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.     

Analysis of pile‐raft foundations under complex loads in layered soils

Considering there is hardly any concerted effort to analyze the pile‐raft foundations under complex loads (combined with vertical loads, horizontal loads and moments), an analysis method is proposed in this paper to estimate the responses of pile‐raft foundations which are subjected to vertical loads, horizontal loads and moments in layered soils based on solutions for stresses and displacements in layered elastic half space. Pile to pile, pile to soil surface, soil surface to pile and soil surface to soil surface interactions are key ingredients for calculating the responses of pile‐raft foundations accurately. Those interactions are fully taken into account to estimate the responses of pile‐raft foundations subject to vertical loads, horizontal loads and moments in layered soils. The constraints of the raft on vertical movements, horizontal movements and rotations of the piles as well as the constraints of the raft on vertical movements and horizontal movements of the soils are considered to reflect the coupled effect on the raft. The method is verified through comparisons with the published methods and FEM. Then, the method is adopted to investigate the influence of soil stratigraphy on pile responses. The study shows that it is necessary to consider the soil non‐homogeneity when estimating the responses of pile‐raft foundations in layered soils, especially when estimating the horizontal responses of pile‐raft foundations. The horizontal loads and the moments have a significant impact on vertical responses of piles in pile‐raft foundations, while vertical loads have little influence on horizontal responses of piles in pile‐raft foundations in the cases of small deformations. The proposed method can provide a simple and useful tool for engineering design. Copyright © 2013 John Wiley & Sons, Ltd.     

Vertical response of pile raft foundations subjected to tunneling‐induced ground movements in layered soil

A simplified analysis method has been developed to estimate the vertical movement and load distribution of pile raft foundations subjected to ground movements induced by tunneling based on a two‐stage method. In this method, the Loganathan–Polous analytical solution is used to estimate the free soil movement induced by tunneling in the first stage. In the second stage, composing the soil movement to the pile, the governing equilibrium equations of piles are solved by the finite difference method. The interactions between structural members (such as pile–soil, pile–raft, raft–soil, and pile–pile) are modeled based on the elastic theory method of a layered half‐space. The validity of the proposed method is verified through comparisons with some published solutions for single piles, pile groups, and pile rafts subjected to ground movements induced by tunneling. Good agreements between these solutions are demonstrated. The method is also used for a parametric study to develop a better understanding of the behavior of pile rafts influenced by tunneling operation in layered soil foundations. Copyright © 2011 John Wiley & Sons, Ltd.     

考虑应力扩散时桩端土性对单桩沉降影响分析方法

基于虚土桩模型,分析了层状地基中桩端土性对单桩沉降特性的影响。首先,以虚土桩扩散角反映桩端土层应力扩散效应,将桩端一定锥角范围内由桩端至基岩面的土体视为虚土桩,并根据其变截面特性,将虚土桩沿纵向划分为有限个微元段。然后,对桩及虚土桩桩侧土体采用理想弹塑性荷载传递模型,利用荷载传递法,推导了层状地基中以桩侧土塑性发展深度为变量的单桩荷载-沉降递推计算方法,并进一步得到了桩身轴力及桩侧摩阻力递推计算式。在此基础上,给出了荷载传递模型参数选取方法,并分析了虚土桩临界深度的影响因素及由实测荷载-沉降曲线反演虚土桩扩散角的可行性。最后,利用该方法分析了桩端沉渣和软弱下卧层对荷载-沉降曲线的影响。结果表明,考虑桩端土层应力扩散效应时,通过计算得到的桩顶及桩端荷载-沉降曲线与实测曲线吻合较好;当桩端存在沉渣或软弱下卧层时,采用虚土桩模型的单桩沉降计算方法可以在一定程度上反映沉渣特性及软弱下卧层埋深等因素对桩顶荷载-沉降曲线的影响。     

砂土地基斜桩水平承载特性p-y曲线法

基于Reese建议的砂土p-y曲线,提出了适用于斜桩水平承载特性分析的修正p-y曲线及相应的有限元分析方法。根据桩前土体受力特点,假设桩前浅层土体达到极限破坏状态时桩侧被动破坏区为一个三维楔形体,采用极限平衡分析方法导出了斜桩桩侧极限土抗力表达式,对比研究了斜桩倾角、桩-土界面摩擦系数等因素对极限土抗力的影响规律,并给出了斜桩初始地基反力模量确定方法。在此基础上,修正Reese提出的直桩p-y曲线,提出了砂土地基中斜桩p-y曲线确定方法。利用虚功原理,导出了斜桩水平承载特性分析的有限单元列式,编制相应的分析程序,通过算例验证了方法的合理性和有效性。最后,采用修正的p-y曲线法对一离心机模型试验进行了模拟分析,结果表明,计算与试验结果吻合得很好     

