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Mitigation of Railway Traffic Induced Vibrations: The Influence of Barriers in Elastic Half-Space

Mitigation of Railway Traffic Induced Vibrations: The Influence of Barriers in Elastic Half-Space Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2009, Article ID 956263, 7 pages doi:10.1155/2009/956263 Research Article Mitigation of Railway Traffic Induced Vibrations: The Influence of Barriers in Elastic Half-Space 1 2 2 1 Michele Buonsanti, Francis Cirianni, Giovanni Leonardi, Adolfo Santini, and Francesco Scopelliti Department of Mechanics and Materials (MECMAT), Mediterranean University of Reggio Calabria, Via Graziella, Feo di Vito, 89100 Reggio Calabria, Italy Department of Information Science, Mathematics, Electronics, and Transportations (DIMET), Mediterranean University of Reggio Calabria, Via Graziella, Feo di Vito, 89100 Reggio Calabria, Italy Correspondence should be addressed to Giovanni Leonardi, giovanni.leonardi@unirc.it Received 23 January 2009; Revised 6 May 2009; Accepted 9 May 2009 Recommended by Yehia Bahei-El-Din In this paper, the problem of vibrations induced by trains and their propagation through the soil is studied. Particular attention is focused on the vibration induced by trains in motion and on the effects of such vibrations on the foundations of buildings in proximity of the tracks. The interaction between propagating waves induced by trains in motion and buildings foundations is a problem which does not admit a straightforward analytical solution; thus a solution is given by the use of a model based on the finite elements method. Firstly, we analyze the theoretical aspects of the problem by considering constant or harmonic loads moving along a straight railway track; then, we define a transfer function soil-railway and the response function of the entire system. The study aims to address the wave propagation in an elastic semi-space and the presence in the ground of a discontinuity element, such as a barrier of a given depth is considered. The efficiency variation of barriers is analyzed in function of the different materials used, and different numerical simulations are analyzed in order to study how the wave propagation and the track-soil interaction are influenced by the membrane, seen as damping barrier. Copyright © 2009 Michele Buonsanti et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction to constant or harmonic loads, produced by the movement of trains. The foundations of structures are subject to vibrations due Numerical aspects are developed in detail in Lombaert to moving masses, as vehicles in transit, vibrations which et al. [3] and Fiala et al. [4], whileinHubertetal. [5], a can cause damage locally to the foundations, as they can also detailed analysis using the Boundary Element Method, in travel up to the structures in elevation (Figure 1). the dominion of frequency and of time, is developed on a In this paper we consider the effects of vibrations due to model of elastic semispace. In the work of Andersen and the transit of locomotives on a railway, analyzing how the Nielsen [6], a model for the reduction of vibrations of the presence of material discontinuity, in an elastic semispace ground through the use of barriers dipped in the ground has an influence on the transmission of waves generated by a and laid between rail track and structure is proposed. In train passing by. The presence of barriers, made in polymeric the paper, after some general concepts of theory of the material, rubber, or in concrete, seems to cause reflection and vibrations, the aim is set on the problem in homogenous refraction of the vibrations generated in the ground. There elastic and isotropic semispace, where a two-dimensional are some references in scientific literature. Francois et al. [1], constitutional discontinuity is localized, representing the approached interaction between ground and infrastructures, artificial barrier in the hypothesis set for different materials in presence of vibrations generated by road traffic, while (rubber, polyurethane, and concrete). Sheng et al. [2], modelled the propagation of vibrations due 2 Advances in Acoustics and Vibration Building damage Vehicle response Excitation of Building response dynamic forces Track structure response Ground surface response Surface waves Foundation response Propagation of Secondary body waves stress waves Primary body waves Figure 1: Transmission of induced vibrations. which happens in the semispace, regarding the conditions of quiet, and anyway without the influence of a traveling load. Refraction wave In particular, we observe as the nature of the material