A 2.5D coupled FE-BE model for the prediction of railway induced vibrations

A 2.5D coupled FE-BE model for the prediction of railway induced vibrations P. Galv´ına,b, S. Fran¸coisa, M. Schevenelsa, E. Bonginic, G. Degrandea, G. Lombaerta aDepartment of Civil Engineering, K.U.Leuven, Kasteelpark Arenberg 40, B-3001 Leuven, Belgium bEscuela T´ecnica Superior de Ingenieros, Universidad de Sevilla, Camino de los Descubrimientos s/n, E-41092 Sevilla, Spain cSNCF, Direction de l’Innovation et de la Recherche, rue de Londres 45, 75379 Paris Cedex 8, France Abstract Ground vibrations induced by railway traffic at grade and in tunnels are often studied by means of two-and-half dimensional (2.5D) models that are based on a Fourier transform of the coordinate in the longitudinal direction of the track. In this paper, the need for 2.5D coupled finite element-boundary element models is demonstrated in two cases where the prediction of railway induced vibrations is considered. A recently proposed novel 2.5D methodology is used where the finite element method is combined with a boundary element method, based on a regularized boundary integral equation. In the formulation of the boundary integral equation, the Green’s functions of a layered elastic halfspace are used, so that no discretization of the free surface or the layer interfaces is required. In the first case, two alternative models for a ballasted track on an embankment are compared. In the first model, the ballast and the embankment are modelled as a continuum using 2.5D solid elements, whereas a simplified beam representation is adopted in the second model. The free field vibrations predicted by both models are compared to those measured during a passage of the TGVA at a site in Reugny (France). A very large difference is found for the free field response of both models that is due to the fact that the deformation of the cross section of the embankment is disregarded in the simplified representation. In the second case, the track and free field response due to a harmonic load in a tunnel embedded in a layered halfspace are considered. A simplified methodology based on the use of the full space Green’s function in the tunnel-soil interaction problem is investigated. It is shown that the rigorous finite element-boundary element method is required when the distance between the tunnel and the free surface and the layer interfaces of the halfspace is small compared to the wavelength in the soil. Key words: 2.5D modelling, BEM-FEM coupling, dynamic soil–structure interaction, embankment, underground traffic Email address: pedroga[email protected]s (P. Galv´ına,b) Preprint submitted to Soil Dynamics and Earthquake Engineering April 3, 2010 1. Introduction The recent deployment of the high-speed train (HST) network in Europe, the USA, and Asia has stimulated the development of several numerical models for the prediction of railway induced vibrations. The most general framework is offered by the three-dimensional (3D) finite element (FE) method and 3D coupled finite element-boundary element (FE-BE) methods. These methods allow accounting for the full coupling between the track and the soil and can be used to study singular points in the track such as transition zones and switches. Furthermore, a time domain formulation offers the possibility to account for non-linear constitutive behaviour in the FE part of the model. Mohammadi and Karabalis [1] have presented a frequency domain FE-BE formulation where the railway track is modelled as a group of rigid sleepers resting on a viscoelastic halfspace. Andersen and Nielsen [2] have presented a FE-BE model in the frequency domain to investigate the free field response produced by vertical and horizontal moving loads on an embankment and the reduction of vibrations by means of barriers or soil improvement along the railway track. Celebi [3] has used a BE model to compute the transfer function between a point load on the track and the free field response and to study the mitigation of railway induced vibrations by open trenches. The track is represented by an equivalent homogeneous layer that represents a concrete slab or an asphalt layer over a base and frost protection layer. O’Brien and Rizos [4] have