Semi-analytical solutions to one-dimensional consolidation for unsaturated soils with symmetric semi-permeable drainage boundary

This paper presents semi-analytical solutions to Fredlund and Hasan’s one-dimensional consolidation for unsaturated soils under symmetric semi-permeable drainage boundary conditions. Two variables are introduced to transform two coupled governing equations of pore-air and pore-water pressures into an equivalent set of partial differential equations, which are easily solved by the Laplace transform. Then, the pore-air and pore-water pressures, and soil 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 solution is more applicable to various types of drainage boundary conditions, and in a good agreement with existing solutions from the literature. Furthermore, several numerical examples are provided to investigate the consolidation behavior of an unsaturated single-layer soil with traditional drainage boundary (single or double), and single-sided and double-sided semi-permeable drainage boundaries. Finally, it illustrates the changes in pore-air and pore-water pressures and soil settlement with time at different values of symmetric semi-permeable drainage boundary conditions parameters. In addition, parametric studies are conducted by the variations of pore-air and pore-water pressures at different ratios of air-water permeability coefficient and the depth.     

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.     

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.     

半渗透边界下非饱和土砂井地基固结特性

以往的非饱和土砂井地基研究中未考虑砂井的涂抹作用,均按理想砂井进行研究。然而在实际工程中,施工的扰动会导致砂井内壁附近的土体渗透性减弱,从而影响地基的固结。针对这种情况,本文以均布荷载作用下非饱和土砂井地基为模型,考虑涂抹作用,并将其形成的边界假设为半渗透边界;引入Bessel函数,根据超孔隙压力的耦合控制方程推导了半渗透边界条件下非饱和土砂井地基在Laplace变换域内的固结半解析解,并且采用Crump方法进行Laplace逆变换,得到了时间域内的解;将该砂井地基模型退化为理想砂井模型,与现有文献结果对比,验证了本文研究方法的可靠性。最后通过算例来考察半渗透系数对非饱和土砂井地基固结特性的影响。研究表明,半渗透系数对砂井地基的固结有着重要的影响,且半渗透系数越大,超孔隙压力消散越快。     

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.     

Analytical solution to one-dimensional consolidation in unsaturated soils under loading varying exponentially with time

This note presents an analytical solution to one-dimensional consolidation in unsaturated soils with a finite thickness under confinement in the lateral direction and vertical loading varying exponentially with time. The boundary conditions are that the top surface is permeable to water and air and the bottom is impermeable to water and air. The transfer relationship between the state vectors at the top surface and any depth is gained by applying the Laplace transform and Cayley-Hamilton mathematical methods to the governing equations of water and air, Darcy’s law and Fick’s law. The excess pore-air and pore-water pressures and settlement in the Laplace-transformed domain are obtained by using the Laplace transform with the initial and boundary conditions. By performing the inverse Laplace transforms, the analytical solutions of the excess pore-air and pore-water pressures at any depth and settlement are obtained in the time domain.     

A semi-analytical solution to consolidation of unsaturated soils with the free drainage well

Based on Fredlund’s one-dimensional consolidation equation for unsaturated soil, Darcy’s law and Fick’s law, a semi-analytical solution was presented to the free drainage well with a finite thickness under application of uniform vertical loading and the boundary of the top and bottom surfaces impermeable to water and air. According to the polar governing equations of water and air phases and the boundary and initial conditions, the excess pore-air and pore-water pressures and the soil layer settlement in the Laplace transformed domain are obtained by performing the Laplace transform and utilizing the Bessel functions. Crump’s method is used to perform the inversion of Laplace transform in order to obtain numerical solutions in the real time domain. Finally, a typical example is given to illustrate the changes in the excess pore-air and pore-water pressures and soil layer settlement with time factor at different ratios of air–water permeability coefficient and/or different distances from the well.     

