomposition method and the variable separation method are introduced to solve the governing equations of soil vibration in which the vertical and radial displacement components are coupled. On this basis, the impedance function at the top of the pile segment is derived by invoking the force and displacement continuity conditions at the soil–pile interface as well as the bottom of pile segment. The impedance function at the pile head is then obtained by means of the impedance function transfer method. By means of the inverse Fourier transform and convolution theorem, the velocity response in the time domain can also be obtained. The reasonableness of the assumptions of the soil-layer interactions have been verified by comparing the present solutions with two published solutions and a set of well-documented measured pile test data. A parametric analysis is then conducted using the present solutions to investigate the influence of the transverse inertia effect on the dynamic response of an intact pile and a defective pile for different design parameters of the soil–pile system.     

Analysis of laterally loaded rigid monopiles and poles in multilayered linearly varying soil

A new method for calculation of head displacement and rotation of laterally loaded rigid monopiles and poles in multilayered heterogeneous elastic soil is presented. The analysis considers the soil as a layered elastic continuum in which the modulus vary linearly with depth within each layer. Rational pile and soil displacement fields are assumed, and the interaction between the pile and soil is taken into account by using the principle of virtual work. Two sets of equilibrium equations, one describing the pile displacement and rotation and the other describing the displacements in the soil, are obtained and solved analytically and numerically following an iterative algorithm. The new method produces pile responses as accurate as those obtained from three-dimensional finite element analysis but does not require any elaborate input for geometry and mesh.     

层状地基中轴向受荷单桩的边界元法分析

以层状地基内部作用一竖向集中力时的广义Mindlin解作为边界单元法的基本解,对层状地基中的轴向受荷单桩进行了分析,对基本解的奇异性处理方法进行了改进。考虑了桩的可压缩性和长径比对桩-土荷载传递规律和沉降特性的影响,编制了计算程序,并进行了数值分析和计算。结果表明,该方法具有较快的计算速度和良好的计算精度。     

A Winkler model approach for vertically and laterally loaded piles in nonhomogeneous soil

An investigation is made to present analytical solutions provided by a Winkler model approach for the analysis of single piles and pile groups subjected to vertical and lateral loads in nonhomogeneous soils. The load transfer parameter of a single pile in nonhomogeneous soils is derived from the displacement influence factor obtained from Mindlin's solution for an elastic continuum analysis, without using the conventional form of the load transfer parameter adopting the maximum radius of the influence of the pile proposed by Randolph and Wroth. The modulus of the subgrade reaction along the pile in nonhomogeneous soils is expressed by using the displacement influence factor related to Mindlin's equation for an elastic continuum analysis to combine the elastic continuum approach with the subgrade reaction approach. The relationship between settlement and vertical load for a single pile in nonhomogeneous soils is obtained by using the recurrence equation for each layer. Using the modulus of the subgrade reaction represented by the displacement influence factor related to Mindlin's solution for the lateral load, the relationship between horizontal displacement, rotation, moment, and shear force for a single pile subjected to lateral loads in nonhomogeneous soils is available in the form of the recurrence equation. The comparison of the results calculated by the present method for single piles and pile groups in nonhomogeneous soils has shown good agreement with those obtained from the more rigorous finite element and boundary element methods. It is found that the present procedure gives a good prediction on the behavior of piles in nonhomogeneous soils. Copyright © 2011 John Wiley & Sons, Ltd.     

BEM analysis of the interaction factor for vertically loaded dissimilar piles in saturated poroelastic soil

This paper focuses on an analysis by the boundary element method (BEM) of the pile-to-pile interaction for pile groups with dissimilar piles of different pile lengths embedded in saturated poroelastic soil. The behaviour of the poroelastic homogeneous soil is governed by Biot’s consolidation equations. The pile–soil system is decomposed into extended soil and fictitious piles. Considering the compatibility of vertical strain between fictitious piles and soil, the second kind of Fredholm integral equations were obtained to predict the axial force and settlement along pile shafts numerically. For the analysis of the interaction factor, two loading conditions for a two-dissimilar-pile system were proposed: (a) only one pile is loaded and (b) each pile is subjected to a load proportional to the pile length. Furthermore, the two-pile system was extended to pile groups with a rigid cap to capture the optimum design where each pile shares the same loading at the pile heads. The optimum results require shortening the peripheral piles and elongating internal piles, and the consolidation effect needs to be considered due to the adjustment of loading distribution among piles.     

A three‐dimensional displacement approach for analysis of laterally loaded piles in nonhomogeneous soil

An analytical approach using the three‐dimensional displacement of a soil is investigated to provide analytical solutions of the horizontal response of a circular pile subjected to lateral loads in nonhomogeneous soil. The rocking stiffness coefficient of the pile shaft in homogeneous soil is derived from the analytical solution taking into account the three‐dimensional displacement represented in terms of scalar potentials in the elastic three‐dimensional analysis. The lateral stiffness coefficient of the pile shaft in nonhomogeneous soil is derived from the rocking stiffness coefficient taking into account the rocking rotation of a rigid pile shaft. The relationship between horizontal displacement, rotation, moment, and shear force of a pile subjected to horizontal loads in nonhomogeneous soil is obtainable in the form of the recurrence equation. The formulation of the lateral displacement and rotation of the pile base subjected to lateral loads in nonhomogeneous soils is presented by taking into account Mindlin's equation and the equivalent thickness for soil layers in the equivalent elastic method. There is little difference between lateral, rocking, and couple stiffness coefficients each obtained from both the two‐dimensional and three‐dimensional methods except for the case of Poisson's ratio near 0.5. The comparison of results calculated by the current method for a pile subjected to lateral loads in homogeneous and nonhomogeneous soils has shown good agreement with those obtained from analytical and numerical methods. Copyright © 2017 John Wiley & Sons, Ltd.     

