Three Dimensional Vibration Transmission of a Plate on a Layer of Porous Medium Due to a Moving Load

Three-dimensional (3D) transmission of vibration in an infinite elastic thin plate on a layer of poroviscoelastic medium, due to a harmonic, rectangular moving load, is investigated theoretically based on Biot’s theory. The material of the medium is idealized as a uniform, fully saturated poroviscoelastic layer on bedrock. By introducing four scalar potential functions and Helmholtz decomposition theorem, analytical solutions of stress, displacement, and pore pressure with and without thin plate are derived using Fourier transform technique. Numerical results are obtained with the help of inverse Fourier transform and are used to analyze the influence of load velocity, porosity, permeability, relative stiffness of plate versus ground, and the thickness of plate on the vibration. Furthermore, the results are compared with the available dynamic response results of a non-moving load on a layer of viscoelastic material.     

Dynamic effects of moving load on a poroelastic soil medium by an approximate method

The problem of the dynamic response of a fully saturated poroelastic soil stratum on bedrock subjected to a moving load is studied by using the theory of Mei and Foda under conditions of plane strain. The applied load is considered to be the sum of a large number of harmonics with varying frequency in the form of a Fourier expansion. The method of solution considers the total field to be approximated by the superposition of an elastodynamic problem with modified elastic constants and mass density for the whole domain and a diffusion problem for the pore fluid pressure confined to a boundary layer near the free surface of the medium. Both problems are solved analytically in the frequency domain. The effects of the shear modulus, permeability and porosity of the soil medium and the velocity of the moving load on the dynamic response of the soil layer are numerically evaluated and compared with those obtained by the exact solution of the problem. It is concluded that for fine poroelastic materials, the accuracy of the present method against the exact one is excellent.     

移动荷载作用下饱和土地基中的波动特性分析

基于Biot波动方程,经过Fourier变换和逆Fourier变换后可获得波数-频率域以及时间-空间域的解析解。通过数值分析的手段研究了移动荷载作用下饱和多孔弹性地基中波的传播特性。重点就弥散曲线、多谱勒效应、波的成分和动力响应频率等几个特性进行了分析,发现饱和土地基由于比弹性地基多了一项流体介质,波动特性明显差异于弹性介质。     

高速荷载下多孔饱和地基的动力响应

研究高速荷载作用下梁与多孔饱和半空间的动力响应。由Fourier变换求解多孔饱和固体的动力基本方程，根据梁与半空间的接触条件得出多孔饱和半空间上梁的垂直位移的表达式。文中的数值算例考虑了荷载移动速度对梁的动力位移的影响，并与相应的弹性半空间问题作了对比。从算例中可以发现荷载移动速度对动力位移有很大的影响，当移动速度与半空间的表面波速相近时，地面会当产生很大的振动，同时还发现当速度大于介质的剪切波速时，多孔饱和半空间上梁的动力响应与弹性半空间上梁的动力响应有很大的差别。     

移动简谐力作用下三维多孔饱和半空间的动力问题

研究了移动荷载作用下多孔饱和地基的动力问题.应用Fourier变换求解该问题的控制偏微分方程,考虑了荷载的移动速度及振动频率对多孔饱和地基动力响应的影响,重点研究了移动速度达地基表面波速时多孔饱和半空间的振动问题(马赫效应),并与相应的弹性介质的解答进行了比较.结果显示当移动速度与多孔饱和半空间的表面波速相近时,地基会产生很大的振动;当移动速度大于表面波速时,多孔饱和半空间的动力响应与弹性半空间的动力响应有较大的差别.     

A dynamic analysis of elastic circular plate on saturated poroelastic half space

In this paper the low frequency vibrations of an elastic circular plate on a saturated poroelastic half space are studied by the analytical method. First the governing equations of the dynamic problem for a saturated poroelastic medium are solved by means of Hankel transform. Then the dual integral equations of vertical forced vibration of an elastic plate on saturated poroelastic half space are established according to the mixed boundary-valued condition. By applying Abel transform the dual integral equations are reduced to a Fredholm integral equation of the second kind. Numerical examples are given at the end of the paper.     

Response of railway track system on poroelastic half-space soil medium subjected to a moving train load

Based on the dynamic poroelastic theory of Biot, dynamic responses of a track system and poroelastic half-space soil medium subjected to moving train passages are investigated by the substructure method. The whole system is divided into two separately formulated substructures, the track and the ground, and the rail is described by introducing the Green function for an infinitely long Euler beam subjected to the action of moving axle loads of the train and the reactions of the sleeper. Sleepers are represented by a continuous mass and the effect of the ballast is considered by introducing the Cosserat model for granular medium. 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 train speed. Computed results show that the shape of the rail displacements of the elastic and poroelastic soil medium are in good agreement with each other of the low train velocity, but the result of the poroelastic soil medium is significantly different to that of the elastic soil medium for the high train velocity which is higher than Rayleigh-wave speed in the soil. The influence of the soil intrinsic permeability on soil responses is discussed with great care in both time domain and frequency domain. The dynamic responses of the soil medium are considerably affected by the fluid phase as well as the load velocity.     

