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.     

Dynamic response of a poroelastic stratum to moving oscillating load

The dynamic response of a poroelastic stratum subjected to moving load is studied. The governing dynamic equations for poroelastic medium are solved by using Fourier transform. The general solutions for the stresses and displacements in the transformed domain are established. Based on the general solutions, with the consideration of boundary conditions, the final expressions of stresses and displacements in physical domain are put forward for the three-dimensional single-layer medium. Some numerical solutions for the stresses, displacements and pore fluid pressure are presented and reveal that the response of a poroelastic stratum varies obviously with the moving velocity.     

Vibration of Timoshenko beam on hysteretically damped elastic foundation subjected to moving load

The vibration of beams on foundations under moving loads has many applications in several fields, such as pavements in highways or rails in railways. However, most of the current studies only consider the energy dissipation mechanism of the foundation through viscous behavior; this assumption is unrealistic for soils. The shear rigidity and radius of gyration of the beam are also usually excluded. Therefore, this study investigates the vibration of an infinite Timoshenko beam resting on a hysteretically damped elastic foundation under a moving load with constant or harmonic amplitude. The governing differential equations of motion are formulated on the basis of the Hamilton principle and Timoshenko beam theory, and are then transformed into two algebraic equations through a double Fourier transform with respect to moving space and time. Beam deflection is obtained by inverse fast Fourier transform. The solution is verified through comparison with the closed-form solution of an Euler-Bernoulli beam on a Winkler foundation. Numerical examples are used to investigate:(a) the effect of the spatial distribution of the load, and(b) the effects of the beam properties on the deflected shape, maximum displacement, critical frequency, and critical velocity. These findings can serve as references for the performance and safety assessment of railway and highway structures.     

Transient Waves Due to Mechanical Loads in Elasto-Thermo-Diffusive Solids

This paper deals with the study of transient waves in a homogeneous isotropic, solid half-space with a permeating substance in the context of the theory of generalized elasto-thermodiffusion. The half-space is assumed to be disturbed due to mechanical loads acting on its boundary. The model comprising of basic governing differential equations and boundary conditions has been solved by employing Laplace transform technique. Noting that the second sound effects are short lived, the small time approximations of solution for various physical quantities have been obtained and the results are discussed on the possible wave fronts. In case of continuous and periodic loads acting at the boundary, the displacement is found to be continuous at each wave front while it is discontinuous in case of impulsive load. The temperature and concentration fields are found to be discontinuous at all the wave fronts. The displacement, temperature change and concentration deviation due to impulsive, continuous and periodic mechanical loads have also been evaluated in the physical domain at all times by employing numerical inversion technique of integral transform. The computer simulated numerical results have been presented graphically in respect of displacement, temperature change and concentration deviation for brass. A significant effect of mass diffusion has been observed on the behaviour of mechanical and thermal waves.     

基于新型混合场积分方程的半空间三维均匀介质目标电磁散射分析

利用介质半空间格林函数,将一种混合场积分方程(JMCFIE)推广应用到有耗半空间的均匀介质的散射分析中,以减少离散矩阵方程迭代求解时所需的迭代步数,提高矩量法(MoM)对半空间介质目标散射分析的求解效率.数值算例验证了方法的准确性和有效性.     

The effect of critically moving loads on the vibrations of soft soils and isolated railway tracks

The dynamic response of the railway track is strongly influenced by the underlying soil. For a soft soil and very high train speeds or for a very soft soil and regular train speeds, the train speed can be close to the speed of elastic waves in the soil. This paper presents a detailed study of the so-called “moving-load effect”, i.e. an amplification of the dynamic response due to the load movement, for the tracks on soft soil. The analysis is carried out by evaluating the related integrals in the wavenumber domain. The influence of the load speed is quantified for a large set of parameters, showing that the effect on the soil vibration is reduced with increase of the frequency, track width and inverse wave velocity. Therefore, the moving-load effect associated with vibratory train loads is negligible whereas the amplification associated with the moving dead weight of the train can be significant. The strong moving-load effect on a perfectly homogeneous soil, however, can be strongly diminished by a layered or randomly varying soil situation. This theoretical result is affirmed by measurements at a test site in Germany where the trains run on a very soft soil at a near-critical speed. The results for soft soils are compared with experimental and theoretical results for a stiff soil. It is found that the influence of the stiffness of the soil is much stronger than the moving-load effect. This holds for the soil vibration as well as for the track vibration which both show a minor dependence on the load speed but a considerable dependence on the soil stiffness in theory and experiment.Railway tracks can include soft isolation elements such as rail pads, sleeper shoes and ballast mats. For these types of isolation elements and normal soil conditions, the influence of the load speed is usually negligible. There is only one isolation measure for which the moving load may be effective: a track which is constructed as a heavy mass-spring system. The resonance of this track system is shifted to lower frequencies and amplitudes for increasing train speed. A critical train speed can be reached if the mass-spring system has a marginal bending stiffness along the track.     

