Analytical solution for the 1D consolidation of unsaturated multi-layered soil

The 1D consolidation of unsaturated multi-layered soil is studied based on the theory proposed by Fredlund and Hasan, and an analytical solution for a typical boundary condition is obtained by assuming all material parameters remain constant during consolidation. In the derivation of the analytical solution, the eigenfunction and eigenvalue for the multi-layered problem are first derived through the transfer matrix method. Then, by using the method of undetermined coefficients and the orthogonal relation of the eigenfunction, the analytical solution is obtained. The present method is applicable to various types of boundary conditions. Finally, numerical examples are provided to investigate the consolidation behavior of unsaturated multi-layered soil.     

Two-dimensional meshfree modelling of contaminant transport through saturated porous media using RPIM

This paper presents a new numerical tool to model the two-dimensional contaminant transport through saturated porous media using a meshfree method, called radial point interpolation method (RPIM) with polynomial reproduction. In RPIM, an approximate solution is constructed entirely in terms of a set of nodes and no characterisation of the interrelationship of the nodes is needed. The advection–dispersion equation with sorption is considered to illustrate the applicability of the RPIM. The Galerkin weak form of the governing equation is formulated using 2D meshfree shape functions constructed using thin plate spline radial basis functions. MATLAB code is developed to obtain the numerical solution. Three numerical examples are presented and the results are compared with those obtained from the finite element method and analytical solutions. In order to test the practical applicability and performance of the RPIM, two case studies of contaminant transport through landfill liners are presented. A good agreement is obtained between the results of the RPIM and the field investigation data.     

Analytical layer-element solution to axisymmetric consolidation of multilayered soils

After the application of a Laplace–Hankel transform, the governing equations of Biot’s consolidation are solved analytically by using the eigenvalue approach. Then the analytical layer-element of a single soil layer can be obtained in the transformed domain by synthesizing the generalized displacements and stresses, which are both expressed by six arbitrary constants. The elements of the analytical layer-element only contain negative exponential functions, which leads to a considerable improvement in computation efficiency and stability. The global stiffness matrix equation of multilayered soils is further obtained by assembling the interrelated layer-elements, and the actual solution is achieved by numerical inversion of the Laplace–Hankel transform after the solution in the transformed domain is obtained. Numerical examples are given to demonstrate the accuracy of this method and to study the influence of the layered soil properties and time history on the consolidation behavior.     

渗透各向异性圆形围堰稳态渗流场解析解

对渗透各向异性土层中的圆形围堰渗流场进行解析研究。将圆形围堰周围渗流场分为3个区域,通过坐标变换以及将边界条件齐次化后,用分离变量法分别得到柱坐标系下3个区域的水头分布级数解形式,结合区域间的连续条件,并利用贝塞尔函数正交性得到渗透各向异性土层中圆形围堰稳态渗流场的显式解析解,与Plaxis2D软件的计算结果进行对比,验证了解析解的正确性。基于解析解,进一步推导出了涌水量、出逸比降公式,并与不同方法的计算结果作了对比,最后对稳态渗流情况下的水压力进行了计算分析。结果表明：相比于其他近似法,解析解求取的流量、出逸比降以及渗流情况下水压力与数值计算结果高度吻合;渗流水压力会随着围堰半径、纵横向渗透系数比值的增大而减小,而后逐渐趋近于一个稳定值。     

隧道围岩流变参数的粘弹性位移反演与验证

将解析反演和数值反演相结合,对试验毛洞的断面形状和所处应力场做出适当的简化,利用对应性原理求解出洞室位移的Kelvin粘弹性解,并以此去拟合试验洞不同位置处的实测位移而得到多组流变参数;分别将其作为FLAC3D数值模型蠕变模式的输入参数,得出各自对应的实际断面形状和实际应力场条件下试验洞的蠕变位移数值解。然后,利用BP神经网络,对各组流变参数和其对应的试验洞同一位置处的蠕变位移数值解进行训练,建立起两者之间的非线性关系;利用训练好的网络,依据实测蠕变位移值得出了围岩流变参数;最后,利用隧道实测位移数据对反演的流变参数进行了数值正分析验证。     

Steady‐state analytical solutions of flow and deformation coupling due to a point sink within a finite fluid‐saturated poroelastic layer

