A simple analytical solution to one‐dimensional consolidation for unsaturated soils

This paper presents a simple analytical solution to Fredlund and Hasan's one‐dimensional (1‐D) consolidation theory for unsaturated soils. The coefficients of permeability and volume change for unsaturated soils are assumed to remain constant throughout the consolidation process. The mathematical expression of the present solution is much simpler compared with the previous available solutions in the literature. Two new variables are introduced to transform the two coupled governing equations of pore‐water and pore‐air pressures into an equivalent set of partial differential equations, which are easily solved with standard mathematical formulas. It is shown that the present analytical solution can be degenerated into that of Terzaghi consolidation for fully saturated condition. The analytical solutions to 1‐D consolidation of an unsaturated soil subjected to instantaneous loading, ramp loading, and exponential loading, for different drainage conditions and initial pore pressure conditions, are summarized in tables for ease of use by practical engineers. In the case studies, the analytical results show good agreement with the available analytical solution in the literature. The consolidation behaviors of unsaturated soils are investigated. The average degree of consolidation at different loading patterns and drainage conditions is presented. The pore‐water pressure isochrones for two different drainage conditions and three initial pore pressure distributions are presented and discussed. Copyright © 2013 John Wiley & Sons, Ltd.     

软土的一维次固结双曲线流变模型研究

软土的次固结对于软土的变形量非常重要。在软土次固结计算中,通常以次固结系数作为计算参数,但该系数不能反映荷载变化的影响,而且只适用于正常固结土。由于工程中所遇到的大都为超固结土,对超固结土的次固结计算还缺少合理的方法。通过室内试验开展了软土次固结沉降相关研究,进行了多组原状软土试样的分级加载次固结试验,试验结果表明,对于软土次固结计算,建议从次固结应变与时间关系的角度,采用双曲线形式进行拟合,并分别对正常固结和超固结状态的次固结应变参数进行分析,建立了一维次固结的经验模型公式,提出了一种考虑压力对次压缩影响的次固结沉降计算方法,并将该方法应用于现场工程项目,验证了该方法可以适用于正常固结土和超固结土,使得软土次固结沉降量的计算能够更好地反映实际工程问题。     

Compression and consolidation anisotropy of some soft soils

The compression and consolidation anisotropy of 11 soft soils were studied by conducting oedometer tests on sets of duplicate undisturbed specimens prepared in the vertical and horizontal directions from adjacent sections of carefully sampled borehole cores. The one-dimensional compression, yield and creep characteristics of the various silts, clays and amorphous peaty material tested were similar for the vertical and horizontal directions. The exception was the structured, coarse fibrous peaty material which was strongly cross-anisotropic. Drainage occurred more rapidly in the horizontal direction with horizontal-to-vertical permeability ratios r _k of 1.0–1.7. Higher r _k values were associated with more marked fabrics, namely for clays with fine sand partings, fibrous organic inclusions or fine root-holes and the laminated silts. The r _k value was for practical purposes independent of the stress level. Brendan C. O’Kelly: Formerly Scott Wilson Consulting Engineers, UK     

软土的弹塑性损伤模型在BiOt固结有限元中应用

在Biot固结理论基础上，引入弹塑性损伤本构关系，建立了固结分析有限元方程。对某试验路堤进行了损伤模型计算与弹塑性模型计算的结果比较。结果表明：损伤模型的计算结果大于弹塑性模型的计算结果；损伤随着加荷时间的增长而增大；路基中心线和路基坡脚下的土损伤程度较大。     

An analytical solution of one-dimensional consolidation for soft sensitive soil ground

An analytical solution of one-dimensional consolidation for soft sensitive soil ground is presented. The moving boundary is introduced to indicate the notable change of consolidation behaviour of sensitive soil with the increase of stress level. It is assumed that the soil structure of the upper subsoil gradually destroys downwards with the dissipation of pore pressure, and the coefficient of consolidation as well as the coefficient of permeability of the upper subsoil become small, which hinders the dissipation of pore pressure of the lower subsoil. The consolidation degree curve obtained from the present solution is found to lie between the two curves obtained from Terzaghi one-dimensional consolidation solution with the parameters of the undisturbed soil and the remolded soil. The calculated results provide a new explanation for a general phenomenon in the consolidation of soft sensitive soil ground, as that for high loadings the consolidation is longer than for small ones. It should be pointed out from this study that both the deep mixing method and the long vertical drains methods are effective techniques for improving deep sensitive soil ground. Copyright © 2004 John Wiley & Sons, Ltd.     

