Theory of one dimensional consolidation of two-layered soil with partially drained boundaries

In this paper, a fully explicit analytical solution is presented for one-dimensional consolidation of two-layered soils with partially drained boundaries. It can be shown that the available analytical solutions so far developed for one dimensional consolidation of soil are the special cases of the solution presented herein. According to the solution, some diagrams are prepared and the relevant consolidation behavior of layered soils associated with partially drained boundaries is discussed in detail. Furthermore, the applications of the solution are illustrated in an example problem.     

复杂工况单向固结理论分析及可视化

以Terzaghi单向固结理论为基础,对均质土层、分层土层、土层荷载增加、土层厚度变化情况下的孔隙水压力进行了理论分析。建立了以孔隙水压力为未知量的差分形式的控制方程,并通过泰勒级数展开,推导建立了具有二阶精度的边界方程;同时采用MATLAB软件编制相应的通用程序,得到Terzaghi单向固结理论孔隙水压力的数值解。此外,基于按作用消散的平均孔隙水压力的定义通过已经得到的孔隙水压力的数值解计算得到各不同复杂工况下的平均固结度,并且利用MATLAB软件强大的数值计算能力和图形处理功能,实现可视化,对于实际工程中的平均固结度的求解及通过平均固结度来推断地面沉降量有一定的指导和参考价值。     

An explicit one-dimensional consolidation solution with semi-permeable drainage boundary for unsaturated soil

Existing solutions for analyzing one-dimensional (1-D) consolidation of unsaturated soil are only derived to cater to two extreme drainage conditions (fully drained and undrained). This study presents a new explicit solution for 1-D consolidation of unsaturated soil with semi-permeable drainage boundary. Based on the assumptions of two independent stress variables and the governing equations proposed by Fredlund, the eigenfunction expansion method is adopted to develop an explicit analytical solution to calculate excess pore-water and pore-air pressures in an unsaturated soil when it is subjected to external loads. The developed general solutions are expressed in terms of depth, z, and time, t. For the semi-permeable drainage boundary, eigenvalues and eigenfunctions in the space domain are developed. The technique of Laplace transform is used to solve the coupled ordinary differential equations in the time domain. The newly derived explicit solution is verified with the existing semi-analytical method in the literature, and an excellent agreement is obtained. Compared with the semi-analytical solution, the newly derived analytical solution is more straightforward and explicit so that this solution is relatively easier to be implemented into a computer program to carry out a preliminary assessment of 1-D consolidation of unsaturated soil.     

深厚吹填砂港区堆场地基沉降预估方法

根据大面积深厚吹填砂工程的特点,分析太沙基一维固结理论在分层地基,分级加载等复杂工程条件下应用的可行性,提出了相应的简化计算模型,并编制一维固结计算程序。结合上海某围海成陆港区堆场工程,对其吹填砂层及软弱下卧层的固结沉降进行了计算分析,通过计算结果和离心机模型试验结果的对比分析,对计算模型进行修正,提出了大面积吹填砂工程地基沉降预估方法。     

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。     

A study on one‐dimensional consolidation of layered structured soils

The governing equations for one‐dimensional consolidation of layered structured soils under time‐dependent loading are established. Using simplified k‐σ′ and m_v‐σ′ models, n‐layered structured soils are transformed into (n + 1) or (n + 2)‐layered soils in which the thickness of upper and lower layers are gradually changing. The approximate solutions for the governing equations are then obtained under two types of boundary conditions, and the computer program is developed. Based on the solutions and computer program, the consolidation behavior of layered structured soils with soft interlayer is studied. It is shown that the permeability and compressibility of the soft interlayer have the greatest influences on the rate of settlement and rate of the dissipation of excess pore water pressure. Copyright © 2015 John Wiley & Sons, Ltd.     