used for the realization of the barrier vary, how the conditions Material 2 ϑ of propagation of the waves vary, and their influence, in an equilibrium state, on the contour conditions of an eventual foundation laid under the barrier. Material 1 ϑ ϑ i 1 2. Vibrations Theory Reflection wave In regard to the fundamental aspects in the theory of the Incident wave vibrations, in all this paper we follow the lines expressed in Hartog [7] and Clough and Penzien [8]. Our problem can have a first general organization in the context of Figure 2: Incident wave and reflection. the response for harmonic type loads. It is possible to characterize the following fundamental equations in the case of systems without damping: mu ¨(t) + ku(t) = p sin ωt,(1) and with damping, mu ¨(t) + cu(t) + ku(t) = p sin ωt. (2) The solution of (1) is admitted, from the combination of a complementary solution with a particular solution, in the shape 0 π 2π 3π 4π p β ωh/β u(t) = A sin ωt + B cos ωt + 1 − β2 sin ωt. (3) Figure 3: Amplification waves for Z /Z = 4. 1 2 In the case of (2) the general solution is of the type −ξωt u(t) = e (A sin ω t + B cos ω t) + D D Once the case is defined in its theoretical formulation, a numerical simulation with the support of computer applica- × 1 − β sin ωt − 2ξβ cos ωt . 2 2 1 − β + 2ξβ tion is carried out. What we want to analyze is the variation (4) of the mechanism of propagation of the vibration waves |A(ω)| Advances in Acoustics and Vibration 3 Barrier Figure 5: Finite element mesh of the two-dimensional model. Wheel load Q Figure 4: Modelling of the railway embankment. Speed v Distance y Thefirsttermof (4) represents the transitory result, while the second term represents the stationary result, which has the pulsation of the solicitation but is out of sync with the last wave. The factor of dynamics amplification D is Depth z defined as the relationship betw.een the amplitude of the stationary response and the static movement produced from Figure 6: Deflection of the track structure from one wheel. the external load. In a generalized manner D is function of the damping ratio and of the frequency relationship and usually it tends to infinite when damping is absent. vector k at constant speed (phase speed): v =,(7) 3. Theoretical Aspects in the Model Characterization any system of waves can be considered as a superimposition of monochromatic plane waves that move with the group In an elastic semispace we consider that the source of velocity: disturbance (train) generates a set of waves which are distributed in the plane system and therefore we can classify dω the waves plane and classify our problem as a motion of v = . (8) dκ waves within a semi-infinite body, which is assumed, for simplicity, elastic, homogenous, and isotropic. This type of Equation (8) represents the speed with which the energy is phenomenon is shown in its generality and specificity in the transported and, if v is different from v , the constant phase f g case of the seismic waves as treated in Boschi and Dragoni surfaces move from an extremity to another in the package [9]. We then consider a longitudinal elastic plane wave that of waves while this last one is the same one in motion. propagates in x-direction, if we assume the movement u Given such assumption, we consider the elastic semispace associated to it as a periodic function of the time, u can be as composed of different homogenous elements. Let’s assume placed in the form to form a put in elastic discontinuity, in the cross-sectional sense to the semispace, so to be placed in cross-sectional sense to the wave movement. This eliminates the direct flow u(x, t) = x a cos kx − ωt + ϕ,(5) of the waves from the source to the site of the structure placing itself as element of filter to the vibrations generated where a is the amplitude, k the wave number, ϕ a constant. from the movement of the train. Regarding the interaction Using the relation between the exponential and trigono- with the wave, or the set of waves, the discontinuity assumes metrical functions, and the complex notation, each compo- the role of reflection element and refraction for two- nent of the movement, for a monochromatic wave, can be dimensional waves. If we consider two elastic semispaces defined as joined with an element of material discontinuity, we may say that ρ, α, β are, respectively, the density and the speed of the ∗ ∗ ∗ i(kx−ωt) j elastic waves in the first semispace while ρ , α , β are the u(x, t) = Ae ,(6) same amounts in the second semispace (Figure 2). Refraction and reflection of the elastic waves will happen that represents an oscillation of amplitude A and wavelength on the discontinuity