presented a direct time domain approach to study the transient response of a track-soil system due to the passage of trains where the sleepers are considered to be rigid and rails are coupled to the sleepers in the vertical direction only. Galv´ın and Dom´ınguez [5, 6] and Galv´ın et al. [7] have presented a time domain formulation to study the ground motion due to a train passage. The model has been validated by means of field measurements [5, 6] and has been used to study the influence of the ballast on a concrete underpass structure [5] and the vibration isolation by a floating slab track system [7]. The main disadvantage of 3D FE or FE-BE models is their very high computational cost. As an alternative to full 3D models, so-called two-and-a-half dimensional (2.5D) models have been proposed for the prediction of railway induced vibrations. The basic assumption underlying the 2.5D methodology is that the geometry of the coupled track-soil system is invariant in the longitudinal direction of the track. This allows for a Fourier transform with respect to the coordinate along the track and leads to a solution in the frequency-wavenumber domain where the original 3D problem is replaced by a 2D problem for each wavenumber. The 2.5D methodology results in a considerable reduction of the time required to set up the model as well as the computation time. Aubry et al. [8] have applied a 2.5D procedure to study the response of an infinitely long beam, coupled to an elastic halfspace, due to a moving load. The methodology has been applied by Sheng et al. [9, 10] to an infinite layered beam model for the track, coupled to a layered halfspace. More recently, this model has been elaborated to account for dynamic train-track interaction [11, 12]. In the model presented by Sheng et al. [9, 10, 11, 12], the tractions at the interface between the 2 [9, 10, 11, 12, 17, 19, 20]. The transfer functions between the track and the free field are compared and subsequently used to predict free field vibrations during the passage of the TGVA at a site in Reugny (France). Predictions are compared to experimental results that have been obtained within the frame of a benchmark study organized by SNCF. In the second case, the track and free field response due to a harmonic load in a tunnel embedded in a layered halfspace are studied. The results of the FE-BE method are compared to those obtained from a simplified methodology presented by Hussein et al. [37] that allows for an approximate solution at reduced computational cost. 2. The 2.5D coupled FE-BE model The dynamic interaction between a railway track and the underlying soil (figure 1a) or a tunnel and the surrounding soil (figure 1b) is a problem of dynamic soil–structure interaction. A domain decomposition method is used to solve the problem, where the subdomain Ωbrepresents the structure and the subdomain Ωsthe soil. The soil–structure interaction problem is solved by enforcing continuity of displacements and equilibrium of stresses on the interface Σbs between both subdomains. It is assumed that the geometry of the track or the tunnel is invariant with respect to the coordinate yin the longitudinal direction. The soil is modelled as a horizontally layered halfspace and, therefore, invariant with respect to the direction eyas well. The dynamic soil-structure interaction problem is assumed to be linear and all equations are elaborated in the frequency domain. The dynamic equilibrium equation of the structure is discretized by means of 2.5D finite elements. (a) Ωb Σbs Ωs x y z (b) Ωb Σbs Ωs x y z Figure 1: The 2.5D coupled FE-BE models: (a) a ballasted track at grade and (b) a tunnel in a halfspace The equilibrium equation for the dynamic soil-structure interaction problem is formulated in a variational form. For any virtual displacement field vbimposed on the structure Ωb, the sum of the virtual work of the internal and the inertial forces is equal to the virtual work of the external loads: −ω2ZΩb vb·ρbubdΩ+ZΩb ǫb(vb) : σb(ub)dΩ = ZΩb vb·ρbbbdΩ+ZΓbσ vb·tnb bdΓ+ ZΣbs vb·tnb b(ub)dΓ (1) 4 where ubis the displacement vector in the structure, ρbbbdenotes the body force in the domain Ωb, and tnb b=σb·nbis the traction vector on a boundary with unit outward normal