Vibration of a beam on continuous elastic foundation with nonhomogeneous stiffness and damping under a harmonically excited mass

In this paper, a method of analysis of a beam that is continuously supported on a linear nonhomogeneous elastic foundation and subjected to a harmonically excited mass is presented. The solution is obtained by decomposing the nonhomogeneous foundation properties and the beam displacement response into double Fourier summations which are solved in the frequency–wavenumber domain, from which the space–time domain response can be obtained. The method is applied to railway tracks with step variation in foundation properties. The validity of this method is checked, through examples, against existing methods for both homogeneous and nonhomogeneous foundation parameters. The effect of inhomogeneity and the magnitude of the mass are also investigated. It is found that a step variation in foundation properties leads to a reduction in the beam displacement and an increase in the resonance frequency for increasing step change, with the reverse occurring for decreasing step change. Furthermore, a beam on nonhomogeneous foundation may exhibit multiple resonances corresponding to the foundation stiffness of individual sections, as the mass moves through the respective sections along the beam

Dynamic response of a laterally-loaded infinite rigid cylinder embedded in a saturated poroelastic medium

In this paper, an analytical solution for the response of a rigid cylinder embedded in a full-space poroelastic medium subjected to a dynamic lateral load is derived. The problem is idealised as a two-dimensional problem. The solution is obtained using Biot’s theory for acoustic waves. In this solution, the displacements of the solid skeleton and the pore pressure are expressed in terms of three scalar potentials. These potential correspond to the wave velocities of the slow and fast compressional wave and to the shear wave. The governing equation for the dynamic motion is expressed in the frequency domain using Fourier transformation and the potentials are shown to be given by Holmholtz equations

A comparison of ground vibration due to ballasted and slab tracks

With the development of non-ballasted track forms (often referred to as slab tracks) over the few last decades, it is important to understand their behaviour with respect to ground-borne vibration compared with the traditional ballasted tracks. This is important in deciding between the use of the two track forms. The present work aims to quantify the differences between slab tracks and ballasted tracks numerically by using the MOTIV model. This is a general and fully coupled three-dimensional model that works in the wavenumber-frequency domain. It can predict the vibration levels of the track and the ground due to the gravitational loading of a passing train and the wheel and rail unevenness. A comparative analysis between the two track types is presented in terms of ground vibration with emphasis given to the influence of the stiffness and inertial parameters of the two track forms. It is shown that, for the same fastener stiffness there are only small differences in ground vibration behaviour, with the mass of the track slab leading to reductions of 1–3 dB at frequencies above 16 Hz. However, if softer rail fasteners are used in the slab track, as is usual, this leads to further reductions above 63 Hz. The critical velocity on soft soil is also considered. Although there is little difference between the different tracks for a homogeneous ground, for grounds with a soft surface layer the critical velocity is increased by the slab bending stiffness. The maximum rail displacement is also smaller for a slab track than the equivalent ballasted track.The work described here has been supported by the EPSRC under the programme grant EP/M025276/1, ‘The science and analytical tools to design long life, low noise railway track systems (Track to the Future)’ and the MOTIV project (Modelling of Train Induced Vibration), grants EP/K005847/2 and EP/K006002/1. All data published in this article are openly available from the University of Southampton repository at https://doi.org/10.5258/SOTON/D1001.Scopu

The effect of vertical-lateral coupling of rails including initial curvature

An understanding of the dynamic behaviour of railway tracks at high frequencies, in both vertical and lateral directions, is important for the assessment of rolling noise. Although many analytical models can be found in the literature, these mostly focus on the vertical vibration of the track. Studies of the lateral vibration are less common while the coupling between the vertical and lateral directions has received very little attention. In this paper, a model of a beam on an elastic foundation is introduced that accounts for the coupling of the vertical and lateral vibration behaviour. The model allows for the effects of beam curvature, asymmetry of the cross-section, shear deformation, rotary inertia and warping. Consideration is given to the fact that the loads at the rail head, as well as those exerted by the railpads at the rail foot, may not be applied through the centroid of the section. The track is subjected to a non-moving harmonic load and the solution is obtained in the wavenumber domain using the Fourier transform method. Results are presented as dispersion curves for the free rail and are validated with the aid of a Finite Element software. Closed form analytical expressions are derived, using contour integration, for the forced response. The track vibration decay rates are also presented and analysed as a means of assessing the noise performance of the rail and the influence of vertical-lateral coupling

A comparison between the use of straight and curved beam elements for modelling curved railway tracks

A major environmental concern related to railway traffic is vibration. A lot of re- search has been carried out to understand vibration of straight tracks, with less attention been paid to curved tracks. Modelling the dynamic behaviour of a curved railway track is important to understand the physics of generation and propagation of vibration fromtrains at non-straight sections of tracks. Modelling is also important to assess the current and any alternative track designs from an environmental point of view. In this paper a curved track is modelled and the effect of curvature is investigated. Two models have been developed and their results have been compared. In the first, the curved track is modelled using straight beam elements. In the second curved beam elements are used. For both, the Euler-Bernoulli beam theory has been adopted to describe their bending behaviour. The elements have 12 degrees of freedom accounting for displacements and rotations in the lateral, transverse and longitudinal directions. The excitation comes from an axle traversing the rails with subcritical velocity, accounting for the wheel-rail contact forces. The describedmodels are solved using the Finite Element Method. The time domain response of the versine of the curved track due to the passage of the axle is computed. A comparison is made on the efficiency of the two models for different curve radii and frequencies. The two models provide very similar results showing that the piecewise straight beam approximation represents the behaviour of the curved track accurately. Also the curved beam model used in this study shows some limitations for the specific application and therefore the straight element method is recommende

