Horizontal vibrations of a disk on a poroelastic half-space

This paper analytically examines the horizontal vibration of a rigid disk on a saturated poroelastic half-space. The pressure-solid displacement form of the harmonic equations of motion for asymmetric dynamic problem are developed from the form of the equations originally presented by Biot. Making use of a new method the solution of the above equations is obtained. According to the mixed boundary -value conditions, the dual integral equations of the horizontal vibration of a rigid disk on a saturated poroelastic half-space are established. By appropriate transforms, it is shown that the dual integral equations can be reduced to a pair of Fredholm integral equations of the second kind, whose solutions are then computed. Numerical results for the horizontal dynamic compliance coefficient are given at the end of this paper.     

Vertical dynamic response of a disk on a saturated poroelastic half-space

This paper considers the vertical dynamic response of a disk on a saturated poroelastic half-space. Firstly the pressure-solid displacement form of the harmonic equations of motion for a poroelastic solid are developed from the form of the equations originally presented by Biot. These equations are solved by a new method. Then the mixed boundary value problem for the vertical harmonic vibration of a disk on a poroelastic half-space is studied. The two types of drainage conditions at the surface of the poroelastic half-space are considered: (a) the surface of the poroelastic half-space is assumed to be completely pervious both within and exterior to the plate; (b) The interface between the plate and the poroelastic half-space is assumed to be impervious and the exterior region is assumed to be pervious. By using the Hankel transform techniques, the paper develops the governing dual integral equations. These governing integral equations are further reduced to systems of standard Fredholm integral equations of the second kind by Abel transform.     

Time harmonic point load and dynamic contact problem of contacting fluid and poroelastic half-spaces

The dynamic response of contacting fluid and fluid-saturated poroelastic half- spaces to a time-harmonic vertical point force or a point pore pressure is investigated. The solutions are formulated using the boundary conditions at the fluid-porous medium interface. The point load solutions are then used to solve the dynamic problem of the vertical vibration of a rigid disc (both permeable and impermeable discs are included) on the surface of the poroelastic half-space. The contact problems are solved by integrating the point force and point pore pressure solutions over the contact area with unknown discontinuous force and pore pressure distributions, which are determined from the boundary conditions. The solutions are expressed in terms of dual integral equations, which are converted to Fredholm integral equations of the second kind and solved numerically. Selected numerical results for the vertical dynamic compliance coefficient for the cases with or without fluid overlying the poroelastic half-space are presented to show the effects of the fluid. The influence of the permeability condition of the disc on the compliance of the poroelastic half-space is investigated. The displacement, vertical stress, pore pressure in the poroelastic half-space and water pressure in the fluid half-space are also examined for different poroelastic materials and frequencies of excitation. The present results are helpful in the study of the dynamic response of foundations on the seabed under seawater.     

A coupled fluid layer–rigid disk–poroelastic half-space vibration problem

This paper proposes a coupled fluid layer–foundation–poroelastic half-space vibration model to study how still water affects foundations operating underwater. As an example, we consider the problem of the vertical vibration of a rigid disk on a poroelastic half-space covered by a fluid layer having a finite depth. The solution of the disk vibration problem is obtained using the boundary conditions at the free surface of the fluid layer and the boundary conditions at the fluid layer–poroelastic medium interface. The solution is expressed in terms of dual integral equations that are converted into Fredholm integral equations of the second kind and solved numerically. Selected numerical results for the vertical dynamic impedance coefficient are examined based on different water depths, poroelastic materials, disk permeabilities and frequencies of excitation. Based on the numerical results, it is proposed that the hydrodynamic pressure caused by the foundation vibration is the intrinsic reason that the existence of a fluid layer has such a great effect on the dynamic characteristics of the foundation. In many cases, the hydrodynamic pressure caused by the foundation vibration cannot be ignored when designing dynamic underwater foundations. These results are helpful in understanding the dynamic response of foundations under still water without water waves, such as foundations in pools, lakes and reservoirs.     

Torsional response of a rigid circular foundation on a saturated half-space to SH waves

An analytical approach is used to study the torsional vibrations of a rigid circular foundation resting on saturated soil to obliquely incident SH waves. Biot’s poroelastic dynamic theory is considered to characterize the saturated soil below the foundation, which is solved by Hankel transform later. In order to consider the scattering phenomena caused by the existence of the foundation, the total wave field in soil is classified into free-field, rigid-body scattering field and radiation scattering field. According to the classification of wave field and the mixed boundary-value conditions between the soil and the foundation, torsional vibrations of the foundation are formulated in two sets of dual integral equations. Then, the dual integral equations are reduced to Fredholm integral equation of the second kind to be solved. Combining with the dynamic equilibrium equations of the foundation, the expressions for the torsional vibrations of the foundation are obtained. Numerical results are presented to demonstrate the influence of excitation frequency, incident angle, the torsional inertia moment of the foundation and permeability of the saturated half-space on the torsional vibrations of the foundation.     

