Parameter calibration in the modelling of railway traffic induced vibrations by use of Barkan

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1 Parameter calibration in the modelling of railway traffic induced vibrations by use of Barkan F.M.B. Galanti, A. Koopman & G. Esposito TNO Building and Construction Research, Delft, The Netherlands ABSTRACT: Reliable prediction of vibration levels in the surroundings of railway links is dependent on an accurate estimation of model parameters. An important modelling parameter is damping, but because of difficulties in its estimation, particularly in the case of soils, calibration by use of in-situ vibration tests is often wanted. In the case of FEM, such an optimization is not straightforward. As an intermediate step, the Barkan formula, which explicitly handles damping, can be used as a common ground to which both measurements and FEM modelling results are translated. In this article such a side step is investigated. 1 INTRODUCTION Due to increased demands in reliable surface transport, there has been a considerable increase in the past two decades in the number of high speed railway links. A direct drawback of such railway links, especially in dense urban environments, is the generation of noise and vibration not only during the exploitation phase but also during the prior construction phase. Consequently, it has become necessary to estimate the levels of noise and vibration in the vicinity of (future) railway lines. Based on the results of such estimates, alternatives may need to be selected for the construction method and for the design of the railway line, whereby in the latter, noise and vibration mitigation measures may be taken into consideration. Considering only the vibration problem, estimates or predictions of vibration levels can be made using a variety of analysis tools ranging from empirical models to advanced models such as the finite element method. In a recent Delft Cluster project, the accuracy of the existing prediction strategies was investigated. The results of the project highlighted that vibration predictions are characterized by a large uncertainty, de Wit & Waarts (2003). The project also demonstrated that the uncertainty does not significantly decrease with the complexity of the used prediction tool. Sophisticated multi-dimensional numerical models seem to yield the same level of uncertainty as that associated with engineering judgment. Nevertheless, advanced models probably offer the best modelling options, but may be difficult to tune due to the necessity to make a number of simplifying assumptions and due to the large number of modelling parameters. In literature, explicit treatment of the reliability of vibration prediction models has received so far almost no attention. Details as to total model uncertainty or as to what part of a model leads to the greatest uncertainty are not available. Vibration prediction models therefore can only be used to provide a rough indication of the response of the structures in the surroundings of a railway line. In order to promote further application of vibration prediction models in engineering practice it is necessary to gain insight as to the reliability which can be expected from the use of a specific model. As a first step in this direction, this study will focus on the estimation of the value of damping to be assumed in vibration prediction models, since this is one of the most important parameters and also one of the most difficult to determine. The underlying model is based on an an approach consists whereby separate modules for the vibration source, the transmission path and the receiver, are used. Any mechanical interaction between the modules is neglected, see Fig. 1. Source Path Receiver Figure 1. Typical components of a vibration prediction model. The present study will focus on the response of the transmission path model, due to the passage of a train, without considering the response of the receiver. The practical case which is considered is that of a section of high speed rail track in the vicinity of Waremme, Belgium. This section of the high speed

2 rail track has been the subject of a an extensive vibration measurement campaign, consisting of falling weight tests and measurements of field vibrations during several passages of a Thalys TGV, Esposito (2003) and Esposito (2004). A procedure will be outlined with which damping parameters can be calibrated on the basis of a comparison of finite element models of the transmission path and field measurements. 2 CALIBRATION PROCEDURE The modelling scheme under investigation consists of a source model for trains (called TRINT ), FEM transfer models (either 2D plane strain or 3D), and a procedure to combine source with transfer, as described by Koopman & Courage (2001). A considerable source of modelling uncertainty are the soil parameters needed for the transfer models. Of these, material damping is the most important. Damping is both hard to determine from geotechnical investigations (like SPT s) and therefore often not available, and at the same time very influential on the model outcome. In this article, therefore, the focus lies on dealing with material damping. When modelling ground-borne vibration, often there is a need and an opportunity to update the modelling with some sort of in-situ vibration measurements. In those cases, the question arises how to perform an update and how to set up a