A holistic approach for the design and assessment of railway tracks

In spite of the global financial crisis, considerable investments are being made in railway infrastructure in the UK and many countries around the world. Improvements in the quality and capacity of current services and the development of new railway infrastructure are needed to meet the increasing demand for transferring more people and goods in a more sustainable way. In particular, the performance of the track system is crucial to the successful and cost-effective operation of the railway. This has motivated much scientific research with the aim of better understanding the performance of the railway system, including both existing railway tracks and improved tracks for the future. Much current research on railway track focuses on individual aspects of the design and performance, e.g. track settlement, rail fatigue, ballast degradation, ride quality, maintenance, and noise and vibration. However to achieve substantial advances in railway track design, it is important to consider all these aspects in an integrated way. Changes that can benefit one aspect should not be allowed to have a negative impact on others. To facilitate this, a single tool should be developed or the computational tools that consider individual aspects of the design need to be integrated. The resulting tool can therefore be used to assess the behaviour of railway tracks in a holistic manner. A preliminary version of such a holistic tool is presented here. In this version, fast running models and empirical relationships are put together in order to calculate the performance of a railway track with regard to ride quality, ground-borne noise and vibration and rolling noise. Results for practical case studies are presented and discussed. The paper also highlights the limitations of the preliminary version and the future plans to achieve a reliable and comprehensive tool

Effect of rail unevenness correlation on the prediction of ground-borne vibration from railways

This paper presents the influence of rail unevenness correlation on the predicted track and ground vibration. The study is based on an integrated railway model in the wavenumber-frequency domain with varying complexity describing the dynamic system of a ballasted track on layered elastic half-space. In order to investigate how ground vibration levels are influenced by taking into account different correlation levels between the two rails, the traction variation across the track-ground interface is included and the track model is discretised laterally including both rails separately and allowing for the pitching motion of the sleepers. The paper presents the effect of the different modelling approaches on the response predictions and compares the dynamic response calculated for a range of model/excitation parameter

Predictions of the dynamic response of piled foundations in a multi-layered half-space due to inertial and railway induced loadings

In this paper, the dynamic pile-soil-pile interaction (PSPI) in a multi-layered half-space is investigated for the prediction of the response of piled foundations due to railway vibrations. Two methods of modelling piled foundations in a multi-layered half-space are presented. The first is an efficient semi-analytical model that calculates the Green’s functions of the multi-layered half-space soil using the thin layer and the dynamic stiffness matrix methods. The second is a fully-coupled model that utilises the boundary element (BE) method to simulate the soil, where the Green’s functions are calculated using the ElastoDynamics Toolbox (EDT). The paper aims to investigate the accuracy and the efficiency of the semi-analytical model by comparing the predictions of the two methods. A set of comparisons is performed, including the driving point response of a single pile and the interaction between two piles. The comparisons reveal that, at most frequencies, the semi-analytical model can predict the driving point response and the dynamic interaction with acceptable accuracy and computational efficiency. The model is then used for predicting the response of a pile-group due to the vibration field generated by a railway in varying distance from the piles. The vibration field generated by the railway is modelled as the superposition of the response due to harmonic loadings generated at the wheel-rail interface and the vibration response is examined at different points on the free surface away from the piles. The comparisons highlight the efficiency and accuracy of the semi-analytical model and illustrate its practical application

Modelling of train-induced vibration

This paper reports on recent developments in techniques for modelling ground vibration from railways. The modelling considers both surface and underground railways, and accounts for the main dynamic systems involved, i.e. tracks (both ballasted and slab), tunnels and multi-layered ground. Results are presented to illustrate the modelling capabilities and the efficiency of computations for the models proposed. The work presented is part of the MOTIV project (Modelling of Train Induced Vibration), which is a collaboration between the Universities of Southampton and Cambridge. Future development of models and plans within the project are also addressed

Modelling the dynamic pile-soil-pile interaction in a multi-layered half-space

Within the context of railway ground-borne vibration, the dynamic pile-soil-pile interaction remains an area that has not been sufficiently investigated. Whilest a number of researchers have scrutinised the vibration response of piled-foundations, their approaches exhibit a compromise between computation time and solution accuracy. In this paper, two models of piled-foundations in a multi-layered half-space are presented; one is an efficient semi-analytical model and another is a fully-coupled boundary element model. The pile is simulated, in both models, by an elastic bar for axial loading and an Euler-Bernoulli beam for transverse loading. A set of comparisons has been performed, including the driving point response of a single pile, the interaction between two piles and the 'far-field' response due to axial and transverse loading on the pile's head. The comparisons reveal that the semianalytical model predicts at most frequencies the driving point response and the dynamic interaction with acceptable accuracy and computational efficiency. It is also highlighted that the semi-analytical model with its present assumptions does not accurately approximate the 'far-field' respons