A new simplified Hypothesis B method for calculating the consolidation settlement of ground improved by vertical drains

Vertical drains are widely used in soft ground improvements to accelerate the consolidation process. This paper develops a new simplified Hypothesis B method for calculating the consolidation settlement of a soil layer improved by vertical drains under the instant and ramp loadings. As a comparison, the traditional Hypothesis A method is also used to calculate the settlement. Then, a fully coupled finite element consolidation analysis is utilized to examine and verify this simplified method and Hypothesis A method. For the instant loading, Carrillo‐Barron method and Zhu‐Yin method are used to obtain the average degree of consolidation for vertical drain system. Typical parameters, such as over‐consolidation ratio (OCR), smear zone, and space ratio of vertical drains, are considered. It is found that the calculation results from the new simplified method in this study agree well with finite element simulations, and relative errors are in the range of 0.1% to 12.3%. Comparatively, there are obvious differences between the calculated results from Hypothesis A method and finite element results. Carrillo‐Olson method and Zhu‐Yin method are utilized to obtain the average degree of consolidation for the vertical drain system to consider the ramp loading. Equivalent time is determined from half of the construction period to calculate the creep compression under the ramp loading. The accuracy of this simplified Hypothesis B method using both Carrillo‐Olson method and Zhu‐Yin method is acceptable with the relative errors less than 9.4%.     

成层竖向排水井地基固结分析

实际工程中竖井地基具有成层性,有时竖井也并未完全打穿软土层。在竖井打设区满足竖井等应变固结理论、下卧层满足一维太沙基固结理论假设的基础上,将现有竖井地基固结理论推广到成层未打穿竖井地基情况。利用边界条件和竖直向连续条件,确定该系统的正交关系,并给出了其固结解答,该解具有广泛的适用性。通过对竖井打设区和下卧层层数的变化,即可获得现有关于简单未打穿竖井地基的固结解答。将该解答编制成应用程序,对一算例进行了分析。结果表明,平均固结度按孔压定义和按变形定义是不相同的,硬表层的存在会加快其下土层的固结。     

成层软黏土地基的固结沉降计算分析

针对实际工程中软黏土地基的成层特性及竖井有时并未打穿软土层情况,采用太沙基一维固结理论和巴伦固结理论的基本假定,利用固结度联系孔压与变形,提出一种简化成层软黏土地基的固结沉降计算方法,对竖向排水地基的固结度及沉降等进行预测。对计算的几个关键问题进行探讨：如固结系数的反演选择、竖井未打穿土层时,竖井底部成层地基固结度计算、压缩土层厚度选取等问题。在上述理论的基础上,用FORTRAN语言编写排水竖井固结程序CONSOL,在工程实践中得到应用,取得良好的效果。     

砂墙地基二维固结自由应变解

将含竖向排水体地基的三维固结变形问题等效为平面应变问题进行数值分析时,砂墙地基二维固结解析解答是合理确定其等效固结计算参数的重要依据。为辨析现有砂墙地基等应变固结近似解答的适用性,针对微单元土体严格的二维固结微分方程,考虑对地基固结有重要影响的井阻作用,以及涵盖完全透水和不完全透水的更一般边界面排水条件,推求得到了其自由应变解答。并与现有解答进行对比分析,同时,分析了泊松比效应以及水平和竖向排水对地基固结的影响。结果表明,现有砂墙地基的等应变固结解答虽然近似,但已有足够精确;砂墙地基以水平向固结为主,竖向固结几乎可以忽略不计;地基固结速率随着泊松比的增大而增大,在将竖向排水体等效为砂墙时,应考虑其作用影响。     