The fundamental solution to equations of linear magnetic hydrodynamics in a moving medium

The fundamental solution to a system of linear differential equations of magnetic hydrodynamics in a moving medium is obtained. Using the Fourier-Laplace transform, the Green tensor function is calculated as a sum of dyadics. In this way, the integral equations of magnetic hydrodynamics can easily be derived. Particular forms of the fundamental solution that are important in applications are analyzed.     

Galerkin methods for natural frequencies of high-speed axially moving beams

In this paper, natural frequencies of planar vibration of axially moving beams are numerically investigated in the supercritical ranges. In the supercritical transport speed regime, the straight equilibrium configuration becomes unstable and bifurcate in multiple equilibrium positions. The governing equations of coupled planar is reduced to two nonlinear models of transverse vibration. For motion about each bifurcated solution, those nonlinear equations are cast in the standard form of continuous gyroscopic systems by introducing a coordinate transform. The natural frequencies are investigated for the beams via the Galerkin method to truncate the corresponding governing equations without nonlinear parts into an infinite set of ordinary-differential equations under the simple support boundary. Numerical results indicate that the nonlinear coefficient has little effects on the natural frequency, and the three models predict qualitatively the same tendencies of the natural frequencies with the changing parameters and the integro-partial-differential equation yields results quantitatively closer to those of the coupled equations.     

Seismic effects of incident P waves on an embedded foundation in poroelastic half-space

Dynamic vibrations of a circular rigid foundation, which is embedded in poroelastic soil and subjected to incident P waves, are studied by semi-analytical methods in this present work. The motion of the soil is governed by Biot's dynamic poroelastic theory. A set of potentials are introduced to represent the incident waves, and the scattering waves caused by the foundation are considered based on the decomposition of the total wave field in soil. The soil along the vertical side of the foundation is assumed to be composed of series of infinitesimally thin poroelastic layers, while the soil under the foundation base is regarded as the poroelastic half-space and to be independent of the overlying soil. The interaction problem is solved by Hankel transforms. Then, combining the boundary conditions along the contact surface between the soil and the foundation and the dynamic equilibrium equation of the foundation, expressions of the vertical and rocking vibration amplitudes of the embedded foundation excited by the incident P waves are acquired. Numerical results are presented to demonstrate the influences of embedded depth, foundation mass, pore water in the soil and incident angle on the vibrations of the foundation.     

Dynamic vertical interaction of a foundation–soil system generated by seismic waves

Based on Biot's dynamic poroelastic theory, a foundation–soil interaction model is established to investigate the vertical vibrations of a rigid circular foundation on poroelastic soil excited by incident plane waves, including the fast P waves and SV waves. Scattering waves caused by the foundation and fluid–solid coupling due to the pore water in the soil are also considered in the model. The solution of the vertical vibrations of the foundation subjected to seismic waves are obtained by solving two sets of dual integral equations derived from the mixed boundary-value conditions. The different vertical vibrations of foundation rest on elastic and saturated half-space are compared. The influences of incident angle, permeability of soil and foundation mass on the vertical vibrations of the foundation are then discussed. The results show that resonant phenomenon of the foundation is observed at certain excitation frequencies; the effects of the pore water on the foundation vertical vibrations are significant. In addition, significant differences are found when the foundation is excited by P waves and SV waves, respectively.     

3D analysis of in-filled trench as passive barriers for ground vibration isolation

The three-dimensional (3D) problem of the ground vibration isolation by an in-filled trench as a passive barrier is studied theoretically. Integral equations governing Rayleigh wave scattering are derived based on the Green’s solution of Lamb problem. The integral equations are solved accurately and efficiently with an iteration technique. They are used to evaluate the complicated Rayleigh wave field generated by irregular scatterers embedded in an elastic half-space solid. The passive isolation effectiveness of ground vibration by the in-filled trench for screening Rayleigh wave is further studied in detail. Effects of relevant parameters on the effectiveness of vibration isolation are investigated and presented. The results show that a trench filled with stiff backfill material gets a better isolation effect than a soft one, and increasing the depth or width of the in-filled trench also improves its screening effectiveness. The effectiveness and the area of the screened zone are surging with the increase in the length of the in-filled trench. Supported by the National Natural Science Foundation of China (Grant Nos. 50678128 and 50538010) and the Research Fund for PhD Student of Chinese College (Grant No. 20050247030)     