surface, longitudinal ones, and of cross- λ = 2π/κ, which propagates in the direction of the wave sectional ones which can transform one in the other when 4 Advances in Acoustics and Vibration 100 0.002 0.0019 0.0018 0.0017 0.0016 0.0015 0 0.0014 00.025 0.05 0.075 0.10.125 0.15 0.175 0.20.225 0.25 00.51 1.52 2.53 3.5 Time (s) (m) Concrete Rubber Figure 7: Load distribution of a moving wheel travelling at speed Polyurethane No barrier of 200 km/h. Figure 9: Trend of the velocity with the increase of the distance from embankment at the depth of 1 m. 0.3 0.2 −2 ×10 4.2 0.1 3.8 3.6 −0.1 3.4 3.2 −0.2 00.51 1.52 2.53 3.5 (m) −0.3 −0.9 −0.6 −0.30 0.30.60.9 Concrete Rubber Horizontal velocity (mm/s) Polyurethane No barrier (a) Figure 10: Trend of the acceleration with the increase of the 0.3 distance from embankment at the depth of 1 m. 0.2 We assume that given a monochromatic wave incident 0.1 with frequency ω,numberofwave k and travelling unitary amplitude in the semispace in direction-x ,itproducesone cross-sectional movement, in direction x expressed in the −0.1 form: −0.2 ( ) [ ( )] u x , t = exp ωt + k x − h , x ≥ h, (10) 3 1 3 3 −0.3 after the phenomenon of refraction and reflection the field of −0.9 −0.6 −0.30 0.30.60.9 movements takes the form: Horizontal velocity (mm/s) u(x , t) = exp[ωt + k (x − h)] 3 1 3 (b) (11) + R exp[ωt − k (x − h)], x ≥ h, 2 3 3 Figure 8: Nodes velocity (m/s) responses for a train speed of 200 km/h. where R is a coefficient that holds account of all the waves that travel in positive direction x . The contour conditions require that traction on free surface x is equal to zero, x = 3 3 inciting on the surface that separates the two semispaces. In 0, and that the movement and the traction are continuous accordance with Graff [10], the coefficient of reflection or on the surface of discontinuity x = h. In agreement with refraction c assumes the form: ij Graff [10], given Z , Z as stiffness of the two semispace 1 the 2 and A(ω) the coefficient of the exponential, the module of the latter supplies the amplification of the waves: 2ρ β cos θ i i i c = . (9) ij ∗ ∗ ρ β cos θ + ρ β cos θ i i i j j j |A(ω)|= . sqr cos ωh/β + (Z /Z ) sin ωh/β j 2 1 j The contour conditions, on plan x = 0 are the continuity of (12) stress and the movement. Load (kN) Vertical velocity (mm/s) Vertical velocity (mm/s) Velocity magnitude (m/s) Acceleration magnitude (m/s ) Advances in Acoustics and Vibration 5 Table 1: Properties of structural materials. Mechanical Rail Sleeper Traditional HMA Protective Embankment characteristics (UIC60) (concrete) ballast subballast layer Density ρ 7850 2400 1250 2200 2000 1000 (kg/m ) Modulus E 210000 30000 130 6000 160 80 (MPa) Poisson’s Ratio ν 0.30 0.15 0.30 0.40 0.45 0.50 Table 2: Properties of barriers. Mechanical Density ρ Modulus E Poisson’s Dumping characteristics (kg/m ) (MPa) ratio ν ratio η Concrete 2500 25000 0.15 0.04 Polyurethane 1170 25 0.50 0.08 (elastomer) Rubber chip 500 20 0.25 0.10 Equation (12) is a periodic function of the frequency ω with the embankment) is considered, considering the passage of a maximums in correspondence of the values: high speed locomotive (ETR 500) with 4 wheels for car. The effects of discontinuity is estimated considering three π j different materials whose properties are shown in Table 2. ω = (2n − 1) . (13) 2 h 4.2. Finite Element Analyses of Train-Induced Ground Vibra- Figure 3 shows the diagram of amplification of the waves. tions. The finite element simulations of train-induced ground vibrations was developed, using the ADINA soft- ware, considering a two-dimensional finite element model 4. Modeling and Simulation: perpendicular to the track (two-dimensional perpendicular The Track-Subsoil Model model). The two-dimensional model is composed of 7517 ele- In this paragraph we show a numerical analysis, using ments and 8533 nodes (Figure 5). The elements consisted a software for defined elements, on a two-dimensional of four-node solid element for ballast, HMA subballast, modeling of the studied system. protective layer and embankment and nine-node solid element for rail and sleeper. Furthermore, the rail pad was 4.1. Model Characteristics. In the proposed model the rail modeled with parallel discrete springs and dampers (spring track is the conventional one, and the elements that compose constant K = 11.2 · 10 N/m and damper constant c = it are (Figure 4) 12 · 10 Ns/m). For these elements, under the accepted hypothesis of (i) a protective compacted layer 30 cm thick of sand/ viscose-elastic behavior, the input parameters for the mate- gravel (called, in the Italian railways, supercompact); rials characterization, are the modulus of