This short paper presents, in a summarized way, non-linear connections for a lattice model of a railway, with the intention of simulating ballast compaction at transition zones.@en

This paper studies the effectiveness of adding auxiliary rails as a mitigation measure for degradation in transition zones of railway tracks. More specifically, it investigates the settlement mechanisms counteracted by the additional rails. Results show that when the system’s response is in the quasi-static regime, adding auxiliary rails over the soft part of the transition zone is beneficial while adding them over the whole transition zone is not. Furthermore, the auxiliary rails have a beneficial impact also when the system’s response is in the dynamic regime; the beneficial effect is caused by the improved load distribution to the supporting structure and not from counteracting the dynamic response amplification that occurs at transition zones. While this mitigation measure has been previously investigated, the contribution of this study lies in a more in-depth analysis of the mechanism through which auxiliary rails can mitigate the degradation at transition zones.@en

This paper presents a semi-analytical solution for the 3D problem of a cylindrical tunnel embedded in an elastic half-space subject to plane harmonic compressional and shear waves. Both the tunnel and soil are modelled as an elastic continuum. Conformal mapping is employed to transform the original physical domain with boundary surfaces of two different types onto an image domain with surfaces of the same type, which makes the problem easier to solve. The total wave field in the half-space consists of incident and reflected (from the half-space surface) plane waves, as well as directly and secondary scattered cylindrical waves, while the total wave field in the tunnel consists of refracted cylindrical waves. The secondary scattered waves, generated when the cylindrical waves directly scattered from the tunnel meet the half-space surface, are represented by cylindrical waves that originate from an image source, which is in line with the spirit of the method of images. The unknown amplitude coefficients of the cylindrical waves are determined from the boundary and continuity conditions of the tunnel–soil system by projecting those onto the set of circumferential modes, which results in a set of algebraic equations. Results show that the present method converges for a small number of circumferential modes. We observe very good agreement between the obtained results and those in literature. In a systematic evaluation, we demonstrate that the method works well for the frequency band of seismic waves, as well as for the complete considered ranges of the tunnel/soil stiffness ratio, the embedded depth of the tunnel, the vertical incident angle and the tunnel thickness. However, the results obtained for a moderate tunnel–soil stiffness contrast under the incident compressional wave are inaccurate when Hankel functions are used to represent the cylindrical waves in the tunnel, which is due to the refracted shear waves in the tunnel transitioning from propagating to evanescent (in the 3D case). These inaccuracies can be perfectly overcome by representing the waves in the tunnel by Bessel functions. We also find that the present method generally works better for the incident compressional wave than for the incident shear wave, as the condition number of the matrix (related to the mentioned algebraic equations) is often larger in the latter case. In view of engineering practice, we conclude that the tunnel is safer when the surrounding soil is stiffer, the tunnel is thicker and the vertical incident angle is larger. Finally, the present method, which is in general fast, elegant and accurate, can be used in preliminary design so as to avoid pronounced resonances, and to assess stress distributions and ground vibrations.@en

The study of periodic systems under the action of moving loads is of high practical importance in railway, road, and bridge engineering, among others. Even though plenty of studies focus on periodic systems, few of them are dedicated to the influence of a local inhomogeneous region, a so-called transition zone, on the dynamic response. In railway engineering, these transition zones are prone to significant degradation, leading to more maintenance requirements than the rest of the structure. This study aims to identify and investigate phenomena that arise due to the combination of periodicity and local inhomogeneity in a system acted upon by a moving load. To study such phenomena in their purest form, a one-dimensional model is formulated consisting of a constant moving load acting on an infinite string periodically supported by discrete springs and dashpots, with a finite domain in which the stiffness and damping of the supports is larger than for the rest of the infinite domain; this model is representative of a catenary system (overhead wires in railway tracks). The identified phenomena can be considered as additional constraints for the design parameters at transition zones such that dynamic amplifications are avoided.