半透水边界下分数阶黏弹性饱和土一维固结特性分析

为描述饱和土体的流变特性,引入分数阶导数Kelvin-Voigt黏弹性模型,采用解析方法对半透水边界下的分数阶黏弹性饱和土一维固结特性进行了研究。分别对骤加恒载下饱和土一维固结微分方程和分数阶Kelvin-Voigt黏弹性本构方程进行Laplace变换,并联立求解得到了双边半透水边界条件下分数阶黏弹性饱和土在Laplace变换域内的解析表达式。通过Crump方法实现Laplace数值反演,得到时间域内的半解析解。将所得到的解分别退化为分数阶黏弹性饱和土一维固结半解析解和双边半透水黏弹性饱和土一维固结半解析解,结果与已有文献半解析解相同,验证了提出的双边半透水边界条件下分数阶黏弹性饱和土一维固结解的可靠性。通过算例考察了半透水边界条件和分数阶黏弹性饱和土参数对一维固结特性的影响。研究表明,双边半透水边界下分数阶黏弹性饱和土一维固结发展过程与半透水边界条件、分数阶次和黏滞系数有关,且土体的压缩模量对饱和土一维固结最终沉降量有显著影响。     

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。     

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。     

循环荷载下双曲线模型修正土体一维固结解

假定土体的应力-应变关系满足双曲线模型、渗透系数的降低与压缩系数的减小成正比以及初始有效应力沿深度均匀分布,推导出任意荷载下单层地基的非线性一维固结方程。结合具体的定解条件采用分离变量法求出土体有效应力的通解,并进一步求出超静孔隙水压力、按沉降定义的平均固结度Us以及按孔隙水压力定义的平均固结度Up的通解。根据通解求出梯形低频循环荷载作用下的解,并找出循环荷载作用下土体固结性状的影响因素。基于循环荷载作用下的解,采用FORTRAN语言编制了计算程序,计算并分析考虑土体本构关系非线性时的固结性状,以及反映土体本构特征的参数E和n对固结性状的影响。计算结果表明：①Us稍大于Up;②距排水平面越近,附加有效应力受加载方式的影响越明显,距排水平面越远则相反;③E和n对Us的影响很小,可以忽略不计;④E和n对Up和附加有效应力的影响较大且不容忽略。     

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.     

地下水向水平井三维流动的有限层分析

基于地下水向水平井三维流动数学模型,利用半解析数值分析原理推导了层状各向异性承压含水层有限层方程,给出了水平井源汇项计算公式,编制了相应的计算程序,实现了有限层方程的求解。通过算例的解析解和有限差分解验证了有限层法及程序的正确性,对比分析了有限层法与有限差法的计算效率。结果表明,有限层法适于层状各向异性承压含水层中地下水向水平井三维流动问题的模拟分析,且具有较高的计算效率。     

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.     

Behaviour of piles in consolidating soil

The development of negative skin friction on circular piles in a consolidating layered soil is investigated by an elasto-plastic load transfer theory which accounts for slippage of the soil. The elastic load transfer theory is premised on the compatibility condition that the vertical displacement of the ‘pile’ is equal to the summation of the vertical displacement of the layered soil due to the consolidation of the upper soil layer and the vertical displacement in the ‘soil layers’ along the pile’s centroidal axis caused by a system of pile-soil interactive forces. Slippage of the soil is accounted for by imposing the shear strength of the clay layer as the limit of the pile-soil interface shear stress. The saturated upper clay is consolidating under a uniform surcharge in accordance with Terzaghi’s one-dimensional consolidation theory. The validity of the proposed solution is confirmed by comparison with field measurements. Extensive parametric studies with regard to the effect of pile-soil slip on pile behaviour are presented.     

Numerical solution for consolidation and desiccation of soft soils

The consolidation and desiccation behaviour of soft soils can be described by two time‐dependent non‐linear partial differential equations using the finite strain theory. Analytical solutions do not exist for these governing equations. In this paper, we develop efficient numerical methods and software for finding the numerical solutions. We introduce a semi‐implicit time integration scheme, and show numerically that our method converges. In addition, the numerical solution matches well with the experimental result. A boundary refinement method is also developed to improve the convergence and stability for the case of Neumann type boundary conditions. Interface governing equations are derived to maintain the continuity of consolidation and desiccation processes. This is useful because the soil column can undergo desiccation on top and consolidation on the bottom simultaneously. The numerical algorithms has been implemented into a computer program and the results have been verified with centrifuge test results conducted in our laboratory. Copyright © 2001 John Wiley & Sons, Ltd.     

