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- Ground vibration (11)
- Finite-element boundary-element method (4)
- Layered soil (4)
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- Hammer impact (3)
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Simple and advanced boundary-element method for the soil and its application to railway dynamics
(2007)
Train passages induce static and dynamic forces on the track, the train-induced vibrations propagate through the soil and excite neighbouring buildings. The problem of train vibrations is divided into the parts emission, which is the excitation by railway traffic (the present contribution), transmission, which is the wave propagation through the soil, and immission, which is the transfer into a building, - The calculation of the axle loads are based on the vehicle-track-soil interaction. This interaction uses the dynamic stiffness of the vehicle (the inertia of the wheelset) and the dynamic stiffness of the track-soil System. Based on various time consuming finite-element boundary-element calculations, an approximate track-soil model has been established. The vehicle-track-soil analysis yields several transfer functions between the various geometric or stiffness irregularities and the axle loads of the train. Geometric irregularities of the vehicle (the wheels) and the track (rail surface and track alignment) are the simplest components. Geometric irregularities of the subsoil (trackbed irregularities) have to be transferred to effective irregularities at rail level. The bending stiffness of the track is filtering out the short-wavelength contribution. Stiffness irregularities occur due to random variations in the bailast or the subsoil, which must also be transferred to effective track irregularities, and due to the discrete rail support on sleepers. The axle loads due to the effective track errors from stiffness variations have their specific vehicle-track transfer function. - All necessary formula for the prediction of axle-load spectra will be presented. The prediction method is compared with axle-box measurements at a Standard ballasted track. Moreover, ground Vibration measurements at numerous sites are exploited for the axle-load spectra and the Validation of the prediction method.
The computation of the wave propagation in homogeneous and layered soils can be performed by a numerical integration in wavenumber domain. The numerical difficulties of an infinite integral and an integrand with poles can be solved. But if this computation must be repeated for many distances, many frequencies, many loads, or many soil models, it becomes a time consuming task which is not acceptable for a user-friendly prediction tool for railway induced ground vibration. Therefore, an approximate method for the computation of the wave field has been developed. The computation consists of several steps. At first, an approximate dispersion profile is calculated according to rules which have been derived from exact solutions. Secondly, the dispersion is used to achieve the amplitude for a certain frequency and a certain distance by calculating the approximate solution of a corresponding homogeneous half-space. Thirdly, three layer corrections are added which include lowfrequency near-field effects, high-frequency far-field effects, and a resonance amplification around the layer frequency. This procedure yields the wave field due to a point load. For a train load, many of these point-load responses have to be summed up, and a frequencydependant reduction factor has to be multiplied to incorporate the effect of the load distribution along and across the track. - The prediction method is applied to real sites, and the appropriate soil models are identified by approximating the measured transfer functions (frequency-dependant amplitudes) which is presented as an alternative to the approximation of the dispersion (frequency-dependant wave velocities). These examples demonstrate the general behavior of layered soils: the low amplitudes of the stiff half-space at low frequencies, the high amplitudes of the softer layer at high frequencies, the strong increase of amplitudes and a possible resonance amplification at mid frequencies. The material damping of the layer yields a strong attenuation of the amplitudes with the distance for high frequencies. The response depends strongly on the resonance or layer frequency which is shown for different layer depths and velocities always in good agreement with measurements. The layer frequency can be of immense influence if train-speed effects are analysed in a layered soil. The good agreement with many measurements in this contribution as well as in the references validates the prediction of ground vibration based on the theory of a layered half-space.
Offshore wind energy towers are dynamically loaded by waves and wind. Pile foundations provide stiffness and damping and should be properly calculated. A combined finite-element boundary-element method for the dynamic interaction of flexible structures and the soil has been developed. The flexible structures such as single piles or complete wind energy towers are modeled by the finite element method whereas the homogeneous or layered soil is modeled by the boundary element method which uses the Green’s functions for interior loads in the layered half-space to establish the dynamic stiffness matrix of the soil. Soils with a stiffness that is continuously increasing with depth can be modeled as multi-layer soils with step-wise increasing stiffness. The effects of different parameters such as the stiffness of the soil, the axial and bending stiffness of the pile, and the radius of the cylindrical contact area will be analysed for the different components of excitation (vertical, horizontal, rotation and coupling). The results can be determined as specific power laws which are different for the different load cases and for the different soil models (Winkler support, homogeneous continuum, continuum with increasing stiffness). The dynamic effect of radiation damping will be analysed by the frequency-dependent compliance functions. A clear layering of the soil can cause noticeable changes in the dynamic compliances as reductions of the stiffness and the damping in certain frequency ranges (below and around layer resonance frequencies). The distribution of the displacements along the pile help to explain the observed laws. An example of an offshore wind energy tower has been modeled and calculated for wind, wave and weight loads. The resonances of the tower are usually limited by the radiation damping which is strongest for a soft soil.
