Analytical layer‐element method for non‐axisymmetric consolidation of multilayered soils

A numerically efficient and stable method is developed to analyze Biot's consolidation of multilayered soils subjected to non‐axisymmetric loading in arbitrary depth. By the application of a Laplace–Hankel transform and a Fourier expansion, the governing equations are solved analytically. Then, the analytical layer‐element (i.e. a symmetric stiffness matrix) describing the relationship between generalized displacements and stresses of a layer is exactly derived in the transformed domain. Considering the continuity conditions between adjacent layers, the global stiffness matrix of multilayered soils is obtained by assembling the inter‐related layer‐elements. Once the solution in the Laplace–Hankel transformed domain that satisfies the boundary conditions has been obtained, the actual solution can be derived by the inversion of the Laplace–Hankel transform. Finally, numerical examples are presented to verify the theory and to study the influence of the layered soil properties and time history on the consolidation behavior. Copyright © 2011 John Wiley & Sons, Ltd.     

Analytical layer element solutions for deformations of transversely isotropic multilayered elastic media under nonaxisymmetric loading

This paper presents the analytical layer element solutions for deformations of transversely isotropic elastic media subjected to nonaxisymmetric loading at an arbitrary depth. The state vectors for the nonaxisymmetric problem are deduced through the substitution of the Hu Hai‐chang solutions into the basic equations for the transversely isotropic elastic media. From the state vectors, the analytical layer element of a single layer is obtained in the Hankel transformed domain. The analytical layer element is an exact and symmetric stiffness matrix whose elements are without positive exponential functions, which can not only simplify the calculation but also improve the stability of computation. On the basis of the continuity conditions between adjacent layers, the global stiffness matrix is obtained by assembling the interrelated layer elements. The solutions for the multilayered elastic media in the transformed domain are obtained by solving the algebraic equation of the global stiffness matrix, which satisfies the boundary conditions. The actual solutions in the physical domain are further obtained by inverting the Hankel transform. Finally, some cases are analyzed to verify the solutions and evaluate the influences of the transversely isotropic character and stratified character of the media on the load–displacement responses. The numerical results show that the variations of the elastic properties between layers have a great effect on the displacements of the multilayered media. Copyright © 2014 John Wiley & Sons, Ltd.     

Axisymmetric thermal consolidation of multilayered porous thermoelastic media due to a heat source

This paper presents an analytical layer element solution to axisymmetric thermal consolidation of multilayered porous thermoelastic media containing a deep buried heat source. By applying the Laplace–Hankel transform to the state variables involved in the basic governing equations of porous thermoelasticity, the analytical layer elements that describe the relationship between the transformed generalized stresses and displacements of a finite layer and a half‐space are derived. The global stiffness matrix equation is obtained by assembling the interrelated layer elements, and the real solutions in the physical domain are achieved by numerical inversion of the Laplace–Hankel transform after obtaining the solutions in the transformed domain. Finally, numerical calculations are performed to demonstrate the accuracy of this method and to investigate the influence of heat source's types, layering, and the porous thermoelastic material parameters on thermal consolidation behavior. Copyright © 2015 John Wiley & Sons, Ltd.     

Consolidation analysis of saturated multi‐layered soils with anisotropic permeability caused by a point sink

This paper presents the analytical layer‐element method to analyze the consolidation of saturated multi‐layered soils caused by a point sink by considering the anisotropy of permeability. Starting from the governing equations of the problem, the solutions of displacements and stresses for a single soil layer are obtained in the Laplace–Hankel transformed domain. Then, the analytical layer‐element method is utilized to further derive the solutions for the saturated multi‐layered soils in the transformed domain by combining with the boundary conditions of the soil system and continuity conditions between adjacent layers. The actual solutions in the physical domain can be acquired by the inversion of Laplace–Hankel transform. Numerical results are carried out to show the accuracy and stability of the proposed method and evaluate the influence of sink depth and anisotropic permeability on excess pore pressure and surface settlement. Copyright © 2011 John Wiley & Sons, Ltd.     

Thermo-mechanical performance of layered transversely isotropic media around a cylindrical/tubular heat source

We develop a new numerical model based on a precise integration method to investigate the coupled thermo-mechanical performance of layered transversely isotropic media around a cylindrical/tubular heat source. To obtain the relational matrices of the extended precise integration method, we first convert the governing equations of the problem into ordinary differential matrix equations through the Laplace–Hankel transform. Then, the cylindrical heat source is divided into a series of plane heat sources, and the plane temperature load term is added to the state vector between layer elements. By combining the layer elements, we build a layered transversely isotropic numerical model containing a cylindrical heat source in the transformed domain. Finally, we solve the model in the transformed domain and obtain the solution of the problem in the real domain through the Laplace–Hankel transform inversion. The accuracy of this method is verified by comparing the solutions with the results of the analytical method and the finite element method. Then, we study the influence of the anisotropy of thermal parameters, the embedded depth, the length/radius ratio, the type of heat source and the stratification of the medium on the thermo-mechanical coupled performance.