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This article presents a set of close approximations to model an elastic half-space supporting an embankment, which are evaluated in the context of phase velocity spectra, i.e. in terms of the normal wave propagation modes of the embankment-half-space system. The ultimate, intended target of these approximations is in the modeling of vertically-acting loads that may travel over the embankment with some prescribed, constant longitudinal speed. The proposed approximations are analogous to the well-known paraxial approximations that closely mimic a half-space terminating at a plane boundary, but differ from these in that the approximations herein aim at properly modeling not the waves with near normal incidence to the half-space, but waves which propagate at shallow, grazing angles along the longitudinal direction of load motion. Thus, these can be referred to as paralongitudinal approximations. The resulting expressions allow for a very effective simulation of the system at hand for loads moving with subcritical speed and solved in the context of a 2.5D solution method. Such 2.5D formulation considers a continuous model in the longitudinal direction and a discrete model in transverse planes.

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Locations in railway tracks where significant variations of the track properties occur are subject to increased track deterioration. To successfully mitigate this, the mechanisms leading to the increased deterioration need to be understood. To this end, this work presents a non-linear constitutive law for a lattice model able to describe the compaction behaviour of railway ballast. The parameters of the non-linear connections are tuned against lab experiments of cyclic loading tests and direct shear tests. The tuned lattice can be used with different foundation properties provided that the ballast in the track is equivalent to that of the tests. The non-linear lattice model is applied to the case of railway transitions, for which ballast compaction under train loading is studied as a cause of geometry degradation. It is observed that for the studied cases of a culvert crossing and of a ballast-slab transition, the operation-induced compaction converges monotonously to a stable situation, without leading to significant changes in the vehicle-track interaction. Ballast compaction is therefore insufficient as a stand-alone mechanism to explain a process of progressive degradation of the track geometry. Other mechanisms like autonomous differential settlement at the foundation level must be taken into account in such cases.@en

The procedure presented in this work aims at providing a framework for studying settlement of ballast in zones with stiffness variation of the railway track support. The proposed procedure results from expanding an existing infinite periodic model of a railway track to account for variations in the stiffness of the foundation. Ballast is simulated via a linear lattice, whose dynamic response differs from that of a continuum. The expanded model is composed of three regions: a left region, which is semi-infinite and periodic; a mid-region, of finite length and where the properties of the foundation can change; and a right region, which is also semi-infinite and periodic and whose properties can differ from those of the left region. The equations of motion of the mid-region are written directly in the time domain, with the rail being described by finite elements. On the other hand, the left and right semi-infinite regions are treated semi-analytically in the frequency domain, and afterwards their responses are converted to the time domain, resulting in convolution integrals prescribed at the boundaries of the mid-region that simulate non-reflective boundaries. The final model only contains the degrees of freedom corresponding to the mid-region (which can be as short as the region where the stiffness variation is present), and that leads to faster calculations than if the boundaries were placed further away to dissipate undesired reflections. The method is cast in the time domain, and all elements are assumed to behave linearly. In the future, the model will be expanded to incorporate non-linear behaviour of the ballast. The presented method is validated by means of simple examples, and afterwards applied to a real scenario in which a culvert crosses a railway track. As presented, the method can be used to study the linear dynamics of transitions zones, study mitigation measures, and infer about indicators like force transmitted and energy dissipated, which might be useful to assess the development of settlement of the ballast.@en