measurement for such a tuning. Given, for example, a falling weight, both from reality as from FEM modelling so much vibration output can be derived, and so many discrepancies between model en measurement can be expected, that an update scheme will never be straightforward, especially given the large number of input variables that can be tuned. Of course there are various numerical update procedures available for many-input many-output problems. However, such procedures often are black boxes in character. What is necessary, in any specific project and for the advancement of this field in general, is an update scheme based on physical insight. Then, further steps after tuning, like the design of mitigation measures, will also benefit. Given a model with a given level of detail, a more abstract model can form the basis for physical insight in the results. For ground-borne vibration, the Barkan model often fulfils the role of the more abstract model. In essence, it describes vibration as propagating wave energy and only uses geometry and damping for the energy balance. Relating this to FEM, the geometry factor mainly links to the soil layering in the FEM model and the damping factor links directly to material damping in the FEM model while slightly less directly to the other material parameters. A possible update scheme is using Barkan as a mediator between measurement and FEM model. As stated, damping is often a crucial parameter in a FEM model and Barkan handles it explicitly. The update then revolves around the relevant parameter and physical insight can be gained. First the two Barkan parameters, geometry and material damping, are fitted to the measurements. Obviously such a fit, although not straightforward, is nevertheless considerably less demanding than a fit of a FEM model to measurements. Next the FEM model is fitted to the Barkan model by tuning the material damping properties. The whole update procedure should be performed per frequency band (like octave bands) in order to appreciate the frequency dependency of the physics involved. The updating can then be enhanced further by also involving the other material parameters and/or the soil layering, through exploitation of their differences in frequency dependency effects. In such an updating care should be taken to keep the parameters within physical limits, preferably within the bandwidth of uncertainty established preparing the first modelling phase. A special case form the FEM codes that use Rayleigh damping, which actually most do. Rayleigh damping, although numerically advantageous, imposes a frequency dependency on material damping that is hardly physical. Compensation by an update that involves more than only the material damping then is hardly avoidable. Further enhancement can be gained by performing different types of measurements. For instance, if possible, apart from a falling weight also a train passage by can be used as a source. One then has a point source (falling weight) as well as a line source (train). In such a case first a 2D plane strain FEM model is tuned to a Barkan fit of the train passage. The geometrical Barkan parameter will then be small and full attention can be paid to the damping parameter. Next a 2D axial symmetric or a 3D model is fit to the falling weight test, using the formerly found damping parameter and focussing on the geometrical parameter. Even better would be a coupled approach, having two parameters and using two equations. The extra dimensions that frequency dependency and possibly other material parameters introduce will impede such approach however. 3 TRAIN, TRACK AND SOIL MODEL The section of interest is a double track inside a semi-submerged open tunnel with a concrete base floor, 0.8 m thick, resting on sheet piles and concrete piles, 13 m in length. The soil consists of a top layer of sandy loam about 12.5 m thick resting on a chalk layer. The track and the soil comprise the transmission module which is simulated using both two-

3 dimensional and three-dimensional finite element models with the use of silent boundaries. The model for the soil is used to determine the transfer function between a point on the track and a series of points along a line perpendicular to the track at the surface level. Subsequently, the response due to a train passage could calculated by combining the transfer functions with modelling or measurements of the train loading. A diagram of the two-dimensional model is given in Fig. 2. In this model, a total of 8527 quadrilateral 8-node plane strain elements are used. The model extends over a cross section of 1 m containing the track and surrounding soil up to a depth of 23 m. A half sine unit impulsive load with a duration of 5 ms is applied at in the middle of the rightmost track. A Newmark-beta time integration scheme is used (γ=1/2, β=1/4, unconditionally stable) with a time step of 1 ms. The total simulation time is 0.6 s. Silent boundaries have been used along all the edges of the model. In order to avoid relying too much on the silent boundary s capability to absorb incident waves, the size of the model has been selected in such a way that the total travel distance of waves over the simulation time (including the travel distance after reflection off the boundary) would be too small to affect adversely the response of the area of interest. In this case the area of interest is the free field surface to the right of the open tunnel at a distance