Practical second‐order reliability analysis applied to foundation engineering

A practical and efficient approach of implementing second‐order reliability method (SORM) is presented and illustrated for cases related to foundation engineering involving explicit and implicit limit state functions. The proposed SORM procedure is based on an approximating paraboloid fitted to the limit state surface in the neighborhood of the design point and can be easily carried out in a spreadsheet. Complex mathematical operations are relegated to relatively simple user‐created functions. The failure probability is calculated automatically based on the reliability index and principal curvatures of the limit state surface using established closed‐form SORM formulas. Four common foundation engineering examples are analyzed using the proposed method and discussed: immediate settlement of a flexible rectangular foundation, bearing capacity of a shallow footing, axial capacity of a vertical single pile, and deflection of a pile under lateral load. Comparisons with Monte Carlo simulations are made. In the case of the laterally loaded pile, the friction angle of the soil is represented as a one‐dimensional random field, and pile deflections are computed based on finite element analysis on a stand‐alone computer package. The implicit limit state function is approximated via the response surface method using two quadratic models. Copyright © 2011 John Wiley & Sons, Ltd.     

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.     

静压沉桩桩体受力全过程数值分析

沉桩是十分剧烈的桩-土相互作用过程,沉桩过程中桩体受力形态对桩后期承载力和耐久性有重要影响。本文运用有限单元法对静压沉桩进行数值模拟,通过位移控制法实现沉桩,模拟计算沉桩过程中桩土径向压缩和竖向摩擦,对沉桩全过程桩体受力情况进行分析。结果显示,沉桩过程中桩主要承受竖向压应力,尤其沉桩后期桩顶部受到的竖向压应力较大。采用单因素控制法对影响桩体受力的各项因素进行分析,并对各影响因素进行相关性和敏感性分析。分析表明,静压沉桩过程中桩体受力与摩擦系数、内摩擦角、粘聚力和桩尖角呈正相关,与桩径呈负相关,桩尖角对桩体所受竖向应力敏感性较弱。     

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.     

Analysis of rectangular piles subjected to lateral loads in nonhomogeneous soil using a Winkler model approach

An analytical approach using a Winkler model based on two lateral soil displacement components in a three‐dimensional soil is investigated to provide analytical solutions of horizontal response of a rectangular pile subjected to lateral loads in nonhomogeneous soil. The two lateral displacement components of a soil surrounding the rectangular pile are represented by the Fourier series of displacement potential functions in the elastic three‐dimensional analysis. The lateral stiffness coefficient of the rectangular pile shaft in nonhomogeneous soil is derived from the rocking stiffness coefficient taking into account rocking rotation of a rigid pile shaft. The relationship between horizontal displacement, rotation, moment, and shear force for the rectangular pile subjected to horizontal loads in nonhomogeneous soil is obtainable in the form of the recurrence equation. The formulation of lateral displacement and rotation for a rectangular pile subjected to lateral loads on the pile base in nonhomogeneous soil is proposed by taking into account Mindlin's equation and the equivalent thickness for soil layers in the equivalent elastic method. The difference of lateral behavior between square and circular piles subjected to lateral loads is insignificant. The effect of aspect ratio of the rectangular pile on the lateral behavior is great for the lower stiffness ratio between pile and soil and the larger length–equivalent diameter ratio. The effect of the value of Poisson's ratio of soil on lateral stiffness coefficient is relatively small except Poisson's ratio close to 0.5. The comparison of the results calculated by the current method for a rectangular pile subjected to lateral loads in nonhomogeneous soil has shown good agreement with those obtained from the analytical methods and the finite element method. Copyright © 2015 John Wiley & Sons, Ltd.     

A revised solution for the horizontal vibration of an end‐bearing pile in viscoelastic soil

This note presents a new method to derive closed‐form expressions describing the horizontal response of an end‐bearing pile in viscoelastic soil subjected to harmonic loads at its head. The soil surrounding the pile is assumed as a linearly viscoelastic layer. The propagation of waves in the soil and pile is treated mathematically by three‐dimensional and one‐dimensional theories, respectively. Unlike previous studies of the problem, the formulation presented allows the governing equations of the soil to be solved directly, eliminating the need to introduce potential functions. Accordingly, the dynamic response of the pile is obtained by means of the initial parameter method, invoking the requirement for continuity at the pile–soil interface. It is demonstrated that the derived compact solution matches exactly an existing solution that utilises potential functions to formulate the problem. Copyright © 2016 John Wiley & Sons, Ltd.     

