Dynamic responses of unsaturated half‐space soil to a moving harmonic rectangular load

The problem of the dynamic responses of a semi‐infinite unsaturated poroelastic medium subjected to a moving rectangular load is investigated analytical/numerically. The dynamic governing equations are obtained with consideration of the compressibility of solid grain and pore fluid, inertial coupling, and viscous drag as well as capillary pressure in the unsaturated soil, and they can be easily degraded to the complete Biot's theory. Using the Fourier transform, the general solution for the equations is derived in the transformed domain, and then a corresponding boundary value problem is formulated. By introducing fast Fourier transform algorithm, the unsaturated soil vertical displacements, effective stresses, and pore pressures induced by moving load are computed, and some of the calculated results are compared with those for the degenerated solution of saturated soils and confirmed. The influences of the saturation, the load speed, and excitation frequency on the response of the unsaturated half‐space soil are investigated. The numerical results reveal that the effects of these parameters on the dynamic response of the unsaturated soil are significant.     

Dynamic response of double beam rested on stochastic foundation under harmonic moving load

This paper deals with the dynamic response of infinite double Euler–Bernoulli beam supported by elastic foundation with stochastic stiffness subjected to an oscillating moving load, which is the first research in relevant literature review. In this matter, equations of motion for double beam are formulated in a moving frame of reference. Moreover, by employing the first order perturbation theory and calculating contour integration, the response of double beam is obtained analytically and validated by a stochastic finite element model. Sensitivity analyses on the various parameters of closed form solution such as velocity, load frequency, coefficient of variation of soil foundation and rail and slab bending stiffness show the significant effect of load frequency on the dynamic response of the doubled beam. From practical point of view, the obtained results of the present study can be utilized efficiently in analysis and design of slab track systems. Copyright © 2013 John Wiley & Sons, Ltd.     

Dynamic responses of a poroelastic half‐space from moving trains caused by vertical track irregularities

The vibrations of railway tracks on a poroelastic half‐space generated by moving trains are investigated through a vehicle–track–ground coupling model. The theoretical model incorporates a vehicle, a track, and a fully saturated poroelastic half‐space soil medium. The source of vibration excitation is divided into two components: the quasi‐static loads and the dynamic loads. The quasi‐static loads are related to the static component of the axle loads, whereas the dynamic loads are due to the dynamic wheel–rail interaction. A linear Hertizian contact spring is introduced between each wheelset and the rail to consider the dynamic loads. Biot's dynamic theory is used to characterize the poroelastic half‐space soil medium. Using the Fourier transform, the governing equations for the track–ground system are solved and the numerical results are presented for a single axle vehicle model. The different dynamic characteristics of the elastic soil medium and the saturated poroelastic medium are investigated. In addition, the different roles of the moving axle loads and the roughness‐induced dynamic loads are identified. It is concluded that the vibration level of the free field off the track predicted by the poroelastic soil medium is smaller than that predicted by the elastic soil medium for vehicle speed below the Rayleigh wave speed of the poroelastic half‐space, whereas it is larger for vehicle speed above the Rayleigh wave speed. The dynamic loads play an important role in the dynamic responses of the track–ground system. Copyright © 2010 John Wiley & Sons, Ltd.     

3D dynamic response of a transversely isotropic multilayered medium subjected to a moving load

The dynamic problem of a transversely isotropic multilayered medium is reducible to quasi‐static problem by introducing a moving system that travels synchronously with the load. Based on the governing equations in the moving system, the ordinary differential equations in the Fourier transformed domain are deduced. An extended precise integration method is adopted to solve the ordinary differential equations, and the solution in the physical domain is recovered by the inverse Fourier transform. Numerical examples are performed to verify the accuracy of the presented method and to analyze the influence of material properties and the load characteristic.     

