Lutz Auersch 1. INTRODUCTION. Federal Institute for Materials Research and Testing, D Berlin, Germany

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1 The Use and Validation of Measured, Theoretical, and Approximate Point-Load Solutions for the Prediction of Train-Induced Vibration in Homogeneous and Inhomogeneous Soils Lutz Auersch Federal Institute for Materials Research and Testing, D Berlin, Germany (Received 10 September 2012; provisionally accepted 15 November 2012; accepted 9 March 2013) The layered soil is calculated in the frequency wavenumber domain and the solutions for fixed or moving point or track loads follow as wavenumber integrals. The resulting point load solutions can be approximated by simple formula. Measurements yield the specific soil parameters for the theoretical or approximate solutions, but they can also directly provide the point-load solution (the transfer function of that site). A prediction method for the train-induced ground vibration has been developed, based on one of these site-specific transfer functions. The ground vibrations strongly depend on the regular and irregular inhomogeneity of the soil. The regular layering of the soil yields a cut-on and a resonance phenomenon, while the irregular inhomogeneity seems to be important for high-speed trains. The attenuations with the distance of the ground vibration, due to point-like excitations such as vibrator, hammer, or train-track excitations, were investigated and compared. All theoretical results were compared with measurements at conventional and high-speed railway lines, validating the approximate prediction method. Figure 1. Emission, transmission, and immission of ground-borne vibration; the separation of the three parts of the prediction of railway induced vibration. 1. INTRODUCTION The prediction of railway-induced vibration consists of the following aspects, as seen in Fig. 1. The emission of vibration includes the vehicle-track-soil interaction, the vehicle and track irregularities, and the dynamic axle loads. 1 The transmission of waves through layered soils constitutes the second part, and the immission concerns the transfer of vibration from the soil to the building. 2 This contribution is concentrated on investigating the transmission component, the influence of the layering and the damping of the soil on the vibration amplitudes, and their characteristic variation with the frequency of excitation and the distance from the track. The dynamic problems of layered soils can be calculated in the frequency-wavenumber domain. 3, 4 The transfer function for the propagation of waves through a layered soil is obtained by an (infinite) integral over the wavenumbers. 5, 6 This wavenumber integral method has also been applied to moving forces These methods are specially used for high-speed trains, if the train speed reaches the wave velocity of a very soft soil. For normal train-soil situations, some simplifications of the complete methods are possible. Simplified rules and models have been established from the results of the detailed models and from experimental experience. These have been integrated in a unique, consistent, prediction scheme. 15 Simple models are necessary to get short computer times, and, most importantly, to get a simple input with only a few relevant parameters. Therefore, a complete prediction of railway vibration is available for specialists and non-specialists. Other simplified prediction schemes, for example, 16, 17 are often not so complete, namely because their transmission parts comprise only transfer functions related to a reference vibration amplitude v 0. A full transmission prediction needs a transfer function related to the excitation force F to properly incorporate the different responses of different soils. The excitation force between the track and the soil is the link between the emission and transmission parts of the prediction, and it is calculated by the analysis of the vehicletrack-soil interaction. The emission model should include the soil under the track 1 (see reference 18 for a counterexample, and see reference 19 for the correction). Another simplified but complete prediction scheme has been developed 20 and is now part of the official prediction method of the US. 21 One of the simplifications is the use of fixed instead of moving loads, which has recently been verified. 22 The central part of the prediction is a specific experimental transfer function for a train load. The prediction of this contribution is 52 (pp ) International Journal of Acoustics and Vibration, Vol. 19, No. 1, 2014