Analysis of pile subjected to torsion in multi‐layered soil

A new method of analysis of piles in multi‐layered elastic soil subjected to a torque at the head is developed. The differential equation governing the angle of twist in the pile is derived using the variational principles of mechanics. The method of initial parameters is used to obtain closed‐form solutions of the angle of twist and torque in the pile as a function of depth. The inputs required for the analysis are shear moduli of pile and soil, pile geometry and thickness of soil layers. Copyright © 2013 John Wiley & Sons, Ltd.     

Exploring influence of sectional flexural yielding on experimental pile response analysis and applicability of distributed plastic hinge model in inelastic numerical simulation for laterally loaded piles

This study used model pile load testing and numerical analysis to investigate the experimental analysis results of pile and soil responses for lateral load testing due to the flexural yielding of a pile, and to examine the applicability of the distributed plastic hinge model to the numerical simulation of inelastic pile response. A lateral load test on an aluminum model pile in sand was conducted as an analysis case. The pile was loaded to a large lateral pile-head displacement, a displacement under which some of the pile sections yielded and thus the pile had inelastic flexural deformation. The test results showed that before the pile yielded, the depth of maximum moment increased with increasing load due to soil nonlinearity; after the pile yielded, the depth of maximum moment varied less and the plastic region expanded upward and downward around this depth with increasing pile displacement. In deducing the responses of the pile and soil for the pile-soil system, the actual nonlinear flexural rigidity of the pile section built based on the bending test was essential to retrieve rational ones. In addition, the distributed plastic hinge model was shown to be effective to model the inelastic pile responses and capture the development of plastic zones in the pile.     

碎石土斜坡水平受荷桩承载特性研究

斜坡上的桩基础的承载性能是复杂多变的。对于四川西部山地地形较广泛,且地基覆盖层多为特有的碎石土地层来说,水平受荷桩的相关研究还较少。为了研究碎石土地基斜坡上单桩基础的水平承载特性及桩土间的相互作用,通过现场水平静载荷试验在坡度为0°、15°、30°、45°的条件下,探讨桩身变形、桩身弯矩、土压力的变化。运用FLAC3D有限元分析软件得出水平荷载作用下,碎石土斜坡不同坡度的桩基础与桩周土之间的应力云图、位移云图的变化特点。将数值模拟结果与现场试验结果进行了对比,提出了单桩水平临界荷载和极限荷载在不同坡度区间内取值时的折减系数,为实际工程提供一定的参考。     

Dynamic analysis of an axially loaded pile embedded in elastic-poroelasitc layered soil of finite thickness

This paper presents an analytical solution for determining the dynamic characteristics of axially loaded piles embedded in elastic-poroelastic layered soil of finite thickness. The interface between the elastic and poroelastic soil coincides with the groundwater table level, which is explicitly taken into account in the solution. The pile is modelled as elastic one-dimensional rod to account for the effect of its dynamic characteristics on the response of the soil-pile system. The solution is based on Biot's poroelastodynamic theory and the classical elastodynamic theory, which we use to establish the governing equations of the soil and pile. Accordingly, the pile base resistance, shaft reaction, and the complex impedance of soil-pile system are obtained using the method of Hankel integral transformation. Following the validation of the derived solution, we identify the main parameters affecting the vertical dynamic impedance of the pile via a parametric study. The presented method poses as an efficient alternative for quickly estimating the dynamic characteristics of axially loaded piles, without having to resort to complex numerical analyses.     

A simplified nonlinear analysis method for piled raft foundation in layered soils under vertical loading

This paper presents a simplified nonlinear solution for piled raft foundations in layered soils under vertical loading. Based on the elastic–plastic analysis of a single pile in a layered soil, the shielding effect between a receiver pile and the soil is taken into account to modify the conventional interaction factor between two piles. An approximate approach with the concept of the interaction factor is employed to study the nonlinear behavior of pile groups with a rigid cap. Considering the variation of soil properties, the solution to multilayered elastic materials is used to calculate the settlement of the soil. The interactions between pile–soil–raft are taken into account to determine the stiffness matrix of the piled raft. By solving the stiffness matrix equations, the settlement and the load shared by the piles and raft could be obtained. Compared with results of the available published literatures, the proposed solution provides reasonable results.     

层状地基中基于Laplace变换的桩基横向振动阻抗计算

基于Laplace变换,对层状地基中桩土横向振动阻抗计算问题进行了研究。考虑土层天然分层的特性及桩顶轴向力的参与作用,结合频域内桩-土动力文克尔理论,采用传递矩阵法并通过拉普拉斯变换,将振动微分方程变成代数方程以求解桩的横向振动响应参数,并导出了单桩横向振动阻抗。基于所得解,进一步计算出桩-土-桩水平动力相互作用因子。通过实例分析对比,验证其有效性和可行性。该方法计算工作量小,易于理解,计算结果与已有结果具有良好的一致性,并能保证解的连续性,对桩-土动力相互作用的研究具有一定的实用意义。