非饱和土拟二维平面应变固结问题的解析计算方法

基于Fredlund非饱和土一维固结理论,建立了二维平面应变条件下的固结方程组,并得到了单层非饱和土平面应变条件下的解析解。基于相关理论,假设体变系数和渗透系数都为常量,同时考虑到瞬时加载条件下,沿着土体深度方向上产生均匀或者线性分布的初始超孔隙压力,建立了二阶二元偏微分方程组。求解时,引入函数方法来降低方程的阶数,然后通过分离变量法获得方程的通解。在此基础上,结合一个针对单面排水条件下二维平面应变问题案例,通过与数值解对比,验证了所提方法的正确性。并采用所提方法计算获得了二维平面下超孔隙水压力、气压力沿垂直和水平方向消散的等时线,通过计算对比,分析了不同线性分布情况下,初始超孔隙压力对固结消散过程的影响。研究结果表明：初始超孔隙压力的不同分布对超孔隙气压力消散的影响几乎可以忽略,而对超孔隙水压力消散的影响更大。     

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.     

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。     

复合加筋排水褥垫加固路基有限元计算

复合加筋排水褥垫是针对沿海缺砂地区开发的,具有水平排水与加筋作用的一项新型专利技术,可取代软基加固中的排水砂垫层。采用二维平面应变Biot固结有限元模型对复合加筋排水褥垫软土路堤进行了数值模拟,结果表明复合加筋排水褥垫可以满足水平排水要求,另外与传统的水平排水砂垫层相比,该技术可降低地基附加正应力和超孔隙水压力,有效抑制路基侧向变形的发展,减小路基沉降与不均匀沉降,提高路基稳定性,加筋效果显著。     

One-dimensional self-weight consolidation with continuous drainage boundary conditions: Solution and application to clay-drain reclamation

Traditional consolidation theories cannot provide good predictions of consolidation settlement in land reclamation because of their assumptions that the influence of soil's self-weight is often neglected, and the drainage boundary is considered as fully pervious/impervious. In view of these limitations, an analytical solution is derived for one-dimensional self-weight consolidation problems with a continuous drainage boundary using the finite Fourier sine transform method. Following the classical Terzaghi's small strain theory, the soil's self-weight is considered to produce consolidation settlement in dredged materials with a constant coefficient of consolidation. The continuous drainage boundary can essentially describe the time-dependent variation of drainage capacity at the interface between two adjacent soil layers. By reducing the interface parameters, the effectiveness of the calculation is demonstrated against the Terzaghi's solution. The influence of interface parameters and soil's self-weight stress coefficient on self-weight consolidation is discussed. As expected, the rate of consolidation considering the self-weight stress is faster, although the dependency of consolidation rate on the material property of void ratio is neglected. Moreover, the plane of maximum excess pore-water pressure is estimated as a function of time factor, based on which a design chart is developed to optimize the layout of horizontal drains in land reclamation.     

Consolidation analysis of saturated multi‐layered soils with anisotropic permeability caused by a point sink

This paper presents the analytical layer‐element method to analyze the consolidation of saturated multi‐layered soils caused by a point sink by considering the anisotropy of permeability. Starting from the governing equations of the problem, the solutions of displacements and stresses for a single soil layer are obtained in the Laplace–Hankel transformed domain. Then, the analytical layer‐element method is utilized to further derive the solutions for the saturated multi‐layered soils in the transformed domain by combining with the boundary conditions of the soil system and continuity conditions between adjacent layers. The actual solutions in the physical domain can be acquired by the inversion of Laplace–Hankel transform. Numerical results are carried out to show the accuracy and stability of the proposed method and evaluate the influence of sink depth and anisotropic permeability on excess pore pressure and surface settlement. Copyright © 2011 John Wiley & Sons, Ltd.     

