Consolidation by vertical drains under time‐dependent loading

A solution for the consolidation by vertical drains under time‐dependent loading is presented in this paper. Considering the well resistance and the smear action, the simultaneous basic partial differential equations of the consolidation by vertical drains are obtained for the arbitrary loading method. However, the impulse function method cannot be directly applied to obtain the solution. The partial differential equations and the solution conditions that satisfy the impulse function method are obtained after some mathematical processing. The solution for the consolidation by vertical drains under time‐dependent loading is obtained by virtue of the impulse function method and the solution under instantaneous loading. The solutions under single ramp loading and multi‐ramp loading are obtained and the feasibility of Carrillo's method under time‐dependent loading is discussed. Further, the characteristics of the consolidation by vertical drains under instantaneous loading and time‐dependent loading are discussed. Copyright © 2000 John Wiley & Sons, Ltd.     

考虑幂函数渗透压缩变化的双层砂井地基固结性状

采用Barron轴对称固结及大变形固结问题的某些简化与假定,推导建立了砂井地基大变形固结控制方程,利用建立的双层砂井地基大变形固结方程及编制的计算程序,通过引入软土渗透系数、有效应力与孔隙比之间的幂函数关系k =ced与e=a( )b,对瞬时加载下双层砂井地基固结性状进行算例计算。结果表明：（1）双层软土幂函数渗透关系及压缩关系中诸参数对双层砂井地基固结性状有重要影响：随着两层软土幂函数渗透关系中参数c1、c2的增加（渗透性增加）、或幂函数压缩关系中参数a1、a2的增加,各土层水平径向与竖向孔隙比减小更快,沉降发展速率与超静孔压消散速率也相应增加,且沉降发展速率快于孔压消散速率。（2）两层土在分界面处的孔隙比及平均超静孔压均出现明显的突变,将沿深度分布曲线分成形状不同的两段,表现出不同的固结性状。     

Consolidation of viscoelastic soil by vertical drains incorporating fractional-derivative model and time-dependent loading

This paper presents a general solution to the consolidation system of viscoelastic soil by vertical drains incorporating a fractional-derivative model and arbitrary time-dependent loading. The fractional-derivative Merchant model is introduced to describe the viscoelastic behavior of saturated soil around the vertical drains. Based on this model, the governing partial differential equation of a consolidation system incorporating vertical and horizontal drainage is obtained for the equal strain condition. Then, a general solution to the consolidation system involving arbitrary time-dependent loading is derived using the Laplace transform technique and eigenfunction expansion method. Further, two comparisons are presented to verify the exactness of the proposed solution, and the consolidation behavior involving four time-dependent loadings is illustrated and discussed.     

Consolidation around stone columns. Influence of column deformation

A solution is presented for the radial consolidation around stone columns under constant surcharge load. The solution considers the influence of vertical and radial deformation of the column, either in elastic and elastoplastic regimes. The solution is in terms of the average excess pore pressure in the soil. It is based on previous solutions, initially developed for rigid column, or including only vertical deformation. For elastic column, the solution gives the variation of strains and stresses between the undrained and final states, for which it coincides with the existing elastic solutions. All the results are given in closed form, and both the elastic and plastic deformations of the column lead to an equivalent coefficient of consolidation for the radial flow, which enables the application of the existing methods of integration of the consolidation equation. A parametric study is presented, showing the influence of the main problem features. A design example is used to illustrate the application to practical cases. Copyright © 2008 John Wiley & Sons, Ltd.     

Consolidation of vertical drain with depth-varying stress induced by multi-stage loading

An explicit analytical solution is developed for the consolidation of vertical drain with both radial and vertical drainage by adopting a depth-varying stress induced by multi-stage loading. The well resistance and smear effect are also considered. The smear effect is described by three decay patterns of horizontal permeability towards drains within the smeared zone, including a reduced constant pattern, a linear decay pattern and a parabolic decay pattern. A parameter analysis is performed to investigate the consolidation behavior of the vertical drain. The convergence of the proposed series solution is discussed.     

