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Eingeladener Vortrag
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Ballast mats are an efficient measure to reduce the vibrations near railway lines. The vehicle-track system gets a low eigenfrequency due to the insertion of an elastic ballast mat under the ballast. For frequencies higher than this low vehicle-track eigenfrequency, the forces, which are generating the vibration of the soil, are considerably reduced. In this contribution, a combined finite-element boundary-matrix method is used to calculate a number of completely three-dimensional track models with and without ballast mats. The influence of the important parameters such as the stiffness of the ballast mat, the unsprung vehicle mass, the mass of the track, and the stiffness of the subsoil is investigated. The numerical results are presented as the transfer functions of the total force that is acting on the soil and generating the vibration of the environment. The effectiveness of ballast mats is achieved by division of two of these force functions. The general tendencies for this insertion loss are discussed and a comparison with measurements is given. To come to an improved practical tool for the design of ballast-mat tracks, the finite-element method results are approximated by a simple two-dimensional model of which the solution is given explicitly.
Seismic reponse of three-dimensional structures using a Green's function approach to the soil
(1986)
The soilstructure interaction of elastic plates on homogeneous or layered soils excited by horizontally propagating waves is analysed. Large plates are modelled by a combined finite-element boundary-element method (FEBEM), whereas the response of infinitely long plates is calculated by a numerical integration in the frequencywavenumber domain. The finite-element boundary-element method yields the complete soilplate transfer function of frequency and distance whereas the frequencywavenumber solution of the infinitely long plate can serve as an approximation for long distances on a finitely long plate. The soilplate transfer function starts to decrease strongly at the coincidence frequency, where the bending stiffness equals the plate inertia. A strong decrease follows at mid frequencies and a strong reduction of less than 0.1 of the ground vibration is reached at high frequencies. Rules for the characteristic frequencies are derived from the numerical results clearly indicating the strongest influence of the soil stiffness and the weaker influence of the bending stiffness of the plate. The influence of the mass, length and width of the plate are shown to be limited in case of realistic parameters, but it should be noted that the reduction effects are less effective for layered soils and for nearer observation points.
An integral method to calculate the solution of a homogeneous or layered soil due to a harmonic point load is described. An infinite plate at the surface of the soil can be introduced in this integration in wavenumber domain, too. Finite structures on the soil are calculated by a combined finite element and boundary element method, which makes use of the point load solution of the soil. The compliance functions for a vertical point load and some vibration modes are calculated for realistic parameters of the plate and the soil and for a wide range of frequencies. The influence of the stiffness of the soil and the foundation is investigated, showing that the soil mainly affects the low-frequent response whereas the structural properties are more important at higher frequencies. A rigid approximation of flexible plates is only found at low frequencies, if the elastic length is used as the radius of a rigid disk. At higher frequencies, a characteristic behaviour of the flexible plate of approximately is observed, what is in clear contrast to the compliance of rigid foundations. A plate on a visco-elastic support (Winkler soil) shows similar displacements as a plate on a homogeneous half-space, but the maximal stresses between the plate and the soil are considerably smaller which is found to be more realistic for a plate on a layered soil. For practical applications, a normalized diagram and some explicit formulas of the exact and the approximate solutions of an infinite plate on a homogeneous half-space are given, which is a useful model to represent the soil-structure interaction of flexible foundations.
Seismic response of three-dimensional structures using a green's function approach to the soil
(1986)
Simple and advanced boundary-element method for the soil and its application to railway dynamics
(2007)
Simple and advanced boundary-element method for the soil and its application to railway dynamics
(2007)
This article presents an integrated model for the computation of vehicletrack interaction and the ground vibrations of passing trains. A combined finite element and boundary element method is used to calculate the dynamic compliance of the track on realistic soil whereas multi-body models are used for the vehicle. The dynamic stiffness of the vehicle and that of the track are combined to calculate the dynamic axle loads due to the irregularities of the vehicle and the track as well as those due to sleeper passing excitation. These loads serve as input for the calculation of ground vibration near railway lines in the time and frequency domains.
The theoretical methods and results have been proven by experiments in several respects and at several instances. First, on the occasion of the test and record runs of the Intercity Experimental, there was a very good quality of the vehicle and of the newly built track so that the deterministic parts of the excitationthe static load and the sleeper-passing componentcould clearly be identified, the first being of minor importance apart from the track. Second, simultaneous measurements of the vehicle, the track and the soil at three different track situations were performed where we could verify the different parts of the stochastic excitation and their importance for the ground vibrations. The irregularities of the vehicle are dominant at high frequencies whereas the irregularities of the track are more important at lower frequencies. The comparison of the theory and the measurements also points to the phenomena of the vehicletrack resonance and the scattering of the quasi-static axle impulses by randomly varying soil.