矩形运动荷载作用下软土地基的三维振动研究

通过引入势函数,并利用Helmholtz原理和Fourier变换技术,研究了运动荷载作用下有限层厚软土地基的振动,考虑了矩形分布荷载作用下振动的三维特性,使得分析更符合工程实际,给出了运动荷载作用下饱和黏弹性地基三维振动的应力、位移和孔隙水压力响应的积分形式解答。利用Fourier数值逆变换进行算例分析,讨论了荷载速度对位移及孔隙水压力分布的影响。结果表明,位移幅值随荷载速度的增加而增大,荷载不同位置处孔隙水压力的分布有很大差异。     

Dynamic response of road pavement resting on a layered poroelastic half‐space to a moving traffic load

This paper is dedicated to study the dynamic response of a thin‐plate resting on a layered poroelastic half‐space under a moving traffic load. Based on the dynamic poroelastic theory of Biot, the general solutions of the homogeneous poroelastic foundation are obtained by Fourier translation. By using the transmission and reflection matrices method in the frequency domain, the equivalent stiffness of the layered poroelastic half‐space is presented. Kirchhoff's hypotheses are applied to obtain the vertical displacement of the thin plate. By using the inverse Fourier transform, the time domain solution is obtained. As an example of three layers, the influences of the load velocity, the material properties of poroelastic layers, and the flexural rigidity of the plate on the response of the pavement system are examined. Analyses show that a soft intermediate layer results in the significant increase of vertical displacement of road pavement. Comparison with the existing work validates the present model. Copyright © 2013 John Wiley & Sons, Ltd.     

Effects of linear and exponential heterogeneity on the dynamic response of a moving load in an irregular isotropic half-space: a comparative study

Heterogeneity, being a trivial feature inside the earth or in a geostructure, makes a strong basis for its consideration in the study of geomechanics. Inclusion of the concept of heterogeneity along with irregularity in the medium brings a novelty to the existing literature related to the study of the moving load. The present study investigates the effects of linear and exponential heterogeneity on the dynamic response due to a normal load moving with constant velocity on a rough irregular heterogeneous isotropic half-space in a comparative approach. Expressions for both normal and shear stresses for either case of heterogeneity have been established in closed form. Substantial effects of the affecting parameters such as depth, irregularity factor, maximum depth of irregularity, frictional coefficient, linear heterogeneity parameter and exponential heterogeneity parameter on normal and shear stresses for both the cases of heterogeneity have been observed. Numerical computation has been carried out and the effects of said parameters have been meticulously examined by means of graphs. Moreover, different cases of heterogeneity and homogeneity along with various types of irregularity namely rectangular, parabolic and no irregularity are compared which serve as a focal theme of the study.     

Three‐dimensional finite element modeling of tire/ground interaction

Tire/ground interaction has been an important issue in terramechanics, transportation and pavement engineering. Characterization of tire/ground interaction has been majorly investigated based on empirical approaches and field tests. So far very few dynamic tire/ground interaction models have been presented. This paper presents a three‐dimensional finite element model for tire/ground interaction. The rubber material is modeled as nearly incompressible finite strain hyperelasticity instead of being modeled as a rigid wheel. The tire is also modeled as bias type and steel rebars are embedded within the tire structure. The tire model is combined with the ground model to form a tire/ground interaction model. Both single tire/ground interaction and four‐tire vehicle/pavement interaction models are presented, which allow us to investigate a lot of issues easily and accurately, such as compaction, pavement response and pavement damage. Numerous simulations are carried out to demonstrate that the dynamic tire/terrain interaction model can be used to predict the dynamic ground response due to moving vehicle. Different tire rolling conditions can be easily incorporated into the tire/ground interaction model, which further substantiates the broad application of the model in transportation and pavement engineering. Copyright © 2011 John Wiley & Sons, Ltd.     