between 5 and m from the source. Material parameters have been selected on the basis of SCPT s and borings carried out at the site. From these tests the elastic modulus and Poisson ratio of the two soil layers have been estimated. The tests, however, gave little indication as to the material damping. This property, which determines part 11.5 Z X F +3.4 m m Figure 2. Two dimensional model. The bottom of the concrete tunnel floor is at z= 2.5 m. The black line denotes the boundary between the soft top layer (loamy sand) and the bottom chalk layer is at z= 9.5 m. of the attenuation of vibrations generated at the source, had to be estimated. The top soil layer has a damping ratio of 3.5%, whilst the bottom layer and all other materials have a damping ratio of 1.2%. From these damping ratios, equivalent Rayleigh damping ratios have been extrapolated in the frequency range between 3 and 100 Hz. The final material parameters used in the analysis are given in Table 1. Table 1. Material properties used in the simulations. Property Material Upper Lower Concrete Steel soil layer soil layer Young's [MPa] modulus Poisson ratio [-] Density [kg/m 3 ] Rayleigh damping Shear wave speed a [s -1 ] b [s] 1.12E E E E-05 [m/s] In order to compare different modelling options, a similar analysis has been carried out using a three dimensional model, see Fig. 3. This model is simply an extrusion of the two-dimensional model, however, with a slightly rougher mesh. The mesh is formed by a total of 7077, 20 node brick elements, and extends by 120 m in depth. Silent boundary elements are placed along the edges of the model except for the edge at z=0, which, out of symmetry considerations, is fixed in the z direction (the load being applied at this edge). Y Z Figure 3. Three dimensional model with silent boundaries. X The first step of the analysis will be that of comparing the response obtained from the two models with the response measured from falling weight tests. From the comparison of the attenuation in the and in the models, the accuracy of the initial estimate of the damping in the soil will be evaluated. As a final step the response of the models due to the passage of a train will be compared with measurements COMPARISON OF MODEL RESULTS WITH FALLING WEIGHT TESTS In order to compare the attenuation of vibration in the models with that which is observed in the s, frequency response functions at various locations along the surface have been evaluated. Plots of the mobility against frequency are given in Fig. 4 for two locations on the free field at 6 m and 48 m distance from the source. From the graphs, it can be observed that the response of the 2D model is less than two orders of magnitude greater than that

4 of the 3D model and that the measured response lies in between that of the two models. The 2D model response is greater than that of the 3D model since the 2D model effectively represents a line loading with infinite extension on an elastic half-space, whilst in the 3D model the loading consists of a concentrated force. It is interesting to note that the model spectra and that of the all show a similar trend at all distances FF01Z - ch. 13 response functions, damping parameters r and α can be estimated. Subsequently the material damping parameter α can be used to determine the damping ratio of the soil, given the following relationship: ( f ) V R ( f ) 2πfζ α = α = (2) where f is the frequency, V R is the Rayleigh wave speed and ζ is the damping ratio. Although frequency dependent, it will be be assumed that the Rayleigh wave speed is constant. 2D model mobility [s/kg] mobility [s/kg] 2D model 3D model frequency [Hz] FF06Z - ch. 22 one-third octave - centre frequency [Hz] D model D model 3D model frequency [Hz] Figure 4. Mobility compared between different models and distance [m] 4.1 Estimation of damping Attenuation of waves traveling from the source is caused by radiation damping and by material damping and is typically described by Barkan s formula: u u ( x) ( x ) 0 r x0 = exp x x [ α( x )] 0 (1) where x is the distance from the source and r and α are respectively the geometrical and material damping parameters. Frequency response functions taken at various distances from the source can be used to evaluate damping, since the amount by which a signal attenuates solely due to material damping is frequency dependent. This dependency can be observed in Fig. 5 which shows the variation of inertance spectra with the distance from the source. By fitting the Barkan model for each single frequency in the Figure 5. Attenuation of inertance frequency response functions for the vertical direction as a function of distance from the source: and 2D model. The first attempts to use the fitting procedure led to results which were rather inconclusive. It was noted that the geometrical damping parameter varied erratically with frequency and that very little attenuation occurred at frequencies below 30 Hz. The latter can be explained by the fact that signals were measured within a section of the surface between 5 and m distance from the source. Since planar Rayleigh waves form only beyond a wavelength s distance from the source and recalling a Rayleigh wave speed for the site of 185 m/s, only Rayleigh waves with a frequency above 37 Hz should be considered. Waves with a frequency below this value (or with a wavelength greater than 5 m) therefore are not fully formed and will not be detected properly.