Vertical and horizontal vibrations of a rigid disc on a multilayered transversely isotropic half-space

A half-space containing horizontally multilayered regions of different transversely isotropic elastic materials as well as a homogeneous half-space as the lowest layer is considered such that the axes of material symmetries of different layers and the lowest half-space to be as depth-wise. A rigid circular disc rested on the free surface of the whole half-space is considered to be under a forced either vertical or horizontal vibration of constant amplitudes. Because of the involved integral transforms, the mixed boundary value problems due to mixed condition at the surface of the half-space are changed to some dual integral equations, which are reduced to Fredholm integral equations of second kind. With the help of contour integration, the governing Fredholm integral equations are numerically solved. Some numerical evaluations are given for different combinations of transversely isotropic layers to show the effect of degree of anisotropy of different layers on the response of the inhomogeneous half-space.     

Seismic response of an embedded pile in a transversely isotropic half-space under incident P-wave excitations

A rigorous mathematical formulation is presented for the analysis of a thin cylindrical shell embedded in a transversely isotropic half-space under vertically incident P-wave excitation. By virtue of a set of ring-loads Green's functions for the shell and a group of dynamic fundamental solutions for the half-space under arbitrary interfacial dynamic loads, the problem is shown to be reducible to a pair of Fredholm integral equations. By utilizing an adaptive-gradient family capable of capturing regular-to-singular solution transitions smoothly, an accurate numerical procedure is developed. To assess the effect of material anisotropy on the dynamic load-transfer process, a set of comprehensive numerical results presented for various material and geometrical conditions. The accuracy of the proposed numerical scheme is confirmed by its comparison with a benchmark solution for the corresponding isotropic problem.     

Three-dimensional simulation of track on poroelastic half-space vibrations due to a moving point load

An analytical approach is used to investigate dynamic responses of a track system and the poroelastic half-space soil medium subjected to a moving point load under three-dimensional condition. The whole system is divided into two separately formulated substructures, the track sub-system and the ground. The ballast supporting rails and sleepers is placed on the surface of the ground. The rail is modeled by introducing the Green function for an infinitely long Euler beam subjected to the action of the moving point load and the reaction of sleepers represented by a continuous mass. Using the double Fourier transform, the governing equations of motion are then solved analytically in the frequency–wave-number domain. The time domain responses are evaluated by the inverse Fourier transform computation for a certain load velocities. Computed results show that dynamic responses of the soil medium are considerably affected by the fluid phase as well as the load velocity.     

Seismic response of foundations embedded in a layered half-space

An integral equation technique to determine the response of foundations embedded in a layered viscoelastic half-space when subjected to various types of seismic waves is presented. The technique is validated by comparison with previous results for rigid hemispherical and cylindrical foundations embedded in a uniform half-space. Illustrative results for rigid cylindrical foundations embedded in layered media are also presented.     

Diffraction of plane SV waves by a shallow circular-arc canyon in a saturated poroelastic half-space

An analytical solution for the scattering and diffraction of incident plane SV waves by a shallow circular-arc canyon in a saturated poroelastic half-space is derived by the wave function expansion method. The solution is utilized to analyze the dependence of the computed surface motions on the incident frequencies, incident angles, porosity, boundary drainage and Poisson's ratio. It is shown that, depending on the incident angles, the surface displacement amplitudes around a canyon in a dry poroelastic half-space and saturated poroelastic half-space can be very different. The surface displacement amplitudes of an undrained saturated poroelastic half-space are close to those of a drained saturated poroelastic half-space. For low porosity, the surface displacement amplitudes of a saturated poroelastic half-space are almost identical to those of a dry poroelastic half-space, and drainage condition has little influence on the surface displacement amplitudes. But for high porosity, the effect of drainage condition becomes significant, and for the same porosity, the displacement amplitudes of an undrained saturated half-space will be larger than those of a drained saturated half-space. Poisson's ratio is also an important factor affecting the surface displacement amplitudes around the canyon, both in drained and undrained conditions, but leads to larger effects for an undrained saturated half-space than for a drained saturated half-space. Large pore pressures are found around the canyon and their amplitudes depend on the incident angles and frequencies. Below the surface, the amplitudes of pore pressures are less than they are at the surface, especially for high frequencies.     