     

Analytical layer-element solution to axisymmetric consolidation of multilayered soils

After the application of a Laplace–Hankel transform, the governing equations of Biot’s consolidation are solved analytically by using the eigenvalue approach. Then the analytical layer-element of a single soil layer can be obtained in the transformed domain by synthesizing the generalized displacements and stresses, which are both expressed by six arbitrary constants. The elements of the analytical layer-element only contain negative exponential functions, which leads to a considerable improvement in computation efficiency and stability. The global stiffness matrix equation of multilayered soils is further obtained by assembling the interrelated layer-elements, and the actual solution is achieved by numerical inversion of the Laplace–Hankel transform after the solution in the transformed domain is obtained. Numerical examples are given to demonstrate the accuracy of this method and to study the influence of the layered soil properties and time history on the consolidation behavior.     

A quasistatic analysis of a plate on consolidating layered soils by analytical layer‐element/finite element method coupling

In this paper, a coupling method between finite element and analytical layer‐elements is utilized to analyze the time‐dependent behavior of a plate of any shape and finite rigidity resting on layered saturated soils. Based on the integral transform techniques together with the aid of an order reduction method, an analytical layer‐element solution is derived from the governing equations for three‐dimensional Biot consolidation with respect to a Cartesian coordinate system and then extended to be the fundamental solution for the layered saturated soil under a point load. The Mindlin plate is modeled by eight‐noded isoparametric elements. The governing equations of the interaction between soil and plate in the Laplace‐Fourier transformed domain are deduced by referring to the coupling theory of FEM/BEM, and the final solution is obtained by applying numerical inversion. Numerical examples concerned with the time‐dependent response of a plate are performed to demonstrate the influence of soil and plate properties on the interaction process. Copyright © 2014 John Wiley & Sons, Ltd.     

任意荷载下双面半透水边界分数阶导数黏弹性饱和土层一维固结

基于Terzaghi一维固结理论,分析了考虑半透水边界条件的分数阶导数黏弹性饱和土层在随时间变化的任意荷载作用下一维固结问题。首先,应用Laplace变换联立求解饱和土层一维固结微分方程和分数阶Kelvin-Voigt黏弹性本构方程,推导出有效应力和沉降在Laplace变换域内的解析解,采用Crump方法进行Laplace逆变换,得到了时间域内的半解析解。然后将本文得到的半解析解分别退化为半透水边界条件下基于黏弹性假设的一维固结半解析解和双面透水边界条件下基于分数阶黏弹性假设的一维固结半解析解,结果与已有文献的半解析解相同,验证了本研究所提出解的可靠性。最后通过算例分别考察了半透水边界参数、分数阶黏弹性模型参数和荷载参数对饱和土层固结沉降的影响。研究表明,半透水边界条件参数、分数阶次与黏滞系数主要影响饱和土层固结的发展快慢,而饱和土层的最终沉降量主要受到土层压缩模量的影响;另外,饱和土层的固结规律与外荷载变化规律一致。     

Time‐dependent behaviour of a rigid foundation on a transversely isotropic soil layer

An analytical solution is presented in this paper to study the time‐dependent settlement behaviour of a rigid foundation resting on a transversely isotropic saturated soil layer. The governing equations for a transversely isotropic saturated soil, within Biot's poroelasticity framework, are solved by means of Laplace and Hankel transforms. The problem is subsequently formulated in the Laplace transform domain in terms of a set of dual integral equations that are further reduced to a Fredholm integral equation of the second kind and solved numerically. The developed analytical solution is validated via comparison with the existing analytical solution for an isotropic saturated soil case, and adopted as a benchmark to examine the sensitivities of the mesh refinement and the locations of truncation boundaries in the finite element simulations using ABAQUS. Particular attention is paid to the influences of the degree of soil anisotropy, boundary drainage condition, and the soil layer thickness on the consolidation settlement and contact stress of the rigid foundation. Copyright © 2009 John Wiley & Sons, Ltd.     

