Analysis of pumping a compressible pore fluid from a saturated elastic half space

A solution method is presented for the consolidation of a saturated, porous elastic half space due to the pumping of a pore fluid at a constant rate from a point sink embedded beneath the surface. It is assumed that the saturated medium is homogeneous and isotropic with respect to both its elastic and flow properties. The soil skeleton is modelled as an isotropic, linear elastic solid obeying Hooke's Law while the pore fluid is assumed to be compressible with its flow governed by Darcy's Law. The solution has been evaluated for a soil with a value of Poisson's ration of 0.25 and for a number of different cases of pore fluid compressibility. It is demonstrated that this compressibility can have a significant influence on the rate of consolidation around the sink. The solutions presented may have application in practical problems such as the extraction of groundwater and other fluids from compressible geological media.     

层状地基中单桩负摩擦问题积分方程解法

利用Biot固结理论和积分方程方法研究了表面有堆载的层状地基中单桩负摩擦问题。根据层状饱和土的圆形载荷基本解得出了单桩在圆形均布载荷作用下在时间域内的第二类Fredholm积分方程组。运用Laplace变换对上述积分方程组进行简化。再结合传递和刚度矩阵传递到各个层中去,对变换域内的积分方程采用Schapery 逆变换方法得到时域内单桩的近似积分方程。求解积分方程组并进行相应的数值逆变换,就可得出层状地基中的单桩在表面圆形均布载荷作用下的位移、轴力、孔压和桩侧摩阻力随时间的变化情况。计算结果表明,桩侧剪力和孔压分层明显。     

Stress analysis of a deep rigid axially-loaded cylindrical anchor in an elastic medium

The elastostatic problem of an infinite elastic medium containing an axially-loaded rigid cylindrical inclusion is investigated. This problem is of interest in connection with the geotechnical study of the time-independent, load-deflection characteristics of deep rigid anchors embedded in cohesive soil or rock media. The problem is formulated by means of Hankel integral transforms and reduced to a system of four coupled sing ular integral equations, where the unknown quantities are the normal and the shear stresses acting on the entire surface of the anchor. Numerical solutions are investigated for various Poisson's ratios and several values of the aspect ratio of radius to length of the cylindrical anchor.     

Poroelastic model for pile–soil interaction in a half‐space porous medium due to seismic waves

In this paper, frequency domain dynamic response of a pile embedded in a half‐space porous medium and subjected to P, SV seismic waves is investigated. According to the fictitious pile methodology, the problem is decomposed into an extended poroelastic half‐space and a fictitious pile. The extended porous half‐space is described by Biot's theory, while the fictitious pile is treated as a bar and a beam and described by the conventional 1‐D structure vibration theory. Using the Hankel transformation method, the fundamental solutions for a half‐space porous medium subjected to a vertical or a horizontal circular patch load are established. Based on the obtained fundamental solutions and free wave fields, the second kind of Fredholm integral equations describing the vertical and the horizontal interaction between the pile and the poroelastic half‐space are established. Solution of the integral equations yields the dynamic response of the pile to plane P, SV waves. Numerical results show the parameters of the porous medium, the pile and incident waves have direct influences on the dynamic response of the pile–half‐space system. Significant differences between conventional single‐phase elastic model and the poroelastic model for the surrounding medium of the pile are found. Copyright © 2007 John Wiley & Sons, Ltd.     

Scattering of elastic wave by a partially sealed cylindrical shell embedded in poroelastic medium

Scattering of an elastic wave by a cylindrical shell embedded in poroelastic medium is investigated theoretically with the assumption that the shell material is also a porous elastic medium. The porous medium is modellized via Biot's theory and the scattering by cylindrical shell is expressed by the definition of scattering matrix. The normal mode expansion technique is employed for analysing the scattering field, and the asymptotic solutions of displacements, stresses and pore pressure are derived. Two limiting cases‐scattering by a poroelastic cylinder in Biot medium and a elastic cylindrical shell in elastic medium are obtained from the general solutions. The dispersion curves of displacement amplitude at the interface of shell and medium is compared with the case of elastic shell. The scattering amplitude associated with the fast, slow and transverse waves are identified by numerical simulation. Furthermore, the influence of the poroelastic property of shell material on scattering amplitude is analysed. Copyright © 2005 John Wiley & Sons, Ltd.     