The Green''''s function of the harmonic horizontal force applied at the interior of the saturated half-space soil

IntroductionThe wave propagation problems in saturated soil are very important for the civil engineering, geophysics and seismology. Biot (1956,1962) established the theory of wave propagation in saturated soil firstly, and hereafter many researchers have used Biot theory to study wave propagation problems in saturated soil. By using integral transform and potential function method, Philippacopoulos (1988) studied the Lamb(s problem of a vertical point force applied to the surface of saturate…     

Seismic response of a cylindrical shell embedded in a layered viscoelastic half-space. I: Formulation

A method to obtain the three-dimensional harmonic response of a infinitely long cylindrical shell of circular cross-section embedded in a layered viscoelastic half-space and subjected to harmonic plane waves impinging at an oblique angle with respect to the axis of the shell is presented. The procedure combines an indirect integral representation for the field in the exterior half-space with a model of the pipeline or tunnel based on Donnell shell theory. The integral representation for the soil is based on the use of moving Green's functions for the layered viscoelastic half-space. The accuracy of the formulation is tested by comparison of results obtained by using different discretizations. Extensive comparisons with previous two- and three-dimensional results for the case of a shell embedded in a uniform half-space and some new numerical results for a shell embedded in a multilayered half-space are presented in a companion paper.     

The Green￠s function of the harmonic horizontal force applied at the interior of the saturated half-space soil

By using integral transform methods, the Green's functions ofhorizontal harmonic force applied at the interior of the saturated half-space soil are obtained in the paper. The general solutions of the Biot dynamic equations in frequency domain are established through the use of Hankel integral transforms technique. Utilizing the above- mentioned general solutions, and the boundary conditions of the surface of the half-space and the continuous con-ditions at the plane of the horizontal force, the solutions of the boundary value problem can be determined. By the numerical inverse Hankel transforms method, the Green's functions of the harmonic horizontal force are obtainable. The degenerate case of the results deduced from this paper agrees well with the known results. Two numerical examples are given in the paper.     

The reflection of plane waves in a poroelastic half-space saturated with inviscid fluid

This paper discusses surface displacements, surface strain, rocking, and energy partitioning during reflection-of-plane waves in a fluid-saturated poroelastic half-space. The medium is modeled by Biot's theory, and is assumed to be saturated with inviscid fluid. A linear porosity-modulus relation based on experimental data on sandstones is used to determine the material parameters for Biot's model. Numerical results in terms of angle of incident waves and Poisson's ratio are illustrated for various porosities and degrees of solid frame stiffness. The results show that the amount of solid frame stiffness controls the response of a fluid-saturated porous system. A poroelastic medium with essentially dry-frame stiffness behaves like an elastic medium, and the influence of pore fluid increases as dry-frame stiffness is reduced. The effects of a second P-wave become noticeable in poroelastic media with low dry-frame stiffness.     

3D scattering of obliquely incident plane SV waves by an alluvial valley embedded in a fluid-saturated,poroelastic layered half-space

The indirect boundary element method is used to study the 3D dynamic response of an infinitely long alluvial valley embedded in a saturated layered half-space for obliquely incident SV waves. A wave-number transform is first applied along the valley’s axis to reduce a 3D problem to a 2D plane strain problem. The problem is then solved in the section perpendicular to the axis of the valley. Finally, the 3D dynamic responses of the valley are obtained by an inverse wave-number transform. The validity of the method is confirmed by comparison with relevant results. The differences between the responses around the valley embedded in dry and in saturated poroelastic medium are studied, and the effects of drainage conditions, porosity, soil layer stiffness, and soil layer thickness on the dynamic response are discussed in detail resulting in some conclusions.     

Lateral dynamic compliance of pile embedded in poroelastic half space

The time-harmonic response of a pile in a poroelastic half space and under lateral loadings is studied. By treating the pile as a one-dimensional structure and the half-space as a three-dimensional poroelastic continuum, the dynamic interaction between a pile and a poroelastic medium is formulated as a Fredholm integral equation of the second kind. Green's functions for a distributed lateral force field acting inside a poroelastic half space is an important ingredient of this paper. Numerical results for lateral dynamic compliance functions are presented to illustrate the dynamic characteristics of a pile in a poroelastic half space.     

