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 solution for one-dimensional consolidation of clayey soils with a threshold gradient

General approximate analytical solutions are developed for one-dimensional consolidation with consideration of the threshold gradient under a time-dependent loading. A comparison is made between the present solution and some available numerical solutions for a particular case, and the results show that the approach employed in this article is reasonable. The influence of the threshold gradient and the loading period on consolidation behaviour is investigated, and the results show that the moving boundary of seepage moves downward gradually. The greater the threshold gradient is, the slower the boundary moves. The excess pore pressure will not be completely dissipated at the end of consolidation, and the larger the threshold gradient is, the greater the residual excess pore pressure is. The average degree of consolidation considering the threshold gradient defined by settlement is different from that defined by pore pressure. Moreover, the greater the threshold gradient is, the larger the average degree of consolidation in terms of strain, whereas the smaller the average degree of consolidation in terms of stress. It is also shown that the longer the loading period is, the longer the time moving boundary takes to reach the bottom of the layer, and the greater the average degree of consolidation is.     

Analytical solution to axisymmetric consolidation of unsaturated soil stratum under equal strain condition incorporating smear effects

This paper proposes closed‐form analytical solutions to the axisymmetric consolidation of an unsaturated soil stratum using the equal strain hypothesis. Following the 1‐dimensional (1D) consolidation theory for unsaturated soil mechanics, polar governing equations describing the air and water flows are first presented on the basis of Fick's law and Darcy's law, respectively. The current study takes into account the peripheral smear caused by an installation of vertical drain. Separation of variables and Laplace transformation are mainly adopted in the analytical derivation to obtain final solutions. Then, the hydraulic conductivity ratio, the radius of influence zone and smear parameters influencing time‐dependent excess pore pressures, and the average degree of consolidation are graphically interpreted. In this study, a comparison made between the proposed equal strain results and the existing free strain results suggests that both hypotheses would deliver similar predictions. Moreover, it is found that the smear zone resulting from vertical drain installations would hinder the consolidation rate considerably.     

多层地基非轴对称Biot固结的理论解

提出了一种新方法来推导多层地基非轴对称Biot固结问题的传递矩阵,相应的理论推导工作量较少。基于柱坐标系下非轴对称Biot固结的基本方程,通过引入中间变量,并对坐标 进行Fourier级数展开,对时间 和坐标 进行Laplace-Hankel变换,得到了6×6阶和2×2阶的两组常微分方程;然后,两组常微分方程进行关于 的Laplace变换和逆变换,得到了单层地基非轴对称Biot固结问题的传递矩阵;结合边界条件和连续条件,运用传递矩阵法得到多层地基非轴对称Biot固结问题在Laplace-Hankel变换域内的解,通过Laplace-Hankel逆变换得到了该问题物理域的真实解答。编制了计算程序,并进行了数值计算与分析。该方法具有计算速度较快的特点,方便工程应用。     

饱和软土成层地基一维非线性固结解析解

假定土体在固结过程中压缩性和渗透性的变化成正比,基于 - 及 - 关系,推导出饱和软土成层地基一维非线性固结解析解,分别给出了按沉降定义和按有效应力定义的每层土平均固结度及整个土层总固结度的计算公式。采用Fortran语言编制了相应的计算程序,将计算得到的结果与已有双层地基一维非线性固结解析解计算结果进行比较,验证该解析解的正确性。利用该程序分析成层地基一维非线性固结性状,分别讨论了初始竖向渗透系数、初始体积压缩系数、荷载值及土层厚度对地基固结性状的影响。分析结果表明：在成层地基一维非线性固结过程中,初始竖向渗透系数对超静孔压的影响较为复杂,对上下层地基固结速率影响不同;初始体积压缩系数增大,超静孔压增大,固结速率变小;所加荷载值越大,超静孔压消散越慢,固结发展越慢;超静孔压消散速率不仅取决于土层厚度,同时取决于各层土渗透性的相对大小。     

Analytical and computational procedure for solving the consolidation problem of layered soils

One‐dimensional consolidation analysis of layered soils conventionally entails solving a system of differential equations subject to the flow conditions at the bounding upper and lower surfaces, as well as the continuity conditions at the interface of every pair of contiguous layers. Formidable computational efforts are required to solve the ensuing transcendental equations expressing the matching conditions at the interfaces, using this method. In this paper, the jump discontinuities in the flow parameters upon crossing from one layer to the other have been systematically built into a single partial differential equation governing the space–time variation of the excess pore pressure in the entire composite medium, by the use of the Heaviside distribution. Despite the presence of the discontinuities in the coefficients of the differential equation, a closed‐form solution in the sense of an infinite generalized Fourier series is obtained, in addition to which is the development of a Green's function for the differential problem. The eigenfunctions of the composite medium are the coordinate functions of the series, obtained computationally through the application of the extended equations of Galerkin. The analysis has been illustrated by solving the consolidation problem of a four‐layer composite, and the results obtained agree very well with the results obtained by previous researchers. Copyright © 2013 John Wiley & Sons, Ltd.     