2 similar but uses the transfer function of a point load, which is easier to measure, and the transfer function for a train load is derived by the superposition of point loads. The whole transmission part of the prediction scheme is checked by measurements and by a comparison of the point load transfer function to the train load response. This article consists of five sections. The detailed wavenumber, the combined finite-element boundary-element, and the vehicle-track interaction methods are described briefly by appropriate references in Section 2. The simplified prediction methods, which are used throughout this contribution, are given with all the necessary formulas in Section 3. Section 4 shows the calculated results regarding the differences between point and train loads, as well as the influence of layered soils on the transfer functions and the predicted train vibration. Section 5 presents and compares the measurements and the predictions for point-load and train excitation, concluding a validation of the approximate prediction method. Section 6 discusses two differences between prediction and measurement namely the influence of a high speed of the train and a high damping of the soil and solutions are presented. Generally, the predictions can be made by the use of theoretical, experimental, or approximate transfer functions. Appendix A completely describes how to calculate the approximate transfer functions. The train induced ground vibration is mainly dominated by the dynamic loads, due to the irregularities of the wheel and track. The rules for the quasi-static excitation are established in Appendix B, demonstrating that it is only of importance for low frequencies, short distances, and high train speeds. 2. DETAILED METHODS OF WAVE PROPA- GATION AND VEHICLE-TRACK-SOIL IN- TERACTION The soil is treated as a horizontally-layered, elastic, hysteretically-damped continuum. Each layer is described by the material parameters shear wave velocity v S, mass density ρ, Poisson ratio ν, damping D (according to G = G 0 (1+i2D), where i is the imaginary unit and G = ρvs 2 as the shear modulus), and the thickness h. The compliance, N(f, k), of the layered soil was calculated in the frequency-wavenumber domain, 3, 4, 23 and the solution v at a distance r from a point force F follows as a wavenumber integral 6 v (r, f) = f F 0 N(f, k)j 0 (kr)k dk; (1) with the Bessel function J 0. This solution was used here to calculate the wave propagation in the soil, but it can also be used to calculate a stiffness matrix of the soil via the boundary element method. The stiffness matrix of the track was calculated by the finite-element method. The stiffness K T S of the track-soil system follows from the coupled finite-element boundary-element method. 24 This stiffness K T S of the track under an axle was combined with the stiffness K V of the vehicle at the wheel set, and the vehicle-track interaction yielded the dynamic axle load F, due Figure 2. Third-of-octave spectra of the combined wheel and track irregularities s (a) and of the resulting excitation force F for a wheelset of 1500 kg mass on a standard track with medium stiff ballast and soil (b), for different train speeds 63, + 80, 100, 125, 160 km/h. 1 to the irregularities s of the track and the wheel 1 F = K T SK V K T S + K V s. (2) 3. APPROXIMATE METHODS FOR THE PRE- DICTION OF TRAIN-INDUCED GROUND VIBRATION The prediction of train-induced ground vibration consists of four steps. First, the excitation force spectrum F (f) of an axle was specified. There are three options: (1) a calculated spectrum, as an input from the emission module or from another detailed analysis (see Fig. 2 1 as an example); (2) an experimental spectrum, as an input from train measurements which can be backcalculated to the excitation forces; and (3) a standard train load of 1 kn per axle and third of octave, 1 which is also in fair agreement with the example in Fig. 2. Second, the frequency-dependent effect of the track width of 2a = 2.6 m was approximated by modifying the force spectrum 25 F { sin a F (f) = /a for a π/2 1/a for a > π/2 ; (3) with a = 2πfa v R, where v R is the velocity of the Rayleigh wave. Third, the transfer function H P of the soil for a point load was specified. Once again, there are three options: (1) the input of a theoretical transfer function from a wavenumber integral, International Journal of Acoustics and Vibration, Vol. 19, No. 1,

3 Figure 3. Transfer functions (left) and response to a standard train load (right) for different soils, homogeneous soil of (a), (e) v S = 100 m/s and (b), (f) v S = 300 m/s, (c), (g) layer on a half-space, v S1 = 100 m/s, v S2 = 300 m/s, h = 2 m, (d), (h) soil with increasing stiffness, v S = m/s, distances 4, 8, 16, + 32, 64 m from the source, and damping D = 2.5%. 54 International Journal of Acoustics and Vibration, Vol. 19, No. 1, 2014