5 With respect to the first problem, the two damping parameters have a similar effect on the attenuation in the Barkan model, hence small variations in the data can easily lead to an interchange of attenuation values between the two parameters. If an assumption is made as to the amount of geometrical damping and if only frequencies corresponding to wavelengths smaller than 5 m are considered, then more realistic material damping values are obtained. In Fig. 6 average damping ratios over one-third octave bands for the and the 3D model are presented. For the 3D model and the falling weight, the exponent for the geometrical damping, r=0.5. Considering the frequency range between 20 and 100 Hz, average damping ratios of 2.7% for the, and 2.8% for the 3D model are obtained. ζ Damping ratio 3D Model Avg. 3D model Avg one-third octave centre frequency [Hz] Figure 6. Damping ratio evaluated per one-third octave for the 2D model (top) and the 3D model (bottom). Nonetheless, no particular trend in the variation of damping ratio with frequency is to be observed both in the model and in the. Variations in damping ratio with frequency, both in the model and the could be associated with the special geometry of the situation which includes the open tunnel. Because of its presence, the open tunnel could lead to a wave field which attenuates more like that caused by a line load. 4.2 Updating Fitting Barkan on the train passage measurements and the related 2D plane strain FEM model, assuming r=0, yields the results shown in Figure 7. On average, the chosen FEM parameters seem to give rise to too much damping, compared to the measurements. Remarkably, this FEM damping is also too high compared to the material damping that was put into the model and to which it is supposed to be closely related. Also, the characteristic frequency dependency of Rayleigh damping ('the bathtubspectrum') is not reproduced by the Barkan fit. As of yet, no explanation has been found for this dampingspectrum that 2D plane strain FEM modelling yields for this case. Therefore, tuning the material damping through the 2D model to the train passage measurements is disregarded at this stage. ζ Damping ratio D Model Avg. 2D model train passage Avg. train passage one-third octave centre frequency [Hz] Figure 7. Damping ratio evaluated per one-third octave from data relating to a high speed train passage The next step is fitting Barkan on the falling weight test and the related 3D FEM model (Figure 6). Initially, the damping found in the 2D case would be used as given and the geometry factor would be fitted. However, in this case the geometry factor must be considered the least uncertain and set to 0.5 while fitting the damping once again. Measurement and modelling resemble each other much more now, especially in the frequency range up to Hz, and further tuning of the model might even be considered superfluous. The damping found is also much closer to the material damping put into the model. Considering that the geometry of the situation is not quite axial symmetric (as stated before) and the set geometry (r=0.5) is therefore unlikely to be very accurate or frequency independent, no bathtub shaped damping spectrum could be expected. Now, the next step would be to combine the traintrack source model with the now verified 3D FEM model to make predictions. A further verification step, which would mainly verify the source model and the way the two models are coupled, would be to compare the results with the train passage measurements again. This step, however, lies outside the scope of this article. 5 CONCLUSIONS Accurate assessment of situations which may lead to the production of noise and vibration requires prediction models which can represent reliably the frequency and amplitude of vibrations. To gain reliability, models can be tuned with in-situ vibration measurements. To facilitate tuning of FEM models,

6 more general models such as the Barkan formula might be of use. - Comparison between measurement and modeling on the basis of the Barkan fits they produce turns out to be quite satisfactory for 3D FEM modeling, for the case under investigation. - Given that damping is a crucially uncertain parameter this makes it possible to fit material damping input for FEM models to in-situ vibration measurements by use of the Barkan formula. - The Barkan fit did not work well in the 2D case, yielding damping ratio s that seem to be too high. - An updating scheme involving both 2D and 3D measurements and modelling as coupled tuning problems could therefore not be put to the test. The actual quality of the investigated updating scheme can only be determined when an updated model is validated by separate measurements, preferably in the changed, future situation for which the modelling was needed. Such a validation is left to be undertaken. ACKNOWLEDGEMENT C.B.M. Blom Holland Railconsult / HSL Zuid. Delft, The Netherlands REFERENCES M.S. de Wit & P.H. Waarts Reliability of vibration predictions in civil engineering applications. ESREL G. Esposito Experimental Determination of the Impedance of a RC Railway Construction in Waremme(BE), High Speed line Brussels Cologne. TNO-report 2003-CI- R0066. G. Esposito Experimental Determination of Vibrations Induced by IC and HS Train Passages. TNO-report CI-R0072. A. Koopman & W.C. Courage Modelling of soil vibrations from railway tunnels Part 1 : TNO. Proceedings of 7th IWRN. H.R. Stuit, W. Gardien, A. Koopman, R.J. Aartsen Benchmark for prediction models for vibrations from railway tunnels. Proceedings of 7th IWRN, 2001