Transition zones in railway tracks are locations with a significant variation of track properties (i.e. foundation stiffness) encountered near structures such as bridges and tunnels. Due to strong amplification of the track’s response, transition zones are prone to rapid degradation. To investigate the degradation mechanisms in transition zones, researchers have developed a multitude of models, some of them being very complex. This study compares three solution methods, namely an integral-transform method, a time-domain method, and a hybrid method, with the goal of solving these systems efficiently. The methods are compared in terms of accuracy, computational efficiency, and feasibility of application to more complex systems. The model employed in this paper consists of an infinite, inhomogeneous, and piecewise-linear 1-D structure subjected to a moving constant load. Although the 1-D model is not particularly demanding computationally, it is used to make qualitative observations as to which method is most suitable for the 2-D and 3-D models, which could lead to significant gains. Results show that all three methods can reach similar accuracy levels, and in doing so, the time-domain method is most computationally efficient. The integral-transform method appears to be efficient in dealing with frequency-dependent parameters, while the time-domain and hybrid methods are efficient in dealing with a smooth nonlinearity. For multi-dimensional models, if nonlinearities and inhomogeneities are considered throughout the depth, the time-domain method is likely to be most efficient; however, if nonlinearities and inhomogeneities are limited to the surface layers, the integral-transform and hybrid methods have the potential to be more efficient than the time-domain one. Finally, although the 1-D model presented in this study is mainly used to assess the three methods, it can also be used for preliminary designs of transition zones in railway tracks.

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This paper investigates the instability of vertical vibrations of an object moving uniformly through a tunnel embedded in soft soil. Using the indirect Boundary Element Method in the frequency domain, the equivalent dynamic stiffness of the tunnel-soil system at the point of contact with the moving object, modelled as a mass-spring system or as the limiting case of a single mass, is computed numerically. Using the equivalent stiffness, the original 2.5D model is reduced to an equivalent discrete model, whose parameters depend on the vibration frequency and the object's velocity. The critical velocity beyond which the instability of the object vibration may occur is found, and it is the same for both the oscillator and the single mass. This critical velocity turns out to be much larger than the operational velocity of high-speed trains and ultra-high-speed transportation vehicles. This means that the model adopted in this paper does not predict the vibrations of Maglev and Hyperloop vehicles to become unstable. Furthermore, the critical velocity for resonance of the system is found to be slightly smaller than the velocity of Rayleigh waves, which is very similar to that for the model of a half-space with a regular track placed on top (with damping). However, for that model, the critical velocity for instability is only slightly larger than the critical velocity for resonance (of the undamped system), while for the current model the critical velocity for instability is much larger than the critical velocity for resonance due to the large stiffness of the tunnel and the radiation damping of the waves excited in the tunnel. A parametric study shows that the thickness and material damping ratio of the tunnel, the stiffness of the soil and the burial depth have a stabilising effect, while the damping of the soil may have a slightly destabilising effect (i.e., lower critical velocity for instability). In order to investigate the instability of the moving object for velocities larger than the identified critical velocity for instability, we employ the D-decomposition method and find instability domains in the space of system parameters. In addition, the dependency of the critical mass and stiffness on the velocity is found. We conclude that the higher the velocity, the smaller the mass of the object should be to ensure stability (single mass case); moreover, the higher the velocity, the larger the stiffness of the spring should be when a spring is added (oscillator case). Finally, in view of the stability assessment of Maglev and Hyperloop vehicles, the approach presented in this paper can be applied to more advanced models with more points of contact between the moving object and the tunnel, which resembles reality even better.