软黏土固结系数确定方法探讨

根据揭普高速原状软土的室内固结试验结果和现场实测结果,对7种固结系数确定方法的原理、精度、使用简便性、优缺点等进行了比较研究。研究结果表明,三点法和时间平方根法是较好的方法,但采用时间平方根法时其固结试验应采用较大的加载比。研究还发现,同一场地单元、同类软土、不同土样的固结系数离散性较大。实际工程中,同一单元地段应进行多组固结试验计算固结系数,剔除较高值者,取余下均值为该地段的固结系数。     

Finite element consolidation analysis of soils with vertical drain

This paper presents a finite element procedure for the analysis of consolidation of layered soils with vertical drain using general one‐dimensional (1‐D) constitutive models. In formulating the finite element procedure, a Newton–Cotes‐type integration formula is used to avoid the unsymmetry of the stiffness matrix for a Newton (Modified Newton) iteration scheme. The proposed procedure is then applied for the consolidation analysis of a number of typical problems using both linear and non‐linear soil models. Results from this simplified method are compared with those from a fully coupled consolidation analysis using a well‐known finite element package. The average degree of consolidation, excess porewater pressure and average vertical effective stress are almost the same as those from the fully coupled analysis for both the linear and non‐linear cases studied. The differences in vertical effective stresses are tolerable except for the values near the vertical drain boundaries. The consolidation behaviour of soils below a certain depth of the bottom of vertical drain is actually one‐dimensional for the partially penetrating case. Therefore, there are not much differences in whether one uses a one‐dimensional model or a three‐dimensional model in this region. The average degree of consolidation has good normalized feature with respect to the ratio of well radius to external drainage boundary for the cases of fully penetrating vertical drain using a normalized time even in the non‐linear case. Numerical results clearly demonstrate that the proposed simplified finite element procedure is efficient for the consolidation analysis of soils with vertical drain and it has better numerical stability characteristics. This simplified method can easily account for layered systems, time‐dependent loading, well‐resistance, smear effects and inelastic stress–strain behaviour. This method is also very suitable for the design of vertical drain, since it greatly reduces the unknown variables in the calculation and the 1‐D soil model parameters can be more easily determined. Copyright © 2000 John Wiley & Sons, Ltd.     

软土一维非线性固结近似解析解

软土非线性固结变形计算目前还主要依赖于数值方法,致使非线性固结理论的工程应用受到极大限制。引入经典的e-lg?' 和e-lgkv非线性关系,在自重应力均匀分布假定下通过变量代换并利用迭代法给出压缩指数Cc与渗透指数Ck比值不等于1时的非线性固结近似解析解。在Cc /Ck趋近1时本文解与其等于1时的差分解及精确解相差无几。但如果Cc /Ck值偏离1,该近似解会存在一定偏差,且偏差值会随Cc /Ck值偏离1的程度和外荷载增加而逐渐增大。在一般工程荷载作用下,如果Cc /Ck值介于0.9～1.1之间,本文解的平均固结度与差分解间最大偏差在2%左右。当Cc /Ck值在0.75～1.25之间时,本文解的平均固结度与差分解最大偏差在5%左右。如果Cc /Ck值在0.5～1.5之间,本文解的平均固结度与差分解间最大偏差在10%左右。当外荷载一定时,土层的非线性固结速率会随着Cc /Ck值的增大而减慢。如果Cc /Ck<1,土层的非线性固结速率会随外荷载的增大而加快;相反,如果Cc/Ck>1,土层的非线性固结速率会随外荷载增大而减慢。     