vector nb. Tractions tnb bare imposed on the boundary Γbσ. Accounting for the equilibrium of stresses on the interface Σbs and using a finite element formulation for the interpolation of the displacement field with respect to the coordinates xand z, equation (1) can be elaborated as follows [22]: −ω2Mbb +K0 bb −ikyK1 bb −k2 yK2 bb +ik3 yK3 bb +k4 yK4 bb +Ks bb(ky, ω)˜ ¯ ub(ky, ω) = ˜ ¯ fb(ky, ω) (2) where K0 bb,K1 bb,K2 bb,K3 bb and K4 bb are the stiffness matrices, Mbb is the mass matrix, ˜ ¯ fb(ky, ω) is the external load vector, and Ks bb(ky, ω) represents the dynamic soil stiffness matrix. A tilde above a variable denotes its representation in the frequency-wavenumber domain. The finite element matrices Mbb and K0 bb to K4 bb in equation (2) are independent of the wavenumber kyand the frequency ωand are only assembled once. Equation (2) is now further elaborated by dividing the finite element degrees of freedom ˜ ¯ ub(ky, ω) into internal degrees of freedom ˜ ¯ ub1(ky, ω) and degrees of freedom ˜ ¯ ub2(ky, ω) on the soil-structure interface:  −ω2  Mb1b1Mb1b2 Mb2b1Mb2b2  +  K0 b1b1K0 b1b2 K0 b2b1K0 b2b2  −iky  K1 b1b1K1 b1b2 K1 b2b1K1 b2b2   −k2 y  K2 b1b1K2 b1b2 K2 b2b1K2 b2b2  +ik3 y  K3 b1b1K3 b1b2 K3 b2b1K3 b2b2  +k4 y  K4 b1b1K4 b1b2 K4 b2b1K4 b2b2   +  0 0 0Ks b2b2(ky, ω)    ˜ ¯ ub1(ky, ω) ˜ ¯ ub2(ky, ω) =  ˜ ¯ fb1(ky, ω) ˜ ¯ fb2(ky, ω) (3) The dynamic soil stiffness matrix ˜ Ks b2b2(ky, ω) is written as: ˜ Ks b2b2(ky, ω) = ZΣbs NT b2˜ ts(Nb2)(x, ky, z, ω)dΓ (4) and computed by means of a 2.5D boundary element method. The boundary element mesh is chosen to match the finite element mesh on the soil-structure interface Σbs. As a result, the boundary element interpolation functions Ns(x, z) correspond to the finite element shape functions Nb2(x, z) on the soil-structure interface. This allows introducing the boundary element traction discretization in equation (4): ˜ Ks b2b2(ky, ω) = ZΣbs NT b2Nb2dΓ˜ ¯ ts(Nb2)(ky, ω) = ˜ Tq˜ ¯ ts(Nb2)(ky, ω) (5) The transformation matrix Tq=RΣbs NT b2Nb2dΓ in equation (5) is independent of wavenumber and frequency. The tractions ˜ ¯ ts(Nb2)(ky, ω) are found by a 2.5D boundary element method, based on the integral equation that relates the displacements in the soil domain to the displacements and tractions on the soilstructure interface. The 2.5D boundary integral equation has been derived by Sheng et al. [29] from the 5 2.5D reciprocal theorem. In this paper, a regularized version [22] of the 2.5D boundary integral equation is applied, leading to the following boundary element system of equations: h˜ T(ky, ω) + Ii˜ ¯ us(ky, ω) = ˜ U(ky, ω)˜ ¯ ts(ky, ω) (6) where ˜ U(ky, ω) and ˜ T(ky, ω) are fully populated unsymmetric boundary element system matrices. The unit matrix Icorresponds to the integral-free term and is not present in the case of a bounded medium. The evaluation of the boundary element system matrices ˜ U(ky, ω) and ˜ T(ky, ω) in equation (6) requires the Green’s displacements ˜uG ij(x, ky, z, ω) and tractions ˜ tG ij (x, ky, z, ω) of the layered halfspace. These fundamental solutions are computed with the direct stiffness method [13, 30] using the MATLAB toolbox EDT 2.1 [31]. As the traction free surface of the halfspace is accounted for in these fundamental solutions, only the interface Σbs between the structure and the layered halfspace has to be discretized. The tractions ˜ ¯ ts(Nb2)(ky, ω) are found by solving the system of equations (6): ˜ ¯ ts(Nb2)(ky, ω) = ˜ U−1(ky, ω)˜ T(ky, ω) + I(7) An expression for the dynamic soil stiffness matrix ˜ Ks b2b2(ky, ω) is found by introducing the solution (7) in equation (5): ˜ Ks b2b2(ky, ω) = ˜ Tq˜ U−1(ky, ω)˜ T(ky, ω) + I(8) Once the equilibrium equation (3) for the dynamic soil-structure interaction problem has been solved, the integral representation theorem is applied to compute the radiated wave field from the tractions ¯ ˜ ts(ky, ω) and displacements ¯ ˜ us(ky, ω) at the soil-structure interface: ¯ ˜ ur(x, ky, z, ω) = ˜ Ur(x, ky, z, ω)¯ ˜ ts(ky, ω)−˜ Tr(x, ky, z, ω)¯ ˜ us(ky, ω) (9) where the matrices ˜ Ur(x, ky, z, ω) and ˜ Tr(x, ky, z, ω) follow from the introduction of the boundary element discretization in the integral representation theorem and the vector ¯ ˜ ur(x, ky, z, ω) collects the displacement components at nrreceiver locations. 