Saturated Poroelastic Half-space Research Articles

A comparison of two numerical models for modeling a twin tunnel in a saturated half-space

In this work an analytical as well as a numerical model are established for computing vibrations induced by a twin tunnel buried in a saturated poroelastic half-space. In the analytical solution the mechanical response for the half-space ground is expressed as a superposition of planar waves from the ground surface and cylindrical waves from two tunnel-soil interfaces. While in the numerical solution, three-dimensional saturated poroelastic finite element and multi-transmitting formula are formulated. The comparisons are made between the two methodologies, and the advantages and disadvantages of the two models are highlighted. Using the analytical one, the influence of the fluid in the soils, the reflection effect of the ground surface and the existence of a neighboring tunnel with or without a source on the mechanical response is investigated. By using the three-dimensional numerical model, the influence of various irregular tunnels on the surface vibration is investigated. The results suggest that: for the displacement and pore pressure response, the difference between one source tunnel and twin source tunnel can be up to 20 dB; the rectangular tunnels lead to the largest response followed by horseshoes and circular ones.

SV-wave scattering by an underwater circular cylindrical canyon in a saturated poroelastic half-space

To study the effects of underwater terrain on plane seismic waves (P-SV wave) propagation, the theoretical solution of ground displacement, strain, rocking, and energy for the underwater terrain is derived based on Biot’s theory. Numerical results in terms of incident angle and Poisson’s ratio are illustrated for various porosities and degrees of solid frame stiffness. The results show that the response of a saturated-fluid porous soil under overlying water is controlled by the solid frame stiffness and Poisson’s ratio. Considering the effect of complex site amplification, the scattering theoretical solution of plane SV wave by a circular-arc canyon with overlying water is first derived based on the free-wave field above obtained. The effects of the incident frequencies, the incident angles, and the circular arc canyon on the ground motion of the soil-water interface are investigated parametrically. In particular, the effect of overlying water on ground motion is studied, and the results indicate that the existence of an overlying water layer absorbs the energy of a seismic wave to a certain extent. Meanwhile, the accuracy of the results is also verified by reducing the complex site to that without an overlying water layer and no canyon. Finally, based on the theoretical solution derived above, the coherence functions and the underground variable seismic motion of the site also are simulated, respectively. Compared with ground motions, the numerical results show that underground motions are smaller, which indicates that the canyon has a certain amplification effect on the seismic motions.

Exact artificial boundary condition and 1D time-domain method for seismic response analysis of layered saturated poroelastic half-space

This study proposed a one-dimensional (1D) time-domain finite element method (FEM) for seismic response analysis of a layered saturated soil half-space under obliquely incident plane waves, based on Biot's u-w wave equations. The spatially two-dimensional problem was transformed into an equivalent 1D time-domain problem along the depth according to Snell's law. An exact artificial boundary condition for saturated poroelastic media was developed to simulate the wave absorption and input effects of the truncated half-space. Using the displacement and stress continuity conditions between the adjacent saturated poroelastic-medium layers and combining all finite elements and the exact artificial boundary condition, the total dynamic equation for the spatially 1D problem was obtained. The 1D dynamic equation was solved using the FEM with explicit time integration. The proposed method was verified by analyzing two engineering sites in the time domain and a comparison with the frequency-domain transfer matrix method with fast Fourier transform.

Energy dissipation of an infinite damping beam supported by saturated poroelastic halfspace

Reproducing the dissipative effects in the non-conservative dynamic system numerically is one of the challenges in the numerical method because that, both the artificial dissipation of the numerical method and the real dissipation of the system are contained in the numerical results. In this paper, a complex structure-preserving numerical approach with tiny artificial dissipation is developed to investigate the energy dissipation in the road-foundation interaction system subjected to a moving load. Simplifying the road as an infinite damping beam with a finite width and the foundation as a saturated poroelastic halfspace with viscosity, the nonlinear coupling damping dynamic model is established. The energy dissipation laws of the road-foundation interaction system are revealed with different parameters of the moving load in the numerical simulations by the complex structure-preserving approach. The complex structure-preserving approach developed in this paper provides a new way to analyze the coupling dissipative problems. In addition, the energy dissipation laws obtained from the numerical results give some suggestions on the road design and the foundation design in engineering.