4 Figure 4. Transfer functions for x = 2.5, 5, 10, + 20, 37, 57 m (left) and response to train excitation for x = 3, 12, 20, + 30, 50 m (right) at site H, homogeneous soil, v S = 225 m/s, D = 4%, measurements (a), (b), and approximate predictions (c), (d). (2) the input of a measured transfer function from hammer or vibrator excitation, and (3) the calculation of an approximate transfer function based on the near and far field asymptote of a homogeneous half-space H P (r, f) = v (r, f) = F f(1 ν) Gr { 1 for r r e Dκr 0 r /r0 for r > r0 ; (4) with r = 2πfr v S and r More information 6 and details may be found in Appendix A. The amplitudes of these asymptotes attenuate with distance r, according to A r 1 (near field) or A r 0.5 (far field) and the additional damping factor e Dr. The most important soil parameters shear stiffness G and damping D are derived from the measurement of the wave velocity and attenuation. 26 The soil model can be defined as a shear wave profile v S (z) or the dispersion v R (f) of the Rayleigh wave (see Appendix A for details). Whereas the approximate method yields the amplitudes for all distances, the transfer functions from a theoretical or experimental input must be approximated for this purpose. The stable law A r q is used for each frequency for practical reasons (see the next section). Fourth, the response to a train load is calculated by adding the responses of all n axle loads along the train length L n v T (x, f) = j=1 H 2 P ( x 2 + y 2 j, f ) F (f); (5) where n = 40 and L = 250 m are used as standard values. It has been found by numerical tests that this train length already yields the response of an infinitely long train for normal soils with material damping and for normal distances up to 100 m. That means that the ground vibration amplitudes which are predicted for trains longer than 250 m are the same as those of a standard train of L = 250 m. 4. THEORETICAL RESULTS FOR POINT AND TRAIN EXCITATION Four different soil models were considered: (1) a soft (v S = 100 m/s) homogeneous soil, (2) a stiff (v S = 300 m/s) homogeneous soil, (3) a layered soil consisting of the soft and stiff soil (v S1 = 100 m/s, v S2 = 300 m/s, h = 2 m), and (4) a soil with continuously increasing stiffness (v S = m/s). The other parameters are kept constant at ρ = 2000 kg/m 3 and ν = 0.33 throughout this contribution. The theoretical results were calculated by the wavenumber integral (1) and presented in Fig. 3, where the transfer functions for a point and the standard train load are confronted. Generally, the amplitudes of the point-load transfer functions increase along with the frequency. The bending down of the curves at high frequencies is due to the material damping. For the homogeneous soils, as seen in Figs. 3(a) and 3(b), the effect of the material damping D = 2.5% was expressed by the exponential factor e Dκr, which can be seen in Eq. (4). The layered soil, portrayed in Fig. 3(c), shows the low-frequency International Journal of Acoustics and Vibration, Vol. 19, No. 1,

5 Figure 5. Transfer functions for x = 2, 4, 8, + 16, 32, 64 m (left) and response to train excitation for x = 3, 10, 17, + 24, 36, 52 m (right) at site F, soil with continuously increasing stiffness v S = m/s, measurements (a), (b), and predictions with measured transfer functions (c), (d). low amplitudes of the underlying stiff half-space and the highfrequency high amplitudes of the soft top layer. Whereas the layered soil seen in Fig. 3(c) shows a strong increase around 16 Hz from the low-frequency low amplitudes to the highfrequency high amplitudes, this increase was smoother for the soil with continuously increasing stiffness, as seen in Fig. 3(d). In general, similar observations were made for the train loads seen in Figs. 3(e) 3(h). In addition to the reduction of the high-frequency amplitudes due to the material damping, there was a stronger high-frequency reduction for the train-track excitation because of the larger excitation area of the track compared to the hammer excitation. This effect of load distribution, which was expressed in Eq. (3), was stronger for soft soils, such as those seen in Fig. 3(e), so that the high-frequency train vibration was not so different for different soils. Another compensating effect for different soils was due to the material damping. Softer soils, as seen in Fig. 3(e), have a stronger damping effect so that the high-frequency far-field amplitudes were smaller than those of stiffer soils, which can be seen in Fig. 3(f). In the case of the inhomogeneous soils, seen in Figs. 3(g) and 3(h), the higher amplitudes of the stiffer soil in Fig. 3(f) can be found at the high-frequency far field. At the same frequencies, the high near-field amplitudes of the soft soil were reached, so that the attenuation of the inhomogeneous soils as seen in Figs. 3(g) and 3(h) was weaker than the attenuation of the soft homogeneous soil seen in Fig. 3(e). As the mid-frequencies had the lowest attenuation with distance, they were dominant in the far field. Moreover, the high-frequency reduction effects also yielded a maximum in the mid-frequency range of the near field. In the case of the layered soil, as seen in Fig. 3(g), this maximum coincides with the resonance frequency of the layer. If the train load in the right side Fig. 3 is compared to the point load on the left side of Fig. 3, the train-induced amplitudes were higher, especially in the low-frequency far-field. Thus, the attenuation with distance was weaker and the curves for different distances came close together in the case of a train excitation. The power law of A r 1 (near field point load) was reduced to A r 1/2 for