elasticity (E), (ii) a 12 cm subballast layer made of bituminous (hot- Poisson’s ratio (v), density (ρ), α and β Rayleigh’s coefficients mix asphalt) concrete; for the definition of the damping matrix. (iii) a traditional ballast layer 35 cm thick; The acting forces are a sequence of axial loads, moving like the train (the loads are similar to those of an ETR 500 (iv) an embankment 60 cm high. locomotive) (Figure 6). The dimensional characteristics of the elements above The forces are applied using a time function that indicated are those requested by the Italian standard (RFI) represents the time history of the force in the considered for high speed lines. node. During the simulations the loads were applied four Moreover, to analyze the propagation process of the times, reproducing the passing of four axes. The loads can vibrations in the soil, it has been considered a thickness of be thought of as triangular pulses distributed over the wheel- the support soil of the embankment equal to a total 5 m. rail surface of contact as shown in Figure 7. The materials properties used in the model were derived All the finite element analyses in this study were per- from tests and available experimentations in literature. formed in the time domain. The time step of the analyses Table 1 lists the physical-mechanical characteristics for the was fixed and set to 0.001 second. soil and track structure. Figure 8 shows the variation of velocity (y-direction and Finally a discontinuity (10 cm thick and 2 m deep) in the z-direction) of the middle node under truck and of a node semispace at a given distance from the rail track (2 m from at the distance of 1.5 m from the embankment. The orbits of 6 Advances in Acoustics and Vibration 2.5E − 03 4.5E − 02 1.6E − 04 4E − 02 1.4E − 04 2E − 03 3.5E − 02 1.2E − 04 3E − 02 1.5E − 03 1E − 04 2.5E − 02 8E − 05 2E − 02 1E − 03 6E − 05 1.5E − 02 4E − 05 1E − 02 5E − 04 2E − 05 5E − 03 0E +00 0E +00 0E +00 0 50 100 150 200 250 0 50 100 150 200 250 0 50 100 150 200 250 Time (mm/s) Time (mm/s) Time (mm/s) Conceret 1 Conceret 1 Conceret 1 Conceret 2 Conceret 2 Conceret 2 (a) (b) (c) 1.6E − 04 2.5E − 03 4.5E − 02 4E − 02 1.4E − 04 2E − 03 3.5E − 02 1.2E − 04 3E − 02 1E − 04 1.5E − 03 2.5E − 02 8E − 05 2E − 02 6E − 05 1E − 03 1.5E − 02 4E − 05 1E − 02 5E − 04 2E − 05 5E − 03 0E +00 0E +00 0E +00 0 50 100 150 200 250 0 50 100 150 200 250 0 50 100 150 200 250 Time (mm/s) Time (mm/s) Time (mm/s) Rubber 1 Rubber 1 Rubber 1 Rubber 2 Rubber 2 Rubber 2 (d) (e) (f ) 2.5E − 03 4.5E − 02 1.6E − 04 4E − 02 1.4E − 04 2E − 03 3.5E − 02 1.2E − 04 3E − 02 1E − 04 1.5E − 03 2.5E − 02 8E − 05 2E − 02 1E − 03 6E − 05 1.5E − 02 4E − 05 1E − 02 5E − 04 2E − 05 5E − 03 0E +00 0E +00 0E +00 0 50 100 150 200 250 0 50 100 150 200 250 0 50 100 150 200 250 Time (mm/s) Time (mm/s) Time (mm/s) Polyurethane 1 Polyurethane 1 Polyurethane 1 Polyurethane 2 Polyurethane 2 Polyurethane 2 (g) (h) (i) Figure 11: Trend of displacement, velocity, and acceleration in two nodes located before and after the barrier. vertical to horizontal particle motions were mostly counter 6. Conclusions clockwise; this could indicate the presence of Rayleigh waves The results of the proposed elaborations at the finite elements [11]. have been finalized to the assessment of the incidence of the barrier on the vibration state induced from the passage 5. Results of a high speed locomotive. The response of three different materials for the barrier was confronted and the following The graphs of the outcomes of the various analyses are shown conclusions can be made: in Figures 9 and 10. To estimate the effectiveness of the barrier we have esti- (i) concrete seems to provide a better reduction of mated the dynamic characteristics (speed and acceleration) the vibration. In spite of the greater density of the before and after the discontinuity, in correspondence of an material it involves an increase of the reflection eventual foundation plan of a building placed at a distance of phenomena and a consequent increase of the phe- 1.5 meters from the barrier. nomenon at the top side of the barrier (Figures 9 and As a general result of the analyses, as illustrated in Figures 10); 9 and 11, for the considered loads, it is concluded that, when (ii) polyurethane and rubber chip materials seem to the load is travelling with the speed of 200 Km/h, concrete respond in a similar way to the solicitations; however barriersprovide amoreefficient vibration shielding than the their damping contribution, in the analyzed geomet- rubber and polyurethane barriers. ric configuration, do not contribute meaningfully. Figure 11 shows, for the three kinds of barrier, the course of the