Application of Biot’s Poroelasticity to Seismic Analysis of Subway Stations in a Saturated Poroelastic Half-space: Effects of Viscous Coupling

ABSTRACT This paper presents a parametric investigation on the effects of viscous coupling on the seismic responses of subway stations in a saturated poroelastic half-space which is modeled as a two-phase medium by the Biot’s theory. Moreover, an equivalent single-phase model for the saturated poroelastic half-space is introduced, and comparisons between the single-phase solutions and the two-phase solutions are made. The results are obtained by a finite element-indirect boundary element coupling method. The study shows that the viscous coupling has significant influences on the seismic internal forces and deformation of the subway stations, and the single-phase approximation can cause large errors.

Aseismic Performance Analysis of Composite Lining Embedded in Saturated Poroelastic Half Space

Abstract Earthquakes have been found to seriously threaten the structure of tunnels. To study the seismic performance of a composite lining embedded in a saturated poroelastic half space under an i...

An analytical model for calculating vibrations from twin tunnels in a saturated poroelastic half-space

In this paper a semi-analytical solution is proposed for modelling vibrations from a twin tunnel in a saturated poroelastic half-space. The twin tunnels are simulated as two hollow cylinders made of a viscoelastic material. The half-space containing the two circular cavities is described as a two-phase poroelastic material. The wave field in the half-space with two cylindrical cavities is taken as the combination of down-going plane waves from the ground surface and outgoing cylindrical waves from each tunnel. With the help of transformation and translation properties of the plane and cylindrical waves the boundary condition at each scattering surface can be expressed with the appropriate coordinates. Numerical results suggest that the reflection effect of the ground surface has limited influence on the response of the tunnels, but has a greater influence on the free-field response near the surface. Compared with the single tunnel solution, the wave field above the twin tunnels is greatly distorted.

The influence of pore-fluid in the soil on ground vibrations from a tunnel embedded in a layered half-space

A computationally efficient semi-analytical solution for ground-borne vibrations from underground railways is proposed and used to investigate the influence of hydraulic boundary conditions at the scattering surfaces and the moving ground water table on ground vibrations. The arrangement of a dry soil layer with varying thickness resting on a saturated poroelastic half-space, which includes a circular tunnel subject to a harmonic load at the tunnel invert, creates the scenario of a moving water table for research purposes in this paper. The tunnel is modelled as a hollow cylinder, which is made of viscoelastic material and buried in the half-space below the ground water table. The wave field in the dry soil layer consists of up-going and down-going waves while the wave field in the tunnel wall consists of outgoing and regular cylindrical waves. The complete solution for the saturated half-space with a cylindrical hole is composed of down-going plane waves and outgoing cylindrical waves. By adopting traction-free boundary conditions on the ground surface and continuity conditions at the interfaces of the two soil layers and of the tunnel and the surrounding soil, a set of algebraic equations can be obtained and solved in the transformed domain. Numerical results show that the moving ground water table can cause an uncertainty of up to 20 dB for surface vibrations.

The equivalent stiffness of a saturated poroelastic halfspace interacting with an infinite beam under a moving point load

The equivalent stiffness of a saturated poroelastic halfspace that supports an infinite Euler-Bernoulli beam under a moving point load has been investigated in this paper. The smooth contact condition is assumed for the interface of beam and halfspace, however, with an improved continuity condition, i.e. the continuities of contact normal stress and contact vertical displacements are imposed across the width of the beam. The equivalent stiffness of the halfspace has been evaluated using a contour integration approach such that contributions of the Rayleigh pole and the dispersion branches can be explicitly taken into account. Influences of the continuity condition and the permeability on the equivalent stiffness have been studied. It is found that the imaginary part of the equivalent stiffness is nonzero since the viscous coupling between two phases of the saturated halfspace has been considered. This observation fundamentally differentiates the present paper to the work by Shi and Selvadurai (2016) where an infinite permeability has been assumed for the halfspace, i.e. null viscous coupling.