the train load. At high frequencies, the exponential attenuation due to the damping was almost the same for the point and the train load, but the additional power law A r 1/2 (far field point load) was reduced to A r 1/4 for the train load. 5. COMPARISON OF THEORETICAL, AP- PROXIMATE, AND MEASURED IMPULSE AND TRAIN VIBRATION (VALIDATION OF THE PREDICTION METHOD) Figures 4 9 present the predictions of the train-induced vibration in four consecutive steps. The measured and calculated transfer functions are shown as Figs. (a) and (c), and the measured and predicted train vibrations are portrayed as Figs. (b) and (d). The excitation spectra are derived from the measurements usually in the range of 0.3 to 3 kn, which is around the standard value of F = 1 kn. The approximate, three exper- 56 International Journal of Acoustics and Vibration, Vol. 19, No. 1, 2014

6 Figure 6. Transfer functions for x = 3, 7, 15, + 32, 64 m (left) and response to train excitation for x = 6, 12, 24, + 48, 72 m (right) at site C, 2-layer soil, v S1 = 150 m/s, v S2 = 450 m/s, h = 1.5 m, D = 2.5%, measurements (a), (b), and theoretical predictions (c), (d). Table 1. Parameters and transfer functions used for the different sites. Site Soil type Soil model v S (m/s) D (%) Transfer function H Sand Homogeneous Approximate F Sand/gravel Increasing Measured C Clay gravel 2 layer 150, Theoretical S Clay rock 3 layer 125, 250, Theoretical G Sand Homogeneous Measured N Sand Homogeneous Measured W Clay rock 2 layer 270, Detailed imental and two theoretical transfer functions, were used for the predictions at six different sites (see Table 1). The measuring points for the impulse and for the train measurements were in the same range of 2 to 64 m distance from the source, but differ slightly for the different sources and sites. The different wave velocities of 225, , , , 170, and m/s, as well as the different layering and damping (D = 1 5%) yielded specific transfer functions for each site. Homogeneous soils are at sites H and G; sites F and N have an increasing stiffness; and a two layer and a three layer soils are at sites C and S in Table 1. The following characteristics of the different soil models were found to be in agreement between theory and measurement: the weak increase of the amplitudes for the homogeneous half-space (Sites H and G, Figs. 4, 8(a), and 8(c)), the stronger but steady increase of amplitudes for the soils with increasing stiffness (Sites F and N, Figs. 5, 9(a), and 9(c)), the sudden strong increase around 32 Hz (Site C, Figs. 6(a) and 6(c)), and 20 Hz (Site S, Figs. 7(a), and 7(c)) for the two and the three-layer soil. Figures 8 and 9 compare the results of two soils with extremely different material damping. The soil at Site G has a very low damping of D = 1% (Figs. 8(a) and 8(c)). The midfrequency train-induced amplitudes in both theoretical and in experimental results showed almost no attenuation with distance, as can be seen in Figs. 8(b) and 8(d). The soil at Site N has a high damping of D = 5%, and, correspondingly, a strong attenuation at high frequencies for both impulse and train excitation, as can be seen in Fig. 9. These two examples demonstrate that the soil-specific response to train excitation can be predicted well from the measured hammer excitation. All theoretical transfer functions in Figs. 4 9(c) represent well the measured ones, as seen in Figs. 4 9(a). The prediction of the train induced vibration in Figs. 4 9(d) yield narrower bands of spectra, which are modified by some characteristic excitation components. The predicted narrower bands of spectra generally agree with those of the train measurements seen in Figs. 4 9(b). Therefore, it can be concluded that the prediction method was validated by these seven examples seen in Figs. 4 7 and 11. Some minor differences are discussed in the next section and possible modifications of the prediction are shown. The attenuation with distance of the impulse, the predicted and measured train vibration for these six and some more 1, 6, 27, 28 sites, has been evaluated by the approximate power laws A r q, seen in Fig. 10. The measured attenuation at high frequencies is usually stronger with attenuation powers q I = for impulses and q T = for trains, International Journal of Acoustics and Vibration, Vol. 19, No. 1,

