Consolidation of a Finite Transversely Isotropic Soil Layer on a Rough Impervious Base

A semianalytical solution to axisymmetric consolidation of a transversely isotropic soil layer resting on a rough impervious base and subjected to a uniform circular pressure at the ground surface is presented. The analysis uses Biot’s fully coupled consolidation theory for a transversely isotropic soil. The general solutions for the governing consolidation equations are derived by applying the Hankel and Laplace transform techniques. These general solutions are then used to solve the corresponding boundary value problem for the consolidation of a transversely isotropic soil layer. Once solutions in the transformed domain have been found, the actual solutions in the physical domain for displacements and stress components of the solid matrix, pore-water pressure and fluid discharge can finally be obtained by direct numerical inversions of the integral transforms. The accuracy of the present numerical solutions 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. Further, some numerical results are presented to show the influence of the nature of material anisotropy, the surface drainage condition, and the layer thickness on the consolidation settlement and the pore pressure dissipation.     

Numerical Analysis of Geomaterials within Theory of Porous Media

It is widely accepted that the mechanical behavior of saturated geomaterials is largely governed by the interaction of the solid skeleton with the fluids present in the pore structure. This interaction is particularly strong in quasi-static and dynamic problems and may lead to the catastrophic loss of strength known as liquefaction, which frequently occurs under earthquake loading. In this work, numerical simulations of saturated granular deposits under transient loads are presented to illustrate the performance of a u-p-U finite-element method formulation and the versatility of the numerical implementation. Closed-form solutions based on both a Biot formulation and modern theories of mixtures are compared with numerical results. In addition, centrifuge experimental results are correlated with numerical simulations. A companion paper presents the details of the theoretical formulation and the numerical implementation within the finite-element method.     

Implementation of Porous Media Formulation for Geomaterials

The mechanical behavior of saturated geomaterials is largely governed by the interaction of the solid skeleton with the fluids present in the pore structure. Traditional geotechnical analyses, commonly based on simplified effective stress theories, fail to fully describe the behavior of saturated porous materials. Hence, it has become necessary to use more robust and complete formulations. In this context, the use of multiphase theories appears to be an alternative and more appropriate approach. In this work, the governing equations of a porous media interacting with immiscible porous fluids are presented in the light of the theory of mixtures. A generalized Galekin procedure is devised to establish the coupled mixed finite-element equation set with u-p-U form. An unconditionally stable implicit solution procedure is used for the time domain numerical solution. Finally, a recently developed constitutive model based on the fuzzy set plasticity concept is described and implemented in the finite-element tool. A companion paper focuses on applications of this theory, and case studies are used to evaluate the proposed formulation.     

Forced Vertical Vibration of Circular Plate in Multilayered Poroelastic Medium

This paper considers the vertical vibrations of an elastic circular plate in a multilayered poroelastic half space. The plate is subjected to axisymmetric time–harmonic vertical loading and its response is governed by the classical thin-plate theory. The contact surface between the plate and the multilayered half space is assumed to be smooth and either fully permeable or impermeable. The half space under consideration consists of a number of layers with different thicknesses and material properties and is governed by Biot’s poroelastodynamic theory. The vertical displacement of the plate is represented by an admissible function containing a set of generalized coordinates. Contact stress and pore pressure jump are established in terms of generalized coordinates through the solution of flexibility equations based on the influence functions corresponding to vertical and pore pressure loading. Solutions for generalized coordinates are obtained by establishing the equation of motion of the plate through the application of Lagrange’s equations of motion. Selected numerical results are presented to portray the influence of various parameters on dynamic interaction between an elastic plate and a multilayered poroelastic half space.     