A new analytical model for consolidation with multiple vertical drains

Various analytical theories of consolidation for soils with vertical drains have been proposed in the past. Most conventional theories are based on a cylindrical unit cell that contains only a single vertical drain. This paper described a new analytical model where a vertical drain located at the centre (the ‘inner vertical drain’) and is surrounded by two or three vertical drains (the ‘outer vertical drains’), the number of which depends on whether the configuration is triangular or rectangular. Both types of drains are combined into a cylindrical unit cell, and the water is assumed to flow both inwards to the inner vertical drain and outwards to the outer vertical drains distributed around the circumference. The outer radial boundary of the unit cell is regarded as a permeable boundary, with a drainage capacity of two or three separate vertical drains for triangular and rectangular configurations, respectively. The smear effects and the drainage resistances for both the inner and outer vertical drains are considered in the analysis as well. In this way, the equations governing the consolidation process with multiple vertical drains are derived, and the corresponding analytical solutions are obtained for instantaneously loading, ramp loading and multi‐stage of instantaneously loading and multi‐stage of ramp loading. The present solutions are finally compared with several conventional solutions for a single vertical drain in the literature. The results show that the present model predicts the same average degree of consolidation as conventional models do, which verifies the correctness of this new model. Finally, the settlement predicted by the present solution is compared with the measured settlement from a field test at the Port of Brisbane, Australia, which shows a good agreement between them. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

Trend of settlement in primary and secondary consolidations

The trend/rate of settlement of any load on clay can be measured at any time without knowing the past history of load increment. It can be a source of some useful information like quick evaluation of consolidation characteristics, time-compression data of the present, past and future, type and stage of consolidation, drainage conditions, time of load increment, etc. Basic mathematical/graphical properties of vertical, radial, 3D and secondary consolidations are studied from this angle. Two types of 3D consolidations are discussed. First is the combined vertical and radial consolidation with vertical drains. Second is the general 3D consolidation in Cartesian co-ordinates where drainage occurs through all the six phases of a parallelopipe-shaped sample. But the settlement in both cases is allowed in the vertical direction only. It is shown that primary compression after 60% consolidation, with whatever simple or complicated drainage conditions, essentially gives a straight line on a linear plot of settlement versus rate of settlement while secondary consolidation is a straight line on semi-log plot of settlement versus rate of settlement.     

Determination of Coefficient of Radial Consolidation Using Steepest Tangent Fitting Method

Barron (Trans ASCE 113:718–724, 1948) theoretical relationship between degree of consolidation (U _r ) and time factor (T _r ) is generally used to determine the coefficient of radial consolidation c _r . Several researchers have proposed different laboratory techniques for obtaining the value of c _r . However, the usual approach is to compare some salient features of the theoretical U _r –T _r relationship with the compression, δ and time, t based on the laboratory data. In this paper, rapid consolidation test procedure originally proposed by Su (J Soil Mech Found Div Am Soc Civ Eng No. 95: Proc. Pap. 1729 (1–9), 1958) to determine the coefficient of vertical consolidation (c _v ) is extended to evaluate Barron’s coefficient of radial consolidation c _r for the case of equal strain. The characteristic feature of the proposed method is the identification of the steepest tangent from the δ–log t plot. The slope of steepest tangent is independent of the drain spacing ratio (n) with a value of 0.847. This approach is rapid, and it predicts more reliable values of c _r as it is less influenced by the secondary compression. In addition, this method is more versatile and can be applied for consolidation trends which do not always exhibit the typical S-shape.     

任意附加应力下竖井径竖向耦合固结等应变解

竖向排水体包括砂井和塑料排水板已广泛应用于加速软黏土地基的固结。不同形式的轴对称条件下含竖向排水体均质各向同性地基的固结解析解答也相继被提出,而各种解答均假定地基中附加应力在竖向排水体的深度范围内是均匀分布的。显然,当竖向排水体深度与地面堆载区宽度的比值相对较高,或竖向排水体的影响区域靠近地面堆载区的边缘时,这个假定并不适用。为此,在现有径竖向耦合固结等应变解的基础上,提出任意分布的地基附加应力条件下的解答,并分析附加应力分布形式和线性加载速率对地基固结的影响。结果表明,均匀分布的附加应力条件下的解答低估了地基的固结速率,而瞬时加载条件下的解答则高估了地基的固结速率。     

Investigation on the Characteristics of Pore Water Flow During CRS Consolidation Test

Several experimental methods have been proposed for consolidation test such as: constant loading rate, constant gradient and constant rate of strain (CRS). Unfortunately, there are no unique criteria for performing the CRS consolidation test. Also, there are considerable differences among the standards proposed for the test. In the present paper, authors have tried to find the reasons for differences among standards, by studying the basic assumptions made on the characteristics of the pore water flow during consolidation process. In the course of study, CRS consolidation tests were carried out under different strain rates on remolded samples. The results of the tests have indicated that the Darcy’s law is not valid throughout the CRS test and therefore, any consolidation equation based on Darcy’s law would not provide accurate results. The results of the current experiments also showed that with regard to the applied strain rates, there are three different flow regimes governing the process named as: pre-linear (non-Darcy flow), linear (Darcy flow) and post-linear (non-Darcy flow). Experimental results also showed that distinction between boundaries of linear and nonlinear flow is possible from the excess pore pressure developed during the tests.     