An exact steady‐state closed‐form solution is presented for coupled flow and deformation of an axisymmetric isotropic homogeneous fluid‐saturated poroelastic layer with a finite radius due to a point sink. The hydromechanical behavior of the poroelastic layer is governed by Biot's consolidation theory. Boundary conditions on the lateral surface are specifically chosen to match the appropriate finite Hankel transforms and simplify the transforms of the governing equations. Ordinary differential equations in the transformed domain are solved, and then the analytical solutions in the physical space for the pore pressure and the displacements are finally obtained by using finite Hankel inversions. The analytical solutions at some special locations such as the top and bottom surfaces, lateral surface, and the symmetrical axis are given and analyzed. And a case study for the consolidation of a water‐saturated soft clay layer due to pumping is conducted. The analytical solution is verified against the finite element solution. Meanwhile, an analysis of coupled hydromechanical behavior is carried out herein. The presented analytical solution is an exact solution to the practical poroelastic problem within an axisymmetric finite layer. It can provide us a better understanding of the poroelastic behavior of the finite layer due to fluid extraction. Besides, it can be applied to calibrate numerical schemes of axisymmetric poroelasticity within finite domains. Copyright © 2017 John Wiley & Sons, Ltd.     

A Simple Method for Analysis of Point Anchored Rockbolts in Circular Tunnels in Elastic Ground

Summary. A simple analytical method for the analysis of point anchored rockbolts is presented in this paper. The solution has been derived for elastic ground and rockbolts, for plane strain conditions, and for tunnels with circular cross section. The method provides accurate results for the rockbolts’ loads and displacements and explicitly includes the connection of the rockbolts to the surrounding ground. The addition of such details to a Finite Element numerical model is critical; otherwise the solution obtained may be dependent on the discretization used and on the stiffness of rockbolts and ground. As an alternative to including details of the rockbolt head and anchor point in the numerical model, which could be computationally very expensive, an equivalent spring constant is proposed. The spring constant is obtained by matching numerical with analytical results for a simple case, but keeping the geometry, material properties, and discretization unchanged.     

Solutions for the deformations and stability of elastoplastic hollow cylinders subjected to boundary pressures

In this paper, a closed-form solution is presented for the stress and displacement distributions throughout a hollow cylinder subjected to uniform pressures acting on its internal and external boundary surfaces under plane strain conditions. The material is assumed to be elastoplastic, obeying a Mohr–Coulomb failure criterion, and exhibiting dilatant plastic deformation according to a non-associated flow rule. The newly developed analytical solution is verified through comparison with the solutions obtained from an infinite boundary problem (for which a closed-form solution exists), and numerical analyses using the program FLAC. The solution is also compared with the results of a borehole collapse test on a thick-walled hollow cylinder of synthetic shale. The analytical solution can be used to calculate the stress and displacement distributions around boreholes and other cylindrical cavities under both infinite and finite boundary conditions under both drained and undrained conditions. Copyright © 1999 John Wiley & Sons, Ltd.     

Prediction of swelling rocks strain in tunneling

Swelling deformations leading to convergence of tunnels may result in significant difficulties during the construction, in particular for long term use of tunnels. By extracting an experimental based explicit analytical solution for formulating swelling strains as a function of time and stress, swelling strains are predicted from the beginning of excavation and during the service life of tunnel. Results obtained from the analytical model show a proper agreement with experimental results. This closed-form solution has been implemented within a numerical program using the finite element method for predicting time-dependent swelling strain around tunnels. Evaluating effects of swelling parameters on time-dependent strains and tunnel shape on swelling behavior around the tunnel according to this analytical solution is considered. The ground-support interaction and consequent swelling effect on the induced forces in tunnel lining is considered too. Effect of delay in lining installation on swelling pressure which acting on the lining and its structural integrity, is also evaluated. A MATLAB code of “SRAP” is prepared and applied to calculate all swelling analysis around tunnels based on analytical solution.     

波流共同作用下多孔介质海床动力响应的解析解

本文提出一个评价波流共同作用下多孔介质海床的孔隙水压力和有效应力动力响应的u-p近似模型。与过去相关研究主要不同处是在于本文着重考虑了潮流在该问题解析解中的作用。本文提出的解析解首先分别与实测数据和COMSOL数值解对比验证。接着以此解析解为基础进行了一系列关于流的作用的参数分析。本文首先研究了潮流对于不同海床深度的影响以及不同波流荷载下海床的动力响应问题,数值结果显示了潮流和非线性波对孔隙水压力和有效应力的重要作用。本文最后研究了不同潮流方向对海床动力响应的影响,结果显示由不同的潮流方向引起的孔隙水压力的相对变化可以达到静水压力的15%。     