Elastic consolidation around a point sink embedded in a half-space with anisotropic permeability

The complete solution is presented for the transient effects of pumping fluid from a point sink embedded in a saturated, porous elastic half-space. It is assumed that the medium is homogeneous and isotropic with respect to its elastic properties and homogeneous but anisotropic with respect to the flow of pore fluid. The soil skeleton is modelled as a linear elastic material obeying Hooke's law, while the pore fluid is assumed to be incompressible with its flow governed by Darcy's law. The solution has been evaluated for a particular value of Poisson's ratio of the solid skeleton, i.e. 0.25, and the results have been presented graphically in the form of isochrones of excess pore pressure and surface profile for the half-space. The solutions presented may have application in practical problems such as dewatering operations in compressible soil and rock masses.     

Interaction of a cylindrical thin‐walled pile with an inhomogeneous isotropic elastic medium

This paper presents a rigorous analysis for the static interaction of a cylindrical thin‐walled pile with an inhomogeneous isotropic elastic half‐space under vertical, horizontal, and torsional forces individually applied at the top of pile. The inhomogeneity is specified with the exponential variation of shear modulus along depth of the embedding medium, and the Poisson's ratio is assumed to be constant. By means of a set of Green's functions for pile and soil medium and satisfying the compatibility conditions between the 2 interacting media, the formulation is reduced to coupled Fredholm integral equations. Using the adaptive‐gradient elements, capable of capturing the singular stress transfer at both ends of the pile, a numerical procedure is developed and utilized for evaluating the relevant integral equations and studying the inhomogeneity effect on the soil‐pile interaction responses. The analysis results have been validated for different soil‐pile modulus ratios under axial load and for a Poisson's ratio of 0.3 under lateral load. The procedure does not consider the nonlinear behavior of the soil medium or plastic yielding in the pile section, and the impact of the unreliable results for the case of high Poisson's ratio is not examined.     

On Garg's solution of Biot's equations for wave propagation in a one-dimensional fluid-saturated elastic porous solid

Garg's approximate analytical solutions of Biot's equations for wave propagation in a fluid-saturated elastic porous solid of infinite extent subjected to a velocity boundary condition of a Heaviside function at one end are examined for small and large drag. Garg's approximations were apparently introduced to facilitate exact inversion of Laplace transforms of certain quantities. The approximate solutions are compared with carefully evaluated numerical inverses of the Laplace transform solutions for different soils with widely varying properties. It is seen that for most soils (clays, silts and, sands) the error in Garg's approximate solutions in insignificant, and the solutions can be used as benchmarks for verifying numerical analysis procedures such as finite element codes.     

Variational elastic solution for axially loaded piles in multilayered soil

Most analytical or semi‐analytical solutions of the problem of load‐settlement response of axially loaded piles are based on the assumption of zero radial displacement. These solutions also are only applicable to piles embedded in either a homogeneous or a Gibson soil deposit. In reality, soil deposits consist of multiple soil layers with different properties, and displacements in the radial direction within the soil deposit are not zero when the pile is loaded axially. In this paper, we present a load‐settlement analysis applicable to a pile with circular cross section installed in multilayered elastic soil that accounts for both vertical and radial soil displacements. The analysis follows from the solution of the differential equations governing the displacements of the pile–soil system obtained using variational principles. The input parameters needed for the analysis are the pile geometry and the elastic constants of the soil and pile. We compare the results from the present analysis with those of an analytical solution that considers only vertical soil displacements. The analysis presented in this paper also provides useful insights into the displacement and strain fields around axially loaded piles. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical solution for the dilation of a fluid-saturated porous elastic sphere due to a point source of fluid at the centre of the sphere