3. A ballasted track on an embankment In this section, the proposed 2.5D FE-BE model is used to predict the track and free field vibrations at a site in Reugny (France) situated along the high speed railway line LGV Atlantique. The railway track in Reugny is a classical ballasted track, situated on top of an embankment [32]. The rails have a UIC60 cross section and are continuously welded. The rails are supported by rail pads and fixed with clips on twin block concrete sleepers [32] with a spacing of d= 0.60 m. The concrete twin block sleepers have a total length lsl = 2.41 m and are composed of two tied concrete blocks with a length lbl = 0.84 m, a width bbl = 0.29 m at the base, and a height hbl = 0.22 m under the rail. The total mass of the sleepers msl = 250 kg. The 6 track is supported by a ballast layer with a thickness hbof about 0.30 m. The density of the ballast layer is situated in a range between 1400 kg/m3and 1700 kg/m3. The embankment has a width of about we1 = 6 m at the top supporting the railway track, a width of we2 = 13 m at the soil’s surface, and a height he= 2 m. Two alternative track models are considered. For both models, the geometry of the coupled track-soil system is assumed to be invariant with respect to the longitudinal direction of the track, so that the 2.5D methodology can be applied. In the first model, the ballast and the embankment are modelled as an elastic continuum using 2.5D solid elements (figure 2). The second model is a simplified model where the ballast is represented by distributed springs and dampers while the embankment is modelled as an Euler-Bernoulli beam (figure 3). Similar simplified models of the ballast and the embankment have been frequently used in the literature [9, 10, 11, 12, 17, 19, 20]. In the following, the dynamic characteristics of the track and the soil that have been provided by the SNCF [32] within the frame of a benchmark study are used to define the parameters of both models. Next, the track receptance and free field mobility obtained by both models are compared. Finally, a comparison is made of the free field vibrations during the passage of the TGV Atlantique (TGVA). 3.1. Track model 1 lsl y1y2 ur1 krp crp ur2 usl βsl we2 rail rail pad sleeper ballast hb he we1 embankment x y FF02z Figure 2: Cross section of model 1 of the ballasted track on the embankment. Figure 2 shows the cross section of the first model with the mesh of the 2.5D solid elements that are used to model the ballast and the embankment. The rails are represented by Euler-Bernoulli beams with a bending stiffness ErIrand a mass ρrArper unit length. The rail displacements are denoted as ur1(y, t) and ur2(y, t). The positions of the rail are determined by y1= 1.145 m and y2= 2.580 m, with y2−y1equal to the track gauge rd. The UIC 60 rails have a bending stiffness ErIr= 6.45 ×106N/m2and a mass per unit length ρrAr= 60.34 kg/m for each rail. The internal energy dissipation in the rail is modelled by a loss factor ηr= 0.05. The rail pads are modelled as continuous spring-damper connections. The rail pad 7 stiffness krp of a single rail pad is used to calculate an equivalent stiffness krp =krp/d = 130 ×106N/m2. A loss factor ηrp = 0.23 is used to account for internal energy dissipation in the rail pad. The concrete sleepers are assumed to be rigid in the plane of the track cross section, so that the vertical sleeper displacements along the track are determined by the vertical displacement usl(y, t) and rotation βsl(y, t) at the centre of gravity of the sleeper. The sleepers are modelled as a uniformly distributed mass msl =msl/d of 417 kg/m. The sleeper’s rotational inertia ρslIsl =ρslIsl/d has been estimated as 298 kgm2/m taking into account the excentric position of the two blocks. The ballast bed is modelled as an elastic continuum, using 88 2.5D solid elements [22]. The Young’s modulus of the ballast bed is computed from the given vertical ballast stiffness per sleeper kb= 180×106N/m taking into account the support area of the twin block sleeper 2lblbbl = 0.48 m2. This results in a ballast stiffness Kb= 370 ×106N/m3or a Young’s modulus Eb=Kbhb= 111 ×106N/m2. Additionally, a Poisson’s ratio