考虑非达西流的弱透水层固结计算

由于成岩作用的影响,深部弱透水层中的黏性土较为坚硬、致密,在这种黏土层中的孔隙渗流规律有偏离达西定律的现象,而在现有的地面沉降研究中,对深部弱透水层的固结变形计算始终沿用以达西定律为基础的固结方程进行,这显然是与实际情况不相符的。鉴于此,以孔隙渗流为非达西流为基础,对太沙基一维固结方程进行了重新推导,并采用半解析法对非达西流-太沙基一维固结方程进行了求解。分别采用达西流-太沙基固结方程和非达西流-太沙基固结方程对同一固结问题进行了计算,结果表明,考虑非达西流后土层达到某一固结度所需的固结时间较之不考虑非达西流时明显增加了,而且固结时间增长的趋势与实测的增长趋势较为一致。     

Nonlinear radial consolidation of vertical drains under a general time‐variable loading

By incorporating the nonlinear variation of a soil's compressibility and permeability during the process of consolidation, an analytical solution for the radial consolidation of vertical drains has been developed for a general time‐variable loading. The general solution was verified for the cases of instantaneous loading and ramp loading. Detailed solutions were further derived for two special loading schemes: multistage loading and preloading–unloading–reloading. The nonlinear consolidation behavior of a vertical drain subjected to these two types of loading schemes was then investigated by a parametric study. The results show that the loading rate, the ratio of the compressibility index to the permeability index (C_c/C_k), and the initial stress state have a significant influence on the consolidation rate. A smaller value of C_c/C_k, a larger initial stress, or a fast loading rate always leads to a rapid consolidation rate. During the unloading period, a negative excess pore water pressure may occur, and a slower unloading rate may reduce this negative value. Copyright © 2014 John Wiley & Sons, Ltd.     

Biot固结理论中连续性方程形式的讨论

对于Biot固结方程中的连续性方程形式,不同的研究者存在分歧。在详细分析了Biot推导固结方程的过程后,发现其在推导固结方程时用到的弹性应变能密度函数存在错误。为了得到正确的连续性方程,采用两种思路进行了分析：①通过对土体单元变形的分析,得到了考虑土体颗粒和流体可压缩性的连续性方程;②按照Biot推导固结方程的思路,通过修正推导过程中用到的弹性应变能密度函数得到了连续性方程。上述两种思路得到的连续性方程是等价的,同时将上述连续性方程和其他研究者通过质量守恒得到的连续方程进行了对比,结果是一致的,从而确定了连续性方程的正确形式,并澄清了目前在Biot固结方程中的连续性方程上存在的分歧,所得结论可为固结方程的解析或数值计算提供依据。     

Decoupling Conditions for Elasto-plastic Consolidation Question Based on Numerical Modeling Method

INTRODUCTION Theconsolidationofsaturatedsoilisanimpor tantsubjectingeomechanics.Biot(1941)proposed thethree dimensionalconsolidationtheory,whichis basedontheprincipleofeffectivestress,continuity conditionsandequilibriumequations.Theremarka blesuccessofthistheoryistheanalysisofthetime dependenteffectandcouplingeffectofsoilandpore water.Thus,thetheoryisaclassictheoryinsatu ratedsoilstatistics.Itwasalmostunfeasibletoperformanalysesof thistheorywithoutthedevelopmentofhigh speed computersandnu…     

Consolidation of a poroelastic half-space with anisotropic permeability and compressible constituents by axisymmetric surface loading

The fully coupled Biot quasi-static theory of linear poroelasticity is used to study the consolidation of a poroelastic half-space caused by axisymmetric surface loads. The fluid and solid constituents of the poroelastic medium are compressible and its permeability in the vertical direction is different from its permeability in the horizontal direction. An analytical solution of the governing equations is obtained by taking the displacements and the pore pressure as the basic state variables and using a combination of the Laplace and Hankel transforms. The problem of an axisymmetric normal load is discussed in detail. An explicit analytical solution is obtained for normal disc loading. Detailed numerical computations reveal that the anisotropy in permeability as well as the compressibilities of the fluid and solid constituents of the poroelastic medium have significant effects on the consolidation of the half-space. The anisotropy in permeability may accelerate the consolidation process and may lead to a dilution in the theoretical prediction of the Mandel-Cryer effect. The compressibility of the solid constituents may also accelerate the consolidation process.     