Reflection–transmission matrix method for the consolidation of a multilayered saturated soil

The consolidation of the layered saturated soil is an important issue in civil engineering and has been investigated extensively during the past decades. In this study, based on the Biot's theory, the reflection–transmission matrix (RTM) method for treating the layered saturated soil under axisymmetric consolidation is developed. To decouple the governing equations of the Biot's theory, the McNamee displacement functions are introduced, and the general solution for the saturated soil is obtained using the Laplace and Hankel transforms. In order to develop the RTM method for the layered saturated soil, based on the obtained general solution, the static wave vector corresponding to the state vector of the saturated soil and the transform matrix relating the aforementioned two vectors are defined. Also, the transfer matrices corresponding to the two vectors are introduced, and the representations of the RTMs for the static wave vector of the saturated soil are presented. As the state vector, static wave vector, and the transform matrix relating the two vectors are all defined in the global coordinate system, the RTMs obtained in this study thus have a reasonable physical meaning. By using the RTMs for the layered saturated soil, the solutions for the layered saturated soil subjected to external sources are derived. Comparison of results due to the proposed RTM method with some existing results and results due to the transfer matrix method validates the developed RTM method. Some numerical results are obtained based on the proposed RTM method for the layered saturated soil. Copyright © 2016 John Wiley & Sons, Ltd.     

Influence of a point sink on a pile group in saturated multilayered soils with anisotropic permeability

The behavior of a pile group is solved using the finite element method, and the fundamental solution of saturated multilayered soils with anisotropic permeability is obtained by the analytical layer element method. Based on the supposition of no slip occurring at the pile‐soil interface, the governing equations of the interaction between the pile group and the soils due to a point sink are established in the Laplace‐Hankel transformed domain by considering the pile‐soil compatibility condition. Numerical results are presented to study the effect of point sink pumping, the properties of soils, and the geometries of piles on the behavior of the pile group.     

横观各向同性饱和地基三维瞬态Lamb问题

研究了横观各向同性饱和土地基在地表动力荷载作用下的三维瞬态响应。基于饱和多孔介质的三维Biot波动理论,利用Laplace变换,建立圆柱坐标系下横观各向同性饱和土的波动方程;解耦波动方程后,根据算子理论,并借助Fourier展开和Hankel变换技术,得到瞬态荷载作用下,饱和土介质的土骨架位移和应力、孔隙水相对位移和孔隙水压力的一般解;利用一般解,给出横观各向同性饱和地基在地表集中荷载激励下的瞬态Lamb问题的解答。数值算例结果表明,采用各向同性饱和介质的动力学模型,不能准确描述具有明显各向异性特性的饱和土地基的瞬态动力特性。     

A method of computing the consolidation behaviour of layered soils using direct numerical inversion of Laplace transforms

A method is presented for obtaining the consolidation behaviour of a layered soil subjected to strip, circular, or rectangular surface loadings, or subjected to fluid withdrawal due to pumping. The solution method involves applying a Fourier or Hankel transform to the field quantities along with a Laplace transformation. The effect of the Fourier or Hankel transform is to reduce a two- or three-dimensional problem or one involving axial symmetry, to one involving only a single spatial dimension. In cases where the soil is horizontally layered, this has great advantages over conventional methods, such as finite element or finite difference methods, since very little computer storage and data preparation time is required. Solution of the time dependent problem is achieved by applying a Laplace transformation to the field variables, obtaining solutions in Laplace transform space, and then numerically inverting the transformed solutions to obtain the real time behaviour. This eliminates the need for ‘marching type’ schemes where a solution is found from one at a previous time. By direct inversion of the Laplace transform, a solution may be obtained directly at any given time.     

The axisymmetric consolidation of a semi‐infinite transversely isotropic saturated soil

This paper presents an exact analytical solution to fully coupled axisymmetric consolidation of a semi‐infinite, transversely isotropic saturated soil subjected to a uniform circular loading at the ground surface. The analysis is under the framework of Biot's general theory of consolidation. First, the governing equations of consolidation are transformed into a set of equivalent partial differential equations with the introduction of two auxiliary variables. These partial differential equations are then solved using Hankel–Laplace integral transforms. Once solutions in the transformed domain have been obtained, the actual solutions in the physical domain for displacements and stress components of the solid matrix, pore‐water pressure and fluid discharge can be finally obtained by direct numerical inversion. The accuracy of the numerical solutions developed is confirmed by comparison with an existing exact solution for an isotropic and saturated soil that is a special case of the more general problem addressed. Numerical analyses are also presented to investigate the influence of the degree of material anisotropy on the consolidation settlement. Copyright © 2005 John Wiley & Sons, Ltd.     