软土地基电渗固结理论分析与数值模拟

电渗固结是加速软土固结、提高地基承载力的有效技术。传统的电渗固结理论假设土体的物理力学、水力学和电学特性均匀稳定,其理论解答与试验结果差别较大。针对电渗固结处理过程,对土体位移场、渗流场和电场的耦合特征进行了理论分析,根据电荷守恒原理、水流连续原理和Biot固结理论,建立了电渗固结过程的多场耦合控制方程;考虑土体相关特性参数的非线性关系,开发了有限元软件用于分析电渗过程中电场强度、土体位移以及超静孔隙水压力的变化特征。计算结果与理论数据吻合较好,能够反映土体相关特性参数非线性关系对结果的影响。软件能够为电渗固结系统设计提供参考依据。     

变荷载下考虑井阻随时空变化的竖井地基固结解

竖井排水固结法中井阻随时空演变（即由淤堵和弯折所引起的竖井排水能力下降）的现象已引起广泛关注,且变井阻对竖井地基固结速率的影响不容忽略。但目前能同时考虑变荷载及井阻随时间和空间变化的固结解析解还鲜有报道。考虑井阻随时空演变过程,引入实际中广泛采用的单级或多级加载模式,建立了竖井地基固结模型,并应用分离变量法获得固结模型的解析解答。通过与已有的解析解、有限差分解及工程实测值进行对比分析,充分验证了该模型的正确性。通过大量的计算,分析变井阻参数对竖井地基固结性状的影响。结果表明：竖井地基固结速率随竖井最终排水能力的增大而加快,随深度井阻参数及时间井阻参数的增大而减缓,且时间井阻参数的影响更为显著。     

多级线性荷载下饱和软黏土一维大应变固结解析解

在软土地基工程中,荷载往往都是分级施加,随着固结地基强度增加到一定值后,再施加下一级荷载以确保地基稳定。基于谢康和饱和软黏土一维大应变固结解析解,采用数学归纳法推导了任意形式多级线性荷载作用下的解,并编制了相应的通用计算程序,可以方便进行计算分析。通过与线性小应变假定下解的计算对比表明,在实际工程中基于非线性大应变进行计算分析更为合理。     

Semi‐analytical solution to one‐dimensional consolidation for unsaturated soils with semi‐permeable drainage boundary under time‐dependent loading

This paper presents semi‐analytical solutions to Fredlund and Hasan's one‐dimensional consolidation of unsaturated soils with semi‐permeable drainage boundary under time‐dependent loadings. Two variables are introduced to transform two coupled governing equations of pore‐water and pore‐air pressures into an equivalent set of partial differential equations, which are easily solved by the Laplace transform. The pore‐water pressure, pore‐air pressure and settlement are obtained in the Laplace domain. Crump's method is adopted to perform the inverse Laplace transform in order to obtain semi‐analytical solutions in time domain. It is shown that the present solutions are more general and have a good agreement with the existing solutions from literatures. Furthermore, the current solutions can also be degenerated into conventional solutions to one‐dimensional consolidation of unsaturated soils with homogeneous boundaries. Finally, several numerical examples are provided to illustrate consolidation behavior of unsaturated soils under four types of time‐dependent loadings, including instantaneous loading, ramp loading, exponential loading and sinusoidal loading. Parametric studies are illustrated by variations of pore‐air pressure, pore‐water pressure and settlement at different values of the ratio of air–water permeability coefficient, depth and loading parameters. Copyright © 2017 John Wiley & Sons, Ltd.     

FEM validation of a double porosity elastic model for consolidation of structurally complex clayey soils

Laboratory consolidation of structured clayey soils is analysed in this paper. The research is carried out by two different methods. The first one treats the soil as an isotropic homogeneous equivalent Double Porosity (DP) medium. The second method rests on the extensive application of the Finite Element Method (FEM) to combinations of different soils, composing 2D or fully 3D ordered structured media that schematically discretize the complex material. Two reference problems, representing typical situations of 1D laboratory consolidation of structured soils, are considered. For each problem, solution is obtained through integration of the equations governing the consolidation of the DP medium as well as via FEM applied to the ordered schemes composed of different materials. The presence of conventional experimental devices to ensure the drainage of the sample is taken into account through appropriate boundary conditions. Comparison of FEM results with theoretical results clearly points out the ability of the DP model to represent consolidation processes of structurally complex soils. Limits of applicability of the DP model may arise when the rate of fluid exchange between the two porous systems is represented through oversimplified relations. Results of computations, obtained having assigned reasonable values to the meso‐structural and to the experimental apparatus parameters, point out that a partially efficient drainage apparatus strongly influences the distribution along the sample and the time evolution of the interstitial water pressure acting in both systems of pores. Data of consolidation tests in a Rowe's cell on samples of artificially fissured clays reported in the literature are compared with the analytical and numerical results showing a significant agreement. Copyright © 2000 John Wiley & Sons, Ltd.     