Frequency analysis of finite beams on nonlinear Kelvin-Voight foundation under moving loads

The vibration of an Euler-Bernoulli beam, resting on a nonlinear Kelvin-Voight viscoelastic foundation, traversed by a moving load is studied in the frequency domain. The objective is to obtain the frequency responses of the beam and the effects of different parameters on the system response. The parameters include the magnitude and speed of the moving load and the foundation nonlinearity and its damping coefficient. The solution is obtained by using the Galerkin method in conjunction with the multiple scales method (MSM). The governing nonlinear partial differential equations of motion are discretized into sets of nonlinear ordinary differential equations. Subsequently, the solution is calculated for different harmonics by using the MSM as one of the powerful perturbation techniques. The steady-state responses of the main harmonic as well as its two super-harmonics are then obtained. As a case study, a conventional railway track is dynamically simulated and the jump phenomenon in the response is observed for three harmonics. Moreover, a thorough stability analysis of the system is carried out.     

On the durable critic load in creep buckling of viscoelastic laminated plates and circular cylindrical shells

Based on the first order shear deformation theory and classic buckling theory, the paper investigates the creep buckling behavior of viscoelastic laminated plates and laminated circular cylindrical shells. The analysis and elaboration of both instantaneous elastic critic load and durable critic load are emphasized. The buckling load in phase domain is obtained from governing equations by applying Laplace transform, and the instantaneous elastic critic load and durable critic load are determined according to the extreme value theorem for inverse Laplace transform. It is shown that viscoelastic approach and quasi-elastic approach yield identical solutions for these two types of critic load respectively. A transverse disturbance model is developed to give the same mechanics significance of durable critic load as that of elastic critic load. Two types of critic loads of boron/epoxy composite laminated plates and circular cylindrical shells are discussed in detail individually, and the influencing factors to induce creep buckling are revealed by examining the viscoelasticity incorporated in transverse shear deformation and in-plane flexibility.     

Vibration analyses of an inclined flat plate subjected to moving loads

The object of this paper is to present a moving mass element so that one may easily perform the dynamic analysis of an inclined plate subjected to moving loads with the effects of inertia force, Coriolis force and centrifugal force considered. To this end, the mass, damping and stiffness matrices of the moving mass element, with respect to the local coordinate system, are derived first by using the principle of superposition and the definition of shape functions. Next, the last property matrices of the moving mass element are transformed into the global coordinate system and combined with the property matrices of the inclined plate itself to determine the effective overall property matrices and the instantaneous equations of motion of the entire vibrating system. Because the property matrices of the moving mass element have something to do with the instantaneous position of the moving load, both the property matrices of the moving mass element and the effective overall ones of the entire vibrating system are time-dependent. At any instant of time, solving the instantaneous equations of motion yields the instantaneous dynamic responses of the inclined plate. For validation, the presented technique is used to determine the dynamic responses of a horizontal pinned–pinned plate subjected to a moving load and a satisfactory agreement with the existing literature is achieved. Furthermore, extensive studies on the inclined plate subjected to moving loads reveal that the influences of moving-load speed, inclined angle of the plate and total number of the moving loads on the dynamic responses of the inclined plate are significant in most cases, and the effects of Coriolis force and centrifugal force are perceptible only in the case of higher moving-load speed.     

The non-stationary freefield response for a moving load with a random amplitude

This paper presents a stochastic solution procedure for the calculation of the non-stationary freefield response due to a moving load with a random amplitude. In this case, a non-stationary autocorrelation function and a time-dependent spectral density are required to characterize the response at a fixed point in the freefield. The non-stationary solution is derived from the solution in the case of a moving load with a deterministic amplitude. It is shown how the deterministic solution can be calculated in an efficient way by means of integral transformation methods if the problem geometry exhibits a translational invariance in the direction of the moving load. A key ingredient is the transfer function between the source and the receiver that represents the fundamental response in the freefield due to an impulse load at a fixed location. The solution in the case of a moving load with a random amplitude is formulated in terms of the double forward Fourier transform of the non-stationary autocorrelation function. The solution procedure is illustrated with an example where the non-stationary autocorrelation function and the time-dependent standard deviation of the freefield response are computed for a moving harmonic load with a random phase shift. The results are compared with the response in the deterministic case.     