A Critical Reappraisal of the Average Degree of Consolidation

Since its development in 1925, Terzaghi’s expression for the average degree of consolidation has been extensively used in teaching, research and consulting to determine the consolidation settlement that has taken place at a specific time in a geotechnical problem. Recent research by the authors has revealed that the term degree of consolidation is not very meaningful when the initial pore pressure distribution is highly skewed. In view of these findings, a review of all aspects of Terzaghi’s consolidation theory is required, particularly when considering asymmetric initial pore pressure distributions. An exact method for estimating the consolidation settlement of a clay stratum using the mass flux per unit area out of the drainage boundaries is proposed. Graphical representations of both exact and traditional methods reveal identical average degree of consolidation curves. In light of this, Terzaghi’s expression for the average degree of consolidation was re-examined. Upon application of the original governing one-dimensional consolidation equation, it was found that the traditional and exact methods for calculating the average degree of consolidation are identical. Thus, the widespread proclivity for geotechnical engineers to use Terzaghi’s average degree of consolidation equation to estimate the consolidation settlement of a clay stratum has been validated.     

循环荷载作用下竖向排水板加固软黏土的孔隙水压力累积特性研究

采用大型动三轴试验仪进行重塑高岭土试样的循环三轴试验,试样直径为300 mm,高度为600 mm。圆柱体试样中心放置了一块竖向排水板,在循环加载试验进行时和结束后都可进行径向排水。试验结果验证了径向排水可以有效地消散循环荷载引起的孔隙水压力。通过结合径向固结理论与不排水循环加载土体模型,提出了一个循环荷载作用下径向固结模型,用来描述在允许径向排水的情况下软黏土在循环荷载作用下的孔压累积特性。模型中考虑了应力历史和孔隙水压力消散对孔隙水压力生成的影响,并用大型循环三轴试验结果进行验证。研究发现,当施加较大循环荷载时,径向排水减缓了孔隙水压力累积到临界值的速率,因而土体在破坏前可以经历更多次的循环荷载;当施加中等循环荷载时,径向排水可有效阻止孔隙水压力增长至临界值。除此之外,还研究了加载间歇期对孔压累积特性的影响,结果显示3组循环加载结束后,累积孔隙水压力开始减小,表明之后更多的循环加载并不会引起孔隙水压力的累积增长。     

Criteria of Acceptance for Constant Rate of Strain Consolidation Test for Tropical Cohesive Soil

In this study, the rapid consolidation equipment (RACE) was developed as an alternative device to the conventional consolidation test using Oedometer, consuming merely a few hours for the whole precedure to determine the consolidation characteristics of cohesive soil. RACE operates based on the constant rate of strain (CRS) consolidation theory, which is a continuous loading method of testing, requiring a good estimation of the loading rate such that it is ideal for the achievement of steady state condition during testing. The steady state condition is achieved when the c _v values from drained and undrained face of CRS converged with the c_v from Oedometer test. A slightly modification has been made on the normal constant rate of strain (CRS) test by proposing a direct back pressure system to the specimen using a tube to saturate the soil sample. This research has produced a set of criteria for determining the suitable rate for the rapid consolidation test based on the ratio of normalized strain rate, β, and proposed a new coefficient in terms of a ratio of β to clay fraction (CF), as a part of new criteria for testing a fine soil. Four types of sample were tested with different rates of strain using the RACE and their results were compared with those conducted using the Oedometer on the same soil type, from which fairly good agreements were evident in many specimens. It was found from the study that the minimum value of normalized strain rate, β, for the CRS test is 0.005 and for the u _a /σ _v ratio is suggested as 0.01. Also, the maximum β/CF for soils with clay friction lower and higher than 50 % are 0.008 and 0.001, respectively. The minimum β/CF value for both conditions is 0.0001.     

A closed form analytical solution for two‐dimensional plane strain consolidation of unsaturated soil stratum

This paper discusses the excess pore‐air and pore‐water pressure dissipations and the average degree of consolidation in the 2D plane strain consolidation of an unsaturated soil stratum using eigenfunction expansion and Laplace transformation techniques. In this study, the application of a constant external loading on a soil surface is assumed to immediately generate uniformly or linearly distributed initial excess pore pressures. The general solutions consisting of eigenfunctions and eigenvalues are first proposed. The Laplace transform is then applied to convert the time variable t in partial differential equations into the Laplace complex argument s. Once the domain is obtained, a simplified set of equations with variable s can be achieved. The final analytical solutions can be computed by taking a Laplace inverse. The proposed equations predict the two‐dimensional consolidation behaviour of an unsaturated soil stratum capturing the uniformly and linearly distributed initial excess pore pressures. This study investigates the effects of isotropic and anisotropic permeability conditions on variations of excess pore pressures and the average degree of consolidation. Additionally, isochrones of excess pore pressures along vertical and horizontal directions are presented. It is found that the initial distribution of pore pressures, varying with depth, results in considerable effects on the pore‐water pressure dissipation rate whilst it has insignificant effects on the excess pore‐air pressure dissipation rate. Copyright © 2015 John Wiley & Sons, Ltd.