7 Figure 7. Transfer functions for x = 2, 12, 16, + 20, 42 m (left) and response to train excitation for x = 3, 5, 10, + 20, 30, 50 m (right) at site S, 3-layer soil, v S1 = 125 m/s, v S2 = 250 m/s, v S3 = 750 m/s, h i = 2, 4 m, D = 2.5%, measurements (a), (b), and theoretical predictions (c), (d). which can be seen in Figs. 10(a) and 10(b). The attenuation at mid and low frequencies is considerably smaller with q I = for impulses and q T = for trains. The attenuation power q P of the predicted train vibration, seen in Fig. 10(d), has been found to be reduced compared to the attenuation power q I of the theoretical impulse transfer function, Fig. 10(c), as approximately q P = q I 0.5. The attenuation of the measured train vibration is similar to the predicted one, but can be as strong as the impulse attenuation in some cases. Some soils and some trains show a stronger attenuation in the low-frequency near field (see next section). 6. DISCUSSION OF HIGH-SPEED EFFECTS, HIGH-DAMPING EFFECTS, AND MODI- FIED PREDICTIONS The train-induced vibration of one additional site was predicted by a modified prediction method, which included the passage of static loads, in addition to the transfer function of the layered soil and a dynamic force spectrum; see Fig. 11. The passage of static train loads yielded the low-frequency ground vibration, seen in Figs. 11(b) and 11(d), which decayed very rapidly with distance. Considerable near-field amplitudes can be seen for the train with 200 km/h, in Fig. 11; for slower train speeds, the quasi-static effect was not so strong, as seen in Figs The rules for this special high-speed effect are available in Appendix B. The specific high frequency amplitudes around 100 Hz, seen in Figs. 11(b) and 11(d), were due to the parametrical excitation by the sleeper passage with the sleeper-distance frequency. The wider frequency band of this component was a result of the Doppler effect of moving loads. The amplitudes of the low, high, and mid-frequency component can be easily predicted for all train speeds between 100 and 300 km/h; this is done through a detailed prediction method, 28 where the midfrequency component has been attributed to irregular inhomogeneities of the soil. 29 Besides this possible effect of irregular inhomogeneities, it can be concluded from the presented site examples that the effects of regularly inhomogeneous soils, such as layered soils or soils with increasing stiffness, can be found in many measurements. Some of the measured transfer functions showed a certain high-frequency cut-off for the amplitudes; see Figs. 7(a) and 11(a) at about 64 Hz, among other examples. 6, This cutoff can be explained by higher damping values for the high frequencies or the top layers. However, it has also been found and can be seen in Figs. 7(b) and 11(b) that the cut-off effect was not so strong for the train-induced vibration as for the hammer vibration. It can be concluded that in the case of a high-frequency cut-off in the impulse measurements, it is better to use a more regular theoretical transfer function that is derived from the soil measurements at that specific site. For the majority of the measuring sites and frequencies the amplitudes and attenuation laws for train-induced ground vibration are easily predicted by both theoretical and experimental transfer functions. 58 International Journal of Acoustics and Vibration, Vol. 19, No. 1, 2014

8 Figure 8. Transfer functions for x = 4, 10, 18, + 38, 54 m (left) and response to train excitation for x = 4, 8, 16, + 32, 64 m (right) at site G, homogeneous soil, v S = 170 m/s, D = 1%, measurements (a), (b), and predictions with measured transfer functions (c), (d). 7. CONCLUSIONS of the soil. A practical prediction scheme, based on fixed dynamic axle loads and different (theoretical, measured, and approximate) transfer functions of the soil, has been demonstrated for many theoretical and measurement sites, widely validating its worth. A realistic homogeneous or layered soil model is important for to achieve a good prediction. The attenuation of the amplitudes with distance has been evaluated for the power laws A r q, with the exponent q in the wide range of 0.2 to 2.5. The attenuation of the impulse vibration was the strongest, compared to which the predicted train vibration yielded a clearly reduced attenuation power (approximately reduced by 0.5). The measured train vibration showed a similar attenuation as the prediction, but in some cases a little higher attenuation was observed. The mid-frequency train vibration usually had the lowest attenuation, so that this component was dominant in the mid and far fields. The high-frequency amplitudes had the strongest attenuation, which is most sensitive to the soil parameters. Simple and stable laws seem to be best suited for the prediction of these components. Due to the passage of static loads, the low-frequency component can have relevant near-field amplitudes and a strong near-field attenuation in the case of highspeed trains. Hammer- and train-induced vibrations generally depend on the stiffness of the soil. At high-frequencies, the material damping and the wider load distribution of the track led to a reduction of amplitudes, and therefore to a compensation of the stiffness effect so that the high-frequency amplitudes were sensitive to the damping, rather than to the stiffness ACKNOWLEDGEMENTS Thanks to S. Said, W. Schmid, and W. Wuttke for the pleasant cooperation at the measurements. REFERENCES 1 Auersch, L. Theoretical and experimental excitation force spectra for railway induced ground vibration: vehicle-tracksoil interaction, irregularities and soil measurements, Vehicle System Dynamics, 48, , (2010). 2 Auersch, L. Building response due to ground vibration simple prediction model based on experience with detailed models and measurements, International Journal of Acoustics and Vibration, 15, , (2010). 3 Kausel, E. and Roesset, J. Stiffness matrices for layered soils, Bulletin of the Seismological Society of America, 71, , (1981). 4 Wolf, J. Dynamic Soil Structure Interaction, Englewood Cliffs, New Jersey, Prentice Hall, (1985). 5 Jones, D. Surface propagation of ground vibration, PhD Thesis, University of Southampton, (1987). 6 Auersch, L. Wave propagation in layered soil: theoretical solution in wavenumber domain and experimental results International Journal of Acoustics and Vibration, Vol. 19, No. 1,