νb= 0.36, a density ρb= 1550 kg/m3, and a loss factor ηb= 1.00 are assumed for the ballast layer. The embankment is modelled as an elastic continuum using 528 2.5D solid elements. A Young’s modulus Ee= 170 ×106N/m2, Poisson’s ratio νe= 0.36, and a density ρe= 1400 kg/m3are used for the embankment. The soil is modelled as a horizontally layered elastic halfspace [32], with a single layer with a thickness of 2.0 m and a shear wave velocity Cs= 211 m/s on top of a halfspace with a shear wave velocity of 403 m/s. The density ρis equal to 1400 kg/m3for the top layer and equal to 2650 kg/m3for the underlying halfspace. The Poisson’s ratio νis 0.36 for the top layer and 0.16 for the halfspace. The material damping ratio βin both deviatoric and volumetric deformation has a value of 0.05 and 0.06 for the top layer and the halfspace, respectively. 3.2. Track model 2 The cross section of model 2 is shown in figure 3. Compared to model 1 (figure 2), the rails, rail pads, sleeper, and underlying soil are modelled similarly, whereas a simplified model is used for the ballast and the embankment. The ballast bed is now represented by a set of distributed linear springs and dampers. The smeared ballast stiffness kbis computed from the vertical spring stiffness kbper sleeper [N/m] as kb/d and equal to 300 ×106N/m2. The loss factor ηb= 1.00. The equivalent ballast mass mbis computed from the ballast mass mbsituated under each sleeper as mb/d. The ballast mass mbis estimated from the height hbof the ballast layer and a width wb1 =lsl and wb2 = 3 m at the top and the bottom of the ballast layer, respectively, as mb= 0.5ρbhb(wb1 +wb2)bbl. This leads to a value of 608 kg/m for the equivalent ballast mass mb. The embankment is represented by an Euler-Bernoulli beam with a bending stiffness EeIe, a torsional rigidity GeJe, a rotational inertia ρeIpe, and a mass ρeAeper unit length. This implies that the cross section of the embankment is now assumed to be rigid. The embankment has a Young’s modulus Ee= 8 l14 lsl y1y2 ur1 krp crp ur2 usl βsl kbcb rail rail pad sleeper ballast hb he we1 embankment ue βe Figure 3: Cross section of model 2 of the ballasted track on the embankment. 170 ×106N/m2, a shear modulus Ge= 107 ×106N/m2, and a density ρe= 1400 kg/m3. Based on the dimensions of the cross section of the embankment, the following section characteristics have been computed: the area Ae= 19 m2, the bending moment of inertia Ie= 20.67 m4, the polar moment of inertia Ipe= 182.96 m4, and the torsion constant Je= 22.03 m4. At the interface between the embankment and the soil, relaxed boundary conditions are assumed, so that only continuity of the vertical displacements is imposed. Due to the assumption of a beam model, the vertical displacements usz(x, y, z = 0, t) in the soil at the interface Σ are determined by the vertical displacement ue(y, t) and rotation βe(y, t) at the centre of the interface. It is expected that model 1 leads to more accurate results as it allows for a better approximation of the stress distribution at the interface between the embankment and the soil. Steenbergen and Metrikine [21] have shown that this is crucial for an accurate prediction of railway induced vibrations. 3.3. Track receptance and free field mobility In the following, the track receptance and the free field mobility are compared for both models. The track receptance is computed from the solution of equation (3) that governs the dynamic track-soil interaction, considering an impulsive point load on both rails. The response of the outer rail at the point y= 0 where the impulsive point load is applied is obtained subsequently by means of an inverse wavenumber domain transform. Figures 4a and 4b compare the modulus and phase of the track receptance for both models. As an additional reference, the track receptance has also been computed without taking into account the embankment in model 2. In this case, the interface between the ballast and the soil is assumed to be rigid over a width equal to the sleeper length lsl = 2.41 m. This result is also shown in figure 4. The most pronounced difference between the results is found at frequencies below 100 Hz. In this frequency range, the highest value for the track receptance is found