The axisymmetric consolidation of a semi‐infinite transversely isotropic saturated soil

This paper presents an exact analytical solution to fully coupled axisymmetric consolidation of a semi‐infinite, transversely isotropic saturated soil subjected to a uniform circular loading at the ground surface. The analysis is under the framework of Biot's general theory of consolidation. First, the governing equations of consolidation are transformed into a set of equivalent partial differential equations with the introduction of two auxiliary variables. These partial differential equations are then solved using Hankel–Laplace integral transforms. Once solutions in the transformed domain have been obtained, the actual solutions in the physical domain for displacements and stress components of the solid matrix, pore‐water pressure and fluid discharge can be finally obtained by direct numerical inversion. The accuracy of the numerical solutions developed is confirmed by comparison with an existing exact solution for an isotropic and saturated soil that is a special case of the more general problem addressed. Numerical analyses are also presented to investigate the influence of the degree of material anisotropy on the consolidation settlement. Copyright © 2005 John Wiley & Sons, Ltd.     

非饱和土路基毛细作用的数值与解析方法研究

通过建立非饱和土毛细作用的孔隙分布的分形模型,推导获得了非饱和土毛细水的最大上升高度,同时基于Richards水分运动微分方程,引入边界及初始条件,基于Laplace变换,得到了毛细作用下非饱和土路基湿度变化的解析解;同时引入算例,将所提出的解析方法计算结果与未简化参数的数值计算结果进行了对比分析;最后考虑不同因素的影响进行了非饱和土路基毛细作用下的湿度变化分析。分析结果表明：解析求解获得的路基湿度变化趋势和未进行参数简化的数值法求解结果基本一致,证明解析解法是合理可信的;路基填筑的初始含水率越大,填土的初始吸力越小,毛细水上升的高度及湿度变化量也相应越小;透水性能较好的路基填土毛细水上升速度较快,但上升高度较小,毛细水可以在较短时间内上升到最大高度;路基的填土类型不同,路基在毛细作用下的湿度变化状态也不同,需要针对不同的填土路基进行相应的防排水措施。     

Two-dimensional modelling of contaminant transport through saturated porous media using the radial point interpolation method (RPIM)

A new numerical tool is presented which models the two-dimensional contaminant transport through saturated porous media using a meshfree method called the radial point interpolation method (RPIM) with polynomial reproduction. In RPIM, an approximate solution is constructed entirely in terms of a set of nodes and no characterisation of the interrelationship of the nodes is needed. An advection-dispersion equation with sorption is considered to illustrate the applicability of the RPIM. The Galerkin weak form of the governing equation is formulated using two-dimensional meshfree shape functions constructed using thin plate spline radial basis functions. A computer program is developed for the implementation of the RPIM procedure. Three numerical examples are presented and the results are compared with those obtained from the analytical solution and finite element method. The experimental results are also used to validate the approach. The proposed RPIM has generated results with no oscillations and they are insensitive to Peclet constraints.     

饱和土体一维复杂非线性固结特性研究

结合大量宁波软土室内固结与压缩试验数据,采用偏微分有限元软件FLEXPDE,同时考虑土体固结过程中材料和几何的复杂非线性,进行了土体固结特性研究。在压缩指数和渗透指数不同比值条件下,进行了解析解推导和有限元数值分析,得出了不同条件下土体有效应力和孔隙比沿高度的变化规律。结果表明,压缩指数和渗透指数比值不等于1时,由于数值解法充分考虑土体的复杂非线性,其结果与实际更加符合;而在比值等于1时所得出的结果与Gibson得出的结果一致;考虑土体材料和几何复杂非线性使得有效应力及固结度预测偏于安全。     

尾矿库浸润线解析解及适用性分析

目前求解尾矿库浸润线大多采用数值模拟和模型试验的方法,尚没有建立起完善的尾矿库浸润线的解析解。鉴于此,建立了尾矿库浸润线的理论模型,采用能量方程和达西渗透定律推导了尾矿库浸润线的微分方程,并建立了上下游边界条件方程。在此基础上,求解了尾矿库浸润线的解析解（由于在解析解的推导过程中引入了杜比假定,该解析解更适应于库底底坡较缓的尾矿库浸润线的预测）,并与数值模拟结果进行了对比分析,验证了解析解的正确性。考虑到尾矿库的复杂性,分析了尾矿砂各向异性、尾矿库地质分层和底坡地形对尾矿库浸润线分布的影响,探讨了所提出的解析解对复杂尾矿库浸润线求解的适应性,通过尾矿库实例验证了文中浸润线解析解的有效性。     