Consolidation of sensitive clay with vertical drain

The coefficient of consolidation is one of the most important parameters that control the rate of consolidation. Conventional consolidation theories assume that the coefficient of consolidation is constant during the whole consolidation process. In the case of sensitive clay, the coefficient of consolidation is strongly dependent on the level of effective stress of clay. With the dissipation of pore water pressure and the increase of effective stress, the soil structure of the upper subsoil is gradually destroyed downwards and its coefficient of consolidation becomes smaller. At the same time, the coefficient of permeability of the vertical drains drops down because of the kinking or bending effect. The destructured upper subsoil and the kinking of the vertical drain hinder the dissipation of the pore pressure in the lower subsoil. This paper presents a model to describe the above important phenomena during the consolidation of sensitive clay with vertical drain. The solution for proposed model can be obtained by extending the closed‐form solution of the consolidation of double‐layered ground with vertical drain by the interactive method introducing the concept of the moving boundary and the reduction of discharge capacity of vertical drain. The numerical results obtained from the finite element method package PLAXIS (Ver. 7.2) are adopted to compare those obtained from the present algorithm. Moreover, the rationality of the moving boundary is explained by the distributions of the excess pore water pressure in natural soil zone along the radial direction. Wenzhou airport project is taken as a case study in this paper. The results for the sensitive soil with decaying sand drain agree very well with the in situ measured data. The calculated results can properly explain two general phenomena observed in the consolidation of soft sensitive soil ground with vertical drains: one is that the time to achieve the same consolidation degree is much longer under heavy loading than that under light loading; the other is that the consolidation speed is much slower in the lower subsoil than in the upper subsoil. Finally, it is indicated that the vertical drains can decrease the hindrance effect of the destructured subsoil. Copyright © 2007 John Wiley & Sons, Ltd.     

Vertical drain consolidation analysis in one,two and three dimensions

A new method, the η or ‘eta’ method, for modeling consolidation by vertical and horizontal drains is presented. The approach is applicable in one, two and three dimensional as well as axisymmetric cases. Material and geometry properties are familiar from unit cell vertical drain analysis and are consistent across dimensions. An uncoupled finite element method (FEM) program is used to test the efficacy of the new approach. Because drains are not explicitly modeled in the finite element mesh, mesh complexity and computational time are greatly reduced. Unlike existing plane strain matching methods there is no special transformation of permeability or drain properties. The analyses conducted indicate that the η method provides an efficient and consistent means of modeling drains in any dimension.     

Analytical solutions for advective–dispersive solute transport in double‐layered finite porous media

Analytical solutions for advection and dispersion of a conservative solute in a one‐dimensional double‐layered finite porous media are presented. The solutions are applicable to five scenarios that have various combinations of fixed concentration, fixed flux and zero concentration gradient conditions at the inlet and outlet boundaries that provide a wide number of options. Arbitrary initial solute concentration distributions throughout the media can be considered via explicit formulations or numerical integration. The analytical solutions presented have been verified against numerical solutions from a finite‐element‐based approach and an existing closed‐form solution for double‐layered media with an excellent correlation being found in both cases. A practical application pertaining to advective transport induced by consolidation of underlying sediment layers on contaminant movement within a capped contaminated sediment system is presented. Comparison of the calculated concentrations and fluxes with alternative approaches clearly illustrates the need to consider advection processes. Consideration of the different features of contaminant transport due to varying pore‐water velocity fields in primary consolidation and secondary consolidation stages is achieved via the use of non‐uniform initial concentration distributions within the proposed analytical solutions. Copyright © 2010 John Wiley & Sons, Ltd.     

Analysis of soil consolidation by vertical drains with double porosity model

The soil around a drain well is traditionally divided into smeared zone and undisturbed zone with constant hydraulic conductivity. In reality, hydraulic conductivity of the soil changes continuously and it may not be always appropriate to approximate its distribution with two zones. In this study, the horizontal hydraulic conductivity of the soil is described by an arbitrary function of radial distance. The horizontal flow under equal strain condition is analysed for a soil–drain system with a circular or regular polygonal boundary. It is found that the horizontal flow can be generally characterized with a linear equation in which the flow rate of water through soil–drain interface is proportional to the difference between the average excess pore pressure in the soil and the excess pore pressure in the drain well. The water exchange between the drain and the soil is analogous to that between fractures and matrix in a double porosity system, a popular conceptual model of fracture rocks. On the basis of this characterization, a simplified approach to analyse soil–drain systems is developed with one‐dimensional double porosity model (DPM). Analytical solutions for both fully and partially penetrating drains are derived. The solution for partially penetrating drains is compared with both numerical and approximate analytical results in literature. Copyright © 2004 John Wiley & Sons, Ltd.     