The Federal Institute of Material Research and Testing (BAM) has collected some experience with the testing of damaged, repaired and newly constructed railway tracks. The experimental methods are hammer testing of the track at different positions, hammer testing of the soil, measurement of train passages, and in all cases, measurements with geophones at different positions. The measured signals are evaluated for wave velocities (dispersion of the soil or the track-soil system), for transfer functions (transfer admittances of the soil, compliances of the track in amplitude and phase), and one-third octave band spectra of the track response to hammer and train excitation. These methods are applied at different stages of the track construction. Before track construction, wave velocities and transfer functions of the sub-soil can indicate problems with soft soils. After track construction, a check of the acceptable state of the track can be done by comparison of many excitation positions and track sites. After a track damage (a lose sleeper or a lose plate of a slab track) and after its repair, the sensitivity of the different measurement quantities to different track errors and the achieved improvement of the repair can be determined. The contribution shows examples of all these track situations.
Experiments have been performed at a test site with six different tracks with under-ballast plates. Hammer excitations of the soil and the tracks as well as train passages have been measured. The experimental observations are as follows. 1. The natural soil is stiff gravel whereas the railway dam consists of softer material. 2. The track compliance indicates a soft ballast if no train is present to provide a confining pressure. 3. The track response to the train passages can be split into a low-frequency region which is ruled by the static loads and a high-frequency region which is ruled by dynamic loads. 4. The track responses to hammer and track excitation indicate the presence of many voids between the sleepers and the ballast. 5. The ground vibrations are highly influenced by the soil. Due to the stiff soil at the site, the hammer and train induced spectra have a considerable high-frequency content. 6. A reduction of the ground vibration has been observed in a low-frequency range. The mitigation effects of an under-ballast plate are also investigated by calculations of a wavenumber domain model. The under-ballast plate has an effect at low frequencies where it distributes the static load over a longer track section. The impulse of the axle passage is longer and the frequencies are lower due to the plate stiffness. The axle impulses could yield a low-frequency ground vibration in an irregular soil with a randomly varying stiffness. This low-frequency part of the ground vibration (the scattered axle impulses) seem to be reduced by the under-ballast plate.
Explosion-induced ground vibrations have been measured at several places. Results about the wave propagation are shown in this contribution. The particle velocities of the soil have been measured at up to 1000 m distance from the explosion and are presented as time records (seismograms) and one-third octave band spectra (transfer functions). The results are compared with the results of hammer impacts. The seismograms clearly show different wave types, compressional waves of the air, the water and the soil, and the Rayleigh wave. The hammer impacts yield good results up to 100 m and incorporate higher frequencies at about 50 Hz, whereas the explosion results in a ground vibration with frequencies around 10 Hz and a longer range of influence. Explosion and hammer excitations are evaluated for the wave velocities of the soil by using the wavenumber and the spatial auto-correlation method. The attenuation of the ground vibration amplitudes A with distance r can well be presented by a power law A ~ r -q. This type of amplitude-distance law and the corresponding power q > 1 are substantiated in the contribution. The influence of the charge weight W is evaluated as an additional power law A ~ W -p for each measuring site. The power is found quite similarly around q 0.6 as all sites have a medium soft soil such as sand and clay. The obtained amplitude-charge-distance law can be used to predict the explosion-induced ground and building vibrations at other sites.
Three measurement campaigns of train-induced ground vibrations are evaluated for the vehicle-track-soil interaction. Ground vibrations, track vibrations and vehicle vibrations have been measured for train passages and impulse excitation and compared with theoretical results. The soil and the track-soil system are calculated by wavenumber integrals. The influence of the vehicle is introduced by a substructure method. By comparing theory and measurement the different components of excitation force and ground vibration can be analysed, the quasi-static excitation, track-alignment errors, the out-of-roundness of wheels, the wheel and rail roughness, and moreover, scattered axle impulses and ineffective high-frequency parts of the wheelset accelerations and forces.
Vibration of normal apartment, office and production buildings, which are excited by technically induced ground vibrations are considered. Many wavelengths of the Rayleigh waves of the soil fit into the foundation dimensions. The related high discretization effort can nowadays be realized with detailed soil-structure interaction method. The combined finite-element boundary-element method is used here as a detatiled method. Simplified method can be used with less computation time, but these methods must be calibrated by exact results. One simplification is to extent the structure to infinity and to solve the problem by wavenumber domain methods. Another simplification is the use of a Winkler soil instead of the continuous soil. Usually, the Winkler parameters are not only soil parameters but depend also on the rigid or flexible foundation structure. Substructure methods use commercial FEM software for the building part. The contribution will show some detailed and some simplified results on large structural elements such as foundation plates, walls, storey plates on columns as well as results on complete buildings. The reduction of the ground vibration by stiff elements and the amplification due to floor or building resonances are discussed which are the most important phenomena of the soil-building interaction.