移动环形荷载作用下饱和土中圆形衬砌隧洞动力响应研究

基于Biot理论,研究了饱和土中带有衬砌的圆形隧洞在移动环形荷载作用下的动力响应。假定衬砌为弹性体,土体为饱和多孔介质,引入两类势函数来表示土体、孔隙水和衬砌的位移,使隧洞的控制方程解耦。结合边界条件及连续条件,通过傅立叶变换得到频率-波数域中衬砌和土体的应力、位移和孔隙水压力解答,最后用傅立叶积分逆变换得到时-空域中的数值解。计算并比较了3种隧洞模型（弹性土体隧洞、饱和土体隧洞和饱和土衬砌隧洞）的动力响应分析。数值分析结果说明：（1）移动荷载速度对3种隧洞动力响应均具有较大影响;（2）弹性土体隧洞和饱和土体隧洞的动力响应具有明显区别,所以在富水地区的隧洞动力响应中土体应该视为饱和土体;（3）衬砌对隧洞动力响应有较大影响,故隧洞的动力分析中不能忽略衬砌作用。     

Three‐dimensional time‐harmonic fundamental solutions for a fluid‐saturated poroelastic half‐space with partially permeable free surface

By virtue of a pair of scalar potentials for the displacement of the solid skeleton and the pore fluid pressure field of a saturated poroelastic medium, an alternative solution method to the Helmholtz decomposition is developed for the wave propagation problems in the framework of Biot's theory. As an application, a comprehensive solution for three‐dimensional response of an isotropic poroelastic half‐space with a partially permeable hydraulic free surface under an arbitrarily distributed time‐harmonic internal force field and fluid sources is developed. The Green's functions for the poroelastic fields, corresponding to point, ring, and disk loads, are reduced to semi‐infinite complex‐valued integrals that can be evaluated numerically by an appropriate quadrature scheme. Analytical and numerical comparisons are made with existing elastic and poroelastic solutions to illustrate the quality and features of the solution. Copyright © 2016 John Wiley & Sons, Ltd.     

One‐dimensional dynamics of saturated incompressible porous media: analytical solutions and influence of inertia terms

The complexity of formulations for the hydromechanical coupled mechanics of porous media is typically minimised by simplifying assumptions such as neglecting the effect of inertia terms. For example, three formulations commonly employed to model practical problems are classified as fully dynamic, simplified dynamic and quasi‐static. Thus, depending on the porous media conditions, each formulation will have advantages and limitations. This paper presents a comprehensive analysis of these limitations when solving one‐dimensional fully saturated porous media problems in addition to a new solution that considers a more general loading situation. A phase diagram is developed to assist on the selection of which formulation is more appropriate and convenient regarding particular cases of porosity and hydraulic conductivity values. Non‐dimensional formulations are proposed to achieve this goal. Results using the analytical solutions are compared against numerical values obtained with the finite element method, and the effect of porosity is investigated. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

Three‐dimensional steady‐state response of a railway system on layered half‐space soil medium subjected to a moving train

The paper presents a three‐dimensional study on the steady‐state response of a track system and layered half‐space soil medium subjected to the load induced by the passages of a moving train with the substructure method. Practically, due to the ground water table being several meters beneath the ground surface, the soil profile can be divided into two layers: the upper layer modeled by an elastic medium and the lower layer by a fully saturated poroelastic medium governed by Biot's theory. In the meanwhile, the rails are regarded as an infinitely long Euler beam, and the sleepers are represented by a continuous mass. The effect of the ballast is accounted for by introducing the Cosserat model for granular medium, and the train is described by a series of moving axle point loads, depending on the geometry of the train. The influences of the thickness, the mass and the rigidity of the elastic layer and the mass of the ballast on rail's displacement responses are carefully investigated. Numerical results show that the influences of these parameters are significant for high train velocity, while vanishes for low velocity. Copyright © 2008 John Wiley & Sons, Ltd.     

An explicit representation of steady state response of a beam on an elastic foundation to moving harmonic line loads

A closed‐form deflection response of a beam rest is presented in this paper using the integral transform method. The theory of linear partial differential equations is used to represent the deflection of beam subjected to a moving harmonic line load in integration form. The solution is finally carried out using the inverse Fourier transform. To evaluate the integration analytically, poles of the integrand are identified with the help of algebraic equation theory. Residue theorem is then utilized to represent the integration as a contour integral in the complex plane. Closed‐form deflections and numerical results are provided for different combinations of load frequency and velocity. Copyright © 2002 John Wiley & Sons, Ltd.     