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Transition radiation is emitted when a source moves along a straight line with constant velocity and acts on or near an inhomogeneous medium [1]. It occurs, for example, when trains cross areas with substantial variation of track properties (e.g., foundation stiffness) encountered near rigid structures such as bridges; these areas are called transition zones. Apart from potentially giving rise to vehicle instability, transition radiation has been addressed as one of the causes of track and foundation degradation due to the often strong amplification of the stress and strain fields. This leads to a high frequency of maintenance required for transition zones in areas with soft soils, which can be 3–8 times higher than for the regular parts of the railway track. Wave radiation is also generated due to the periodic variation of the support stiffness (i.e., periodically placed sleepers). The periodically supported beam interacting with a vehicle has been addressed in studies on vehicle instability and on the resonant behaviour of the system [2]. Moreover, it has been shown that the periodic supports play a role in the fatigue and corrugation of both wheel and rail [3]. However, most studies consider either the local variation of the foundation stiffness (i.e., transition zones) or the periodic variation of the support stiffness, and not the combination of the two. Furthermore, studies that consider both scales make use of complex supporting structures and vehicle models such that the influence of the discrete supports on the transition radiation is not clear. This work aims at studying the influence of accounting for the discrete supports on the transition radiation (inside transition zones) and on the plastic deformation that develops in the supporting structure. To this end, a 1-D model is formulated, consisting of an infinite Euler-Bernoulli beam discretely supported by nonlinear springs and dashpots whose characteristics locally vary in space, interacting with a moving loaded oscillator. The solution is obtained using a time-stepping method (i.e., Newmark-β) for the temporal dimension of the system while the finite element method is used to discretise the spatial dimension. The infinite extent of the system is ensured through a set of non-reflective boundary conditions obtained using the Floquet theory. The model presented here can be used for preliminary designs of transition zones in railway tracks. Given the stiffness dissimilarity, the optimum length of the transition zone and the train’s maximum velocity can be obtained such that the damage in the railway track is minimized.@en

We introduce the open-source package MercuryDPM, which we have been developing over the last few years. MercuryDPM is a code for discrete particle simulations. It simulates the motion of particles by applying forces and torques that stem either from external body forces, (gravity, magnetic fields, etc.) or particle interactions. The code has been developed extensively for granular applications, and in this case these are typically (elastic, plastic, viscous, frictional) contact forces or (adhesive) short-range forces. However, it could be adapted to include long-range (molecular, self-gravity) interactions as well. MercuryDPM is an object-oriented algorithm with an easy-to-use user interface and a flexible core, allowing developers to quickly add new features. It is parallelised using MPI and released under the BSD 3-clause licence. Its open-source developers’ community has developed many features, including moving and curved walls; state-of-the-art granular contact models; specialised classes for common geometries; non-spherical particles; general interfaces; restarting; visualisation; a large self-test suite; extensive documentation; and numerous tutorials and demos. In addition, MercuryDPM has three major components that were originally invented and developed by its team: an advanced contact detection method, which allows for the first time large simulations with wide size distributions; curved (non-triangulated) walls; and multicomponent, spatial and temporal coarse-graining, a novel way to extract continuum fields from discrete particle systems. We illustrate these tools and a selection of other MercuryDPM features via various applications, including size-driven segregation down inclined planes, rotating drums, and dosing silos. Program summary: Program Title: MercuryDPM Program Files doi: http://dx.doi.org/10.17632/n7jmdrdc52.1 Licensing provisions: BSD 3-Clause Programming language: C++, Fortran Supplementary material: http://mercurydpm.org Nature of problem: Simulation of granular materials, i.e. conglomerations of discrete, macroscopic particles. The interaction between individual grains is characterised by a loss of energy, making the behaviour of granular materials distinct from atomistic materials, i.e. solids, liquids and gases. Solution method: MercuryDPM (Thornton et al., 2013, 2019; Weinhart et al., 2016, 2017, 2019) is an implementation of the Discrete Particle Method (DPM), also known as the Discrete Element Method (DEM) (Cundall and Strack, 1979). It simulates the motion of individual particles by applying forces and torques that stem either from external forces (gravity, magnetic fields, etc.) or from particle-pair and particle–wall interactions (typically elastic, plastic, dissipative, frictional, and adhesive contact forces). DPM simulations have been successfully used to understand the many unique granular phenomena – sudden phase transitions, jamming, force localisation, etc. – that cannot be explained without considering the granular microstructure. Unusual features: MercuryDPM was designed ab initio with the aim of allowing the simulation of realistic geometries and materials found in industrial and geotechnical applications. It thus contains several bespoke features invented by the MercuryDPM team: (i) a neighbourhood detection algorithm (Krijgsman et al., 2014) that can efficiently simulate highly polydisperse packings, which are common in industry; (ii) curved walls (Weinhart et al., 2016) making it possible to model real industrial geometries exactly, without triangulation errors; and (iii) MercuryCG (Weinhart et al., 2012, 2013, 2016; Tunuguntla et al., 2016), a state-of-the-art analysis tool that extracts local continuum fields, providing accurate analytical/rheological information often not available from experiments or pilot plants. It further contains a large range of contact models to simulate complex interactions such as elasto-plastic deformation (Luding, 2008), sintering (Fuchs et al., 2017), melting (Weinhart et al., 2019), breaking, wet and dry cohesion (Roy et al., 2016, 2017), and liquid migration (Roy et al., 2018), all of which have important industrial applications.