复杂荷载作用下考虑下卧层三维渗流的未打穿竖井地基固结分析

针对任意复杂变荷载作用下未打穿的竖井地基,通过在竖井底面以下土层中设置虚拟竖井,使其能够合理考虑下卧层土体三维渗流问题,运用Laplace变换,求得频域内竖井地基的固结解。通过Laplace逆变换,得到了任意荷载作用下竖井地基的平均固结度、孔压消散曲线、沉降曲线。结合具体算例,对影响竖井地基固结的主要影响因素进行了详细分析,并将现有未打穿竖井地基平均固结度近似计算方法的精度和适用的范围进行了对比,得到了一些有益于工程实践的结论。     

土工织物与塑料排水板联合处理软基的效果分析

基于平面应变条件下的Biot固结理论,采用非线性有限元方法对土工织物加筋与塑料板排水联合使用处理软土路基的效果和机制进行了探讨。对比计算表明：在相同固结时间内加设排水板后路堤的竖向沉降大于无排水板情况下的相应值,路堤坡趾处的隆起变小,路堤的侧向位移也变小。塑料排水板极大地缩短了孔压的消散时间,加快了软土的固结进程,使软土地基的承载力得到较快的提高,从而提高了路堤的稳定性。加设排水板后筋材所分担的抗拉力也比无排水板情况下的抗拉力低。为了更好地发挥土工织物的加固效果,土工织物应尽量布置在高应力区域。加设排水板可以缩短路堤各固结阶段的静置时间而达到优化设计。     

盾构施工引起地表固结沉降问题的研究

将隧道周围土体视为均质连续各向同性的饱和弹性介质,采用保角变换的方法将含有隧道的半无限平面映射为同心圆环计算域。根据Terzaghi-Rendulic二维固结理论,建立隧道在不透水的情况下周围土体超孔隙水压力分布的控制方程。然后,采用分离变量法计算得到土体超孔隙水压力分布的解析解,最后,根据弹性理论计算得出隧道中线上方地表固结沉降的计算公式。结合算例,分析盾构施工扰动程度、土体渗透系数、土体弹性模量及隧道埋深等因素对隧道中心上方地表处固结沉降的影响。研究结果表明,地表固结沉降的增加值与隧道外侧初始超孔隙水压力值C0的变化量成正比例关系,施工扰动程度越大所引起的固结沉降越大;土体的渗透系数越大固结沉降速度越快,但土体的渗透系数与最终的地表固结沉降量无关;土体的弹性模量越大,最终的地表固结沉降量越小;隧道埋深越深,地表固结沉降所需时间越长,最终的地表固结沉降量也越大。     

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.     

轴对称多层可压缩渗透各向异性岩基固结分析

从轴对称固结基本方程出发,通过对时间t、坐标r的Laplace-Hankel变换,再对坐标z的Laplace变换,得到三元一次方程组,解此方程组,并进行Laplace逆变换,得到了单层可压缩渗透各向异性岩基轴对称固结问题的传递矩阵,然后利用传递矩阵法,结合层间连续性条件和边界条件,得到了多层可压缩渗透各向异性岩基轴对称固结问题在积分变换域内的解。最后应用Laplace-Hankel逆变换技术得到轴对称固结问题在物理域内的理论解。编制了相应的计算程序,并进行了数值计算与分析,讨论了可压缩性和渗透各向异性对岩基固结的影响,结果表明：可压缩性越大,岩基瞬时沉降越大;渗透各向异性对固结过程影响明显,但对初始和最终沉降的影响很小。     

横观各向同性土体中压力隧洞的应力和位移计算

基于Biot固结理论,考虑了土体和孔隙流体压缩性,通过对控制方程的解耦,得到在横观各向同性饱和土体中圆形隧洞边界上作用随时间变化的轴对称荷载或流体压力所引起的应力、位移和孔隙水压力场在拉普拉斯变换域中的解析表达式,运用拉普拉斯数值逆变换进行算例分析,得到在时间域中的解,讨论了单级加载和循环加载对计算结果的影响,并与瞬时加载条件下的结果进行了比较。同时也分析了土体的横观各向同性性质对应力、位移和孔隙水压力场的影响。     

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.