for model 1 where the continuum model for the embankment allows accounting for the deformation of the cross section. In model 2, the deformation 9 of the cross section of the embankment is disregarded and a substantially lower track receptance is found. A better agreement with model 1 is found when the embankment is disregarded in model 2. (a) 0 50 100 150 200 0 0.5 1 x 10−8 Frequency [Hz] Modulus [m/N] (b) 0 50 100 150 200 −4 −3 −2 −1 0 Frequency [Hz] Phase [rad] Figure 4: (a) Modulus and (b) phase of the track receptance computed by model 1 (dark grey line), model 2 (grey line), and model 2 without embankment (light grey line). (a) 0 50 100 150 0 1 2 3 4x 10−7 Frequency [Hz] Mobility [m/sN] (b) 0 50 100 150 0 2 4 6 x 10−8 Frequency [Hz] Mobility [m/sN] (c) 0 50 100 150 0 0.5 1 x 10−8 Frequency [Hz] Mobility [m/sN] (d) 0 50 100 150 0 1 2 x 10−9 Frequency [Hz] Mobility [m/sN] Figure 5: Free field mobility at (a) 2 m, (b) 12 m, (c) 32 m, and (d) 72 m from the outer rail computed from model 1 (dark grey line), model 2 (grey line), and model 2 without embankment (light grey line). Based on the solution of equation (3), the radiated wave field due to an impulsive point load on both rails is computed. The free field mobility is obtained as the inverse wavenumber domain transform of the free field velocity in the wavenumber-frequency domain. Figure 5 compares the free field mobility at a distance of 2 m, 12 m, 32 m, and 72 m from rail 2 as computed with model 1 and model 2. The point located at 2 m from the outer rail is situated on the embankment (figure 2). The free field mobility for the case where the embankment is omitted in model 2 is shown as well. A very large difference is found between the results of model 1 with the solid embankment model and model 2 where the embankment is represented by an Euler-Bernoulli beam. The assumption of an Euler- Bernoulli model for the embankment leads to high tractions near the edges of the embankment, whereas in the case of the solid embankment model, the tractions are distributed more smoothly along the interface. When the wavelength in the soil is very large compared to the width of the interface between the embankment and the soil, the influence of the traction distribution on the free field mobility is small. In the present case 10 x z 2 m depth t R Figure 10: Cross section of the tunnel. 4.2. The track receptance and track-soil transfer functions In the following, the track receptance and transfer functions between the track and the soil are computed by solving equation (3) that governs the dynamic interaction between the track, the tunnel and the soil. The load consists of an impulsive load on both rails. Figures 11a and 11b show the modulus and phase of the track receptance and the corresponding response of the slab in the frequency range between 1 Hz and 100 Hz . At frequencies below the resonance frequency of the slab mass on the stiffness of the resilient support, the rail and slab move in phase. At the resonance frequency, a peak is observed in the modulus of the response of the slab and the rail (figure 11a), while the phase equals π/2. At higher frequencies, the rail is dynamically uncoupled from the slab and the response of the slab becomes much smaller. (a) 20 40 60 80 100 −200 −190 −180 −170 −160 −150 Frequency [Hz] Displacement [dBref m/N] (b) 20 40 60 80 100 −4 −3 −2 −1 0 Frequency [Hz] Phase [rad] Figure 11: (a) Modulus and (b) phase of the track receptance (black line) and the corresponding response of the slab (grey line). Figure 12a shows the displacement at a point {0,0,−10 m}Tlocated in the halfspace at a height of 10 m above the tunnel axis and a point {0,0,0}Tat the free surface, which is at a height of 20 m above the tunnel axis. The vertical displacements show clear oscillations as a function of the frequency due to interference between different types of waves that are radiated into the soil by the tunnel. Gupta et al. 17 [23] have shown that for a tunnel embedded in a full space, the frequency spacing between the oscillations can be computed from the shear wave velocity Cs, the longitudinal wave velocity Cpand the distance rbetween the observation point and the source as CsCp/(r(Cp−Cs)). In the present case of a layered