High-speed train-induced ground motion and interaction with structures

High speed train-induced ground motion and interaction with structures P. Galv´ın, J. Dom´ınguez ∗ Escuela Superior de Ingenieros, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain Abstract A general numerical model for the analysis of soil motion due to high-speed train passage and effects on nearby surface and underground structures is presented in this paper. Soil, ballast and structures are represented using a 3-D time domain boundary element approach. Material damping has been included in the time domain formulation. Equilibrium and compatibility between soil and underground or surface structures is established in a rigorous manner in order to take into account soil-structure interaction. In contrast to other existing approaches, effects on a particular structure and the influence of non-uniform soil conditions along the track can be evaluated. Numerical results are obtained for different situations and some of them are compared with existing experimental records. The experimental values are to a large extent reproduced by the present numerical approach. Ballast influence and effects of high-speed train passage on a concrete underpass structure are analyzed. Key words: High-speed train, Transient dynamic, BEM, Experimental validation 1 Introduction Many new high speed train (HST) lines are being constructed in Europe, Asia and the USA. The growing interest in the analysis of train-induced in recent years calls for additional studies to be conducted in this area. Problems such as train vibrations, soil free field vibrations produced by train traffic, and dynamic effects on structures situated near the track, are much more important ∗Corresponding author. Tel.: +34 954487293; fax: +34 954487295. Email address: [email protected] (J. Dom´ınguez). Preprint submitted to Elsevier Preprint 29 June 2007 in the case of high speed trains than for conventional ones. A thorough analysis of these effects is required in order to ensure security and comfort in the trains and to avoid eventual problems for nearby constructions which may be affected by vibrations induced by waves transmitted through the soil. Particularly serious would be situations in which the train speed may be higher than that of the surface waves in the underlying soil. Such a possibility was completely unthinkable for conventional trains but it is now something that should be taken into account when high speed trains operate at locations with soft soils or underground discontinuities that may result in relatively low surface wave speed. Worthy of note here is the record speed of 574.8 km/h (159.7 m/s) set by a French train travelling between Paris and Strasburg in April 2007. The study of these problems requires employing comprehensive models that can take into account significant factors related to the characteristics of the train, the track, and the local soil properties. Nearby structures should also be modelled in case where the effects on these structures are being analyzed. The final objective of this paper is to make gains in being able to accurately predict soil wave propagation due to the passage of HST and to forecast vibrations that will affect existing structures and those to be constructed. Different analytical, semi-analytical or numerical methods for the analysis of HST-induced vibrations have been developed in recent years. One of the important aspects of the problem is modelling the force transmitted by the train to the soil through the rail beam and the sleepers. Particularly noteworthy was the model presented by Krylov [1,2] and Krylov and Fergurson [3] during the nineties. In this model, they consider a quasi-static force transmitted by the sleepers and combine it with Green’s functions of a half-space in order to represent the wave propagation through the soil. More advance models of similar type were presented in [4–6]. Several papers have dealt with train track dynamics with ground interaction in the last decade. Dieterman and Metrikine [7,8] used a beam model on an equivalent one-dimensional spring system for the half-space. The model was replaced by a stratum by Metrikine and Popp [9]. Focusing on the track system, some other papers take into account the fact that rails are supported discretely by the ground and that sleepers have a significant inertia effect on the rails. Metrikine and Popp [10] and Vostroukhov and Metrikine [11] considered these factors by taking a general solution for the differential equation. Takemiya