简谐荷载作用下饱和土体中圆形衬砌隧道三维动力响应分析

研究了全空间饱和土体中圆形衬砌隧道在径向简谐点荷载作用下的三维动力响应,将衬砌用无限长圆柱壳来模拟,土体用Biot饱和多孔介质模型来模拟,引入两类势函数来表示土骨架的位移和孔隙水压力,并利用修正Bessel方程来求解各势函数,结合边界条件,得到频率-波数域内衬砌和土骨架位移、孔隙水压力的解答,最后进行Fourier逆变换得到时间-空间域内的响应。通过算例分析了荷载振动频率和土体渗透性对土体和衬砌位移响应及土体孔压的影响。结果表明,饱和土体和弹性土体的位移响应具有明显区别。随着荷载频率的增大,土体和隧道位移幅值减小,土体孔压幅值增大;随着土体渗透性增大,土体位移及孔压幅值减小。     

Dynamic response to fluid extraction from a poroelastic half‐space

In this study, the dynamic response of a poroelastic half‐space to a point fluid sink is investigated using Biot's dynamic theory of poroelasticity. Based on Biot's theory, the governing field equations are re‐formulated in frequency domain with solid displacement and pore pressure. In a cylindrical coordinate system, a method of displacement potentials for axisymmetric displacement field is proposed to decouple the Biot's field equations to three scalar Helmholtz equations, and then the general solution to axisymmetric problems are obtained. The full‐space fundamental singular solution for a point sink is also derived using potential methods. The mirror‐image method is finally applied to construct the fundamental solution for a point sink buried in a poroelastic half‐space. Furthermore, a numerical study is conducted for a rock, that is, Berea sandstone, as a representative example. Copyright © 2013 John Wiley & Sons, Ltd.     

移动荷载作用下饱和土中单桩的动力响应

根据Biot动力理论,采用Fourier和Hankel变换方法得到了半空间饱和土受移动载荷及土体内受垂直简谐载荷作用下频域内基本解。根据虚拟桩法,得到了移动载荷作用下桩基的第2类Fredholm积分方程,并应用IFFT方法得到时间、空间域内单桩的动力响应。数值结果表明,移动荷载会引起桩身的负摩擦力;桩身最大轴力、孔压随移动荷载速度增加而增大;此外,在桩上端部会出现孔压集中现象。     

Reflection–transmission matrix method for the consolidation of a multilayered saturated soil

The consolidation of the layered saturated soil is an important issue in civil engineering and has been investigated extensively during the past decades. In this study, based on the Biot's theory, the reflection–transmission matrix (RTM) method for treating the layered saturated soil under axisymmetric consolidation is developed. To decouple the governing equations of the Biot's theory, the McNamee displacement functions are introduced, and the general solution for the saturated soil is obtained using the Laplace and Hankel transforms. In order to develop the RTM method for the layered saturated soil, based on the obtained general solution, the static wave vector corresponding to the state vector of the saturated soil and the transform matrix relating the aforementioned two vectors are defined. Also, the transfer matrices corresponding to the two vectors are introduced, and the representations of the RTMs for the static wave vector of the saturated soil are presented. As the state vector, static wave vector, and the transform matrix relating the two vectors are all defined in the global coordinate system, the RTMs obtained in this study thus have a reasonable physical meaning. By using the RTMs for the layered saturated soil, the solutions for the layered saturated soil subjected to external sources are derived. Comparison of results due to the proposed RTM method with some existing results and results due to the transfer matrix method validates the developed RTM method. Some numerical results are obtained based on the proposed RTM method for the layered saturated soil. Copyright © 2016 John Wiley & Sons, Ltd.