Mathematical modelling of the ascent of free fluid through relatively strong rock, deep in the Earth's mantle, presents a challenge in geomechanics. Here the medium is considered as fluid-saturated, porous, elastic and bounded, and the fluid enters at a point source. An explicit finite difference method is developed for the numerical solution to the problem of the dilatation of a fluid-saturated porous elastic sphere due to a point fluid source of constant strength at the centre of the sphere. A cubic spline interpolant is used to evaluate a definite integral which occurs in the boundary condition for the pore fluid pressure at the surface of the sphere. The numerical solutions for the dilatation and pore fluid pressure are compared with analytical solutions and the absolute and relative errors of the numerical solutions are calculated. When the fluid source is switched on, the pore fluid pressure starts to decrease, reaches a minimum value and then steadily increases. The initial time rate of decrease of the pore fluid pressure is independent of the radial distance from the source. It decreases as the radius of the sphere increases and vanishes for a point fluid source in an infinite porous elastic medium.     

Analytical solutions of a finite two‐dimensional fluid‐saturated poroelastic medium with compressible constituents

An analytical solution is proposed for transient flow and deformation coupling of a fluid‐saturated poroelastic medium within a finite two‐dimensional (2‐D) rectangular domain. In this study, the porous medium is assumed to be isotropic, homogeneous, and compressible. In addition, the point sink can be located at an arbitrary position in the porous medium. The fluid–solid interaction in porous media is governed by the general Biot's consolidation theory. The method of integral transforms is applied in the analytical formulation of closed‐form solutions. The proposed analytical solution is then verified against both exact and numerical results. The analytical solution is first simplified and validated by comparison with an existing exact solution for the uncoupled problem. Then, a case study for pumping from a confined aquifer is performed. The consistency between the numerical solution and the analytical solution confirms the accuracy and reliability of the analytical solution presented in this paper. The proposed analytical solution can help us to obtain in‐depth insights into time‐dependent mechanical behavior due to fluid withdrawal within finite 2‐D porous media. Moreover, it can also be of great significance to calibrate numerical solutions in plane strain poroelasticity and to formulate relevant industry norms and standards. Copyright © 2014 John Wiley & Sons, Ltd.     

Effect of compressible parameters on vertical vibration of an elastic pile in multilayered poroelastic media

This paper mainly investigates the influences of compressible parameters on the vertical vibration of a pile embedded in layered poroelastic soil media. The pile is treated as a 1D elastic bar by the finite element method, and fundamental solutions for the layered poroelastic soils due to a vertical dynamic load are obtained by the analytical layer element method. Based on the compatibility conditions, the pile-soil dynamic interaction problem is solved. The numerical scheme has been compiled into a Fortran program for numerical calculation. Influences of the pile-soil stiffness ratio, compressible parameters, vibration frequency and the soil stratification are discussed.     

Analysis of rectangular piles subjected to lateral loads in nonhomogeneous soil using a Winkler model approach

An analytical approach using a Winkler model based on two lateral soil displacement components in a three‐dimensional soil is investigated to provide analytical solutions of horizontal response of a rectangular pile subjected to lateral loads in nonhomogeneous soil. The two lateral displacement components of a soil surrounding the rectangular pile are represented by the Fourier series of displacement potential functions in the elastic three‐dimensional analysis. The lateral stiffness coefficient of the rectangular pile shaft in nonhomogeneous soil is derived from the rocking stiffness coefficient taking into account rocking rotation of a rigid pile shaft. The relationship between horizontal displacement, rotation, moment, and shear force for the rectangular pile subjected to horizontal loads in nonhomogeneous soil is obtainable in the form of the recurrence equation. The formulation of lateral displacement and rotation for a rectangular pile subjected to lateral loads on the pile base in nonhomogeneous soil is proposed by taking into account Mindlin's equation and the equivalent thickness for soil layers in the equivalent elastic method. The difference of lateral behavior between square and circular piles subjected to lateral loads is insignificant. The effect of aspect ratio of the rectangular pile on the lateral behavior is great for the lower stiffness ratio between pile and soil and the larger length–equivalent diameter ratio. The effect of the value of Poisson's ratio of soil on lateral stiffness coefficient is relatively small except Poisson's ratio close to 0.5. The comparison of the results calculated by the current method for a rectangular pile subjected to lateral loads in nonhomogeneous soil has shown good agreement with those obtained from the analytical methods and the finite element method. Copyright © 2015 John Wiley & Sons, Ltd.     