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In this work we propose a model for the analysis of the dynamic behaviour of a ballasted track that combines discrete and continuous elements. The rail is modelled via an Euler-Bernoulli beam, the periodically spaced sleepers are represented with lumped masses, and the ballast is simulated using a lattice (regular network of elastically connected lumped masses). All elements are assumed to behave linearly, and the lattice can be supported by a flexible or a rigid foundation, simulating soil or a hard rock. The equations of motion of each component are presented and the coupled system is solved semi-analytically in the frequency domain. The time domain response can be calculated afterwards by means of a numerical inverse Fourier transform. Dispersion curves and time responses are produced for the case of a ballasted track on a stiff soil. These responses are compared with the scenario in which ballast is modelled as lumped supports, and the scenario in which the force is applied directly to the ballast (no superstructure). It is observed that the simpler models fail to capture the vibration modes in which energy is concentrated in the ballast, and that the superstructure significantly alters the response of the track, increasing its critical velocity and changing the deformed shape of the ballast. The model herein proposed can be used to assess the dynamic characteristics of the track (critical speeds, energy propagation, vehicle-track interaction, etc.) and will serve as framework for a development of a tool for assessment of the settlement behaviour of ballast.

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This paper addresses the dynamic response of an infinitely long cylindrical structure embedded in an elastic half-space. The structure has a circular cross-section and its axis is parallel to the half-space surface. Excitation can be incident body waves or forces applied on the surface of the half-space and/or the structure. The model can be used to assess the integrity of structures when acted upon by seismic waves, to predict ground-borne vibration due to circulation of vehicles, and to infer about the safety of vehicles during earthquake events. Because the half-space and the structure surfaces possess different symmetries, the solution is not straightforward. In order to circumvent this difficulty, the physical domain is conformally mapped onto an auxiliary domain with a cylindrical symmetry, in which the free surface of the half-space and the surface of the structure are located at concentric cylindrical surfaces. The solution of the original boundary value problem is finally obtained by solving a set of algebraic equations. Truncation of the summation over circumferential modes is needed in the numerical implementation. Convergence tests, validations and comparisons of stresses and motions for twoand three-dimensional cases are presented and discussed as well as the advantages and disadvantages of the proposed method. Additionally, the effect of the presence of the tunnel is analysed by considering a limiting case of the half-space with just a cylindrical cavity of the same radius as the outer radius of the tunnel. @en

A method is presented to accurately capture the 3D interaction phenomena of a foundation pile embedded in soil, in a computationally efficient non-local 1D model. It is shown how to extract the global stiffness kernels from simulations with the 3D inhomogeneous continuum, and implement them in a 1D non-local Winkler-type model that can subsequently serve as a stand-alone, condensed substructure. The presented method for obtaining the kernels removes the need to assume certain distributions for these functions. We show that the method is very versatile, and yields accurate results for a wide range of pile geometries and soil stiffness profiles, over the full depth of the embedded pile. For the dynamic case, the discretized global stiffness kernels (matrices) become complex-valued (they include soil stiffness, inertia and damping), and the 1D model proves to be capable of mimicking also the out-of-phase part of the response. The method being straightforward and fast, the engineering community is served the benefits of accuracy (3D model) and speed (1D model), without the need of empiric tuning.