Analytical solutions of linear finite‐ and small‐strain one‐dimensional consolidation

Analytical solutions are presented for linear finite‐strain one‐dimensional consolidation of initially unconsolidated soil layers with surcharge loading for both one‐ and two‐way drainage. These solutions complement earlier solutions for initially unconsolidated soil layers without surcharge and initially normally consolidated soil layers with surcharge. Small‐strain solutions for the consolidation of initially unconsolidated soil layers with surcharge loading are also presented, and the relationship between the earlier solutions for initially unconsolidated soil without surcharge and the corresponding small‐strain solutions, which was not addressed in the earlier work, is clarified. The new solutions for initially unconsolidated soil with surcharge loading can be applied to the analysis of low stress consolidation tests and to the partial validation of numerical solutions of non‐linear finite‐strain consolidation. They also clarify a formerly perplexing aspect of finite‐strain solution charts first noted in numerical solutions. Copyright © 2004 John Wiley & Sons, Ltd.     

Vertical deformation of rigid foundations of arbitrary shape on layered soil media

A numerical procedure is described for the analysis of vertical deformation of smooth, rigid foundations of arbitrary shape on homogeneous and layered soil media. The contact area at the interface of the foundation and soil medium is approximated by square subdivisions. The response of the system is then obtained from the superposition of the influence of the individual subdivisions. The flexibility influence coefficients are based on equivalent smooth, rigid circular areas with the same contact area as the square subdivisions. For foundations on a homogeneous, isotropic elastic half-space, the flexibility coefficients are given analytically by the integrated forms of the Boussinesq's solution. For a layered soil medium, the flexibility coefficients are determined from an axisymmetric finite element analysis which is essentially two dimensional. Thus, there is no necessity for a full three-dimensional finite element analysis. Comparison with solutions obtained using the integral transform technique for smooth, rigid rectangular foundations on a homogeneous, isotropic elastic half-space shows good agreement. Parametric solutions are presented for the response of rectangular foundations on some ‘typical’ soil profiles. The use of a simplified method to estimate the settlement of rectangular foundations on a layered soil medium by superposing solutions for homogeneous, elastic strata is discussed.     

A 3D composite finite element used to model a vertical drain and its surrounding remoulded soil

This work addresses an enrichment technique for the three-dimensional (3D) finite element (FE) analysis of a vertical drain foundation because (1) 1D and 2D simulations are insufficient to integrally describe the consolidation behaviour and (2) drains are small both in spacing and size, resulting in enormous computational costs for a traditional 3D FE analysis. Based on the idea of the semi-analytical finite element method (FEM), which combines general FEM with the high accuracy of a closed-form solution, a new spatial element that contains a drain well and its neighbouring smear zone is presented. This new combined element is depicted by eight global independent nodes and two local dependent nodes, and a classical analytical theory is introduced to set up the relationship between the two kinds of nodes. Because permeability diversity between the drain and the smear zone is considered, both the effects of smearing and well resistance are taken into account with the composite element method (CEM). A detailed derivation of the CEM is performed using the weighted residual method. The accuracy of the proposed method is validated with a totally penetrating, single-drain ground analysis for seven calculation conditions. Additionally, the proposed CEM saves 1/4–1/2 mesh elements and helps to avoid slender elements for the FEM analysis of the drained foundation.     

Probabilistic analyses of soil consolidation by prefabricated vertical drains for single‐drain and multi‐drain systems

Natural soils are one of the most inherently variables in the ground. Although the significance of inherent soil variability in relation to reliable predictions of consolidation rates of soil deposits has long been realized, there have been few studies that addressed the issue of soil variability for the problem of ground improvement by prefabricated vertical drains. Despite showing valuable insights into the impact of soil spatial variability on soil consolidation by prefabricated vertical drains, available stochastic works on this subject are based on a single‐drain (or unit cell) analyses. However, how the idealized unit cell solution can be a supplement to the complex multi‐drain systems for spatially variable soils has never been addressed in the literature. In this study, a rigorous stochastic finite elements modeling approach that allows the true nature of soil spatial variability to be considered in a reliable and quantifiable manner, both for the single‐drain and multi‐drain systems, is presented. The feasibility of performing an analysis based on the unit cell concept as compared with the multi‐drain analysis is assessed in a probabilistic context. It is shown that with proper input statistics representative of a particular domain of interest, both the single‐drain and multi‐drain analyses yield almost identical results. Copyright © 2016 John Wiley & Sons, Ltd.     