Effect of compressible parameters on vertical vibration of an elastic pile in multilayered poroelastic media

This paper mainly investigates the influences of compressible parameters on the vertical vibration of a pile embedded in layered poroelastic soil media. The pile is treated as a 1D elastic bar by the finite element method, and fundamental solutions for the layered poroelastic soils due to a vertical dynamic load are obtained by the analytical layer element method. Based on the compatibility conditions, the pile-soil dynamic interaction problem is solved. The numerical scheme has been compiled into a Fortran program for numerical calculation. Influences of the pile-soil stiffness ratio, compressible parameters, vibration frequency and the soil stratification are discussed.     

Axisymmetric consolidation of saturated multi-layered soils with anisotropic permeability due to well pumping

Axisymmetric consolidation of multi-layered soils induced by groundwater extraction from a dewatering well is studied with consideration of the anisotropic permeability and the well length. Laplace-Hankel transforms are utilized to solve the governing equations. The analytical layer-element method is used to build relationships between displacements, stresses, excess pore pressure and seepage velocity in the transformed domain. The real solutions can be obtained by the inversion of Laplace-Hankel transforms. A series of parametric studies, especially the length of a dewatering well and the combined effect of pumping and recharging, are conducted to analyze the consolidation behaviors of layered soils.     

Analytical solution for one‐dimensional consolidation of unsaturated soils using eigenfunction expansion method

This paper introduces an exact analytical solution for governing flow equations for one‐dimensional consolidation in unsaturated soil stratum using the techniques of eigenfunction expansion and Laplace transformation. The homogeneous boundary conditions adopted in this study are as follows: (i) a one‐way drainage system of homogenous soils, in which the top surface is considered as permeable to air and water, whereas the base is an impervious bedrock; and (ii) a two‐way drainage system where both soil ends allow free dissipation of pore‐air and pore‐water pressures. In addition, the analytical development adopts initial conditions capturing both uniform and linear distributions of the initial excess pore pressures within the soil stratum. Eigenfunctions and eigenvalues are parts of the general solution and can be obtained based on the proposed boundary conditions. Besides, the Laplace transform method is adopted to solve the first‐order differential equations. Once equations with transformed domain are all obtained, the final solutions, which are proposed to be functions of time and depth, can be achieved by taking an inverse Laplace transform. To verify the proposed solution, two worked examples are provided to present the consolidation characteristics of unsaturated soils based on the proposed method. The validation of the recent results against other existing analytical solutions is graphically demonstrated. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

考虑桩体固结变形的散体材料桩复合地基固结解析计算

将单个散体材料桩截面和单桩影响范围内的桩周土截面作为一个研究单元考虑,由固结过程中孔隙水排出量等于单元体体积减小量,并引入平均超孔隙水压力的概念,考虑桩体的固结变形,推得散体材料桩复合地基的固结方程。通过等应变假设和初始边界条件,由分离变量法对该固结方程进行求解,得到了桩体和桩间土的平均超孔隙水压力、平均固结度、复合地基整体固结度。通过某工程实例计算,将计算结果与已有解析解进行了比较分析。当散体材料桩复合地基的井径比较大时,两者计算结果十分相近;当井径比较小时,两者差别较大。     

Analytical solution for 1D consolidation under haversine cyclic loading

Presented and discussed in this paper is an exact analytical solution of the nonhomogeneous partial differential equation governing the conventional one‐dimensional consolidation under haversine repeated loading. The derived analytical solution to the 1D consolidation equation is compared with the numerical solution of the same consolidation problem via FEM. The series solution takes into account the frequency of repeated loading through a dimensionless time factor T₀. The paper reveals that an increase in the frequency of imposed repeated haversine loading (a decrease in period of repeated loading) causes an increase in the number of cycles required to achieve the steady state, whereas the effect of frequency on the maximum excess pore water pressure at the bottom of a clay layer with permeable top and impermeable bottom for the range of frequencies studied is generally insignificant. The effective stress at the bottom of the clay deposit with permeable top and impermeable bottom increases with time but with some fluctuations without changing the sign. These fluctuations become more pronounced for increasing values of T₀. An increase in T₀ also causes an increase in maximum effective stress. Copyright © 2013 John Wiley & Sons, Ltd.     