Wave Attenuation over a Rigid Porous Medium on a Sandy Seabed

In this study, an analytic solution of wave interaction with a rigid porous medium above a poro-elastic sandy bottom is derived to investigate the attenuation of the surface wave and the wave-induced soil response. In the model, both inertial and damping effects of the flow are considered in the rigid porous region using the potential theory, while the consolidation theory is adopted in the sand region. A new complex dispersion relation involving parameters of the rigid porous and the poro-elastic medium is obtained. The analytic solutions are verified by some special cases, such as wave interaction with a porous structure over an impermeable bottom or wave interaction with a poro-elastic medium only. Numerical results indicate that the wave attenuation is highly dependent upon the thickness of the rigid porous layer, the soil stiffness, and their respective coefficients of permeability. Increasing the thickness of the rigid porous layer will shorten the wavelength of the surface wave regardless of the sand coarseness. The pore pressure in fine-sand is larger than in coarse sand, with both decaying with wave progression. It is also found that increasing the thickness of the rigid porous medium will effectively reduce the pore pressure in the sand. For the applications, an extended hyperbolic mild-slope equation is finally obtained, based on the basic analytic solutions. Examples of the wave height transformation over submerged permeable breakwaters on a slope sandy seabed are given. The simulated results show that the wave decay of the coarse sand seabed is larger than those of fine-sand and impermeable seabeds when waves pass after the submerged porous breakwater. The wave damping versus the friction factor for various height of the submerged breakwater is discussed.     

Unsaturated-Flow Simulation with Green Element Models

Flows in variably saturated media are of profound interest to numerical analysts, engineers, and scientists because of not only the challenge they pose as a result of their highly nonlinear constitutive relations but also their importance in many fields of engineering such as drainage, irrigation, hydrology, environmental, soil, and petroleum engineering. In this paper, the Picard and Newton-Raphson (N-R) algorithms are incorporated into the Green element method (GEM) to simulate these flows. The GEM offers a viable means of implementing the singular boundary integral theory so that the theory is more generally applicable and computationally efficient. Here GEM discretizes the integro-differential equation in space with suitable polygonal elements and in time with a difference scheme, and the system of nonlinear discretized element equations are linearized by the Picard and N-R algorithms. Calculations carried on three numerical examples of infiltration into unsaturated soils in two spatial dimensions indicate better convergence of the N-R algorithm than the Picard algorithm at comparable computational cost.     

Dynamic Axial Load Transfer from Elastic Bar to Poroelastic Medium

The time-harmonic response of a cylindrical elastic bar (pile) partially embedded in a homogeneous poroelastic medium and subjected to a vertical load is considered. The bar is modeled using 1D elastic theory valid for long bars in the low-frequency range, and the porous medium using Biot's 3D elastodynamic theory. The bar is bonded to the surrounding medium along the contact surface. The problem is formulated by decomposing the bar∕porous medium system into a fictitious bar and an extended porous medium. A Fredholm's integral equation of the second kind governs the distribution of axial force in the fictitious bar. The integral equation involves kernels that are displacement and strain influence functions of a poroelastic half-space subjected to a buried, uniform vertical patch load. The governing integral equation is solved by applying numerical quadrature. The solutions for axial displacement and axial force of the bar, and the pore pressure are also derived. Selected numerical results for vertical impedance, axial force, and pore pressure profiles are presented to portray the influence of bar stiffness and length∕radius ratio, frequency of excitation, and poroelastic properties.     

渗透系数对饱和土强夯效果影响的数值模拟

采用FLAC3D数值软件建立考虑强夯非线性、大变形和流固耦合特性的饱和土地基三维强夯计算模型;应用所建立的计算模型进行渗透系数在0.0001～0.1 cm/s范围内的数值模拟计算,分析渗透系数变化时的位移、孔压、密度和塑性体积的变化特性.从数值计算结果证明了饱和土地基夯击瞬间加固效果与渗透性能有关,渗透系数越大,瞬间夯击效果越好.     

Unilateral Contact Problem for Aging Viscoelastic Beams

The frictionless unilateral contact problem of a viscoelastic Bernoulli–Euler beam resting on a viscoelastic soil is studied. The mathematical formulation involves equilibrium equations, compatibility equations, and constitutive laws, with an aging integral-type form. The unilateral nature of the contact is imposed through a compatibility inequality, which allows the determination of the contact imprint at each time. Further, the governing integro-differential equation for the unknown contact pressure is derived. As special cases, the elastic Winkler-type soil and the rigid soil conditions are discussed. A numerical approach is presented, which employs the finite difference method along space and an adaptive step-by-step algorithm along time. The procedure allows for time discontinuities of both external loads and contact pressure. Several selected numerical examples are presented and the influence of the most important material and geometrical parameters are shown. For the simplest situations, it was possible to compare the results obtained with known analytical solutions.     