displacement, velocity, and acceleration in time. The results obtained were given with the use of different Theresults giveninFigures 9 and 11 supply useful materials for the barrier, in identical geometrical conditions, information for the making of barriers, especially when it in size and position of the proposed damping barriers. A is required to respect regulation’s limit for vibration, as follow up to the presented study, requiring further analyses, expressed in terms of displacements and velocities (i.e., is in course, to estimate the influence of the position and of Italian regulation UNI 9916/2220). the thickness of the barriers on their dampening ability. Displacement Displacement Displacement Velocity (m/s) Velocity (m/s) Velocity (m/s) 2 2 2 Acceleration (m/s ) Acceleration (m/s ) Acceleration (m/s ) Advances in Acoustics and Vibration 7 References [1] S. Francois, L. Pyl, H. R. Masoumi, and G. Degrande, “The influence of dynamic soil-structure interaction on traffic induced vibrations in buildings,” Soil Dynamics and Earthquake Engineering, vol. 27, no. 7, pp. 655–674, 2007. [2] X. Sheng, C. J. C. Jones, and M. Petyt, “Ground vibration generated by a load moving along a railway track,” Journal of Sound and Vibration, vol. 228, no. 1, pp. 129–156, 1999. [3] G. Lombaert, G. Degrande, and D. Clouteau, “Numeri- cal modelling of free field traffic-induced vibrations,” Soil Dynamics and Earthquake Engineering, vol. 19, no. 7, pp. 473– 488, 2000. [4] P. Fiala, G. Degrande, and F. Augusztinovicz, “Numerical modelling of ground-borne noise and vibration in buildings due to surface rail traffic,” Journal of Sound and Vibration, vol. 301, no. 3–5, pp. 718–738, 2007. [5] W. Hubert, K. Friedrich, G. Pflanz, and G. Schmid, “Frequency- and time-domain BEM analysis of rigid track on a half-space with vibration barriers,” Meccanica, vol. 36, no. 4, pp. 421–436, 2001. [6] L. Andersen and S. R. K. Nielsen, “Reduction of ground vibration by means of barriers or soil improvement along a railway track,” Soil Dynamics and Earthquake Engineering, vol. 25, no. 7–10, pp. 701–716, 2005. [7] J.P.D.Hartog, Mechanical Vibrations, Dover, Oxford, UK; Oxford University Press, Oxford, UK, 1975. [8] R.W.Clough andJ.Penzien, Dynamics of Structures, McGraw- Hill, Singapore, 1985. [9] E. Boschi and M. Dragoni, Sismologia, UTET, Torino, Italy, [10] K. F. Graff, Wave Motion in Elastic Solids, Dover, Oxford, UK, [11] L. Hall, “Simulations and analyses of train-induced ground vibrations in finite element models,” Soil Dynamics and Earthquake Engineering, vol. 23, no. 5, pp. 403–413, 2003. 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Mitigation of Railway Traffic Induced Vibrations: The Influence of Barriers in Elastic Half-Space

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Hindawi Publishing Corporation
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Copyright © 2009 Michele Buonsanti et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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10.1155/2009/956263
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Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2009, Article ID 956263, 7 pages doi:10.1155/2009/956263 Research Article Mitigation of Railway Traffic Induced Vibrations: The Influence of Barriers in Elastic Half-Space 1 2 2 1 Michele Buonsanti, Francis Cirianni, Giovanni Leonardi, Adolfo Santini, and Francesco Scopelliti Department of Mechanics and Materials (MECMAT), Mediterranean University of Reggio Calabria, Via Graziella, Feo di Vito, 89100 Reggio Calabria, Italy Department of Information Science, Mathematics, Electronics, and Transportations (DIMET), Mediterranean University of Reggio Calabria, Via Graziella, Feo di Vito, 89100 Reggio Calabria, Italy Correspondence should be addressed to Giovanni Leonardi, giovanni.leonardi@unirc.it Received 23 January 2009; Revised 6 May 2009; Accepted 9 May 2009 Recommended by Yehia Bahei-El-Din In this paper, the problem of vibrations induced by trains and their propagation through the soil is studied. Particular attention is focused on the vibration induced by trains in motion and on the effects of such vibrations on the foundations of buildings in proximity of the tracks. The interaction between propagating waves induced by trains in motion and buildings foundations is a problem which does not admit a straightforward analytical solution; thus a solution is given by the use of a model based on the finite elements method. Firstly, we analyze the theoretical aspects of the problem by considering constant or harmonic loads moving along a straight railway track; then, we define a transfer function soil-railway and the