软土非线性固结计算若干表格及应用

为方便非线性固结理论在实际工程中的应用,基于软黏土非线性一维固结理论,给出了能综合考虑软土压缩性和渗透性的非线性特性、自重应力沿深度实际分布、逐渐加荷等影响的软土非线性固结计算若干表格,并通过算例说明了其应用。为工程计算和相关理论研究提供了工具和参考。     

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.     

软土中单桩桩侧承载力时效性理论解

以有效应力原理为基础,结合考虑应变软化、桩径和弹性区孔压分布等因素的桩周土固结解,推导出软土中单桩桩侧承载力时效性的理论解。与传统桩的承载力时效性简化分析方法不同,该理论解不仅考虑了多土层地基中各土层刚度指数的差异对固结过程的影响,还考虑了各土层因固结系数不同对孔压消散速率的影响。通过现场不同间歇期的单桩载荷试验测试结果验证了其合理性     

袋装砂井爆夯处理软土地基的数值模拟方法及现场试验验证

袋装砂井爆夯法处理软土地基是利用炸药在设置有排水通道的软土中爆炸产生冲击和振动而使土体加固的方法。针对该法进行理论研究,提出了一种将袋装砂井爆夯处理软土地基的三维问题转化为二维平面应变问题的数值模拟方法：袋装砂井转化为等价砂墙;利用等效冲量原理,炮孔爆炸压力则转化为等效压力墙。数值模拟中考虑了土体骨架变形与孔隙水非达西渗流的耦合。对数值模拟的现场试验验证分析表明,沉降数值分析的结果与铁路宁启线软基处理现场测试结果具有很好的可比性。所提出的数值分析方法可模拟袋装砂井爆夯处理软土地基的超静孔隙水压产生和消散以及土体沉降变形的动态过程。     

Simulation of long‐term consolidation behavior of soft sensitive clay using an elasto‐viscoplastic constitutive model

The phenomenon of excess pore water pressure increase or stagnation and continuing large ground deformation in soft sensitive clay following the completion of construction of embankment is simulated for a case study at Saint Alban, Quebec, Canada. The present model employs an updated Lagrangian finite element framework and is combined with an automatic time increment selection scheme. The simulation based on an elasto‐viscoplastic constitutive model considers soil‐structure degradation effect. It is shown that without consideration for the microstructural degradation effect, it is not possible to reproduce the field responses of soft sensitive clay even during the construction of the embankment. When the soil‐structure degradation effect is considered, the present model can offer reasonably accurate prediction for the consolidation behavior of soft sensitive clay, including the so‐called anomalous pore water pressure generation and continuing large deformation even after the end of construction, which has been posing numerous uncertainties on the long‐term performance of earth structures. Copyright © 2013 John Wiley & Sons, Ltd.     

Comparison between iterative solution of symmetric and non‐symmetric forms of Biot's FEM equations using the generalized Jacobi preconditioner

Finite element discretization of Biot's consolidation equations can produce a symmetric indefinite system (commonly used in geomechanics) or a non‐symmetric system. While this difference appears to be minor, however, it will require the selection of entirely different Krylov subspace solvers with potentially significant impact on solution efficiency. The former is solved using the symmetric quasi‐minimal residual whereas the latter is solved using the popular bi‐conjugate gradient stabilized. This paper presents an extensive comparison of the symmetric and non‐symmetric forms by varying the time step, size of the spatial domain, choice of physical units, and left versus left–right preconditioning. The generalized Jacobi (GJ) preconditioner is able to handle the non‐symmetric version of Biot's finite element method equation, although there are no practical incentives to do so. The convergence behaviour of GJ‐preconditioned systems and its relation to the spectral condition number or the complete spectrum are studied to clarify the concept of ill‐conditioning within the context of iteration solvers. Copyright © 2007 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.