Implicit integration of a mixed isotropic–kinematic hardening plasticity model for structured clays

In recent years, a number of constitutive models have been proposed to describe mathematically the mechanical response of natural clays. Some of these models are characterized by complex formulations, often leading to non‐trivial problems in their numerical integration in finite elements codes. The paper describes a fully implicit stress‐point algorithm for the numerical integration of a single‐surface mixed isotropic–kinematic hardening plasticity model for structured clays. The formulation of the model stems from a compromise between its capability of reproducing the larger number of features characterizing the behaviour of structured clays and the possibility of developing a robust integration algorithm for its implementation in a finite elements code. The model is characterized by an ellipsoid‐shaped yield function, inside which a stress‐dependent reversible stiffness is accounted for by a non‐linear hyperelastic formulation. The isotropic part of the hardening law extends the standard Cam‐Clay one to include plastic strain‐driven softening due to bond degradation, while the kinematic hardening part controls the evolution of the position of the yield surface in the stress space. The proposed algorithm allows the consistent linearization of the constitutive equations guaranteeing the quadratic rate of asymptotic convergence in the global‐level Newton–Raphson iterative procedure. The accuracy and the convergence properties of the proposed algorithm are evaluated with reference to the numerical simulations of single element tests and the analysis of a typical geotechnical boundary value problem. Copyright © 2007 John Wiley & Sons, Ltd.     

Modelling crack propagation in concrete structures with a two scale approach

A simplified computational technique based on a refined global–local method is applied to the failure analysis of concrete structures. The technique distinguishes the scale of the structure, modelled with large size finite elements, from the scale at which material non‐linearity occurs due to progressive cracking and macro‐crack propagation. The finite element solution is split into two parts: a linear elastic analysis on a coarse mesh over the entire structure and a non‐linear analysis over a small part of the structure where a dense finite element grid is employed. In the non‐linear calculation, a non‐local damage model is implemented. These two computations are coupled with the help of an iterative scheme. The size and location of the region where a non‐linear analysis is performed, is adapted to follow the development of the damage zone. Numerical examples of mode I fracture of concrete specimens with straight and curved cracks are presented. Copyright © 2003 John Wiley & Sons, Ltd.     

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.     

Finite element analysis of consolidation of layered clay soils using an elastic visco-plastic model

This paper presents a general one-dimensional (1-D) finite element (FE) procedure for a highly non-linear 1-D elastic visco-plastic (1-D EVP) model proposed by Yin and Graham for consolidation analysis of layered clay soils. In formulating the 1-D FE procedure, a trapezoidal formula is used to avoid the unsymmetry of the stiffness matrix for a Newton (modified Newton) iteration scheme. Unlike many other 1-D FE approaches in which the initial in situ stresses (or stress/strain states) are considered indirectly or even not considered, the initial in situ stress/strain states are taken into account directly in this paper. The proposed FE procedure is used for analysis of 1-D consolidation of a clay with published test results in the literature. The FE modelling results are in good agreement with the measured results. The FE model and procedure is then used to analyse the consolidation of a multi-layered clay soils with a parametric study on the effects of the variations of creep parameters in Yin and Graham's 1-D EVP model. It is found that the creep parameters ψ/V and t₀ have significant influence on the compression and porewater pressure dissipation. For some boundary conditions, changes of parameters in one layer will have some effects on the consolidation behaviour of another layer due to the different consolidation rates. Finally, the importance of initial stress/strain states is illustrated and discussed. Copyright © 1999 John Wiley & Sons, Ltd.     