直角坐标下横观各向同性饱和半空间体动力响应问题的解析解

基于双重Fourier变换技术，成功求解了直角坐标系下，横观各向同性弹性饱和多孔介质的三维Biot动力方程，得到了以固体骨架位移分量和孔隙流体压力为基本未知量的积分形式一般解．进而，用一般解给出了饱和介质的总应力分量表达式。在此基础上，研究了在任意分布的表面竖向和水平谐振力作用下，横观各向同性饱和半空间体的动力响应问题。数值结果表明，采用各向同性饱和介质的动力学模型，不能准确描述具有明显各向异性特性的饱和土地基的动力性能。     

Dynamic displacements of an infinite beam on a poroelastic half space due to a moving oscillating load

This investigation is concerned with the dynamic displacements of a beam on a poroelastic half space under a periodic oscillating load of constant velocity. The governing equations for the proposed analysis are solved using Fourier transform. The expression for the vertical displacement is obtained according to the contact condition between a beam and a half space. The effects of the moving velocity and vibration frequency of the load on the dynamic displacement are considered in the numerical examples. The results show that the load velocity has significant influence on dynamic displacement. It is also noted that large differences exist between the dynamic responses for a beam on a poroelastic half space and on an elastic half space when the load velocity is larger than the shear wave speed of the medium. The reported work is supported by the National Natural Science Foundation of China (Project No. 10372073).     

受移动简谐力作用的多孔弹性半平面问题

研究了匀速移动的振动荷载作用下半无限多孔饱和固体中产生的应力和孔隙水压力．应用Fourier变换求解该问题的控制偏微分方程,考虑了荷载的移动速度及振动频率对多孔饱和固体中应力与孔隙水压力的影响,并与相应的弹性介质的解答进行了比较．结果显示多孔饱和半平面中应力和孔隙水压力随荷载的移动速度与振动频率的增加而增大,多孔饱和固体在移动荷载下的动力响应与相应的单相弹性固体的动力响应有较大的差别。     

高速移动简谐荷载下层状多孔饱和固体的动力响应

本文研究了匀速移动的振动荷载作用下层状多孔饱和固体的动力响应。应用Fourier变换求解该问题的控制偏微分方程,首先考虑了荷载的移动速度及振动频率对单层多孔饱和固体中应力与孔隙水压力的影响,并与相应的弹性介质的解答和半平面解答进行了比较;然后运用精确刚度矩阵法求解了层状多孔饱和固体的应力响应,并与单层问题进行了比较。结果显示层状多孔饱和固体中应力和孔隙水压力随荷载的移动速度的增加而明显变化,层状多孔饱和固体在移动荷载下的动力响应与相应的单相弹性固体和半平面固体的动力响应有较大的差别。     

Effect of imperfect bonding on axisymmetric elastodynamic response of a lined circular tunnel in poroelastic soil due to a moving ring load

A two-dimensional elasticity analysis for steady-state axisymmetric dynamic response of an arbitrarily thick elastic homogeneous hollow cylinder of infinite length, which is imperfectly bonded to the surrounding fluid-saturated permeable formation, subject to an axially moving ring load, is presented. The problem solution is derived by using Biot’s dynamic theory of poroelasticity in conjunction with double Fourier transformation with respect to time (frequency) and axial coordinate (axial wave number). The analytical results are illustrated with numerical examples in which a concrete tunnel lining of uniform wall thickness is imperfectly bonded to a surrounding water-saturated poroelastic formation of soft/stiff frame characteristic. Numerical solutions for the radial shell mid-plane and formation displacements are calculated by analytical (numerical) inversion of the Fourier transformation with respect to the frequency (axial wave number). Primary attention is focused on the influence of bonding condition at the liner/soil interface, formation material type, and load velocity on the system’s dynamic response. Limiting cases are considered and good agreements with the solutions available in the literature are obtained.     

Steady state responses of poroelastic half-space soil medium to a moving rectangular load

The dynamic steady state responses of a poroelastic half-space soil medium subjected to a moving rectangular load are investigated analytically/numerically. The full dynamic poroelastic theory of Biot is employed, under the assumption of an incompressible solid grain and neglecting the apparent mass density. Using the Fourier transform, the governing equations of motion are then reduced to a system of four coupled ordinary differential equations which are solved semi-analytically. Soil vertical displacements, accelerations and pore water pressures induced by moving load are calculated. Computed result shows that load velocity and intrinsic permeability of the soil medium shows an apparent effect on its dynamic responses and pore water pressures.     

Layer-element analysis of multilayered saturated soils subject to axisymmetric vertical time-harmonic excitation

The analytical layer-elements for a single poroelastic soil layer and the underlying half-space are established using an algebraic manipulation and Hankel transform. According to the boundary conditions and adjacent continuity conditions of general stresses and displacements, a global matrix equation in the transform domain for multilayered saturated soil media is assembled and solved. Solutions in the frequency domain can be further obtained with an inverse Hankel transform. Numerical examples are used to examine accuracy of the present method and demonstrate effects of soil parameters and load conditions on dynamic responses of the multilayered poroelastic saturated soils.     