and Bian [12] make an extension of [10], by including a layered soil analysis and use a discrete Kelvin model for the sleepers’ complex frequencydependent stiffness. Different authors have also presented approaches for the analysis of soil induced vibrations due to moving loads (road or train traffic) using Green’s 2 functions for a layered half-space. Most of them are based on the so-called two- and-a-half-dimensional fundamental solution, obtained by Tadeu and Kausel [13], combined with a boundary element (BE) discretization. Some of these works were presented by Clouteau et al [14], Lombaert and Degrande [15,16], Lombaert et al. [17,18], Degrande et al. [19], Degrande [20], and Degrande and Schillemans [21]. Andersen and Nielsen [22] presented a boundary element method (BEM) formulation for the analysis of the response of an elastic half-space to a moving surface force. They used a frequency domain full space Green’s function formulated in a frame of reference and obtained the time-domain solution by means of Fourier transformation. A time-domain BEM using a fundamental solution in a frame of reference was proposed by Chouw and Pflanz [23]. Numerical procedures for the analysis of ground vibrations produced by HST based on the combination of BEM and FEM have also been presented by several authors [24,25]. Auersch [26] has recently presented a rather comprehensive model using a combined FEM-BEM model for soil and track and a multi-body model for the vehicle, to calculate the dynamic compliance. Sheng et al. [27] also use a FEM/BEM approach for the analysis of ground vibration produced by trains. These papers and some others [28–30] are based on the idea that the ground and structures such as tunnels and tracks, are homogeneous in the track direction. This assumption, which simplifies the analysis to a large extent, is realistic for the study of free field motion and vehicle vibrations in many parts of a line; however, the analysis of ground vibrations in zones with local discontinuities in the soil properties or the effects on structures such as underpasses, buildings and auxiliary structures around the track, require of a fully three-dimensional model. Situations with important dynamic soil-structure interaction effects also required of a comprehensive model where the interaction is taken into account by coupling both domains rigorously. In this paper, a general and fully three-dimensional BEM model for the analysis of the soil motion and the effects of HST passage on nearby surface and underground structures is presented. Soil, ballast and structures are represented using 3-D time domain BEM. Structures can also be represented by a FE model. A full-space fundamental solution is used in combination with quadratic BE. Special attention is given to stabilization algorithms and element subdivision to improve efficiency, stability and accuracy. Internal material damping is introduced in the BE formulation in a simple and efficient manner. The formulation presented is more general than other existing approaches. It permits full coupling between the soil and nearby structures; it is able to consider embankment, ballast and other local effects, as well as coupling with structures that brake uniformity of the geometry along the track direction. In the following, the time domain BEM formulation is summarized first. Second, internal damping is introduced in the formulation. Third, an 3 analysis of the ground motion induced on a homogeneous half-space by the HST passage is carried out. Fourth, the effects of ballast and embankment geometry are studied, and the obtained results are validated by comparison with those obtained by other authors and existing experimental records. Last of all, the effect of HST passage on an underpass structure is analyzed. Motion at different points and loads on the structure are obtained. Effects of HST passage on different types of structures, while taking 3-D and soil-structure interaction effects into rigorous consideration, will be presented in an upcoming paper. To the authors’ knowledge, no previous studies include the employment of a fully numerical model in the analysis of HST passage effects on local structures, taking dynamic soil-structure interaction rigorously into account. Some of the basic numerical aspects of the BE approach presented in this paper were tested in a previous paper by the same authors [31] using much simpler preliminary problems. 