The aim of this contribution is a practice-oriented prediction of environmental building vibrations. A Green's functions method for layered soils is used to build the dynamic stiffness matrix of the soil area that is covered by the foundation. A simple building model is proposed by adding a building mass to the dynamic stiffness of the soil. The vertical soil-building transfer functions with building-soil resonances are calculated and compared with a number of measurements of technically induced vibrations of residential buildings. In a parametrical study, realistic foundation geometries are modeled and the influence of incompressible soil, deep stiff soil layering, soft top layers, and increasing soil stiffness with depth is analyzed. All these special soil models reduce the resonant frequency compared to a standard homogeneous soil. A physically motivated model of a naturally sedimented soil has a stiffness increasing with the square root of the depth and yields a foundation stiffness that decreases with foundation area considerably stronger than the relatively insensitive homogeneous soil. This soil model is suited for the Berlin measuring sites and reproduces satisfactorily the experimental results.
The effect of critically moving loads on the vibrations of soft soils and isolated railway tracks
(2008)
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 massspring 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 massspring system has a marginal bending stiffness along the track.
The propagation of waves through homogeneous or layered soil is calculated based on half-space theory. The moving dynamic loads of a train are approximated by fixed dynamic loads and the wave field can be calculated if the spectrum of the dynamic train loads is known. In addition to this dynamic wave field, there are three different components at three different frequency ranges which are caused by the passage of the static loads:
the regular static component at low frequencies,
the irregular static component at medium frequencies,
the sleeper-passing component at high frequencies.
For each of these components, an approximate solution is presented. The calculated wave field is compared with measurements of different trains at different sites. The measurement of impulse and harmonic point load excitation verifies the soil dynamic base of the method.
A finite-element boundary-element software for the dynamic interaction of flexible structures and the soil has been extended for pile foundation. The boundary element method for the soil uses the Green´s functions of the layered half-space which have been generalised for interior loads. Pile groups of 10 to 20 piles of different arrays are analysed and compared with single piles. Simplified models have been developed for a user-friendly, practice oriented prediction software for railway induced ground and building vibration.
A method is presented which allows to calculate the wave-field in a homogeneous or layered soil in case of a dynamic interior load. The wave propagation along the surface, the distribution of the response over the depth, the horizontal propagation at different depths and the vertical downward propagation are shown and compared with the simpler surface solution of the half-space and the interior solution of the full-space. The complete wave-field (Green's function) is applied to the dynamic behaviour of piles and pile groups by use of a boundary element formulation. The stiffness, damping and – typically for piles – mass of different groups of piles are presented. Different group effects occur for lines, circles, grids, parallels and crosses of piles, which can be regarded as oscillations around average values. Moreover, the piles and pile groups behave almost like a damper for most of the frequencies. A building on a pile group that is excited by ground vibration due to surface or interior loads shows a reduction of the wave-field due to kinematic and inertial soil–building interaction effects. The results presented lead to simplified descriptions of the wave-field due to interior loads and of the soil–pile–building interaction which can be used for the prediction of technically induced vibration.
Transfer admittance spectra of technically induced surface wave fields are analyzed in theory and experiments. Theoretical admittance spectra of layered soils are obtained by integration in wavenumber domain and compared with experimental admittances due to hammer or vibrator excitation. The admittance spectra are strongly influenced by the layering and damping of the soil. Deep stiff-soil layers yield a low-frequency cutoff, whereas a strong damping yields a high-frequency cutoff. A sharp cutoff in a narrow frequency band, which is measured at some sites, can be explained by a damping that increases with frequency, such as viscous material or scattering damping.
Excitation force spectra are necessary for a realistic prediction of railway-induced ground vibration. The excitation forces cause the ground vibration and they are themselves a result of irregularities passed by the train. The methods of the related analyses - the wavenumber integration for the wave propagation in homogeneous or layered soils, the combined finite-element boundary-element method for the vehicle-track-soil interaction - have already been presented and are the base for the advanced topic of this contribution. This contribution determines excitation force spectra of railway traffic by two completely different methods. The forward analysis starts with vehicle, track and soil irregularities, which are taken from literature and axle-box measurements, calculates the vehicle-track interaction and gets theoretical force spectra as the result. The second method is a backward analysis from the measured ground vibration of railway traffic. A calculated or measured transfer function of the soil is used to determine the excitation force spectrum of the train. A number of measurements of different soils and different trains with different speeds are analysed in that way. Forward and backward analysis yield the same approximate force spectra with values around 1 kN for each axle and third of octave.