Analytical solutions for nonlinear consolidation of soft soil around a shield tunnel with idealized sealing linings

This paper presents the analytical solutions for nonlinear consolidation of soft soil around a shield tunnel with idealized sealing linings. By introducing the empirical relation between permeability and compressibility, along with the conformal transformation, the governing equations of nonlinear consolidation are established, and the corresponding analytical solutions are derived. Then, the Terzaghi consolidation solutions are derived from the degenerate governing equation of nonlinear consolidation. Through the predictions of different consolidation theories in both completely permeable and impermeable lining conditions, the influences of a tunnel acting as a drain and impacting the dissipation of pore pressure, degree of consolidation, long-term ground settlements and ground settlement rates are investigated. During the early stages of consolidation, the case studies reveal that the predictions made by this study strongly agree with the field data when a completely permeable lining is applied. This study confirms that a tunnel acting as a drain can accelerate the consolidation of soil and enlarge soil deformation due to consolidation. During long term consolidation, a notable nonlinearity of the soil consolidation is exhibited by a small and gradually decreasing settlement rate, showing agreement with the tendency of field data from the impermeable conditions.     

Analytical solution for the 1D consolidation of unsaturated multi-layered soil

The 1D consolidation of unsaturated multi-layered soil is studied based on the theory proposed by Fredlund and Hasan, and an analytical solution for a typical boundary condition is obtained by assuming all material parameters remain constant during consolidation. In the derivation of the analytical solution, the eigenfunction and eigenvalue for the multi-layered problem are first derived through the transfer matrix method. Then, by using the method of undetermined coefficients and the orthogonal relation of the eigenfunction, the analytical solution is obtained. The present method is applicable to various types of boundary conditions. Finally, numerical examples are provided to investigate the consolidation behavior of unsaturated multi-layered soil.     

多层横观各向同性地基轴对称固结的传递矩阵解

从横观各向同性地基轴对称Biot固结的基本方程出发,通过关于t的Laplace变换和关于r的Hankel变换,得到关于z的一阶常微分方程组。然后,对变换域内的基本未知量进行线性化处理,建立了变换域内的基本状态变量在z = 0处和任意深度处z的显式关系。利用传递矩阵法,结合层间连续性条件和边界条件,得到了多层横观各向同性地基的Biot固结轴对称问题的解答。该解答能避免随着层数增加而需要求解大型方程组的困难,明显地提高了计算效率。     

变荷载下软土一维流变固结解析理论

研究了变荷载作用下考虑土的流变特性时的饱和软土层的一维固结问题,通过Laplace变换得到了瞬时加载条件下的解析解,进而根据瞬时加载条件下的解,通过积分的方法,得到了变荷载条件下的解析解。并且根据解析解的结果,通过编程计算作出了几个主要影响因素对固结过程的影响曲线图。结果表明：?1的取值对固结曲线的形状有显著影响;随?0的减小,固结度减小,固结速率减缓。     

非饱和土固结理论新进展

总结了国内外多年来在非饱和土固结理论领域研究的主要理论成果,内容包括国内外非饱和土固结理论研究的概况,涉及线性、非线性、弹塑性和结构性等方面,非饱和土固结理论的研究具有非常好的应用前景。但非饱和土固结理论还很不成熟,有待广大科研工作者的进一步研究。     

连续渗透边界下非饱和土竖井地基固结解析解EI北大核心CSCD

基于非饱和土轴对称固结理论和等应变假设,引入连续渗透边界条件,采用边界条件齐次化、本征函数法,推导得到瞬时均布荷载下非饱和土竖井地基三维固结解析解。通过与双面完全渗透边界条件下已有解析解进行对比,验证了所得解析解的正确性。对所得解进行算例分析发现:通过设置合理上、下界面参数,所得解可用于模拟实际上、下边界透水透气任意分布的情况,弥补了目前无法考虑上、下边界渗透性介于渗透与不渗透之间或不对称分布的不足;在井径比及井深适当的前提下,当径、竖向渗透系数比大于2时,竖向渗流对于超孔隙压力消散的影响较小;当考虑竖向渗流时,上述影响随着上、下边界渗透性能的增强而愈加明显。     

Analytical 3D transient elastodynamic fundamental solution of unsaturated soils

Unsaturated soils are considered as porous continua, composed of porous skeleton with its pores filled by water and air. The governing partial differential equations (PDE) are derived based on the mechanics for isothermal and infinitesimal evolution of unsaturated porous media in terms of skeleton displacement vector, liquid, and gas scalar pressures. Meanwhile, isotropic linear elastic behavior and liquid retention curve are presented in terms of net stress and capillary pressure as constitutive relations. Later, an explicit 3D Laplace transform domain fundamental solution is obtained for governing PDE and then closed‐form analytical transient 3D fundamental solution is presented by means of analytical inverse Laplace transform technique. Finally, a numerical example is presented to validate the assumptions used to derive the analytical solution by comparing them with the numerically inverted ones. The transient fundamental solutions represent important features of the elastic wave propagation theory in the unsaturated soils. Copyright © 2010 John Wiley & Sons, Ltd.