2 Time domain BEM for 3-D problems The three dimensional time domain BE formulation for transient problems is very briefly summarized in this section. This topic is more thoroughly addressed in Dom´ınguez [32] and Marrero and Dom´ınguez [33]. The integral representation of the displacement uat a point ion the boundary of an elastic body, at time twith zero body forces and zero initial conditions can be written as [32]: ci lkui k(xi, t) = Zt+ 0ZΓulk(x, t −τ;xi)pk(x, τ)dΓ(x)dτ −Zt+ 0ZΓplk(x, t −τ;xi)uk(x, τ)dΓ(x)dτ (1) where ukand pkstand for the kcomponent of the displacement and traction, respectively; ulk(x, t −τ;xi) and plk(x, t −τ;xi) are the fundamental solution displacement and traction tensors, respectively, at point xdue to a point load at xi. The coefficient ci lk depends only on the boundary geometry at point i. Displacements and tractions over the boundary are approximated from their nodal values at each time step umj kand pmj k, the space interpolation functions φj(r) and ψj(r), and the time interpolation functions ηm(τ) and µm(τ). 4 After interpolation of the boundary variables Equation (1) becomes ci lkuni k= n X m=1 Q X j=1 "(ZΓjhZ∆tm ulkµmdτiψjdΓ)pmj k −(ZΓjhZ∆tm plkηmdτiφjdΓ)umj k#(2) where Qis the total number of boundary nodes and Γjare the elements to which node jbelongs. The integrals of ulkµmand plkηmfor each time step are usually called Unm lk and Pnm lk , respectively. Thus, equation (2) becomes ci lkuni k= n X m=1 Q X j=1 "(ZΓj Unm lk ψjdΓ)pmj k−(ZΓj Pnm lk φjdΓ)umj k#(3) which in a more compact form can be written as ci lkuni k= n X m=1 Q X j=1 "Gnmij lk pmj k−ˆ Hnmij lk pmj k#(4) Once the independent term ci lk is included in the system matrix, the integral representation for point iat time t=n∆tbecomes n X m=1 Q X j=1 Hnmij lk umj k= n X m=1 Q X j=1 Gnmij lk pmj k(5) and the system of equations for all the boundary nodes at time t=n∆tcan be written in matrix form as n X m=1 Hnmum= n X m=1 Gnmpm(6) Once the boundary conditions are applied, Equation (6) yields a system of equations which can be solved step-by-step to obtain the time variation of the boundary unknowns. Piecewise constant time interpolation functions µm(τ) are used for tractions and piecewise linear functions ηm(τ) for displacements. The time integrals in Equation (2) can be evaluated analytically without much difficulty. In the present paper, nine node rectangular and six node triangular quadratic elements are used. Each side of the element is divided into equal parts in the natural coordinates domain yielding an element subdivision. The spatial integration extends only to those subdivisions whose mid-points are under the effects of the fundamental solution waves according to the causality condition of each term of the fundamental solution. 5 Ensuring that the stepping procedure stable is an important issue in time domain BEM. To do so an approach based on the idea of using a linear combination of equations for several time steps in order to advance one step, is applied in the present paper. A constant velocity is assumed at each time step to predict traction and displacement values in the time step ahead of that being computed. Details of this stabilization approach can be found in [33]. 3 Internal soil damping It is well known that wave amplitudes decrease as they propagate in solids due to material internal damping. Material damping effects in soil motions due to high-speed trains have been experimentally observed by different authors (Madshus and Kaynia [34], Degrande and Schillemans [21], Degrande [35], Auersch [26]). Including viscous material damping in a frequency domain BEM formulation is simple [32]. Using complex valued shear modulus of the type µ(1 + 2ξi), where ξis the viscous damping coefficient, or µ(1 + ζi) where ζis the hysteretic damping coefficient (ζ= 2ξ), internal damping is taken into account in a very simple and straightforward way. However, to consider internal damping in time domain BEM formulations is not that simple. Several authors [36,37] have proposed different approaches where Laplace or another transform type would permit the introduction of internal damping at a high computational cost. Jin et al. [38] proposed a simple approach for 2D problems with a dominant frequency in the system response. The approach, based on the idea that periodic oscillators present a constant amplitude logarithmic decrement per period, is extended here to 3-D problems and used for the analysis of soil motion induced by high-speed train passage. The basic idea is rather simple: a damping term is introduced in the boundary integral equation as ci lkui k(xi, t) = Zt+ 0"ZΓulk(x, t −τ;xi)pk(x, τ)dΓ(x)#exp (−2πξ(t−τ)/T)dτ −Zt+ 0"ZΓplk(x, t −τ;xi)uk(x, τ)dΓ(x)#exp (−2πξ(t−τ)/T)dτ (7) where the damping term is determined from the viscous damping coefficient ξ or the hysteretic coefficient ζ= 2ξ, and the dominant vibration period Tin the response. Once the integral equation is discretized one obtains the following equation for each time step Hnnuu=Gnnpu+ n−1 X m=1 (Gnmpm−Hnmum) exp (−2πξ(n−m)∆t/T) (8) 6 where the right hand side term derived from previous steps is damped by an exponential coefficient with linearly increasing exponent, as time goes on. It should also be mentioned here that the material damping term included in Equations 7 and 8 is consistent with that proposed by Barkan [39] in his classical work where an attenuation law depending on the distance rfrom the source and the dominant period T uξ(r) = uξ(r0)µr r0¶−n exp Ã−2πξ(r−r0)/c T!(9) is assumed. Note that (r/r0)−ncorresponds to radiation damping, which in the BEM case is included in the exact fundamental solution. Results obtained with the model represented by Equations (7) and (8) have been shown to be accurate, as compared to known solutions and experimental results for situations simpler than train-induced ground vibration problems [31]. Even though the present damping model is based on an attenuation law for vibrations with a known period, good results can be obtained using a reference dominant period. In particular, for high-speed train induced vibrations one can use a dominant frequency in the range 40 Hz to 80 Hz depending on the train speed. These values correspond to the intermediate range, containing the relatively high peaks of the experimental frequency spectra shown in Figures 10-12. 4 Homogeneous half-space motion due to train passage A problem of soil vibrations induced by a high-speed train passage will be studied next. The train is considered to be a set of axle loads travelling at constant speed and the soil a uniform viscoelastic half space. 4.1 Axle load distribution and near field discretization The load is assumed to be transmitted to the soil by each couple of wheels through the sleepers according to the load distribution proposed by Krylov [40]. He obtained a quasi-static load distribution pattern among several sleepers that transmit the load of an axle. In this model, the track is represented as a beam on a Winkler foundation. The model does not incorporate some sources of vibration such as rail roughness, wheel flats and parametric excitation. The distribution depends on the rail beam stiffness (EI = 12.76 ×106 Nm2is assumed for two rail beams), the ballast stiffness (ks= 250 MPa is 7 assumed) and the mass per unit length including both of the rails and the sleepers (m= 620 kg/m is assumed). The values for the track have been taken from Degrande [20]. The rail beam vertical displacement wis obtained from the solution of the differential equation EI ∂4w ∂y4+m∂2w ∂t2+ksw=Tδ(y−vt) (10) where Tis the axle load. The solution of Equation (10) is given by [40]: w(y−vt) = T 8EIβ3δexp(−βδ|y−vt|)Ãcos βη(y−vt)+ δ ηsin βη|y−vt|!(11) where β= (ks/EI)0.25,δ= [1 −(v/cmin)2]0.25,η= [1 + (v/cmin)2]0.25 and cmin = (4ksEI/m2)0.25 is the wave propagation velocity along the beam. The load P(t) transmitted to the soil by one of the sleepers located at y= 0 is: P(t) = T Nst eff w(vt) wst max (12) where Nst eff is the number of sleepers supporting the load in the case that all the sleepers had the maximum rail beam deflection. It can be seen from this equation that each sleeper transmits a fraction of the axle load Tproportional to the track deflection at time t. Figure 1.(a) shows the load transmitted to the soil by one sleeper when a unit axle load travels at a speed of v= 315 km/h. The load distribution depends to a certain extent on the load velocity. The frequency content corresponding to this load variation with time is shown in Figure 1.(b). Prior to any train passage analysis, and in order to asses the most appropriate BE mesh for the soil surface, a very refined mesh is first assumed for a rather short patch on the soil surface along the track. It includes (Figure 2.(a)) the length corresponding to 36 sleepers and a width of 21.28 m, 960 being the total number of elements. The surface under each sleeper is discretized into six rectangular elements and so is the surface between two sleepers (Figure 3.(a)). The rest of the elements are also quadrilateral with size increasing with distance to the track. The mesh in Figure 2.