A numerical model for the isolation of moving-load induced vibrations by pile rows embedded in layered porous media

A numerical model is developed to analyse the isolation of moving-load induced vibrations using pile rows embedded in a layered poroelastic half-space. Based on Biot’s theory and the transmission and reflection matrices (TRM) method, the free wave field solution for a moving load applied on the surface of a layered poroelastic half-space and the fundamental solution for an harmonic circular patch load are determined. Using Muki and Sternberg’s method, the second kind of frequency domain Fredholm integral equations for the dynamic interaction between pile rows and the layered poroelastic half-space are derived. The time domain solution is recovered via inverse Fourier transform in order to obtain the amplitude reduction ratio and thus assess the vibration isolation efficiency of pile rows. A special case of the present model shows good agreement with an existing solution. Numerical results of this study show that the speed of moving loads has an important influence on the isolation of vibrations by pile rows: the same pile rows can achieve better isolation efficiencies for higher speed loads than for lower speed loads. Pile rows embedded in a two-layered poroelastic half-space with a softer overlying layer usually generate better vibration isolation effects than those with a stiffer overlying layer. Finally, better isolation vibration may be realized by increasing the pile length and decreasing the net spacing between neighboring piles in a pile row.     

层状横观各向同性饱和土的非轴对称动力响应

通过方位角的Fourier变换,将圆柱坐标系下横观各向同性饱和土的Biot非轴对称波动方 程转化为一组一阶常微分方程组. 然后基于径向Hankel变换,建立问题的状态方程;求解状态方程后,得到传递矩阵. 进而利用传递矩阵,结合饱和层状地基的边界条件、排水条件及层间接触和连续条件,求解 了任意震源力作用下层状横观各向同性饱和地基频域动力响应问题. 时域解可通过频率的Fourier积分得到.     

饱和土中端承桩纵向振动特性研究

基于饱和多孔介质理论研究了三维轴对称条件下端承桩在饱和土中的纵向耦合振动. 首先通 过引入势函数对Biot动力固结方程解耦，采用算子分解方法及分离变量法求得饱和土层振动 解，进而利用桩土完全耦合条件得到桩土系统定解. 然后对桩顶的频率和时域响应进行参数 研究，结果表明，桩的长径比和桩土模量比对桩顶动力响应有较大影响，而渗透力对其影响 较小. 最后将幅频及时域反射波理论拟合结果与桩基实测结果加以对比，结果表明理论曲线 与实测曲线规律一致.     

Effects of the dynamic wheel–rail interaction on the ground vibration generated by a moving train

Based on Biot’s fully dynamic poroelastic theory, the dynamic responses of the poroelastic half-space soil medium due to quasi-static and dynamic loads from a moving train are investigated semi-analytically. The dynamic loads are assumed to be generated from the rail surface irregularities. The vehicle dynamics model is used to simulate the axle loads (quasi-static loads) and the dynamic loads from a moving train. The compatibility of the displacements at wheel–rail contact points couple the vehicle and the track–ground subsystem, and yield equations for the dynamic wheel–rail loads. A linearized Hertzian contact spring between the wheel and rail is introduced to calculate the dynamic loads. Using the Fourier transform, the governing equations for the poroelastic half-space are then solved in the frequency–wavenumber domain. The time domain responses are evaluated by the fast inverse Fourier transform. Numerical results show that the dynamic loads can make important contribution to dynamic response of the poroelastic half-space for different train speed, and the dynamically induced responses lie in a higher frequency range. The ground vibrations caused by the moving train can be intensified as the primary suspension stiffness of the vehicle increases.     

爆炸荷载作用下饱和半空间中衬砌隧道弹性动力响应IBIEM模拟

基于Biot两相介质理论,采用一种高精度间接边界积分方程法（IBIEM）研究了饱和半空间中浅埋衬砌隧道在内部爆炸荷载作用下的瞬态弹性动力反应。通过典型算例,给出了爆炸荷载作用下隧道附近地表位移、衬砌动应力、围岩径向位移和衬砌表面孔隙水压的时程响应,并对比分析了饱和半空间和全空间中隧道动力响应的区别。研究表明：覆土层厚度对浅埋隧道-围岩整体动力响应特征具有明显影响;衬砌表面透水状态对爆炸荷载的时程响应的影响不显著;随半空间饱和介质孔隙率增加,围岩受隧道内部爆炸影响程度降低,衬砌承担的爆炸作用增大;当和直达波、衬砌内部反射波的峰值叠加作用时,半空间表面反射波对衬砌隧道拱顶附近响应影响显著,使得衬砌动应力幅值、径向位移相比深埋情况大幅度增加。     

THE AXISYMMETRIC MIXED BOUNDARY-VALUE PROBLEM OF THE VERTICAL VIBRATION OF A RIGID FOUNDATIONON SATURATED LAYERED SOIL SUBGRADE

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