Interpretation of torsional shear results for nonlinear stress–strain relationship

Torsional tests are frequently used in material testing to determine shear stress–strain relationships. In the tests, torque is applied to a solid or hollow cylindrical specimen, whose twist angle is recorded as the response. While the raw test results are readily available in terms of torque–twist sets, converting them into intrinsic stress–strain relationships is not always straightforward, owing to the nonlinear stress–strain behavior and the non‐uniform radial distribution of the stress and strain fields. It follows that the approaches currently used to interpret torsional data are essentially empirical based. The present paper addresses this data interpretation issue from a more rigorous perspective. The paper includes two parts: the first part concerns hollow cylindrical specimens under torsion, where an analytical‐based recursive procedure is developed; and the second part deals with solid torsional specimens, for which an analytical solution given by Nadai is directly employed. Examples pertaining to these two cases are examined and the resulting shear stress–strain curves are compared with those based on a number of popular empirical methods. The proposed analytical approaches can serve as a benchmark for checking the accuracy and validity of those empirical propositions. It is worth noting that, for brittle materials exhibiting softening behavior, only the analytical method gives the correct interpretation. Copyright © 2007 John Wiley & Sons, Ltd.     

液化场地桩-土-桥梁结构地震相互作用简化分析方法

液化场地桩-土-桥梁结构地震相互作用分析属于桩基桥梁抗震设计中的一个关键科学问题,而目前尚缺乏合理的简化分析方法。鉴于此,直接针对振动台试验,基于Penzien模型,建立了液化场地桩-土-桥梁结构地震相互作用的数值分析模型与相应的简化分析方法。通过振动台试验验证了数值建模途径与简化计算分析方法的正确性,可用于液化场地桩基桥梁结构地震反应的分析,并且特别考虑砂层中孔压升高引起的砂土承载力衰减效应,推荐了计算参数的合理选取方法;据此进行了桩径、桩土初始模量比、砂土内摩擦角、上部桥梁结构质量等重要参数对液化场地桩-土地震相互作用影响的敏感性分析。研究表明：在液化场地条件下,随桩径和桩土初始模量比的增大,桩的峰值加速度、峰值位移减小,而桩的峰值弯矩则增大;随砂土内摩擦角增大,桩的峰值加速度、峰值弯矩、峰值应力均增大,而桩的峰值位移则减小;随上部结构配重增大,桩的峰值位移、峰值弯矩均增大。     

Vertical impedance of an end‐bearing pile in viscoelastic soil

This paper presents a new method to derive the analytical solution for the vertical impedance of an end‐bearing pile in viscoelastic soil. The soil is assumed as a homogeneous and isotropic layer, and the pile is considered as a one‐dimensional Euler rod. Considering both the vertical and radial displacements of soil and soil–pile coupled vibration, the governing equations of the soil and pile are established. The volumetric strain of soil is obtained by transformation on the equations of soil and variable separation method. Then the vertical and radial displacements of soil are obtained accordingly. The displacement response and impedance function of pile are derived based on the continuity assumption of the displacement and stress between the pile and soil. The solution is verified by being compared with an existing solution obtained by introducing potential functions. Furthermore, a comparison with two other simplified solutions is conducted. Numerical examples are presented to analyze the vibration characteristics of the pile. Copyright © 2014 John Wiley & Sons, Ltd.     

Analysis of piles subjected to lateral soil movements using a three‐dimensional displacement approach

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 soil movements in nonhomogeneous soil. The lateral stiffness coefficient of the pile shaft in nonhomogeneous soil is derived from the rocking stiffness coefficient that is obtained from the analytical solution, taking into account the three‐dimensional displacement represented in terms of scalar potentials in the elastic three‐dimensional analysis. The relationship between horizontal displacement, rotation, moment, and shear force of a pile subjected to lateral soil movements in nonhomogeneous soil is obtainable in the form of the recurrence equation. For the relationship between the lateral pressure and the horizontal displacement, it is assumed that the behavior is linear elastic up to lateral soil yield, and the lateral pressure is constant under the lateral soil yield. The interaction factors between piles subjected to both lateral load and moment are calculated, taking into account the lateral soil movement. 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 the Mindlin equation and the equivalent thickness for soil layers in the equivalent elastic method. For lateral movement, lateral pressure, bending moment, and interaction factors, there are small differences between results obtained from the 1‐D and the 3‐D displacement methods except a very flexible pile. Copyright © 2015 John Wiley & Sons, Ltd.     