Upper bound limit analysis using linear finite elements and non‐linear programming

A new method for computing rigorous upper bounds on the limit loads for one‐, two‐ and three‐dimensional continua is described. The formulation is based on linear finite elements, permits kinematically admissible velocity discontinuities at all interelement boundaries, and furnishes a kinematically admissible velocity field by solving a non‐linear programming problem. In the latter, the objective function corresponds to the dissipated power (which is minimized) and the unknowns are subject to linear equality constraints as well as linear and non‐linear inequality constraints. Provided the yield surface is convex, the optimization problem generated by the upper bound method is also convex and can be solved efficiently by applying a two‐stage, quasi‐Newton scheme to the corresponding Kuhn–Tucker optimality conditions. A key advantage of this strategy is that its iteration count is largely independent of the mesh size. Since the formulation permits non‐linear constraints on the unknowns, no linearization of the yield surface is necessary and the modelling of three‐dimensional geometries presents no special difficulties. The utility of the proposed upper bound method is illustrated by applying it to a number of two‐ and three‐dimensional boundary value problems. For a variety of two‐dimensional cases, the new scheme is up to two orders of magnitude faster than an equivalent linear programming scheme which uses yield surface linearization. Copyright © 2001 John Wiley & Sons, Ltd.     

Consolidation of double‐layered ground with vertical drains

Vertical drains are usually installed in subsoil consisting of several layers. Due to the complex nature of the problem, over the past decades, the consolidation properties of multi‐layered ground with vertical drains have been analysed mainly by numerical methods. An analytical solution for consolidation of double‐layered ground with vertical drains under quasi‐equal strain condition is presented in this paper. The main steps for the computation procedure are listed. The convergence of the series solution is discussed. The comparisons between the results obtained by the present analytical method and the existing numerical solutions are described by figures. The orthogonal relation for the system of double‐layered ground with vertical drains is proven. Finally, some consolidation properties of double‐layered ground with vertical drains are analysed. Copyright © 2001 John Wiley & Sons, Ltd.     

Analysis of settlement and lateral deformation of soft clay foundation beneath two full-scale embankments

In 1986, the Malaysian Highway Authority constructed a series of trial embankments on the Muar Plain (soft marine clay) with the aim of evaluating the effectiveness of various ground improvement techniques. This study investigates the effect of two such ground improvement schemes: (a) preloading of foundation with surface geogrids and synthetic vertical drains and (b) sand compaction piles. The paper is focused on the finite element analysis of settlements and lateral displacements of the soft foundation. In scheme (a), the numerical predictions are compared with the field measurements. In scheme (b), only the numerical analysis is presented and discussed in the absence of reliable measurements due to the malfunctioning of the electronic extensometer and inclinometer system during embankment construction. The current analysis employs critical state soil mechanics, and the deformations are predicted on the basis of the fully coupled (Biot) consolidation model. The vertical drain pattern is converted to equivalent drain walls to enable plane strain modelling, and the geogrids are simulated by linear interface slip elements. The effect of sand compaction piles is investigated considering both ideal drains and non-ideal drains, as well as varying the pile stiffness. © 1997 by John Wiley & Sons, Ltd.     

A parallel, implicit, cell‐centered method for two‐phase flow with a preconditioned Newton–Krylov solver

A new parallel solution technique is developed for the fully implicit three‐dimensional two‐phase flow model. An expandedcell‐centered finite difference scheme which allows for a full permeability tensor is employed for the spatial discretization, and backwardEuler is used for the time discretization. The discrete systems are solved using a novel inexact Newton method that reuses the Krylov information generated by the GMRES linear iterative solver. Fast nonlinear convergence can be achieved by composing inexact Newton steps with quasi‐Newton steps restricted to the underlying Krylov subspace. Furthermore, robustness and efficiency are achieved with a line‐search backtracking globalization strategy for the nonlinear systems and a preconditioner for each coupled linear system to be solved. This inexact Newton method also makes use of forcing terms suggested by Eisenstat and Walker which prevent oversolving of the Jacobian systems. The preconditioner is a new two‐stage method which involves a decoupling strategy plus the separate solutions of both nonwetting‐phase pressure and saturation equations. Numerical results show that these nonlinear and linear solvers are very effective.