An analytical solution for stresses induced by a post-tensioned anchor in rocks containing two perpendicular joint sets

A new analytical approach for the analysis of stress around a post-tensioned anchor in rock with two perpendicular joint sets is presented. The solution considers nonlinear shear stresses developed along the anchor bond length as well as debonding at the tendon–grout interface. The bearing plate effects are also considered in the analyses. The following assumptions are made: (1) homogeneous and orthotropic elastic rock; (2) half-space rock mass with plane strain conditions; (3) trilinear bond-slip model for the behavior of the tendon–grout interface; (4) an elastic anchor. The employed methodology is to decompose the anchor problem into two problems of simpler loadings: stresses produced under a rigid bearing plate and stresses induced by interfacial shear stresses mobilized along the bond length. Based on the proposed solution, an illustrative example is given and the results show that a large compressive region is formed around the anchor free length. Tensile stress concentrations are also observed around the bond length and near the rock surface outside the bearing plate. A parametric study is also conducted to investigate the effects of joint properties on the induced stresses. The results show that as the number of joints intersecting the anchor axis increases, the magnitude of compressive stresses in the free zone decreases, while the size of compression zone around the anchor increases. To verify the results of the analytical approach, a comparison is made with numerical results obtained by using the finite element method. The analytical solutions compare very well with the results obtained by numerical method.     

边坡负压排水非稳定流研究

自启动负压排水方法是一种新兴的间歇性波动排水方法,为对该方法的实现机理进行理论研究,建立了可供解析研究的负压排水模型。先引入狄拉克函数表述负压排水模型中点汇的数理特性,并对此模型推求出潜水剖面二维流运动方程及潜水面边界条件;再使用二维傅里叶正逆变换求解负压排水模型的定解问题,得到负压排水渗流场解析解。设置两种工况分别计算了自启动负压排水方法下数值模拟和理论解得到的边坡浸润线曲线,并进行了对比分析。结果表明：在不同排水孔位置工况的3组条件中,排水孔位置越深,理论解误差越大,其中排水孔位置为（24,10）时,靠近排水孔区域理论解与数值模拟结果误差最大;在排水孔位置固定、不同初始水头高度工况的3组条件中,左、右水头越高,理论解误差越大,其中左水头为28 m、右水头为8 m时,理论解与数值模拟结果误差最大。该自启动负压排水渗流场解析解在简化条件下的浸润线理论解与数值模拟结果吻合度较高。     

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.     

均质非饱和边坡降雨入渗解析解及在黄土边坡的应用

降雨诱发的黄土边坡失稳非常普遍。建立黄土边坡渗流场计算模型,基于非饱和土渗流控制方程。采用VG函数和Gardner函数分别描述土-水特征曲线和渗透系数曲线,利用行波约化和级数展开法推导降雨入渗解析解。利用数值反演法将模型试验数据对土-水参数拟合,证明了该解析解的有效性。对比分析试验值与解析解在不同工况下体积含水率分布规律,分析结果表明：边坡试验模型中的浅层测点体积含水率试验值与解析解较为接近,试验值在浸润锋下移的过程中所达到的体积含水率峰值相比解析解较小;边坡试验模型中的深层测点体积含水率试验值与解析解在前期存在一定误差,深层测点体积含水率解析解在初期增长速度相较于试验值较快,主要原因是较深土层浸润锋的滞后性。     

Analytical solutions of one-dimensional advection-diffusion equation with variable coefficients in a finite domain

Analytical solutions are obtained for one-dimensional advection-diffusion equation with variable coefficients in a longitudinal finite initially solute free domain, for two dispersion problems. In the first one, temporally dependent solute dispersion along uniform flow in homogeneous domain is studied. In the second problem the velocity is considered spatially dependent due to the inhomogeneity of the domain and the dispersion is considered proportional to the square of the velocity. The velocity is linearly interpolated to represent small increase in it along the finite domain. This analytical solution is compared with the numerical solution in case the dispersion is proportional to the same linearly interpolated velocity. The input condition is considered continuous of uniform and of increasing nature both. The analytical solutions are obtained by using Laplace transformation technique. In that process new independent space and time variables have been introduced. The effects of the dependency of dispersion with time and the inhomogeneity of the domain on the solute transport are studied separately with the help of graphs.