The attenuation of the amplitudes with distance of technically induced surface wave fields is analyzed in theory and experiments. Experimental results of technically induced ground vibration are presented and collected from literature, which show a power-low attenuation A ~ r–q of amplitudes A with distance r and exponents q > 0.5 higher than for elastic surface waves. Additional attenuation effects are analyzed theoretically. The most important effect is due to the material or scattering damping. Each frequency component is attenuated exponentially as A ~ exp(–kr), but for a broadband excitation, the sum of the exponential laws yields a power law with a higher exponent. Some more effects are discussed, for example the dispersion of the Rayleigh wave due to the layering of the soil, which yields a stronger attenuation A ~ r–q–dq, including an additional exponent of dq = 0.5 in case of an impulsive loading.
The dynamics of slab tracks and floating slab tracks are analyzed by multibeam models for the track and by integration in the wave-number domain for the soil, which is modeled as a layered half-space. Frequency-dependent compliances and force transfers are calculated for a great variety of track and soil parameters. The distribution of the load and the displacements along the track is investigated as well as the wave propagation perpendicular to the track and the ground vibration amplitudes. The floating slab track has a dominating plate-mat resonance and a strong high-frequency reduction. A track-soil resonance can also be recognized for an unisolated slab track in the case of layered soils. Generally, there is a strong damping of the track by the soil. The reduction effect of the slab mat is mainly owing to the elimination of this strong damping. The continuous soil yields slightly different rules for the displacements and force densities than those of a Winkler support. The total force transfer from the rail to the soil is the best criterion to judge the effectiveness of a floating slab track in reducing the ground vibration at some distance from the railway line. The total force transfer is easier to calculate than the double Fourier integrals of the ground vibration amplitudes, namely in the far field, and it has the best correlation with the reduction of the ground vibration.
Vehicle, track and ground vibration as well as their interaction are considered in a combined finite-element boundary-element (FEBEM) approach. The layered soil is calculated in frequency wavenumber domain and the solution for fixed or moving point or track loads follow as wavenumber integrals. The soil results from the measurements and the detailed models are approximated by simple formula which are used for the prediction of train-induced ground vibration. The influence of the track and the soil on the train induced ground vibration is analysed by the detailed models. The ground vibrations strongly depend on the regular and random inhomogeneity of the soil. The regular layering of the soil yields a cut-on and resonance phenomenon while the random inhomogeneity yields a scattering of the axle impulses which proved to be important for high-speed trains. The attenuation with distance of the ground vibration due to the point-like excitations such as vibrator or hammer excitations and the train-track excitation are investigated and compared. All theoretical results are compared with measurements at conventional and high-speed railway lines.
The attenuation of wave amplitudes is ruled by the planar, cylindrical or spher-ical geometry of the wave front (the geometric or power-law attenuation) but also by the damping of the soil (an exponential attenuation). Several low- and high-frequency filter effects are derived for the layering and the damping of the soil, for the moving static and the distributed train loads and for a homoge-neous or randomly heterogeneous soil. Measurements of hammer- and train-induced vibrations at five sites have been analysed for these attenuation and filter effects. The measured attenuation with distance can be discribed by gen-eralised power laws and some reasons will be discussed. The theoretical filter effects can well be found in the measurements.
Train-induced ground vibrations are all generated by the vehicle, by static or dynamic vehicle loads. The most important and most accepted excitation are the dynamic wheel loads from the passage over track irregularities. Dynamic wheel loads will be compared from parallel axle-box and ground vibration measurements at more than seven sites. Some low-frequency excitation of ground vibrations, typically between 10 and 30 Hz, cannot be found in the axle-box measurements. Therefore, other vehicle modes, such as rigid bogie modes, flexible carriage modes, rigid and flexible wheelset modes, have been analysed for additional excitation forces. These vehicle dynamics analyses give an explanation for higher axle-box results at high frequencies, but not for the excitation of the higher low-frequency ground-vibration component. Finally, the effect of the moving static train loads will be analysed. For a regular track and soil, the moving static train loads yield the quasi-static response which exists only in the low-frequency nearfield of the track. If the support stiffness is randomly varying along the track, the pulses on the track generate an additional low-frequency component which is called the irregular pulse responses.
This component will be demonstrated by numerical analysis where all axle pulses are superposed in frequency domain.
The attenuation of wave amplitudes is ruled by the planar, cylindrical or spherical geometry of the wave front (the geometric or power-law attenuation) but also by the damping of the soil (an exponential attenuation). Several low- and high-frequency filter effects are derived for the layering and the damping of the soil, for the moving static and the distributed train loads and for a homogeneous or randomly heterogeneous soil. Measurements of hammer- and train-induced vibrations at five sites have been analysed for these attenuation and filter effects. The measured attenuation with distance can be discribed by generalised power laws and some reasons will be discussed. The theoretical filter effects can well be found in the measurements.
