Amplitude reduction of elastic waves by a row of piles in poroelastic soil

This paper studies the displacements behind a row of elastic piles in a homogeneous unbounded poroelastic soil after excitation by the passage of elastic waves. Biot’s theory is employed to describe soil behavior. Using a Fourier–Bessel series expansion with the aid of a translational addition theorem, closed-form expressions for the scattering coefficients are developed by imposing continuity conditions at the pile–soil interfaces. The influence of certain parameters, such as pile rigidity and soil permeability on the screening performance of the pile barrier is investigated. Particular attention is focused on the influence of the soil permeability on the screening effectiveness of a row of piles in the frequency domain.     

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.     

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.     

Poroelastic model for pile–soil interaction in a half‐space porous medium due to seismic waves

In this paper, frequency domain dynamic response of a pile embedded in a half‐space porous medium and subjected to P, SV seismic waves is investigated. According to the fictitious pile methodology, the problem is decomposed into an extended poroelastic half‐space and a fictitious pile. The extended porous half‐space is described by Biot's theory, while the fictitious pile is treated as a bar and a beam and described by the conventional 1‐D structure vibration theory. Using the Hankel transformation method, the fundamental solutions for a half‐space porous medium subjected to a vertical or a horizontal circular patch load are established. Based on the obtained fundamental solutions and free wave fields, the second kind of Fredholm integral equations describing the vertical and the horizontal interaction between the pile and the poroelastic half‐space are established. Solution of the integral equations yields the dynamic response of the pile to plane P, SV waves. Numerical results show the parameters of the porous medium, the pile and incident waves have direct influences on the dynamic response of the pile–half‐space system. Significant differences between conventional single‐phase elastic model and the poroelastic model for the surrounding medium of the pile are found. Copyright © 2007 John Wiley & Sons, Ltd.     

Dynamic responses of a poroelastic half‐space from moving trains caused by vertical track irregularities

The vibrations of railway tracks on a poroelastic half‐space generated by moving trains are investigated through a vehicle–track–ground coupling model. The theoretical model incorporates a vehicle, a track, and a fully saturated poroelastic half‐space soil medium. The source of vibration excitation is divided into two components: the quasi‐static loads and the dynamic loads. The quasi‐static loads are related to the static component of the axle loads, whereas the dynamic loads are due to the dynamic wheel–rail interaction. A linear Hertizian contact spring is introduced between each wheelset and the rail to consider the dynamic loads. Biot's dynamic theory is used to characterize the poroelastic half‐space soil medium. Using the Fourier transform, the governing equations for the track–ground system are solved and the numerical results are presented for a single axle vehicle model. The different dynamic characteristics of the elastic soil medium and the saturated poroelastic medium are investigated. In addition, the different roles of the moving axle loads and the roughness‐induced dynamic loads are identified. It is concluded that the vibration level of the free field off the track predicted by the poroelastic soil medium is smaller than that predicted by the elastic soil medium for vehicle speed below the Rayleigh wave speed of the poroelastic half‐space, whereas it is larger for vehicle speed above the Rayleigh wave speed. The dynamic loads play an important role in the dynamic responses of the track–ground system. Copyright © 2010 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.     

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.     

排桩对柱面SH波散射问题研究：解析求解

采用波函数展开法结合Graf加法定理,利用桩体与土体之间的位移和应力连续边界条件,给出了全空间单排桩对柱面SH波散射解析解,并采用傅里叶逆变换,求得时域结果。该方法考虑了入射波曲率的影响,在频域内分析了柱面SH波入射时排桩散射的频谱规律,给出了时域排桩柱面SH波散射位移云图,讨论了桩数与桩间距对柱面SH波入射时排桩散射的影响。研究表明：在振源距排桩较近（d/a=10）时,低频段（η=0~1.0）排桩对柱面SH波的隔离作用显著;振源距排桩较远时,各频段排桩对柱面SH波的隔离作用均较为显著。桩间距固定时,增加桩数,排桩后方首次出现的最大位移响应相应减小,排桩散射的影响范围随之增大;对柱面SH波进行阻隔时,为提高阻隔效率的同时节约成本,不仅需要考虑桩数,还应考虑波源与桩体、桩间空隙相对位置的影响;排桩分布宽度固定时,由于入射波曲率对排桩散射的影响,减小桩间距,排桩后方首次出现的最大位移响应有可能会出现放大的情况,故应采用合理的桩间距对柱面SH波进行阻隔。     