Indentation on one-dimensional hexagonal quasicrystals: general theory and complete exact solutions

This paper presents a general account of the indentation responses of a one-dimensional hexagonal quasicrystal half-space pressed by an axisymmetric rigid punch. Based on Green's functions of the half-space subjected to point sources on the surface, the mixed boundary value problem is transformed to integral equations and solved exactly using the results of the potential theory method. Explicit expressions for the generalised pressures and indentation forces are derived for three common indenters (cylinder, cone and approximate sphere) in a systematic manner. For conical and spherical indenters, relations between the contact radius and indentation loads are determined. The coupling phonon–phason fields in the half-space under indentation are accurately expressed in terms of elementary functions. Numerical calculations are performed and discussions on related physical phenomena are given. The present exact solutions can serve as benchmarks for approximate or numerical analyses and can guide the experimental characterisation of material properties of quasicrystals.     

Scattering of monochromatic longitudinal waves on a planar crack of arbitrary shape in a fluid-saturated poroelastic medium

Scattering of monochromatic longitudinal waves on a planar crack of arbitrary shape in a saturated poroelastic medium is considered. The medium is described by Biot’s constitutive equations, the crack sides are fluid permeable. The problem is reduced to a two-dimensional integral equation for the crack opening vector. Gaussian approximating functions are used for discretization of this equation. For such functions, the elements of the matrix of discretized problem are combinations of four standard one-dimensional integrals that can be tabulated. As a result, numerical integration is not needed. For regular grids of approximating nodes, this matrix has Toeplitz’s structure, and matrix-vector products can be calculated by the fast Fourier transform technique. The latter accelerates substantially the process of iterative solution of the discretized problem. Calculation of crack opening vectors, differential, and total cross-sections of circular and elliptic cracks are performed for longitudinal incident waves orthogonal to the crack surfaces. Dependencies of these characteristics on the medium permeability and wavefrequency are studied. Comparison of a crack in the poroelastic medium and in a dry elastic medium with the same porosity and skeleton elastic properties is presented.     

Modelling of continuous and discontinuous floating slab tracks in a tunnel using a periodic approach

This paper presents a periodic approach to couple a track and a tunnel-soil system of different periodicity. The periodicity of the track and the tunnel-soil system is exploited using the Floquet transform to efficiently formulate the problem in the frequency-wavenumber domain as well as to limit the discretization effort to a reference cell. The track and the tunnel-soil system are modelled as two separate systems of different periodicity and are coupled in the frequency-wavenumber domain. A coupled periodic finite element-boundary element method is used to model the tunnel-soil system, while a periodic finite element model or an analytical approach is used to model the track.A general analytical formulation to compute the response of three-dimensional periodic media that are excited by moving loads is discussed. It is shown that the response due to moving loads on the track can be calculated from the transfer function of the track-tunnel-soil system and the axle loads.A methodology for computing the transfer functions of the coupled track-tunnel-soil system as well as the computation of dynamic forces accounting for the interaction between the moving vehicle and the periodic track are described. The model accounts for quasi-static forces as well as dynamic forces due to parametric excitation and unevenness excitation.The methodology has been used to assess the vibration isolation efficiency of continuous and discontinuous floating slab tracks. It is concluded that both continuous and discontinuous floating slab tracks have a similar efficiency in the frequency range well above the isolation frequency of the slabs, which is usually higher than the slab passage frequency. In case of discontinuous slab tracks, the parametric excitation is found to be important, which results in a poorer performance of the track at low frequencies.     

Effective solution for Rayleigh-type waves in a randomly inhomogeneous medium

This paper is concerned with the effective solution for Rayleigh-type waves in a randomly inhomogeneous layer overlying a homogeneous half-space. The considerations are confined to the correlation theory. The method is based on the idea of a fundamental matrix for the system of differential equations. By using the Bourret approximation the system of coupled two second-order integro-differential equations for the average displacements is obtained. This system is solved by Laplace transform. By using boundary and continuity conditions the particular solution is obtained.     

Free vibration analysis of continuous rectangular plates

Vibration characteristics of rectangular plates continuous over full range line supports or partial line supports have been studied by using a discrete method. Concentrated loads with Heaviside unit functions and Dirac delta functions are used to simulate the line supports. The fundamental differential equations are established for the bending problem of the continuous plate. By transforming these differential equations into integral equations and using the trapezoidal rule of the approximate numerical integration, the solution of these equations is obtained. Green function which is the solution of deflection of the bending problem of plate is used to obtain the characteristic equation of the free vibration. The effects of the line support, the variable thickness and aspect ratio on the frequencies and mode shapes are considered. By comparing the numerical results obtained by the present method with those previously published, the efficiency and accuracy of the present method are investigated.