Vibration measurements have many causes and many technical and natural sources. Problems can sometimes be solved by short-term measurements, but in many cases, a long-term measurement is necessary. In long-term measurements of days, weeks, months and even years, it is easy to collect a huge quantity of raw data, but at the end, the post-processing of these data can be exhausting (for example one-year vibration data of a wind energy tower). A software has been developed which con-sists of measuring and evaluation routines where the measuring routines can operate different meas-uring systems and different measuring cards. The main advantage of this software is the fact that the interesting evaluations can be integrated in the measuring process so that the characteristics of the vibration can be extracted without storing all the raw data. Only important time segments are stored, for example train passages. The overall concept of the software and the main evaluation routines will be described in some details. Examples of our measurement experience will illustrate the capabilities of the software. 1) Surveying construction work in nearby sensitive buildings (for example an old wind tunnel), including a stable alarm system and meaningful vibration limits. 2) Prediction of train-induced vibration for a planned building to prevent annoyance and to improve the building design. 3) Modal analysis and long term measurements of several single- or multi-span, concrete or steel bridges 4) Modal and wave analysis of coupled floors in a historical building (“Neues Palais” at Potsdam). 5) Soil properties of various measurement sites (different routines to evaluate the dispersion). Moreover, from many projects, amplitudes, frequencies, and attenuation laws have been collected and analysed for the different sources such as vibratory or impact pile driving and ground compaction, demolition work with different machines, blasting in quarries and in tunnel works, bomb and mine clearing.
Usually, geometric irregularities are considered as the main cause of ground vibrations from trains. A varying stiffness of the track, the track support and the soil can also generate ground vibrations. The regular stiffness variation of the track on and between the sleepers results in a deterministic dynamic axle load. The random stiffness variation of the track support yields also dynamic axle loads. The dynamic axle loads are generated by the varying wheel displacements under the static axle load by the acceleration of the unsprung mass of the rail vehicle. The random stiffness variation has a second effect. The pulses from the passage of the static axle loads are superposed regularly to the quasi-static response, but also irregularly to yield a “scattered” part of the axle pulses. The same holds for a random variation of the soil stiffness. All these effects of stiffness variations have been calculated by wavenumber-domain multi-beam track models, a random finite-element soil model and the superposition of axle impulses in a stochastic simulation. The results are confronted with many measurements at different sites. It is concluded that the stiffness variation of the track and the soil generate an important ground vibration component near railway lines.
A simple and fast prediction scheme is presented for train-induced ground and building vibrations. For the emission, finite-element boundary-element or multiple-beam-on-continuous-soil models of the track have been analysed and approximated by faster track-on-Winkler-soil models. The vehicle-track interaction due to irregularities yields the excitation forces. For the transmission of waves in the soil, the wavenumber integral of the compliance of layered soils has been evaluated. The calculation time is reduced for the prediction by using the solution of a homogeneous half-space with a frequency-dependent wave velocity (the dispersion) of the soil. For the immision, many 2 and 3-dimenisonal finite-element building models have been investigated, and a good approximation has been established by a 1-dimensional soil-wall-floor model. In addition, the axle sequence of the train, the quasi-static and the “scattered” response of the soil, and the wave propagation from a tunnel to a pile foundation of a building have been included.
A simple and fast prediction scheme is presented for train-induced ground and building vibrations. For the emission, finite-element boundary-element or multiple-beam-on-continuous-soil models of the track have been analysed and approximated by faster track-on-Winkler-soil models. The vehicle-track interaction due to irregularities yields the excitation forces. For the transmission of waves in the soil, the wavenumber integral of the compliance of layered soils has been evaluated. The calculation time is reduced for the prediction by using the solution of a homogeneous half-space with a frequency-dependent wave velocity (the dispersion) of the soil. For the immision, many 2 and 3-dimenisonal finite-element building models have been investigated, and a good approximation has been established by a 1-dimensional soil-wall-floor model. In addition, the axle sequence of the train, the quasi-static and the “scattered” response of the soil, and the wave propagation from a tunnel to a pile foundation of a building have been included.
Usually, geometric irregularities are considered as the main cause of ground vibrations from trains. A varying stiffness of the track, the track support and the soil can also generate ground vibrations. The regular stiffness variation of the track on and between the sleepers results in a deterministic dynamic axle load. The random stiffness variation of the track support yields also dynamic axle loads which are generated by the acceleration of the unsprung mass (from the varying wheel displacements under the static axle load). The random stiffness variation has a second effect. The pulses from the passage of the static axle loads are superposed regularly to the quasi-static response, but also irregularly to yield a “scattered” part of the axle pulses. The same holds for a random variation of the soil stiffness. All these effects of stiffness variations have been calculated by wavenumber-domain multi-beam track models, a random finite-element soil model and the superposition of axle impulses in a stochastic simulation. The results are confronted with many measurements at different sites. It is concluded that the stiffness variation of the track and the soil generate an important ground vibration component near railway lines.