Analysis of Surface Waves in Saturated Layered Poroelastic Half-Spaces Using the Thin Layer Method

There are multiple modes of surface waves in saturated layered poroelastic half-spaces. The phase velocity and the attenuation of the modes are frequency dependent. The frequency behaviour of the modes can be studied using the layer transfer, stiffness and the transmission/reflection matrix methods. However, it is very difficult to find the complex roots of the determinants because the entries of the matrices involve the complex exponential functions of the wavenumber and the thickness of layer. To overcome this difficulty, the entries in the matrix are expressed in the form of algebraic functions using the thin layer method. Thus, the eigenvalues and eigenvectors can be easily solved using the matrix decomposition techniques instead of the root-searching ones. Some of the eigenvalues correspond to the wavenumbers of the surface waves, and can be picked out based on the characteristics of the surface waves. The frequency behaviour, variations of the pore pressure and the skeleton’s displacements with the depth can be then investigated from the corresponding eigenvalues and eigenvectors, respectively. The method is verified by comparing the analytical and the discrete results in the saturated poroelastic half-space with the permeable surface. The method is applied to appreciate the effects of an impermeable surface on Rayleigh waves (R-waves) and the existence of Stoneley waves in the poroelastic half-space. The frequency behaviour of Rayleigh waves in three typical layered poroelastic half-spaces is also analyzed.

Closed-Form Analytical Solution for Vibrations from a Tunnel Embedded in a Saturated Poroelastic Half-Space

A closed-form analytical solution for ground-borne vibrations from a tunnel embedded in a saturated poroelastic half-space is proposed in this paper. The tunnel modeled as an elastic hollow cylinder is surrounded by the water-saturated poroelastic material. The total wave field in the poroelastic half-space with a cylindrical cavity consists of the outgoing cylindrical waves and the down-going plane waves. In addition to the traction-free condition on the ground surface and the continuous conditions of displacement and stress at the tunnel-soil interface, hydraulic boundary conditions on these two scattering surfaces are required to solve this multiple-scattering problem. The transformation properties between the plane and cylindrical wave functions are employed to apply the boundary conditions expressed in both the rectangular and cylindrical coordinate systems. Numerical results for the displacements of the solid on the ground surface and the pore pressure in the saturated soil attributable to a harmonic point load applied at the tunnel invert are presented. The influence of the soil permeability and the hydraulic boundary at the tunnel-soil interface on the free-field responses is investigated. The proposed solution can serve as a benchmark for other computational methods and as an efficient tool for predicting vibrations from a tunnel embedded in a saturated poroelastic half-space.

Prediction of Ground Vibration from High Speed Trains Using a Vehicle–Track–Ground Coupling Model

The dynamically induced ground vibration from high speed trains (HSTs) is investigated using a semi-analytical vehicle–track–ground coupling model. A multi-body vehicle is adopted along with rail irregularity considered in the model. The soil is simulated as a saturated poroelastic half-space with two elastic layers. The coupling system is solved in the transformed domain by applying the Fourier transform, and the dynamic stiffness matrix method is used to deal with the layered soil. The time-domain solutions are obtained by the inverse fast Fourier transform (FFT). The effects of the vehicle speed, observation location, rail irregularity, subgrade-bed stiffness, and vehicle type on the ground vibration are investigated thoroughly. The results show that all these factors can significantly affect the dynamically induced ground vibration.