水平循环荷载下桩基变形特性的离心模型试验研究

波浪、船舶等长期水平循环荷载作用下,桩基将不可避免地产生附加应力和变形。针对饱和黏土地层,开展离心模型试验研究了船舶系泊水平荷载作用下单桩和群桩的变形特性。发现水平循环加-卸载诱发了桩周土体的塑性变形,进而导致桩身产生了不可恢复的水平位移和弯曲变形。随着循环荷载的增加,单桩和群桩的桩顶最大水平位移和残余水平位移均同时增加,但残余水平位移明显小于最大水平位移。单桩的桩顶残余水平位移与最大位移比值介于0.17～0.22;群桩的桩顶残余水平位移与最大水平位移比值介于0.30～0.84。水平循环加-卸载作用下,桩身残余弯曲应变明显小于最大弯曲应变。单桩的残余弯曲应变与最大弯曲应变比值介于0.13～0.50;群桩的桩身残余弯曲应变与最大弯曲应变比值介于0.23～0.82。群桩前桩的残余和最大弯曲应变明显大于后桩,前桩与后桩的最大弯曲应变、残余应变比值分别高达3.2和3.1。因此,前桩要采取合理的加固和保护措施,以确保桩基长期服役的安全性。     

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.     

半空间饱和土中圆形衬砌对弹性压缩波的散射

借助于Biot 波动理论和弹性波的传播理论,采用复变函数和多级坐标法,对半空间饱和土中圆形衬砌结构对弹性稳态压缩波的散射问题进行求解和分析。利用一个半径很大的圆弧来逼近半空间直边界,将待解问题转化为稳态弹性压缩波在一个大圆孔和一个弹性衬砌结构的散射问题。通过引入势函数,将饱和土的Biot波动方程和衬砌的弹性波动方程解耦成Helmholtz 方程,借助复变函数级数展开便可以预先写出该组Helmholtz方程的通解。然后,通过引用复变量,把饱和土和衬砌结构中的应力、位移及孔压用设定的势函数表示出来,再利用半空间饱和土和衬砌结构的连续性条件和近似直边界的圆弧边界和衬砌内边界的边界条件求解出该组势函数的特解。最后,利用势函数的特解,得到饱和土中的位移,应力和孔压及衬砌结构的位移和应力;变换不同的参数求解衬砌结构内外边界的动应力和孔压的集中系数,通过对算例结果的分析得出一系列有益的结论。     

Group interaction on vertically loaded piles in saturated poroelastic soil

The pile-to-pile interaction was obtained for vertically loaded piles embedded in homogeneous poroelastic saturated soil. Deduced from Biot’s theory, the fundamental functions of the quasi-static development for the force, displacement and pore pressure were acquired in cylindrical coordinates. The pile–soil system was decomposed into extended soil and fictitious piles, and the compatibility condition was set up between the axial strain of the fictitious piles and the corresponding average strain over the extended soil. This approach results in the governing equations, which consist of the Fredholm integral equations of the second kind and the basic unknowns of the axial forces along the fictitious pile shaft. The axial force and settlement along the pile shaft were calculated based on the axial forces of the fictitious piles. The interaction between the piles was investigated under different consolidation conditions through a two-pile model, and two pile interaction factors were obtained. Stemming from the two-pile analysis, numerical analyses on the settlement of the pile groups were conducted to probe pile interaction with consolidation. The conventional solutions for the single-phase soil-pile problem seem to underestimate the interaction factor if the consolidation effect is taken into account as pile settlement continues. The pile-to-pile interaction can also aggravate the percentage of consolidation settlement (PCS), and as the pile number increases, the value of the PCS will also increase. Several key factors, such as the pile stiffness, pile slenderness ratio and pile spacing, are investigated to better understand the impact of consolidation on pile analysis.     