(a) is very dense and, therefore, able to accurately represent the soil behaviour near the loaded area; but, it is not long enough to properly represent the effect of a load that is travelling from far away and whose influence on a given observation point starts long before the load is over this discretized area. It would be necessary to make it 8 several times longer, but this would give way to an extremely large number of elements. A less dense mesh is tested to represent the soil around the track. The new mesh for the same surface is shown in Figure 2.(b). The smallest elements in this case are 1.2 m ×1.25 m rectangles arranged in a way such that two elements represent the area corresponding to two sleepers and two intermediate areas. A row of 5 nodes is on the mid line of each sleeper (Figure 3.(b)). To compare the two meshes in figures 2.(a) and 2.(b), a load of 15×104N travelling at 315 km/h from one end of the mesh to the other is assumed. The time step size is ∆t= 0.003 s. In the first case (Figures 2.(a) and 3.(a)), the pressure corresponding to each sleeper according to Krylov’s distribution is applied to the 39 nodes on each sleeper. In the second (Figures 2.(b) and 3.(b)) this traction is applied to 5 nodes on the same sleepers. It should be taken account that in this second case, as the shape functions extend over the complete element, the area between two sleepers would also be loaded. An equivalent load is given by the relationship between the total load under the sleepers according to Krylov’s distribution in the first mesh, and the total load that would be transmitted to the soil in the simplified BE model where tractions are prescribed at the corresponding nodes. This relation depends on Krylov’s distribution, which in turn depends on the rail beam-ballast-soil properties and the load velocity. In the case considered, with a load speed 315 km/h, the relation is 0.345. Taking into account this correction for the second mesh, surface displacements are evaluated using both meshes. The soil is assumed to be a homogeneous elastic half-space with shear modulus µ= 18.5×106 N/m2; Poisson’s ratio, ν= 0.3 and density, ρ= 1850 kg/m3. Vertical and cross displacement components obtained at points on the mid cross section located at several distances from the track axis show that both meshes give almost identical soil surface displacements. This fact and results for other preliminary tests for different speed values lead to the conclusion that elements such as those pertaining to the coarse mesh can be used to represent the soil surface near the track. 4.2 Homogeneous half-space surface motion The analysis of the soil vibrations induced by a high speed train on an homogeneous viscoelastic half-space is now carried out using the BE mesh of Figure 4 which has a total length of 86.4 m, a width of 25 m, and elements for the track zone of the same type as those in Figure 2.(b). The train is represented by an array of loads corresponding to the axles location and load values of trains Alstom (Thalys) as shown in Figure 5 and Table 1. In 1997, Degrande and coworkers [20,21,35] made a set of experimental mea- 9 is small for t= 0.267 s and t= 0.804 s. Vertical displacements are almost symmetric and their values are almost the same as for greater distances to the underpass. At t= 0.537 s the load is over the underpass and, as compared to the other two cases, the Zdisplacements values under the load and its vicinity are smaller due to the stiffening effect of the concrete plate located just under the loaded surface. Figure 22 shows vertical displacements at three half-space surface reference points for the train passage. The three points are shown in Figure 20 as A1, A2 and A3. Their distances to the track axis is 11.5 m and their distance to the plane of symmetry is 23.4 m, 3.6 m and 0 m, respectively. The bogie passage effect can be seen at the three locations. It should be noted that the half-space surface motion at point A1 (X=−23.4 m) is almost the same as for points with larger values of X. The maximum displacements of the three points shown, take place at point A2 (X=−3.6 m) due to the load transmitted to the soil surface through the concrete walls, whereas point A3 (X= 0 m) at the plane of symmetry take intermediate values. In any case, displacements at the half-space surface are much smaller that those at embankment point closer to the track. Vertical displacements at points B of the embankment at distance Y= 3.5 m from the track axis are shown in Figure 23. Values for three points, B1, B2 and B3 at distances 23.4 m, 3.6 m and 0 m, respectively, are shown. The