response function of the entire system. The study aims to address the wave propagation in an elastic semi-space and the presence in the ground of a discontinuity element, such as a barrier of a given depth is considered. The efficiency variation of barriers is analyzed in function of the different materials used, and different numerical simulations are analyzed in order to study how the wave propagation and the track-soil interaction are influenced by the membrane, seen as damping barrier. Copyright © 2009 Michele Buonsanti et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction to constant or harmonic loads, produced by the movement of trains. The foundations of structures are subject to vibrations due Numerical aspects are developed in detail in Lombaert to moving masses, as vehicles in transit, vibrations which et al. [3] and Fiala et al. [4], whileinHubertetal. [5], a can cause damage locally to the foundations, as they can also detailed analysis using the Boundary Element Method, in travel up to the structures in elevation (Figure 1). the dominion of frequency and of time, is developed on a In this paper we consider the effects of vibrations due to model of elastic semispace. In the work of Andersen and the transit of locomotives on a railway, analyzing how the Nielsen [6], a model for the reduction of vibrations of the presence of material discontinuity, in an elastic semispace ground through the use of barriers dipped in the ground has an influence on the transmission of waves generated by a and laid between rail track and structure is proposed. In train passing by. The presence of barriers, made in polymeric the paper, after some general concepts of theory of the material, rubber, or in concrete, seems to cause reflection and vibrations, the aim is set on the problem in homogenous refraction of the vibrations generated in the ground. There elastic and isotropic semispace, where a two-dimensional are some references in scientific literature. Francois et al. [1], constitutional discontinuity is localized, representing the approached interaction between ground and infrastructures, artificial barrier in the hypothesis set for different materials in presence of vibrations generated by road traffic, while (rubber, polyurethane, and concrete). Sheng et al. [2], modelled the propagation of vibrations due 2 Advances in Acoustics and Vibration Building damage Vehicle response Excitation of Building response dynamic forces Track structure response Ground surface response Surface waves Foundation response Propagation of Secondary body waves stress waves Primary body waves Figure 1: Transmission of induced vibrations. which happens in the semispace, regarding the conditions of quiet, and anyway without the influence of a traveling load. Refraction wave In particular, we observe as the nature of the material used for the realization of the barrier vary, how the conditions Material 2 ϑ of propagation of the waves vary, and their influence, in an equilibrium state, on the contour conditions of an eventual foundation laid under the barrier. Material 1 ϑ ϑ i 1 2. Vibrations Theory Reflection wave In regard to the fundamental aspects in the theory of the Incident wave vibrations, in all this paper we follow the lines expressed in Hartog [7] and Clough and Penzien [8]. Our problem can have a first general organization in the context of Figure 2: Incident wave and reflection. the response for harmonic type loads. It is possible to characterize the following fundamental equations in the case of systems without damping: mu ¨(t) + ku(t) = p sin ωt,(1) and with damping, mu ¨(t) + cu(t) + ku(t) = p sin ωt. (2) The solution of (1) is admitted, from the combination of a complementary solution with a particular solution, in the shape 0 π 2π 3π 4π p β ωh/β u(t) = A sin ωt + B cos ωt + 1 − β2 sin ωt. (3) Figure 3: Amplification waves for Z /Z = 4. 1 2 In the case of (2) the general solution is of the type −ξωt u(t) = e (A sin ω t + B cos ω t) + D D Once the case is defined in its theoretical formulation, a numerical simulation with the support of computer applica- × 1 − β sin ωt − 2ξβ cos ωt . 