9 Figure 9. Transfer functions for x = 2.5, 5, 10, + 20, 35, 45 m (left) and response to train excitation for x = 3, 5, 10, + 20, 30, 50 m (right) at site N, v S = m/s, D = 5%, measurements (a), (b), and predictions with measured transfer functions (c), (d). WAVE 94 Wave Propagation and Reduction of Vibrations, Bochum, , (1994). 9 Dietermann H. and Metrikine A. The equivalent stiffness of a half-space interacting with a beam. Critical velocities of a load moving along a beam, European Journal of Mechanics A/Solids, 15, 67 90, (1996). 10 Grundmann, H., Lieb, M., and Trommer, E. The response of a layered half-space to traffic loads moving along its surface, Archive of Applied Mechanics, 69, 55 67, (1999). 11 Sheng, X. Ground vibration generated from trains, PhD Thesis, University of Southampton, (2001). Figure 10. Measured (top) and calculated (bottom) attenuation of ground vibration amplitudes with distance from the hammer (left) and the train excitation (right), minimum and maximum attenuation powers q according to A x q for the low-, mid- and high-frequency range. of hammer and railway traffic excitation, Journal of Sound and Vibration, 173, , (1994). 7 De Barros F. and Luco J. Response of a layered viscoelastic half-space to a moving point load, Wave Motion, 19, , (1994). 8 Aubry, D., Clouteau, D., and Bonnet, G. Modelling of wave propagation due to fixed or mobile dynamic sources, Proc. 12 Takemiya, H. Simulation of track-ground vibrations due to high-speed train: the case of X-2000 at Ledsgard, Journal of Sound and Vibration, 261, , (2003). 13 Lombaert, G., Degrande, G., Kogut, J., and François, S. The experimental validation of a numerical model for the prediction of railway induced vibrations, Journal of Sound and Vibration, 297, , (2006). 14 Auersch, L. The effect of critically moving loads on the vibrations of soft soils and isolated railway tracks, Journal of Sound and Vibration, 310, , (2008). 15 Rücker, W. and Auersch, L. A user-friendly prediction tool for railway induced ground vibrations: emissiontransmission-immission, Proc. IWRN9 9th International 60 International Journal of Acoustics and Vibration, Vol. 19, No. 1, 2014

10 Figure 11. Transfer functions for x = 2.5, 5, 10, + 20, 30, 45 m (left) and response to train excitation for x = 2.5, 5.5, 9.5, + 20, 30, 50 m (right) at site W, 2-layer soil, v S1 = 270 m/s, v S2 = 1000 m/s, h = 10 m, D = 5%, measurements (a), (b), and theoretical prediction for a high-speed train with v T = 200 km/h (c), (d). Workshop on Railway Noise, München, CD-ROM, 1 9, (2007). 16 Melke, J. Noise and vibration from underground railway lines: proposals for a prediction procedure, Journal of Sound and Vibration, 120, , (1988). 17 Ziegler, A. Vibra-1-2-3: a software package for ground borne vibration and noise prediction, Proc. 6th International Conference on Structural Dynamics (EURODYN 2005), Paris, , (2005). 18 Kouroussis, G., Verlinden, O., and Conti, C. Free field vibrations caused by high-speed lines: measurements and time domain simulation, Soil Dynamics and Earthquake Engineering, 31, , (2011). 19 Kouroussis, G., Verlinden, O., and Conti, C. A two-step time simulation of ground vibrations induced by the railway traffic, Journal of Mechanical Engineering Science, 226, , (2012). 20 Nelson, J. and Saurenman, H. A prediction procedure for rail transportation groundborne noise and vibration, Transportation Research Record, 1143, 26 35, (1987). 21 Hanson, C., Towers, D., and Meister, L. Transit noise and vibration impact assessment, Report FTA-VA for the Federal Transit Administration, HMMH Inc., Burlington, (2006). 22 Verbraken, H., Lombaert, G., and Degrande, G. Verification of an empirical prediction method for railway induced vibrations by means of numerical simulations, Journal of Sound and Vibration, 330, , (2011). 23 Auersch, L. Wave propagation in the elastic half-space due to an interior load and its application to ground vibration problems and buildings on pile foundations, Soil Dynamics and Earthquake Engineering, 30, , (2010). 24 Auersch, L. Dynamics of the railway track and the underlying soil: the boundary-element solution, theoretical results and their experimental verification, Vehicle System Dynamics, 43, , (2005). 25 Auersch, L. The excitation of ground vibration by rail traffic: theory of vehicle-track-soil interaction and measurements on high-speed lines, Journal of Sound and Vibration, 284, , (2005). 26 Auersch, L. and Said, S. Experimental soil parameters by different evaluation methods for impulsive, train and ambient excitation, Proc. 5th International Symposium on Environmental Vibration, Chengdu, China, 3 9, (2011). 27 Auersch, L. and Maldonado, M. Interaction véhiculevoie-sol et vibrations dues aux trains modélisation et vérifications expérimentales, Revue Européenne de Mécanique Numérique, 20, , (2011). International Journal of Acoustics and Vibration, Vol. 19, No. 1,