peak displacement values due to the train passage are of one order of magnitude larger than those for the soil surface and decrease very significantly when the point is over the under-pass structure due to its stiffening effect. Traction values at the underpass deck due to the HST passage are also directly obtained from the model. The time history for two different points C and D of the deck cross track axis of symmetry are shown in Figure 24. Point C is the centre point of the deck (see Figure 20) and point D is on the deck external boundary. As expected, tractions take their maximum values under the sleepers zone and decrease towards the deck external boundary. Additional numerical results for displacements and tractions on the underpass deck corresponding to different load velocities can be found in [41]. 7 Conclusions A three-dimensional time domain boundary element formulation for viscoelastic solids has been applied to the analysis of soil motion due to high-speed train passage. A decaying law, that takes into account the internal soil material damping has been included in the formulation which is more general 16 than previous boundary element formulations and allows for a realistic representation of the problem. As compared to frequency domain formulations, the present BE formulation allows for the consideration of coupling with nearby structures that may have a non-linear behaviour. When compared to formulations based on direct integration of half-space Green’s functions, the present BE formulation has the added advantage of being able to consider the actual geometry, embankment and ballast effects, and other local effects. As compared to 2.5-D solutions the present formulation can take into account local soil discontinuities, underground constructions such as underpasses, and coupling with nearby structures that brake the uniformity of the geometry along the track line. The present approach is limited by the required computational time for large meshes. Nevertheless, the translation property in space and time of the fundamental solution together with large parts of the soil surface mesh which are uniform and repeat themselves along the track direction allow for development of numerical algorithms that significantly reduce the computational time and increase the model extension. Those computational advantages have not been used in the studies presented in this paper. All the numerical examples presented in this paper have been run using a laptop computer. Some of the conclusions drawn from the computed numerical results for the particular situations for which there are in situ measurements of soil surface motion due to high-speed train passage are: (1) The actually measured motion at different distances from the track and different train speed values, are rather accurately reproduced by the present numerical approach. The velocity peak values are accurately evaluated. The model should be improved in order to take into account intermediate frequency motions excited by rail and wheel roughness. (2) Ballast layers have a significant influence on the system response. They should be considered in the model. (3) A half-space with equivalent properties may be used to represent soil and ballast effects; however, a prior assessment of the properties of this half-space is difficult. They can be evaluated by comparative analysis of certain simple preliminary problems studied using the actual geometry and properties of soil and ballast, and the equivalent half-space properties. This paper is intended to develop a general numerical model for the analysis of soil vibrations due to high-speed train passage and their effects on nearby structures. The influence of a concrete underpass structure on the soil motion and the loads on the structure produced by the train passage have been studied. To the authors’ knowledge there is not any previous fully 3-D numerical model for the analysis of HST-induced vibrations and their effects on nearby structures, while taking dynamic soil-structure interaction into account. 17 8 Acknowledgements The authors would like to thank the group of researchers from the University of Leuven for the valuable experimental data available on their web page. Our thanks are extended as well to the JSV reviewers and editor for their valuable comments that have certainly improved the paper. This research is financed by the Ministerio de Educaci´on y Ciencia of Spain under the research project BIA2004-03955-C02-01 and by the Ministerio de Fomento of Spain under the research project ”Evaluaci´on de efectos din´amicos del ferrocarril”. The financial support is gratefully acknowledged. References [1] Krylov V. On the theory of railway-induced ground vibrations. Journal de Physique IV 1994; 4: 769-772. [2] Krylov V. 