A three‐dimensional displacement approach for analysis of laterally loaded piles in nonhomogeneous soil

An analytical approach using the three‐dimensional displacement of a soil is investigated to provide analytical solutions of the horizontal response of a circular pile subjected to lateral loads in nonhomogeneous soil. The rocking stiffness coefficient of the pile shaft in homogeneous soil is derived from the analytical solution taking into account the three‐dimensional displacement represented in terms of scalar potentials in the elastic three‐dimensional analysis. The lateral stiffness coefficient of the pile shaft in nonhomogeneous soil is derived from the rocking stiffness coefficient taking into account the rocking rotation of a rigid pile shaft. The relationship between horizontal displacement, rotation, moment, and shear force of a pile subjected to horizontal loads in nonhomogeneous soil is obtainable in the form of the recurrence equation. The formulation of the lateral displacement and rotation of the pile base subjected to lateral loads in nonhomogeneous soils is presented by taking into account Mindlin's equation and the equivalent thickness for soil layers in the equivalent elastic method. There is little difference between lateral, rocking, and couple stiffness coefficients each obtained from both the two‐dimensional and three‐dimensional methods except for the case of Poisson's ratio near 0.5. The comparison of results calculated by the current method for a pile subjected to lateral loads in homogeneous and nonhomogeneous soils has shown good agreement with those obtained from analytical and numerical methods. Copyright © 2017 John Wiley & Sons, Ltd.     

Influences of axial load on the lateral response of single pile with integral equation method

This article revisits the influences of axial load on the lateral response of single pile with integral equation method. The problem is formulated by decomposing the pile soil system into an extended elastic soil and a fictitious pile, the former of which is analyzed by making use of the fundamental Mindlin's solution for a concentrated horizontal load whereas the latter is modeled by the conventional beam bending theory. According to the rotation compatibility condition between the fictitious pile and the extended soil, a Fredholm integral equation of the second kind is established with the shear strain and rotation angle of the fictitious pile being the basic unknowns. The bending moment and displacement distribution along the pile are subsequently obtained. Comparison with existing solutions validates the accuracy and applicability of the present formulation. The results of parametric analysis indicate that the influences of axial load on the lateral response of single piles could be significant, and in general, the bending moment and horizontal displacement distributions along the pile increase considerably with the increase of axial load. Copyright © 2011 John Wiley & Sons, Ltd.     

传递矩阵法分析层状地基中桩的扭转变形

研究了扭矩作用下单桩的扭转变形.采用积分变换和传递矩阵方法,求解了成层土在内部环形荷载作用下的基本解;利用此基本解并考虑桩土位移协调条件,提出了层状地基中单桩扭转变形分析的解析方法;并按此理论方法对匀质地基模型进行了数值计算,其结果与已有经典解答相当吻合.     

Linear coupled analysis of desiccation shrinkage in a double‐layer partially saturated medium: semi‐explicit solutions

Solutions are presented for the problem of isothermal dessiccation shrinkage in a double‐layer porous partially saturated medium. The rheological model taken into account is linear poroelastic. Hence the analysis is mainly focused on hydromechanical coupling effects and contrasts of mechanical and hydraulic properties between two materials: a low thickness skin comprised between the outer boundary and the reference porous material. Three one‐dimensional ideal structures are taken into account: a wall of finite thickness (cartesian geometry), a thick cylinder and a thick sphere. The solution of the time‐dependent problem is arrived at by applying Laplace transforms to the field variables. Exact solutions are obtained in Laplace transform space using Mathematica^© to solve the field equations whilst taking into account the continuity equations at the interface and the boundary conditions. The Talbot's modified algorithm has been performed to invert the Laplace transform solutions. A bibliographical and numerical study shows that this method is remarkably precise, stable and close to the analytical inversion. Results are presented using poroelastic data representative of a concrete material and involve a strong coupling effect between hydraulical and mechanical behaviours. A first approach elastic modelling of degradation process have been presented using a thin outer layer. Apart from emphasising the semi‐explicit solution utility due to accurate speed calculation, this paper deals with more complex problems than those which can be solved using purely analytical solutions. Copyright © 2001 John Wiley & Sons, Ltd.     