11 28 Auersch, L. Train induced ground vibrations different amplitude-speed relations for different layered soils, Journal of Rail and Rapid Transit, 226, , (2012). 29 Auersch, L. Ground vibration due to railway traffic the calculation of the effects of moving static loads and their experimental verification, Journal of Sound and Vibration, 293, , (2006). 30 Maldonado, M. Vibrations dues au passage d un tramway mesures expérimentales et simulations numériques, PhD Thesis, École Centrale de Nantes, France, (2008). 31 Auersch, L. Technically induced surface wave fields, part II: measured and calculated admittance spectra, Bulletin of the Seismological Society of America, 100, , (2010). 32 Alves Costa, P. Vibraçoes do sistema via-maciço induzidas por tráfego ferroviário modelaçao numérica e validaçao experimental, PhD Thesis, University of Porto, Portugal, (2011). 33 Auersch, L. Simplified methods for wave propagation and soil-structure interaction: the dispersion of layered soil and the approximation of FEBEM results, Proc. 6th International Conference on Structural Dynamics (EURODYN 2005), Paris, France, , (2005). 34 Holzlöhner, U. and Auersch, L. Propagation of shock waves at the surface of heterogeneous soil Grounds, International Journal of Numerical Methods in Geomechanics, 8, 57 70, (1984). 35 Lamb, H. On the propagation of tremors over the surface of an elastic solid, Philosophical Transactions of the Royal Society of London, A203, 1 42, (1903). APPENDIX A: THE APPROXIMATE DISPER- SAL SOIL METHOD A two-step procedure to approximately calculate the transfer function of layered soil has been developed. In the first step, the dispersion of the layered soil was approximated, whereas in the second step, the transfer function of a dispersal soil was approximated. Figure 12. Exact and approximated dispersion of (a) a layer on a half-space (v S1 = 100 m/s, v S2 = 300 m/s, h = 2 m) and (b) a soil with increasing stiffness, λ/3-rule, Rayleigh integral, 34 +multi-layer method. with f 1 = v S1 3h 1, which is extended to a multi-layered situation. A correction for each layer was added to the wave speed v R1 of the top layer as: n 1 v R (f) = v R1 + i=1 (v Ri+1 v Ri )0.5 ( 1 + cos π ) f ; (A.2) 2 f i with f i = v Si 3h i, where h i is the depth where the layer i ends. For very low frequencies, the wave speed of the underlying half-space can be reached. In Fig. 12, the approximate dispersion was compared to the exact dispersion and two other methods: the third-ofwavelength rule and a Rayleigh-integral rule. 34 Only the proposed approximate method yielded good results for the clearly layered soil, as seen in Fig. 12(a). For soils with continuously increasing stiffness, like those seen in Fig. 12(b), all three methods gave a good approximation of the dispersion. Approximate Transfer Functions of Dispersal Soils The amplitudes of a homogeneous soil can be approximated by the asymptotes Exact and Approximate Dispersion The exact dispersion curves of layered, multi-layered, and continuously layered soils 33 have been used to derive an approximate calculation of the Rayleigh wave dispersion v R (f) from the soil profile v S (z). It is based on the approximated dispersion of a layer on a stiffer half-space with the wave velocities of the shear or Rayleigh wave v S1 and v S2 or v R1 and v R2, for example as ( v R (f) = v R1 + (v R2 v R1 ) cos π 2 f f 1 ) ; (A.1) H P (r, f) = v (r, f) = F f(1 ν) Gr { 1 for r r e Dκr 0 r /r0 for r > r0 ; (A.3) with r = 2πfr v S. The first part of Eq. (A.3) describes the static solution, which can be used for the near field and the low frequency response. At r = r0, begins the latter part of Eq. (A.3) begins, Lamb s far-field asymptote, 35 where the Rayleigh wave dominated the far-field, with its wave velocity v R = v S κ 62 International Journal of Acoustics and Vibration, Vol. 19, No. 1, 2014 (A.4)