2 2 1 − β + 2ξβ tion is carried out. What we want to analyze is the variation (4) of the mechanism of propagation of the vibration waves |A(ω)| Advances in Acoustics and Vibration 3 Barrier Figure 5: Finite element mesh of the two-dimensional model. Wheel load Q Figure 4: Modelling of the railway embankment. Speed v Distance y Thefirsttermof (4) represents the transitory result, while the second term represents the stationary result, which has the pulsation of the solicitation but is out of sync with the last wave. The factor of dynamics amplification D is Depth z defined as the relationship betw.een the amplitude of the stationary response and the static movement produced from Figure 6: Deflection of the track structure from one wheel. the external load. In a generalized manner D is function of the damping ratio and of the frequency relationship and usually it tends to infinite when damping is absent. vector k at constant speed (phase speed): v =,(7) 3. Theoretical Aspects in the Model Characterization any system of waves can be considered as a superimposition of monochromatic plane waves that move with the group In an elastic semispace we consider that the source of velocity: disturbance (train) generates a set of waves which are distributed in the plane system and therefore we can classify dω the waves plane and classify our problem as a motion of v = . (8) dκ waves within a semi-infinite body, which is assumed, for simplicity, elastic, homogenous, and isotropic. This type of Equation (8) represents the speed with which the energy is phenomenon is shown in its generality and specificity in the transported and, if v is different from v , the constant phase f g case of the seismic waves as treated in Boschi and Dragoni surfaces move from an extremity to another in the package [9]. We then consider a longitudinal elastic plane wave that of waves while this last one is the same one in motion. propagates in x-direction, if we assume the movement u Given such assumption, we consider the elastic semispace associated to it as a periodic function of the time, u can be as composed of different homogenous elements. Let’s assume placed in the form to form a put in elastic discontinuity, in the cross-sectional sense to the semispace, so to be placed in cross-sectional sense to the wave movement. This eliminates the direct flow u(x, t) = x a cos kx − ωt + ϕ,(5) of the waves from the source to the site of the structure placing itself as element of filter to the vibrations generated where a is the amplitude, k the wave number, ϕ a constant. from the movement of the train. Regarding the interaction Using the relation between the exponential and trigono- with the wave, or the set of waves, the discontinuity assumes metrical functions, and the complex notation, each compo- the role of reflection element and refraction for two- nent of the movement, for a monochromatic wave, can be dimensional waves. If we consider two elastic semispaces defined as joined with an element of material discontinuity, we may say that ρ, α, β are, respectively, the density and the speed of the ∗ ∗ ∗ i(kx−ωt) j elastic waves in the first semispace while ρ , α , β are the u(x, t) = Ae ,(6) same amounts in the second semispace (Figure 2). Refraction and reflection of the elastic waves will happen that represents an oscillation of amplitude A and wavelength on the discontinuity surface, longitudinal ones, and of cross- λ = 2π/κ, which propagates in the direction of the wave sectional ones which can transform one in the other when 4 Advances in Acoustics and Vibration 100 0.002 0.0019 0.0018 0.0017 0.0016 0.0015 0 0.0014 00.025 0.05 0.075 0.10.125 0.15 0.175 0.20.225 0.25 00.51 1.52 2.53 3.5 Time (s) (m) Concrete Rubber Figure 7: Load distribution of a moving wheel travelling at speed Polyurethane No barrier of 200 km/h. Figure 9: Trend of the velocity with the increase of the distance from embankment at the depth of 1 m. 0.3 0.2 −2 ×10 4.2 0.1 3.8 3.6 −0.1 3.4 3.2 −0.2 00.51 1.52 2.53 3.5 (m) −0.3 −0.9 −0.6 −0.30 0.30.60.9 Concrete Rubber Horizontal velocity (mm/s) Polyurethane No barrier (a) Figure 10: Trend of the acceleration with the increase of the 0.3 distance from embankment at the depth of 1 m. 0.2 We assume that given a monochromatic wave incident 0.1 with frequency ω,numberofwave k and travelling unitary amplitude in the semispace in direction-x ,itproducesone cross-sectional movement, in direction x expressed in the −0.1 form: −0.2 ( ) [ ( )] u x , t = exp ωt + k x − h , x ≥ h, (10) 3 1 3 3 −0.3 after the phenomenon of refraction and reflection the field of −0.9 −0.6 −0.30 0.30.60.9 movements takes the form: Horizontal velocity (mm/s) u(x , t) = exp[ωt + k (x − h)] 3 1 3 (b) (11) + R exp[ωt − k (x − h)], x ≥ h, 2 3 3 Figure 8: Nodes velocity (m/s) responses for a train speed of 200 km/h. where R is a coefficient that holds account of all the waves that travel in positive direction x . The contour conditions require that traction on free surface x is equal to zero, x = 3 3 inciting on the surface that separates the two semispaces. In 0, and that the movement and the traction are continuous accordance with Graff [10], the coefficient of reflection or on the surface of discontinuity