Time-harmonic response of transversely isotropic multilayered half-space in a cylindrical coordinate system

With the aid of the analytical layer-element method, a comprehensive analytical derivation of the response of transversely isotropic multilayered half-space subjected to time-harmonic excitations is presented in a cylindrical coordinate system. Starting with the governing equations of motion and the constitutive equations of transversely isotropic elastic body, and based on the Fourier expansion, Hankel and Laplace integral transform, analytical layer-elements for a finite layer and a half-space are derived. Considering the continuity conditions on adjacent layers׳ interfaces and the boundary conditions, the global stiffness matrix equations for multilayered half-space are assembled and solved. Finally, some numerical examples are given to make a comparison with the existing solution and to demonstrate the influence of parameters on the dynamic response of the medium.     

平面P波在饱和半空间中洞室周围的散射(II):数值结果

本文通过数值计算研究了入射平面P波在饱和半空间中洞室周围散射问题,分析了入射波频率和角度、边界渗透条件、孔隙率、泊松比等参数对散射的影响。研究表明,平面P波入射情况下,饱和半空间和单相(干土)半空间中洞室附近地表位移幅值的差别很大,干土情况的水平位移幅值相对较大,饱和情况的竖向地表位移幅值相对较大;由于波在洞室附近的干涉,饱和情况与干土情况的地表位移出现相位漂移。随着孔隙率的增大,洞室附近水平地表位移幅值逐渐减小,竖向地表位移幅值则逐渐增大;当孔隙率较低时,边界渗透条件对地表位移幅值的影响很小,而当孔隙率较大时,边界渗透条件的影响则不可忽视,不透水情况下,水平和竖向地表位移幅值的峰值均相对较大;随着入射频率的增加,孔隙率的影响逐渐增大,而且不透水情况下孔隙率的影响相对较大。随着泊松比的增大,洞室附近水平地表位移幅值逐渐降低,竖向地表位移幅值则逐渐增大;泊松比较小时,边界渗透条件对位移幅值的影响较大,泊松比较大时,边界渗透条件对位移幅值的影响则较小;随着入射频率的增加,泊松比的影响逐渐增大。当孔隙率较小时,半空间地表和洞室表面孔隙水压幅值较小,但空间变化比较剧烈,随着孔隙率的增大,孔隙水压逐渐增大但空间变化逐渐平缓;随着入射频率的增加,孔隙水压幅值逐渐增大,且孔隙水压的空间变化逐渐变得复杂。     

Plain strain soil–structure interaction model for a building supported by a circular foundation embedded in a poroelastic half-space

A simple theoretical model for soil–structure interaction in water saturated poroelastic soils is presented, developed to explore if the apparent building–foundation–soil system frequency changes due to water saturation. The model consists of a shear wall supported by a rigid circular foundation embedded in a homogenous, isotropic poroelastic half-space, fully saturated by a compressible and inviscid fluid, and excited by in-plane wave motion. The motion in the soil is governed by Biot's theory of wave propagation in fluid saturated porous media. Helmholtz decomposition and wave function expansion of the two P-wave and the S-wave potentials is used to represent the motion in the soil. The boundary conditions along the contact surface between the soil and the foundation are perfect bond (i.e. welded contact) for the skeleton, and either drained or undrained hydraulic condition for the fluid (i.e. pervious or impervious foundation). For the purpose of this exploratory analysis, the zero stress condition at the free surface is relaxed in the derivation of the foundation stiffness matrix, which enables a closed form solution. The implications of this assumption are discussed, based on published comparisons for the elastic case. Also, a closed form representation is derived for the foundation driving forces for incident plane (fast) P-wave or SV wave. Numerical results and comparison with the full-scale measurements are presented in the companion paper, published in this issue.     

Scattering of SV waves by a canyon in a fluid-saturated, poroelastic layered half-space, modeled using the indirect boundary element method

The scattering of SV waves by a canyon in a fluid-saturated, poroelastic layered half-space is modeled using the indirect boundary element method in the frequency domain. The free-field responses are calculated to determine the displacements and stresses at the surface of the canyon, and fictitious distributed loads are then applied at the surface of the canyon in the free field to calculate the Green's functions for displacements and stresses. The amplitudes of the fictitious distributed loads are determined from the boundary conditions, and the displacements arising from the waves in the free field and from the fictitious distributed loads are summed to obtain the solution. The effects of fluid saturation, boundary conditions, porosity, and soil layers on the surface displacement amplitudes and phase shifts are discussed, and some useful conclusions are obtained. It is shown that the surface displacement amplitudes due to saturation and boundary conditions, different porosities, or the presence of a soil layer can be very dissimilar, and large phase shifts can be observed. The resulting wavelengths for an undrained saturated poroelastic medium are slightly longer than those for a drained saturated poroelastic medium; and are longer for a drained saturated poroelastic medium than those for a dry poroelastic medium. As porosity increases, the wavelengths become longer; and a layered half-space produces longer wavelengths than a homogeneous half-space.