This paper deals with the prediction of train-induced building vibrations by using a numerical framework. The framework is based on a sub-structure approach, where a sequence of different models are used. The free-field ground vibrations and the track receptance are calculated using 2.5D technique where the railway track is represented by finite elements that couple to a dynamic stiffness of the underlying soil, which in turn is obtained from the Green’s function of a horizontally layered half-space using a layer transfer matrix approach. A planar multi-body model of the train, coupled to the track receptance, is used for calculating the train–track interaction forces as the train runs over an uneven rail. Finally, the building response to the incident wavefield is calculated using a 3D finite element model, accounting for the soil dynamic stiffness.The framework is used to evaluate the vibrations in two buildings with identical layout, one lightweight wooden building and one heavyweight concrete building, due to a passenger train passing by at two different speeds. It was found that the difference in response between the two buildings were small. Compared to the incident wavefield, an amplification of the response inside the building was found in frequency bands around the fundamental natural frequencies of the slabs; however for higher frequencies and in terms of the 1 s running RMS velocity the building response was reduced. Further, it was found that accounting for soil-structure-interaction, as opposed to simply enforcing the free-field displacements at the building foundations, significantly reduced the building response in terms of 1 s RMS velocity.

There are well-known methods for determining natural frequencies and mode shapes of displacement for layered elastic media of finite depth with classical stress-free or rigid boundary conditions. There are also methods for handling more complex problems with specific boundary conditions, such as several finite layers resting over a half-space. However, these existing methods either allow evaluation of only the first few modes or require complex back-propagation analyses to achieve stability. There is currently no straightforward explicitly determined method that can determine the natural frequencies and mode shapes for all frequencies and layer depths. This paper introduces an alternative approach to address this limitation by strategically writing the dynamic stiffness matrices of attached layers. The main advantages of this strategy for the modeler include the ability to consider arbitrary frequencies, layer depths, and the number of layer strata over an elastic half-space. Importantly, there is no need to sub-divide layers, which is a requirement in many other methods. While there are some limitations in terms of computational accuracy and capacity, this methodology remains straightforward to program and compute relevant response outputs for general studies of free or forced vibration. The paper provides explicit entries for the involved matrices and presents computations of wavenumber dispersion diagrams, phase velocity plots, and response data in both the frequency and time-domains. Two case studies in earthquake assessments, one for plane-strain and another for axisymmetry, demonstrate the effectiveness of the methodology. The approach is based on a well-conditioned dynamic stiffness method, specifically developed for this purpose, which allows for the study of deep-layered strata. The computational efficiency of the method allows for fast computations even on regular desktop or laptop computers, with response analysis taking only tenths of a second for a forced response analysis. Numerical evidence of a layer resonance, resulting from the presence of a ZGV (zero group velocity) mode phenomenon, is demonstrated through a case study of a ground profile with layers hundreds of meters in depth. Solutions in both the frequency and time-domains highlight this special case.

For a train speed close to the speed of elastic waves in the soil, often referred to as "critical speed ", largely elevated vibration responses occur. This can be a practical problem for soft soil sites, where the phenomenon may cause excessive vibrations in the track and also at distances far from the track. To ensure the running safety of the train, the long-term quality of the track and to reduce the vibrations in the surroundings, such effects must be avoided. An effective counter-measure is to increase the stiffness of the soil underneath the track, thereby increasing the critical velocity.& nbsp;In this paper, a 2.5D finite element model is used for studying the critical velocity phenomenon and its mitigation through soil stiffening, for a ballasted track on a layered half-space with very soft soil. Soil improvement under the track, in the shape of a solid block or as various number of panels, with varying depth and stiffness is considered. The effect of the soil improvement is evaluated both in terms of the maximum rail and free-field displacements. It is shown that a shallow soil stiffening increases the critical velocity and reduces the rail and free-field response for load speeds near the shear wave velocity of the soft top soil layer. It is also demonstrated that a deep soil stiffening, by use of panels along the track direction, increases the critical velocity further, and may also be efficient in reducing the response for load speeds near the shear wave speed of the underlying half-space.

In the paper, the effect of modeling strategies regarding the dynamic behavior of a railway slab track on a layered half-space is studied. The track is modeled with various degrees of accuracy through the use of either beam theory, shell finite elements or solid finite elements. The underlying soil response is included through a dynamic stiffness, obtained via the Green’s function for a horizontally layered visco-elastic half-space in the frequency–wavenumber domain. The effect of different assumptions regarding the track cross-section behavior and the track–soil interface conditions on the resulting free-field vibrations are studied for a harmonic load moving along the track. First, only the out-of-plane displacements of the slab–soil interface are coupled, i.e. only the vertical contact pressure is accounted for. Secondly, the effect of coupling the slab–soil in-plane displacements on the free-field vibrations is studied. It is found that the in-plane slab–soil coupling significantly affects the vertical vibration in the free-field. It is also found that a beam model of the track yields accurate response levels compared to a solid continuum model in the case of a thick slab, whereas considerable differences are obtained for a thin slab.

In the present paper, the effectiveness of a vibration isolation mat for a railway slab track system is studied using a finite element model of the railway track. The finite elements are formulated in a moving frame of reference following the moving load at a particular speed. The rails are modeled using Bernoulli beams, whereas the track slab and an underlying supporting plate are modeled using Kirchhoff plate elements. The vibration isolation mat is modeled as a continuous visco-elastic layer between the track slab and the supporting plate. The response of the underlying soil is represented through a dynamic stiffness matrix, obtained via the Green’s function for a horizontally layered visco-elastic strata in a moving frame of reference in the frequency– wavenumber domain. The model accounts for the quasi-static excitation caused by the static axle loads of a vehicle, as well as the dynamic excitation caused by the vehicle running over an uneven rail. The free-field response and the insertion loss obtained with the vibration isolation mat is first evaluated for a harmonic load moving along the track. Band-averaged vibration levels and the insertion loss for a fixed point next to the track are then calculated for a train cart, represented by a 10 degree-of-freedom multi-body system, running at different speeds. (Less)

To predict ground-borne vibration levels caused by railway traffic, models for estimating the load from the vibration source, as well as the vibration transmission through the ground, are needed. In the present paper, a finite element formulation in a frame of reference following the moving load, is used for modeling a railway slab track. The response of the underlying soil is represented through a dynamic stiffness matrix, obtained via the Green’s function for a horizontally layered visco-elastic half-space in a moving frame of reference in the frequency–wavenumber domain. The track can be modeled as continuously connected beams, but the use of plate elements allows for more general stress and displacement distributions in the transverse direction of the slab to be resolved. Here, the free-field response due to a harmonic load moving along a slab track, is evaluated and compared using different modeling strategies for the slab. (Less)

Many cities experience an increasing population leading to a need for urban densification.In these cities, unused land close to railways will have to be developed with new residential and office buildings.The infrastructural demand will also increase, resulting in heavily trafficked roads and railways close to where people work and live.Annoyance from trafficinduced vibrations and noise is expected to be a growing issue.To predict the level of vibration and noise in buildings caused by railway and road traffic, calculation models are needed.In the present paper, a simplified prediction model is briefly described.This prediction model is based on the assumption that the ground and railway embankment can be described in an axisymmetric model, to provide the transfer functions between the track and the free-field.In the paper, the error that arise by assuming axisymmetric response is studied by comparing the response in a three-dimensional finite-element model.Transfer functions at several positions in the free-field are compared.