A simplified analysis method for piled raft foundations subjected to ground movements induced by tunnelling

A simplified method of numerical analysis has been developed to estimate the deformation and load distribution of piled raft foundations subjected to ground movements induced by tunnelling 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, the piles as elastic beams, and the soil is treated as interactive springs. The interactions between structural members, pile–soil–pile, pile–soil–raft and raft–soil–raft interactions, are modelled based on Mindlin's solutions for both vertical and lateral forces. The validity of the proposed method is verified through comparisons with some published solutions for single piles and pile groups subjected to ground movements induced by tunnelling. Thereafter, the solutions from this approach for the analysis of a pile group and a piled raft subjected to ground movements induced by tunnelling are compared with those from three‐dimensional finite difference program. Good agreements between these solutions are demonstrated. The method is then used for a parametric study of single piles, pile groups and piled rafts subjected to ground movements induced by tunnelling. Copyright © 2005 John Wiley & Sons, Ltd.     

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.     

Dynamic characteristics of a large‐diameter pile in saturated soil and its application

This study theoretically investigates the dynamic response of an end‐bearing pile embedded in saturated soil considering the transverse inertial effect of the pile. The saturated soil surrounding the pile is described by Biot poroelastic theory, and the pile is represented by a Rayleigh‐Love rod because both the vertical and radial displacements at the soil‐pile interface are considered. The potential function decomposition method and variable separation method are introduced to solve the governing equations of the soil, in which the vertical and radial displacement components are coupled. The governing equation of the pile is solved using the continuity conditions at the pile‐soil interface. Next, the velocity admittance in the frequency domain and the velocity response in the time domain at the pile top are presented based on the Laplace transform and inverse Fourier transform, respectively. Subsequently, the reduced solution is compared with a 1‐dimensional model solution to verify the validity, and the influences of the slenderness ratio of the pile on the transverse inertial effect of the pile are analyzed. Moreover, Poisson ratio, the slenderness ratio of the pile, and the pile‐soil modulus ratio are studied. Finally, the theoretical and measured curves in the engineering project are compared, and the results demonstrate the good application prospects of the solution presented in this article.     

Variational elastic solution for axially loaded piles in multilayered soil

Most analytical or semi‐analytical solutions of the problem of load‐settlement response of axially loaded piles are based on the assumption of zero radial displacement. These solutions also are only applicable to piles embedded in either a homogeneous or a Gibson soil deposit. In reality, soil deposits consist of multiple soil layers with different properties, and displacements in the radial direction within the soil deposit are not zero when the pile is loaded axially. In this paper, we present a load‐settlement analysis applicable to a pile with circular cross section installed in multilayered elastic soil that accounts for both vertical and radial soil displacements. The analysis follows from the solution of the differential equations governing the displacements of the pile–soil system obtained using variational principles. The input parameters needed for the analysis are the pile geometry and the elastic constants of the soil and pile. We compare the results from the present analysis with those of an analytical solution that considers only vertical soil displacements. The analysis presented in this paper also provides useful insights into the displacement and strain fields around axially loaded piles. Copyright © 2011 John Wiley & Sons, Ltd.     

A non‐linear coupled finite element–boundary element model for the prediction of vibrations due to vibratory and impact pile driving

This paper presents a non‐linear coupled finite element–boundary element approach for the prediction of free field vibrations due to vibratory and impact pile driving. Both the non‐linear constitutive behavior of the soil in the vicinity of the pile and the dynamic interaction between the pile and the soil are accounted for. A subdomain approach is used, defining a generalized structure consisting of the pile and a bounded region of soil around the pile, and an unbounded exterior linear soil domain. The soil around the pile may exhibit non‐linear constitutive behavior and is modelled with a time‐domain finite element method. The dynamic stiffness matrix of the exterior unbounded soil domain is calculated using a boundary element formulation in the frequency domain based on a limited number of modes defined on the interface between the generalized structure and the unbounded soil. The soil–structure interaction forces are evaluated as a convolution of the displacement history and the soil flexibility matrices, which are obtained by an inverse Fourier transformation from the frequency to the time domain. This results in a hybrid frequency–time domain formulation of the non‐linear dynamic soil–structure interaction problem, which is solved in the time domain using Newmark's time integration method; the interaction force time history is evaluated using the θ‐scheme in order to obtain stable solutions. The proposed hybrid formulation is validated for linear problems of vibratory and impact pile driving, showing very good agreement with the results obtained with a frequency‐domain solution. Linear predictions, however, overestimate the free field peak particle velocities as observed in reported field experiments during vibratory and impact pile driving at comparable levels of the transferred energy. This is mainly due to energy dissipation related to plastic deformations in the soil around the pile. Ground vibrations due to vibratory and impact pile driving are, therefore, also computed with a non‐linear model where the soil is modelled as an isotropic elastic, perfectly plastic solid, which yields according to the Drucker–Prager failure criterion. This results in lower predicted free field vibrations with respect to linear predictions, which are also in much better agreement with experimental results recorded during vibratory and impact pile driving. Copyright © 2008 John Wiley & Sons, Ltd.