The P-y Response of Laterally Loaded Flexible Piles in Residual Soil

This paper describes the main features related to lateral displacements with depth after successive lateral loading–unloading cycles applied to the top of reinforced-concrete flexible bored piles embedded in naturally bonded residual soil. The bored piles under study have a cylindrical shape, with 0.40-m in diameter and 8.0-m in length. Both bored piles types (P1 and P2) include an embedded steel pipe section in their center as longitudinal steel reinforcements: pile type P1 has another 16 steel rods as steel reinforcement to concrete while pile type P2 has no further steel reinforcement. Pile type P1 has three times as much stiffness (EI) and four and a half times the plastic moment (M_y) than pile type P2. A similar load–displacement performance was observed at initial loads as for small displacements of both piles. At this initial loading stage, the response of the reinforced concrete piles is a function of the soil characteristics and of a linear elastic pile deformation. During this stage, piles can even be understood as probes for evaluating soil reactions. For larger horizontal displacements, after the concrete section starts undergoing large deformations, approaching the ultimate bending moment, pile behavior and consequently the load–displacement relation starts to diverge for both piles. For pile P1 the values of relevant lateral displacements are extended to about 2.5-m in depth, while for pile P2 lateral displacements are mostly constrained to about 2.0-m in depth. Measurements of horizontal displacements of pile P1 against depth recorded with a slope indicator show that, after unloading, lateral loads at distinct stages (small and near failure loads), exhibits a much higher elastic phase of the system response. An analytical fitting model of soil reaction is proposed based on the measured displacements from slope indicator. The integration of a continuous model proposed for the soil reaction agrees fairly well with the measured displacements up to moments close to plastic limit. Results of load–displacement show that the stiffer pile (P1) was able to mobilize twice as much lateral load compared to pile P2 for a service limit displacement of about 20 mm. The paper shows results that enable the isolation of the structural variable through real scale pile load tests, thus granting understanding of its importance and enabling its quantitative visualization in examples of piles embedded in residual soil sites.

     

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.     

Analysis of laterally loaded pile groups in multilayered elastic soil

The paper presents a semi-analytical method of calculating the response of a pile group. The approach is based on tying the displacement at any point of the soil mass around a pile or group of piles to the displacements experienced by the piles themselves. This is done by multiplying the pile displacements by decay functions. Application of the principle of minimum potential energy and calculus of variations to the resulting displacement field formulation leads to the differential equations for the soil and piles. Solution of these differential equations using finite differences and the method of eigenvectors leads to the desired displacement field in the soil and deflection profiles of the piles. The method produces displacement fields that are very close to those produced by the finite element method at a fraction of the cost. To illustrate the ease of application of the method, it is then used to prepare pile group efficiency charts for some typical soil modulus profiles.     

Dynamic response of a pile considering the interaction of pile variable cross section with the surrounding layered soil

Assuming that the pile variable cross section interacts with the surrounding soil in the same way as the pile toe does with the bearing stratus, the interaction of pile variable cross section with the surrounding soil is represented by a Voigt model, which consists of a spring and a damper connected in parallel, and the spring constant and damper coefficient are derived. Thus, a more rigid pile–soil interaction model is proposed. The surrounding soil layers are modeled as axisymmetric continuum in which its vertical displacements are taken into account and the pile is assumed to be a Rayleigh–Love rod with material damping. Allowing for soil properties and pile defects, the pile–soil system is divided into several layers. By means of Laplace transform, the governing equations of soil layers are solved in frequency domain, and a new relationship linking the impedance functions at the variable‐section interface between the adjacent pile segments is derived using a Heaviside step function, which is called amended impedance function transfer method. On this basis, the impedance function at pile top is derived by amended impedance function transfer method proposed in this paper. Then, the velocity response at pile top can be obtained by means of inverse Fourier transform and convolution theorem. The effects of pile–soil system parameters are studied, and some conclusions are proposed. Then, an engineering example is given to confirm the rationality of the solution proposed in this paper. Copyright © 2017 John Wiley & Sons, Ltd.     

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.     

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.     

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.     

Numerical analysis of the geotechnical behaviour of energy piles

Energy geostructures are rapidly gaining acceptance around the world; they represent a renewable and clean source of energy that can be used for the heating and cooling of buildings and for de‐icing of infrastructures. This technology couples the structural role of geostructures with the energy supply, using the principle of shallow geothermal energy. The geothermal energy exploitation represents an additional thermal loading, seasonally cyclic, which is imposed on the soil and the structure itself. Because the primary role of the piles is the stability of the superstructure, this aspect needs to be ensured even in the presence of the additional thermal load. The goal of this paper is to numerically investigate the behaviour of energy pile foundations during heating–cooling cycles. For this purpose, the finite element method is used to simulate both a single and a group of energy piles. The piles are subjected to a constant mechanical load and a seasonally cyclic thermal load over several years, imposed in terms of injected–extracted thermal power. The soil and the pile–soil interface behaviours are reproduced using a thermoelastic‐thermoplastic constitutive model. The thermal‐induced stresses inside the piles and the additional displacements of the foundations are discussed. The group model is used to investigate the interactions between the piles during thermo‐mechanical loading. The presented results are specific to the studied cases but lead to the conclusion that both the thermal‐induced displacements and stresses, despite being acceptable under normal working conditions, deserve to be taken into account in the geotechnical design of energy piles. Copyright © 2014 John Wiley & Sons, Ltd.