12 Figure 14. Response of a homogeneous soil to a moving static load, normalised velocity amplitudes v as a function of the normalised frequencydistance parameter f = fx/v T, linear (top) and logarithmic scales (bottom). Figure 13. Approximate transfer functions of different soils, (a) soil with increasing stiffness, v S = m/s, (b) layer on a half-space, v S1 = 100 m/s, v S2 = 300 m/s, h = 2 m, distances x = 4, 8, 16, + 32, 64 m. Table 2. Parameters of the homogeneous half-space. ν β = v S /v P κ = v S /v R A far r slightly lower than that of the shear wave. This asymptote can be calculated by the expression 35 r where r 0 = (2κ2 1) 2 2κπ κ 2 β 2 (1 β 2 ) r 4κ(1 (6 4β 2 )κ 2 + 6(1 β 2 )κ 4 ; ) β = v S 1 2ν = v P 2 2ν (A.5) (A.6) is the ratio of the shear and compressional wave velocities (see Table 2). The amplitudes of a homogeneous soil can be used to calculate the wave-field of an inhomogeneous soil. The amplitudes of the inhomogeneous soil, with its dispersion v R (f), were approximated by the amplitudes of a homogeneous half-space, but for each frequency, the half-space had a different wave speed the wave speed v R (f). This method worked very well for a soil with continuously increasing stiffness; see Fig. 13(a) as compared to Fig. 3(d). If this method is applied to a clearly layered situation, the following modifications must be included. First, a frequency- dependent resonance amplification such as: V = i2dη η 2 ; (A.7) { f/f0 for f f can be introduced, where η = 0, 2 f/f 0 for f > f 0 f 0 = v S 3h, and D = 2 ρ 1v 1 π ρ 2v 2 are used for the normalized frequency η, the resonance frequency f 0, and the damping D was calculated according to the one-dimensional wave theory. The effect of soft top layers on the low-frequency near-field and the effect of stiff deeper layers on the high-frequency farfield were included by a general procedure. 6 Normally, at a certain frequency f, the half-space amplitudes of the corresponding wave velocity v R (f) are calculated. Corrections to the transfer functions were made if deeper and stiffer soil material yielded greater half-space amplitudes, or softer top layers yielded greater layer amplitudes, which are approximated by A L = A H e ar/h, according to the low frequency behaviour of a layer on a rigid base. The greatest amplitude is always chosen as the correct transfer function. The results of this approximate method for the two-layer soil are presented in Fig. 13(b). The approximate results agree very well with the results calculated with the complete (wavenumber domain) method (Fig. 3(c)), so that a simple method for the prediction of the ground vibration of homogeneous and inhomogeneous soils was established. APPENDIX B: THE PASSAGE OF STATIC LOADS Besides the exact calculation of the soil response to moving static loads, which has been used for Fig. 11, an approximate solution exists for most cases when the train speed is well below the wave velocity of the soil. International Journal of Acoustics and Vibration, Vol. 19, No. 1,

13 Figure 16. Measured response at site W (v S1 = 270 m/s, v S2 = 1000 m/s, h = 10 m) to a train excitation with speed v T = 100, 160, 200, + 250, 280 km/h, at x = 3 m. Figure 15. Response of a homogeneous soil (v S = 300 m/s) to a moving static load of 100 kn, third-of-octave spectra at the distances 2.5, 3.2, 4, + 5, 6.4, 8, 10, 12.5 m, v T = 300 km/h (a), and for train speeds v T = 100, 125, + 160, 200, 250, 320 km/h, at x = 2.5 m (b). The displacements u of the soil due to the passage of a single static load F can be calculated approximately by using the amplitude-distance law of a homogeneous half-space u(t, x) = (1 ν)f 0 2πGr(t, x) = (1 ν)f 0 1 2πGx 1 + (vt t/x) = 2 (1 ν)f 0 2πGx u (t ); (B.1) with t = v T t x, where x is the distance of the observation point to the track, r the time dependent distance of the moving load to the observation point, and v T the train speed. The spectral density of the particle velocity at the observation point was established by the Fourier integral v(f, x) = i2πf (1 ν)f 0 if Gx (1 ν)f 0 if Gx u(t, x)e i2πft dt = u (t )e i2πf t dt x v T = t 2 e i2πf t dt = tion. The spectrum v (f ) has a maximum at f = 0.1, which is usually below f = 4 Hz, the value which is reached for x = 2.5 m, and v = 100 m/s (360 km/h). For higher f, the function has an exponential decay, which could be approximated by v 6e 5.5f. That means that there is a weak attenuation after the maximum and a strong attenuation for higher values of f. Therefore, this function is only of importance for lower values; e.g., for f < 0.4, and that means for low frequencies, low distances, and high train speeds. The effects on the third-of-octave ground velocity spectra are demonstrated in Fig. 15. For increasing train speeds, the spectra are shifted to higher frequencies, as seen in Fig. 15(b). For train speeds lower than 100 km/h, the relevant parts of the quasi-static spectrum are below 4 Hz. The attenuation with distance in Fig. 15(a) is very strong. In addition to the normal low-frequency attenuation of v x 1, there is the strong exponential attenuation due to the function v (f ). The strong attenuation with distance can also be found in the measurements in Fig. 11(b). The shift in frequency with increasing train speed has also been found in experiments as shown in Fig. 16. The spectra for a single axle, as see in Figs. 15(a) and 15(b), are amplified and modified when the number and the sequence of all axles of a train are incorporated in the 25, 29 analysis. (1 ν)f 0 v (f ); Gx (B.2) with f = xf v T. A single dimensionless function, v (f ), presented in Fig. 14, completely describes the quasi-static solu- 64 International Journal of Acoustics and Vibration, Vol. 19, No. 1, 2014