Buckling of Circular Mindlin Plates with an Internal Ring Support and Elastically Restrained Edge

This paper is concerned with the elastic buckling problem of circular Mindlin plates with a concentric internal ring support and elastically restrained edge. In solving this problem analytically, the circular plate is first divided into an annular segment and a core circular segment at the location of the internal ring support. Based on the Mindlin plate theory, the governing differential equations for the annular and circular segments are then solved exactly and the solutions brought together by using the interfacial conditions. New exact critical buckling loads of circular Mindlin plate with an internal ring support and elastically restrained edge are presented for the first time. The optimal radius of the internal ring support for maximum buckling load is also found. An approximate relationship between the buckling loads of such circular plates based on the classical thin plate theory and the Mindlin plate theory is also explored.     

Theory of Plastic Sand Flow with Fluid Pressure Effect

Fundamental principles of elastic–plastic mechanics of soils and rocks are given on the base of the original publications. The solid friction and dilatancy effects are included in the nonstandard form of nonassociative rule of plastic flow. The resulting hyperbolic system of equations is represented for a plane case. The slip surfaces are assumed to be jump tangential discontinuities of a velocity field. The possibility of limit equilibrium at slip surfaces is accounted for. The attempts to account for grain rotations, permitting study of slip surface structure, are discussed. The Biot–Frenkel model of interpenetrating continua is developed for plastic flow of porous saturated matrix. In this case the solid matrix state is determined by the effective stresses and pore pressure diffusion happens in plastically flowing matrix. To illustrate the theory possibilities, solutions for failure and mass sand flow, driven by the pore pressure gradient, are selected. They are important especially for oil/gas reservoirs with a weak matrix, typical for offshore geology.     

Vertical Vibrations of a Rigid Foundation Embedded in a Poroelastic Half Space

This paper considers the steady-state vertical vibrations of a rigid, cylindrical massive foundation embedded in a poroelastic soil. The foundation is subjected to time-harmonic vertical loading and is perfectly bonded to the surrounding soil. The contact surface between the foundation and the soil is assumed to be smooth and fully permeable. Biot’s poroelastodynamic theory is used in the analysis. The soil underlying the foundation base is assumed to be a homogeneous poroelastic half space while the soil along the side of the foundation is assumed to consist of a series of infinitesimally thin layers. The dynamic interaction problem is solved by a simplified analytical method. The accuracy of the present solution is verified by comparisons with existing solutions for both elastodynamic and poroelastodynamic interaction problems. Selected numerical results for the vertical dynamic impedance and response factor of the rigid foundation are presented to demonstrate the influence of nondimensional frequency of excitation, depth ratio, mass ratio, shear modulus of the backfill, and poroelastic material properties on dynamic interaction between an embedded foundation and a poroelastic half space.     

Postbuckling of Beam Subjected to Intermediate Follower Force

This paper investigates the postbuckling behavior of a simple beam under an intermediate follower force acting in the tangential direction to the centroidal axis of the beam. One end of the beam is pinned, while the other end is attached to a roller support. Two approaches have been used in this study. The first approach is based on the elastica theory. The governing equations are derived and solved analytically for the exact closed form solutions that include the equilibrium configurations of the beam, equilibrium paths, and bending moment distribution of the beam. The exact solutions take the form of elliptic integrals of the first and second kinds. In the second approach, the shooting method is employed to solve a set of nonlinear differential equations with the boundary and intermediate conditions. The equations are integrated by using the Runge–Kutta algorithm. The error norms of the end and intermediate conditions are minimized to within a prescribed tolerance error. A comparison study between the analytical elliptic integral solutions and the numerical shooting method solutions show excellent agreement of results. Special features of the solutions are also highlighted.     

Simulating Seismic Response of Cantilever Retaining Walls

Many failures of retaining walls during earthquakes occurred near waterfront. A reasonably accurate evaluation of earthquake effects under such circumstance requires proven analytical models for dynamic earth pressure, hydrodynamic pressure, and excess pore pressure. However, such analytical procedures, especially for excess pore pressure, are not available and hence comprehensive numerical procedures are needed. This paper presents the results of a finite-element simulation of a flexible, cantilever retaining wall with dry and saturated backfill under earthquake loading, and the results are compared with that of a centrifuge test. It is found that bending moments in the wall increased significantly during earthquakes both when backfill is dry or saturated. After base shaking, the residual moment on the wall was also significantly higher than the moment under static loading. Liquefaction of backfill soil contributed to the failure of the wall, which had large outward movement and uneven subsidence in the backfill. The numerical simulation was able to model quite well the main characteristics of acceleration, bending moment, displacement, and excess pore pressure recorded in the centrifuge test in most cases with the simulation for dry backfill slightly better than that for saturated backfill.     