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In this paper the effect of a nearby, semi-infinite, level ice sheet on the frequency domain response of a thin, floating, rigid body is studied using a 2D model. The ice is modeled using a dynamic Euler-Bernoulli beam and the finite depth water layer is described with the Laplace equation and the linearized Bernoulli equation. Eigenfunction matching is used to resolve the interface between the ice covered and open water regions. The body is excited by external loads, generating waves. The waves are partially reflected by the ice edge and these reflected waves influence the body's response. It is this influence that this paper focuses on. Below a certain onset frequency the amplitude of the reflected waves is insignificant and consequently the body remains unaffected by the ice. This frequency is only sensitive to the ice thickness with thinner ice resulting in a higher onset frequency. Above the onset frequency the reflected waves cause quasi-standing waves between body and ice. For frequencies at which half the wavelength of the surface wave in the water is approximately an integer multiple of the gap length, the amplitude of the standing waves is greatly amplified. This can result in (anti-)resonance depending on the phasing between the reflected waves and the body's motion.

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This work reports on experiments that were performed with a freely vibrating cylinder exposed to currents and placed near a plane boundary parallel to the cylinder axis. It is observed that the proximity of the boundary affects the vertical response of the cylinder in two ways: (i) for gaps between 0.75 and 2 diameters (D), the amplitude of oscillation is reduced; (ii) for gaps smaller than 0.75D, the cylinder impacts the boundary, resulting in an increase of amplitudes and frequencies of oscillations as the flow is accelerated. The in-line force acting on the cylinder is also examined, and the dependency of its harmonic components on the flow velocity and distance to the boundary is evaluated. Besides the typical amplification of the mean component inside the lock-in region, it is also observed that as the cylinder is placed closer to the boundary, the harmonic component with the frequency of the vertical oscillations increases, while the component with twice that frequency decreases in similar amount.

Based on the experimental observations, an existing wake-oscillator model for vortex-induced vibrations is enhanced in order to account for the effect of the boundary. The proposed model introduces an effective damper that is activated when the cylinder reaches a certain distance from the boundary, and a damper/spring set representing the rigidity of the boundary and the dissipation of energy due to impact.
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Earthquakes induced by the gas extraction is a problem of serious concern in the northern part of The Netherlands. The earthquakes recorded to date can be classified as minor based on their maximum local magnitude (ML=3.6). However, (i) their shallow focus, (ii) the special in-situ soft soil conditions and (iii) the fact the building stock in the region consists primarily of unreinforced masonry, requires some special attention. For this reason, several studies are initiated to investigate the vulnerability of the masonry structures to withstand earthquakes of minor to moderate magnitudes. This paper discusses a detailed study of a masonry school in the region. A coupled finite element-boundary element (FE-BE) model is developed to study the linear dynamic response of the structure to induced seismicity. The structural part, i.e. the masonry building, is modelled using finite elements whereas the soil is described by boundary elements. The modelling of the layered soil medium using boundary elements reduces the computational demands and avoids the need to incorporate non-reflecting boundaries since the radiation condition at infinity is satisfied in an exact manner. This is particularly important for the relatively long wavelengths associated with the low seismic frequencies. The coupled FE-BE model is validated with a full scale in-situ experiment in which the structure is set into motion by a vibratory device (shaker) which is placed close to the building. To serve this purpose, a special shaker was chosen able to extract significant amplitudes in the frequency range between 2-10Hz, which is considered to be relevant for the shallow-focus earthquakes in the region. The dynamic behaviour of the structure, i.e. natural frequencies and modal shapes, is first identified based on ambient vibration measurements. Subsequently, model predictions based on monochromatic ground excitation were compared with in-situ measurements for validation purposes. It is shown that the coupled FE-BE model is capable of predicting the dynamic response of the actual system for a wide range of frequencies. Finally, the effects of soft soil-structure interaction and the vibrational characteristics of the structure are investigated for a wide range of frequencies. @en