Consolidation analysis of a cross-anisotropic homogeneous elastic soil using a finite layer numerical method

This paper presents a finite layer procedure for Biot's consolidation analysis of layered soils using a cross-anisotropic elastic constitutive model. The program is first verified using published results. Then, using this program, the influences of cross-anisotropy on the immediate settlement, the final settlement, and the consolidation behaviour are investigated by changing one model parameter at one time. The results obtained using the cross-anisotropic elastic model are compared with results using an isotropic elastic model. It is found that the cross-anisotropy has very large influences on the immediate settlement, the final settlement, and the consolidation behaviour. Curves or tables of the immediate settlement coefficients, the final settlement coefficients, and the average degree of consolidation are obtained and presented in the paper. These curves or tables can be easily used to estimate the immediate settlement, the final settlement, and the consolidation settlement of a cross-anisotropic soil. Copyright © 2004 John Wiley & Sons, Ltd.     

饱和土体一维固结三参数概率特性研究

在考虑岩土参数空间变异性的情况下,对软粘土地基一维固结平均固结度的概率特征进行了研究,分析了平均固结度的概率特征对渗透系数、孔隙比和压缩系数三参数变异性的敏感性,进行了岩土参数分布类型对平均固结度影响的正交试验,并根据试验结果进行了方差分析,为进一步开展多层地基固结的可靠性分析打下了基础.     

Semi-analytical solution to one-dimensional consolidation for viscoelastic unsaturated soils

This paper presents a semi-analytical solution to one-dimensional consolidation of viscoelastic unsaturated soils with a finite thickness under oedometric conditions and subjected to a sudden loading. The solution is obtained by using Lee’s correspondence principle based on the semi-analytical solution to one-dimensional consolidation of elastic unsaturated soils. The boundary contains the top surface permeable to water and air and the bottom impermeable to water and air. A typical example is given to show the evolution of excess pore-air and pore-water pressures as well as the total degree of consolidation of the soil layer with time for different ratios of air–water permeability coefficient, elastic modulus and viscoelastic coefficient. The one-dimensional consolidation behavior of viscoelastic unsaturated soil is discussed according to the semi-analytical solution. These results contribute to a better understanding of the consolidation behavior of viscoelastic unsaturated soils.     

地震动下考虑各向异性土体-盾构隧道数值模拟

针对层状土体-盾构隧道地震反应分析,引入了横观各向同性弹塑性模型,建立了横观各向同性介质的双渐近-多向透射边界条件。针对盾构隧道抗震设计的特点,基于横观各向同性弹塑性模型,研制了考虑层状土体各向异性和施工开挖效应,适合于地下结构动力计算的弹塑性动力有限元程序。在程序中对不同的材料采用了不同的本构关系和单元形式,并采用了关联流动法则和多种屈服准则,可同时进行横观各向同性土体与地下结构的平面应力、平面应变和轴对称问题的静动力有限元分析。最后,利用所研制的程序对上海地铁二号线石门一路站附近区间隧道在不同超越概率地震波输入下的地震反应进行了计算。结果表明,在层状土体-地铁盾构隧道的抗震分析中考虑土体各向异性的影响是必要的,所提出的计算模型是合理易行的。     

非等温分布条件下考虑半透水边界时饱和黏土的一维固结解析解

温度的变化会导致土体的物理力学性质改变,且在一些实际工程中,饱和黏土会处于非等温分布状态。为此,针对非等温分布条件下饱和黏土的一维固结问题,考虑了更具普遍性的半透水边界,通过某些假定推导了单级线性加荷形式下饱和黏土一维固结控制方程,并利用分离变量法求解得到了控制方程的解析解。通过将所提解析解分别与已有解析解和有限差分解展开对比分析,验证了所提解答的正确性。基于所提解析解,利用某一算例分析了温度梯度、半透水边界参数及加荷时间对固结性状的影响。结果表明：温度梯度 M 越大,土体的渗透性越大,土体的固结速率越快;半透水边界参数 R1和 R2越大,相同时间内土体的超孔隙水压力越小,土体的平均固结度越大;土体的平均固结度随加荷时间 tc 的增大而减小,这主要是由于加荷阶段所施加的外荷载小于最终荷载,但加荷时间tc的延长可一定程度减小土体中产生的最大超孔隙水压力。