Dynamic response of an infinite beam supported by a saturated poroelastic halfspace and subjected to a concentrated load moving at a constant velocity

• Contact problem of an infinite beam and a poroelastic halfspace is investigated. • Bounding techniques are used to prescribe the boundary conditions on the interface. • Equivalent stiffness of the halfspace is derived using contour integration method. • Two critical velocities are identified for the moving constant load. • Solutions for poroelasticity can be analytically reduced to those for elasticity. The steady-state displacements and moments in a Bernoulli–Euler beam of finite width and infinite extent, resting on a poroelastic halfspace and subjected to a concentrated load moving at a constant velocity, were investigated using the concept of the equivalent stiffness of the halfspace. Expressions for the equivalent stiffness of the saturated poroelastic halfspace interacting with the infinite beam of finite width were derived analytically using a contour integration procedure. The influence of adhesion and drainage effects between the beam and the halfspace surface is accounted for by considering “bounding techniques” for prescribing the boundary conditions at the interface. Comparisons have been made between situations for the elastic and poroelastic halfspace with regard to their equivalent stiffness and the dynamic responses of the beam for different velocities of the moving load.

Torsional vibrations of a cylindrical foundation embedded in a saturated poroelastic half-space

Considering the interactions between an embedded foundation and saturated soil, the torsional vibrations of a cylindrical foundation embedded in a saturated poroelastic medium are analyzed in this paper. Both a rigid foundation and an elastic foundation are considered. Assuming both the side surface and the bottom surface of the foundation are perfectly bonded to soil, the reaction torques that the side soil and bottom soil acting on the foundation can be gained from basic dynamic equations of the poroelastic medium. According to the dynamic equilibrium equations of a foundation under harmonic torque, the torsional vibrations of an embedded cylindrical foundation are presented. Besides, the angular amplitude of the foundation, the equivalent stiffness and damping coefficients of the soil are expressed explicitly. Selected examples are presented to investigate the influence of relevant parameters on the torsional vibrations.

The IBIEM Solution to the Scattering of Plane SV Waves around a Canyon in Saturated Poroelastic Half-Space

The scattering of plane SV waves by a two-dimensional canyon in saturated poroelastic half-space is investigated both in frequency and time domain by indirect boundary integrate equation method (IBIEM), and the amplification effect of displacement and fluid pore pressure are thoroughly discussed. It illustrates that the scattering characteristic strongly depends on the incident frequency and angle, soil porosity, boundary drainage condition, and canyon shape. Compared with the incidence of P waves, the amplification effect is not so large for critical porosity, but more obvious amplification feature can be found for incident SV waves at the critical angle.

Dynamic responses of a saturated poroelastic half-space generated by a moving truck on the uneven pavement

A truck–pavement–ground coupling model was established to study the dynamic responses of a saturated poroelastic half-space generated by a moving heavy truck on the uneven pavement. The ground was simulated as a fully saturated poroelastic half-space governed by Biot’s theory. The overlying pavement was simplified as a Kirchhoff thin plate. With the assumption of a sinusoidal pavement surface, the dynamic wheel–pavement force was obtained through a linear Hertizian contact model. The numerical results showed that this dynamic load could make considerable contributions to the stress and excess pore water pressure responses in the ground. Furthermore, the effective stress path of the soil unit beneath the pavement caused by the moving truck was firstly calculated and presented. It was found that the differences between the total stress path and the effective stress path became significant as the truck speed increased, thus the effective stress path was more suitable than total stress path to reflect the stress history of soil elements in the saturated ground during the passage of high-speed traffics.