halfspace, the interference pattern is more complicated, however, as it results from the interaction between Rayleigh waves and shear and longitudinal waves traveling in the different soil layers. Figure 12b shows the vertical displacement at two points {10 m,0,0}Tand {20 m,0,0}Tlocated at the free surface at a horizontal distance of 10 m and 20 m from the tunnel axis, respectively. The results in figures 12a and 12b show that the largest response is not necessarily found at the smaller distance from the tunnel axis due to the interference and the directivity of the waves emitted by the tunnel. (a) 20 40 60 80 100 −290 −270 −250 −230 −210 Frequency [Hz] Displacement [dBref m/N] (b) 20 40 60 80 100 −290 −270 −250 −230 −210 Frequency [Hz] Displacement [dBref m/N] Figure 12: Vertical displacement in the soil at (a) two points {0,0,−10 m}T(black line) and {0,0,0}T(grey line) located above the tunnel axis and (b) two points {10 m,0,0}T(black line) and {20 m,0,0}T(grey line) at the free surface. 4.3. Simplified methodology The proposed 2.5D methodology allows to rigorously account for the layered structure of horizontally stratified soils. This requires a computational cost which may be higher than for other models where approximate methods are used to account for the layered structure of the soil. Such an approach has recently been proposed by Hussein et al. [37] to extend the capabilities of the PiP model that originally predicts vibrations from underground traffic for a tunnel embedded in a full space [26, 27, 28]. First, the tunnel-soil interaction problem in equation (3) is solved considering the tunnel embedded in a homogeneous full space with the same properties as the layer in which the tunnel is located. This implies that the dynamic stiffness matrix of the soil in equation (5) is computed by solving the boundary element system of equations (6) in terms of the full space Green’s functions. Second, the displacements and tractions at the tunnel-soil interface are used to compute the free field response according to equation (9) using the Green’s function of a layered halfspace [31, 38]. In this subsection, it is investigated under which conditions a similar approach can be followed to reduce the computational cost in the 2.5D framework. First, the results presented in figures 12a and 12b are recomputed with the approximate method, based on the solution of the dynamic tunnel-soil interaction problem in a homogeneous halfspace. Figure 13 compares the vertical displacement in the soil at 10 m and 20 m (the free surface) above the centre of the tunnel previously shown in figure 12a with the approximation based on the full space solution. It can be 18 observed that a relatively good approximation of the solution has been obtained. A similar good agreement (a) 20 40 60 80 100 −290 −270 −250 −230 −210 Frequency [Hz] Displacement [dBref m/N] (b) 20 40 60 80 100 −290 −270 −250 −230 −210 Frequency [Hz] Displacement [dBref m/N] Figure 13: Vertical displacement in the soil at two points (a) {0,0,−10 m}Tand (b) {0,0,0}Tlocated above the tunnel axis computed with the original 2.5D methodology (black line) and the simplified 2.5D methodology (grey line). is observed for the two points {10 m,0,0}Tand {20,0,0}Tat the free surface (figure 14) for which the response has been previously shown in figure 12b. In this case, the proposed simplified methodology allows for a good approximation of the response in the soil at a considerably reduced computational cost. It is clear, however, that the free surface and the layered structure of the soil can only be disregarded in the solution of the tunnel-soil interaction problem when the free surface and the layer interfaces are sufficiently far from the tunnel. In order to demonstrate the limitations of the proposed simplified methodology, a second case is considered where the centre of the tunnel is situated at a depth of 5.5 m below the free surface. In this case, the distance from the tunnel apex to the free surface is 2.5 m and the distance from the tunnel apex to layer interface is 0.5 m. (a) 20 40 60 80 100 −290 −270 −250 −230 −210 Frequency [Hz] Displacement [dBref m/N] (b) 20 40 60 80 100 −290 −270 −250 −230 −210 Frequency [Hz] Displacement [dBref