Torsional dynamic response of a large‐diameter pipe pile in viscoelastic saturated soil

An analytical solution is developed in this paper to investigate the dynamic response of a large‐diameter end‐bearing pipe pile subjected to torsional loading in viscoelastic saturated soil. The wave propagation in saturated soil and pile are simulated by Biot's two‐phased linear theory and one‐dimensional elastic theory, respectively. The dynamic equilibrium equations of the outer soil, inner soil, and pile are established. The solutions for the outer and inner soils in frequency domain are obtained by Laplace transform technique and the separation of variables method. Then, the dynamic response of the pile is obtained on the basis of the perfect contacts between the pile and the outer soil as well as the inner soil. The results in this paper are compared with that of a solid pile in elastic saturated soil to verify the validity of the solution. Furthermore, the solution in this paper is compared with the classic plane strain solution to verify the solution further and check the accuracy of the plane strain solution. Numerical results are presented to analyze the vibration characteristics and illustrate the effect of the soil parameters and the geometry size of the pile on the complex impedance and velocity admittance of the pile head. Finally, the displacement of the soil at different depth and frequency is analyzed. Copyright © 2014 John Wiley & Sons, Ltd.     

Modelling the elastic behaviour of granular materials

A review of the literature indicates that the elastic behaviour of granular materials is isotropic and that Poissony's ratio is constant, whereas Young's Modulus, the bulk modulus and the shear modulus vary with the mean normal stress and the deviatoric stress. A nonlinear, isotropic model for the elastic behaviour is developed on the basis of theoretical considerations involving the principle of conservation of energy. Energy is therefore neither generated not dissipated in closed-loop stress paths or in closed-loop strain paths. The framework for the model consists of Hooke's law, in which Poission's ratio is constant and Young's modulus is expressed as a power function invlving the first invariat of the stress tensor and the second invariant of the deviatoric stress tensor. The characteristics of the model are described, and the accuracy is evaluated by comparison with experimental results from triaxial tests and three-dimensional cubical triaxial tests with a variety of stress paths. Parameter determination from unloading–reloading cycles in conventional triaxial compression tests is demonstrated, typical parameter values are given for granular materials and extension of the model to soils with effective cohesion is described.     

准饱和土中圆柱形衬砌的瞬态动力响应分析

内源瞬态荷载作用下圆柱形孔洞的动力响应解答是土动力学的经典问题之一。已有研究大都假设孔洞周围土体为理想弹性介质或完全饱和多孔介质。然而,实际工程中不存在完全弹性和完全饱和土体。分别视衬砌结构和周围土体为弹性材料和准饱和多孔介质（饱和度 95%）,根据牛顿第二定律、达西定律和Biot波动理论推导出准饱和土体的控制方程。根据边界条件导出衬砌和土体的位移、应力和孔隙压力的Laplace变换空间的解答。利用反Laplace变换数值计算方法,将解答转换为时域解。分析了饱和度对衬砌位移、应力和孔压的影响,结果表明,当95% 99%时,饱和度对径向位移和切向应力的影响较小;99% 100%时,饱和度对径向位移和切向应力的影响较大;但饱和度对孔隙压力的影响远大于对径向位移和切向应力的影响。得出位移、应力和孔压沿径向的衰减规律,当95% 99%时,饱和度对径向位移和切向应力沿径向衰减影响较小,99% 100%时,饱和度对径向位移和切向应力沿径向衰减影响较大,但饱和度对孔压沿径向的衰减影响远大于对径向位移和切向应力沿径向的衰减。