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Acustica-Acta Acustica 1998; 84: 78-90. [41] Galv´ın P. Numerical and experimental study of ground vibrations induced by high speed train passage and effects on structures (in Spanish). PhD Thesis, Universidad de Sevilla, Spain, 2007. 21 −0.1 −0.05 0 0.05 0.1 0 0.1 0.2 0.3 0.4 0.50.5 time [s] force [N] (a) 0 50 100 150 200 250 0 0.005 0.010.01 frequency [Hz] force [N/Hz] (b) Fig. 1. Force transmitted by a single sleeper when a unit axle load moves with speed valong the track: (a) Time history (b) Frequency content. 22 (a) (b) Fig. 2. Discretization of the surface around the track for a length equivalent to 36 sleepers. 23 (a) (b) Fig. 3. BE discretization of the sleeper zone 24 Fig. 4. Soil surface discretization. 25 0 50 100 150 200 250 0 0.2 0.4 0.6 0.8 1x 10−3 frequency [Hz] velocity [m/s/Hz] (a) 0 50 100 150 200 250 0 0.2 0.4 0.6 0.8 1x 10−3 frequency [Hz] velocity [m/s/Hz] (b) 0 50 100 150 200 250 0 0.2 0.4 0.6 0.8 1x 10−3 frequency [Hz] velocity [m/s/Hz] (c) 0 50 100 150 200 250 0 0.2 0.4 0.6 0.8 1x 10−3 frequency [Hz] velocity [m/s/Hz] (d) Fig. 11. Frequency content of the vertical velocity at a point 6 m from the track: (a)v= 315 km/h (b)v= 300 km/h (c)v= 271 km/h (d) v= 256 km/h. Experimental (black line) vs. Numerical (grey line). 32 0 50 100 150 200 250 0 1 2 3 4 5x 10−4 frequency [Hz] velocity [m/s/Hz] (a) 0 50 100 150 200 250 0 1 2 3 4 5x 10−4 frequency [Hz] velocity [m/s/Hz] (b) 0 50 100 150 200 250 0 1 2 3 4 5x 10−4 frequency [Hz] velocity [m/s/Hz] (c) 0 50 100 150 200 250 0 1 2 3 4 5x 10−4 frequency [Hz] velocity [m/s/Hz] (d) Fig. 12. Frequency content of the vertical velocity at a point 8 m from the track: (a)v= 315 km/h (b)v= 300 km/h (c)v= 271 km/h (d) v= 256 km/h. Experimental (black line) vs. Numerical (grey line). 33 3 4 5 6 7 8 9 10 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 distance [m] PPV [m/s] (a) 3 4 5 6 7 8 9 10 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 distance [m] PPV [m/s] (b) 3 4 5 6 7 8 9 10 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 distance [m] PPV [m/s] (c) Fig. 13. Computed PPV as a function of the distance to the track for different damping ratio: ’—’ ξ= 0.02, ’− − −’ξ= 0.04, ’− · −’ξ= 0.06. Train speed: (a)v= 300 km/h (b)v= 350 km/h (c)v= 500 km/h. 34 (a)        Z Y A C (b) Fig. 14. Half-space with 0.8 m thick ballast embankment. BE discretization and geometry. 35 0 0.02 0.04 0.06 0.08 0.1 0 2 4 6 8 10 12 x 10−6 time [s] vertical displacement [m] (a) 0 0.02 0.04 0.06 0.08 0.1 0 2 4 6 8 10 12 x 10−6 time [s] vertical displacement [m] (b) Fig. 15. Vertical displacements at points A (’–’ proposed model vs. ’×’ Adam et al.) and B (’··’ proposed model vs. ’+’ Adam et al.): (a)Case 1 (b)Case 2. 36 (a)    (b) Fig. 16. Half-space with 1.0 m thick ballast embankment. BE discretization and geometry. 37 012345678 −6 −4 −2 0 2 4 6x 10−3 time [s] velocity [m/s] (a) 012345678 −6 −4 −2 0 2 4 6x 10−3 time [s] velocity [m/s] (b) 012345678 −6 −4 −2 0 2 4 6x 10−3 time [s] velocity [m/s] (c) 012345678 −6 −4 −2 0 2 4 6x 10−3 time [s] velocity [m/s] (d) Fig. 17. Vertical velocity at a point 4 m from the track considering a ballast layer: (a)v= 315 km/h (b)v= 300 km/h (c)v= 271 km/h (d) v= 256 km/h. Experimental (black line) vs. Numerical (grey line). 38 012345678 −6 −4 −2 0 2 4 6x 10−3 time [s] velocity [m/s] (a) 012345678 −6 −4 −2 0 2 4 6x 10−3 time [s] velocity [m/s] (b) 012345678 −6 −4 −2 0 2 4 6x 10−3 time [s] velocity [m/s] (c) 012345678 −6 −4 −2 0 2 4 6x 10−3 time [s] velocity [m/s] (d) Fig. 18. Vertical velocity at a point 6 m from the track considering a ballast layer: (a)v= 315 km/h (b)v= 300 km/h (c)v= 271 km/h (d) v= 256 km/h. Experimental (black line) vs. Numerical (grey line). 39 012345678 −6 −4 −2 0 2 4 6x 10−3 time [s] velocity [m/s] (a) 012345678 −6 −4 −2 0 2 4 6x 10−3 time [s] velocity [m/s] (b) 012345678 −6 −4 −2 0 2 4 6x 10−3 time [s] velocity [m/s] (c) 012345678 −6 −4 −2 0 2 4 6x 10−3 time [s] velocity [m/s] (d) Fig. 19. Vertical velocity at a point 8 m from the track considering a ballast layer: (a)v= 315 km/h (b)v= 300 km/h (c)v= 271 km/h (d) v= 256 km/h. Experimental (black line) vs. Numerical (grey line). 40 B1 A1 A2 A3 B2 B3 X Z Y (a) C D (b)            A B (c) Fig. 20. Underpass. BE discretization and geometry. 41