m/N] Figure 14: Vertical displacement in the soil at two points (a) {10 m,0,0}Tand (b) {20 m,0,0}Tlocated at the free surface computed with the original 2.5D methodology (black line) and the simplified 2.5D methodology (grey line). Fig. 15 compares the vertical displacement in the soil at a height of 3.5 m above the centre of the tunnel (at the interface between the layer and the halfspace) and at a height of 5.5 m (the free surface) as computed with both methods. In this case, the agreement between both results is less good and differences higher than 10 dB are observed. An accurate solution of the tunnel-soil interaction problem is therefore only obtained by the original 2.5D methodology. Figure 16 compares the vertical displacement at two points {10 m,0,0}Tand {20,0,0}Ton the free surface. For these points, the simplified methodology results in a better approximation than for the points on top of the tunnel in figure 13. Based on a more extensive parametric study, it has been concluded that the 19 (a) 20 40 60 80 100 −270 −250 −230 −210 −190 Frequency [Hz] Displacement [dBref m/N] (b) 20 40 60 80 100 −270 −250 −230 −210 −190 Frequency [Hz] Displacement [dBref m/N] Figure 15: Vertical displacement in the soil at two points (a) {0,0,−2}Tand (b) {0,0,0}Tlocated above the tunnel axis computed with the original 2.5D methodology (black line) and the simplified 2.5D methodology (grey line) for a tunnel depth of 5.5m. (a) 20 40 60 80 100 −280 −260 −240 −220 −200 Frequency [Hz] Displacement [dBref m/N] (b) 20 40 60 80 100 −280 −260 −240 −220 −200 Frequency [Hz] Displacement [dBref m/N] Figure 16: Vertical displacement in the soil at two points (a) {10 m,0,0}Tand (b) {20 m,0,0}Tlocated at the free surface computed with the original 2.5D methodology (black line) and the simplified 2.5D methodology (grey line) for a tunnel depth of 5.5 m. simplified methodology is valid when the distance between the tunnel and the surface of the halfspace spans a number of wavelengths so that the refracted wave does not significantly modify the tunnel-soil interaction problem. The same applies if the tunnel is located near an interface between two soil layers. 5. Conclusions In this paper, the need for 2.5D coupled finite element-boundary element models is demonstrated in two cases where the prediction of railway induced vibrations is considered. In the first case, two alternative models for a ballasted track on an embankment are considered. In the first model, 2.5D solid elements are used to model the ballast and the embankment as a continuum, whereas in the second model a simplified representation is used. The track receptance and free field mobility are found to differ considerably at higher frequencies where differences in the traction distribution at the interface between the embankment and the soil are resolved by the smaller wavelengths in the soil. A comparison of the predicted and measured free field velocity shows that the continuum model of the ballast and the embankment leads to a relatively good approximation at low frequencies where the quasi-static contribution to the response dominates. The less good agreement at high frequencies may be due to an overestimation of the dynamic axle loads. In the second case, the response of a tunnel embedded in a layered halfspace is considered. The proposed 20 2.5D methodology allows rigorously accounting for the layered structure, but may lead to relatively high computational cost. It has therefore been investigated under which conditions the proposed 2.5D methodology can be replaced by a simplified methodology based on the use of the full space Green’s functions in the solution of the tunnel-soil interaction problem. 6. Acknowledgements The first author would like to thank the Ministerio de Educaci´on y Ciencia of Spain (JC2008-00136) and the Junta de Andaluc´ıa (IAC08-II-3343) for their financial support of his research stay at the Department of Civil Engineering of the K.U.Leuven. The second and third author are holder of a Postdoctoral Fellowship of the Research Foundation - Flanders (FWO-Vlaanderen). The financial support of FWO-Vlaanderen is kindly acknowledged. References [1] Mohammadi M, Karabalis DL. 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