Buckling of Column with Base Attached to Elastic Half-Space

Buckling of a heavy elastic column loaded by a concentrated force at the top is analyzed. It is assumed that the base of the column is fixed to a rigid circular plate that is positioned on a homogeneous, isotropic, linearly elastic half-space. The plate has adhesive contact with the half-space. The constitutive equations for the column are assumed in the form that allows axial compressibility and takes into account the influence of shear stresses. It is shown that eigenvalues of the linearized equations determine the bifurcation points of the full nonlinear system of equilibrium equations. The type of bifurcation at the lowest eigenvalue is examined and is shown that it could be super- or subcritical. The postcritical shape of the column is determined by numerical integration of the equilibrium equations.     

Analytical Method for Predicting the Mobility of Slow-Moving Landslides owing to Groundwater Fluctuations

This paper deals with the active landslides that are controlled by pore water pressure changes owing to groundwater fluctuations. These landslides are usually characterized by low displacement rate with deformations essentially concentrated within a narrow shear zone, above which the unstable soil mass moves as a rigid body. Taking advantage of some original analytical solutions, a method is developed for a preliminary prediction of the landslide mobility. This method is based on a simple sliding-block model and allows the landslide velocity to be readily evaluated once pore water pressure measurements are available. An application to a case study documented in the literature is also shown.     

Assessment of Dynamic Stability of Foundations on Saturated Sandy Soils

This paper deals with the dynamic analysis of foundations in saturated soils. In the first part, a mathematical formulation is briefly outlined in which soil is considered as a two-phase medium comprising the soil skeleton and voids filled with a viscous fluid. Such formulation is suitable for describing a solid-fluid transition associated with the liquefaction phenomenon. Subsequently, the notion of dynamic stability is reviewed and a simple criterion is introduced, leading to the definition of a stability factor. The mathematical framework is illustrated by a numerical example involving a foundation subjected to seismic excitation. The effect of viscosity of liquefied material on the stability of the system is examined.     

Analyses of Inclined Boreholes in Poroelastic Media

Solutions for the borehole problem based on Biot’s poroelasticity theory have shown that drilling through fluid saturated formations, gives rise to time-dependent stress and pore pressure fields in the vicinity of the borehole. As a natural consequence, borehole stability in such formations is of a time-dependent nature. However, existing analyses are based on two limiting cases viz., constant pore pressure and/or no flux boundary conditions at the borehole wall. Adding time dependency to the pore fluid boundary conditions can simulate realistic field conditions such as those observed during hydraulic fracturing, fluid injection or development of filter cake. Analytical solutions for inclined boreholes with time-dependent pore pressure and flux boundary conditions at the borehole wall are presented in this paper. Analysis is carried out for two special cases of the ramp-type pore pressure and linearly reducing flux boundary conditions. Analytical solutions are supplemented with asymptotic solutions for small and large time analysis. The effects of these conditions on stress concentrations near the borehole wall and their implications on borehole stability are examined in detail.     

Effect of Fine Particle Migration on the Small-Strain Stiffness of Unsaturated Soils

The paper presents the results of an experimental investigation of fine particle migration from pore body to the pore throat and toward the contact between particles and its effect on skeleton stiffness of granular materials. We hypothesize that the suspended colloids in the pore fluid migrate and deposit on the contact surface between the skeleton-forming particles and change the magnitude of the soil stiffness. Three specimens were prepared using uniform spherical glass particles that were saturated with deionized water and kaolinite or silt-base slurries. The specimens were drained by evaporation which retained the fines in the soil while increasing the matric suction. Changes in soil dynamic stiffness were evaluated using piezoelectric transducers while the migration of fines and the changes of the properties of the pore fluid were monitored using synchrotron X-ray microcomputed tomography (SMCT) on identical specimens. The wave propagation experiments show that the stiffness of the tested specimens increased at different rates during the drying processes. These measurements were complemented with SMCT scanning analysis that shows an increase in mass density of the remaining slurry as the pore fluid concentrated near the particle contacts. The results indicate that the soil stiffness increase due to the alteration of the pore fluid at the particles’ contact and changes caused at the contact behavior itself. These results provide an insight about parameters that influence soil stiffness which may help in better predictions of stiffness changes in compacted soils during moisture changes.