Dynamic response of a poroelastic half-space to accelerating or decelerating trains

The dynamic response of a fully saturated poroelastic half-space due to accelerating or decelerating trains is investigated by a semi-analytical method. The ground is modeled as a saturated poroelastic half-space and Biot's theory is applied to characterize the soil medium, taking the coupling effects between the soil skeleton and the pore fluid into account. A detailed track system is considered incorporating rails, sleepers and embankment, which are modeled as Euler–Bernoulli beams, an anisotropic Kirchhoff plate, and an elastic layer, respectively. The acceleration or deceleration of the train is simulated by properly choosing the time history of the train speed using Fourier transforms combined with Fresnel integrals in the transformed domain. The time domain results are obtained by the fast Fourier transform (FFT). It is found that the deceleration of moving trains can cause a significant increase to the ground vibrations as well as the excess pore water pressure responses at the train speed 200km/h. Furthermore, the single-phase elastic soil model would underestimate the vertical displacement responses caused by both the accelerating and decelerating trains at the speed 200km/h.

Isolation of Train-Induced Ground-Borne Vibration by Trenches on a Poroelastic Half-Space

The screening effects of trenches on train-induced ground vibration are investigated by an analytical method. The embankment and soil banks beside the trench are modeled as elastic layers with appropriate widths. A fully saturated poroelastic half-space is first utilized in the investigation of the isolation of train-induced ground vibration. Biot’s theory is applied to characterize the saturated poroelastic half-space, considering the coupling between the soil skeleton and the fluid. Based on Fourier transforms and Fourier series, the governing equations are solved. The various performances of the trenches on the single-phase elastic half-space and saturated poroelastic half-space as screening barriers to mitigate the ground vibration are specially investigated. It is found for the vibration generated by the axle train load with supercritical speed that the discrepancy between the screening efficiency of the trenches on the poroelastic half-space and elastic half-space is significant; the single-phase elastic half-space model may significantly underestimate the screening effect of the trench. In addition, an excellent screening effect can be achieved by an open trench to the ground vibrations generated by the dynamic wheel-rail load.

Semi-analytical analysis of the isolation to moving-load induced ground vibrations by trenches on a poroelastic half-space

A semi-analytical model is proposed to investigate the screening efficiency of trenches to moving-load induced ground vibrations. The ground is modeled as a fully saturated poroelastic half-space governed by Biot's dynamic poroelastic theory. The trenches are obtained by placing three rectangular elastic layers with appropriate width upon the poroelastic half-space. By Helmholtz decomposition, the displacement fields of the elastic layers are decomposed into three scalar potentials. Analytical solutions are obtained based on Fourier transform and Fourier series in the transformed domain. The time-domain results are obtained by the fast Fourier transform (FFT). The different performances of trenches on a saturated poroelastic half-space and a single-phase elastic half-space to the moving load-induced ground vibration are identified. It is found that the discrepancy of the screening efficiencies between the two models becomes significant when the load speed approaches the Rayleigh wave speed of the ground surface. Also, some parametric studies for the screening efficiency of the trench on the poroelastic half-space are presented.

Analytical solution for the dynamic response of a saturated poroelastic half-space to harmonic stress loading

A theoretical analysis for the dynamic response of a semi-infinite fluid-bearing porous medium to external harmonic loading is presented in this study based on the decoupled poroelasticity equations of Biot (1962). A corresponding initial and boundary value problem is formulated and the analytical solution for the induced pore pressure and total dilatational stress is determined using the technique of Laplace transforms. To investigate the quantitative impact of inertial effect on the poroelastic response, the problem is also solved analytically in the diffusive model (i.e. inertial terms are ignored). Comparison of the analytical solutions obtained from two different models shows that as inertial effect is accounted for, the response undergoes in a dynamic manner but lags behind the loading by a physical factor equal to the travel time necessary for pressure wave to reach the location that is prescribed. A numerical study is then conducted for water-containing Columbia fine sandy loam at lower excitation frequencies as a representative example. Our numerical results reveal that the dynamic model yields a cyclic response for the induced pore pressure and total dilatational stress with respect to depth, but the diffusive model fails to predict this attribute. Lastly, we find in the dynamic model that effective stress may take on a positive value at some depths due to the existence of time lag in the response of pore fluid to external loading so that the solid skeleton needs to sustain excess fluid pressure. This positive value is crucial for the phenomenon of liquefaction if the loading is substantial enough.