非稳态热传导时层状路面体系的温度响应

利用解析层元法推导温度荷载作用下非稳态热传导时层状路面体系的温度响应解答。从热弹性理论平面应变问题的控制方程出发,借助于Laplace-Fourier积分变换,推导出单层介质及下卧半平面的精确刚度矩阵即解析层元,结合有限层法原理及边界条件,组装并求解总刚度矩阵,得到其在变换域内的解答,最后通过相应的积分逆变换得到物理域内的真实解。由于该法刚度矩阵元素中不含正指数项,计算时不会出现溢出或病态矩阵的现象。编译了相应的计算程序,所得结果与有限元模拟结果吻合较好。在此基础上,对有限深度和半平面两种假定条件下的解答进行对比分析,并分析层状路面体系中位移和温度随时间的变化趋势及沿深度的分布规律。分析表明：温度场具有一定的影响深度,超过此深度,有限深度与半平面理论解答基本一致;温度荷载的影响深度与其强度有关,强度越大,其影响深度越深。     

竖向简谐荷载下二维层状饱和地基的解析层元解

基于Biot固结理论,运用解析层元方法求解竖向简谐荷载作用下二维层状饱和地基的动力响应问题。从直角坐标平面应变问题控制方程出发,通过Fourier-Laplace变换将偏微分方程组转化为常微分方程组,求解得到单层饱和地基的解析层元。结合层间连续条件和边界条件,组装得到多层饱和地基的总刚度矩阵方程,进而求得变换域内的解。借助Fourier-Laplace逆变换的数值积分方法,获得平面应变动力问题在物理域内的解,编制了相应的计算程序,其计算结果与已有文献结果吻合较好。通过算例分析了荷载圆频率、荷载作用深度及地基成层性对地基竖向位移的影响。计算结果表明：随荷载圆频率的增大,地基竖向位移先增加后减小;地基竖向位移在荷载作用点处呈现波峰,且受表层土性的影响较大。     

Fluctuation responses of saturated porous media subjected to cyclic thermal loading

Normalized, coupled governing equations for one-dimensional thermal consolidation problems are established. The non-dimensional coefficients of thermal consolidation and thermal diffusivity are defined accordingly. An analytical solution is deduced by using the Laplace transform and the Gauss–Legendre method of Laplace transform inversion. The responses of saturated porous media subjected to cyclic thermal loading are studied. The evolution of temperature, pore pressure and displacement from instantaneous state to quasi-steady state, with elapsed time, are analysed. The characteristics of cyclic fluctuation and the attenuation of the field variables with increased depth are also analysed. The influences of the permeability of media on thermal responses are discussed.     

Extended precise integration method for axisymmetric thermo‐elastic problem in transversely isotropic material

This paper presents a numerical solution for the analysis of the axisymmetric thermo‐elastic problem in transversely isotropic material due to a buried heat source by means of extended precise integral method. By virtue of the Laplace–Hankel transform applied into the basic governing equations, an ordinary differential matrix equation is achieved, which describes the relationship between the generalized stresses and displacements in transformed domain. An extended precise integration method is introduced to solve the aforementioned matrix equation, and the actual solution in the physical domain is acquired by inverting the Laplace–Hankel transform. Numerical examples are carried out to demonstrate the accuracy of the proposed method and elucidate the influence of the character of transverse isotropy, the anisotropy of linear expansion coefficient, the anisotropy of thermal diffusivity, and medium's stratification on the thermo‐elastic response. Copyright © 2015 John Wiley & Sons, Ltd.     

层状饱和地基上刚性圆板的扭转振动及参数分析

用解析的方法首次研究了层状饱和地基上刚性圆板的扭转振动特性.运用Hankel变换求解饱和介质动力问题的控制方程,按混合边值条件建立了层状饱和地基上刚性圆板扭转振动的第二类Fredholm积分方程.数值算例给出了动力柔度系数和扭转角幅值随无量纲频率的变化曲线,与单相弹性及匀质饱和介质情况进行了对比分析,并进行了参数研究.     

Axisymmetric consolidation of a poroelastic soil layer with a compressible fluid constituent due to groundwater drawdown

Axisymmetric consolidation of a poroelastic soil layer with a compressible fluid constituent induced by groundwater drawdown was studied based on Biot’s axisymmetric consolidation theory. Laplace and Hankel transforms were employed to solve the governing equation. Explicit analytical solutions are obtained in the Laplace–Hankel transform domain when groundwater drawdown is induced by a constant pumping well. Based on the solutions, numerical computations were performed to study the influences of the compressibility of the fluid constituent on the consolidation behavior of the soil layer.