x = h. In agreement with refraction c assumes the form: ij Graff [10], given Z , Z as stiffness of the two semispace 1 the 2 and A(ω) the coefficient of the exponential, the module of the latter supplies the amplification of the waves: 2ρ β cos θ i i i c = . (9) ij ∗ ∗ ρ β cos θ + ρ β cos θ i i i j j j |A(ω)|= . sqr cos ωh/β + (Z /Z ) sin ωh/β j 2 1 j The contour conditions, on plan x = 0 are the continuity of (12) stress and the movement. Load (kN) Vertical velocity (mm/s) Vertical velocity (mm/s) Velocity magnitude (m/s) Acceleration magnitude (m/s ) Advances in Acoustics and Vibration 5 Table 1: Properties of structural materials. Mechanical Rail Sleeper Traditional HMA Protective Embankment characteristics (UIC60) (concrete) ballast subballast layer Density ρ 7850 2400 1250 2200 2000 1000 (kg/m ) Modulus E 210000 30000 130 6000 160 80 (MPa) Poisson’s Ratio ν 0.30 0.15 0.30 0.40 0.45 0.50 Table 2: Properties of barriers. Mechanical Density ρ Modulus E Poisson’s Dumping characteristics (kg/m ) (MPa) ratio ν ratio η Concrete 2500 25000 0.15 0.04 Polyurethane 1170 25 0.50 0.08 (elastomer) Rubber chip 500 20 0.25 0.10 Equation (12) is a periodic function of the frequency ω with the embankment) is considered, considering the passage of a maximums in correspondence of the values: high speed locomotive (ETR 500) with 4 wheels for car. The effects of discontinuity is estimated considering three π j different materials whose properties are shown in Table 2. ω = (2n − 1) . (13) 2 h 4.2. Finite Element Analyses of Train-Induced Ground Vibra- Figure 3 shows the diagram of amplification of the waves. tions. The finite element simulations of train-induced ground vibrations was developed, using the ADINA soft- ware, considering a two-dimensional finite element model 4. Modeling and Simulation: perpendicular to the track (two-dimensional perpendicular The Track-Subsoil Model model). The two-dimensional model is composed of 7517 ele- In this paragraph we show a numerical analysis, using ments and 8533 nodes (Figure 5). The elements consisted a software for defined elements, on a two-dimensional of four-node solid element for ballast, HMA subballast, modeling of the studied system. protective layer and embankment and nine-node solid element for rail and sleeper. Furthermore, the rail pad was 4.1. Model Characteristics. In the proposed model the rail modeled with parallel discrete springs and dampers (spring track is the conventional one, and the elements that compose constant K = 11.2 · 10 N/m and damper constant c = it are (Figure 4) 12 · 10 Ns/m). For these elements, under the accepted hypothesis of (i) a protective compacted layer 30 cm thick of sand/ viscose-elastic behavior, the input parameters for the mate- gravel (called, in the Italian railways, supercompact); rials characterization, are the modulus of elasticity (E), (ii) a 12 cm subballast layer made of bituminous (hot- Poisson’s ratio (v), density (ρ), α and β Rayleigh’s coefficients mix asphalt) concrete; for the definition of the damping matrix. (iii) a traditional ballast layer 35 cm thick; The acting forces are a sequence of axial loads, moving like the train (the loads are similar to those of an ETR 500 (iv) an embankment 60 cm high. locomotive) (Figure 6). The dimensional characteristics of the elements above The forces are applied using a time function that indicated are those requested by the Italian standard (RFI) represents the time history of the force in the considered for high speed lines. node. During the simulations the loads were applied four Moreover, to analyze the propagation process of the times, reproducing the passing of four axes. The loads can vibrations in the soil, it has been considered a thickness of be thought of as triangular pulses distributed over the wheel- the support soil of the embankment equal to a total 5 m. rail surface of contact as shown in Figure 7. The materials properties used in the model were derived All the finite element analyses in this study were per- from tests and available experimentations in literature. formed in the time domain. The time step of the analyses Table 1 lists the physical-mechanical characteristics for the was fixed and set to 0.001 second. soil and track structure. Figure 8 shows the variation of velocity (y-direction and Finally a discontinuity (10 cm thick and 2 m deep) in the z-direction) of the middle node under truck and of a node semispace at a given distance from the rail track (2 m from at the distance of 1.5 m from the embankment. The orbits of 6 Advances in Acoustics and Vibration 2.5E − 03 4.5E − 02 1.6E − 04 4E − 02 1.4E − 04 2E − 03 3.5E − 02 1.2E − 04 3E − 02 1.5E − 03 1E − 04 2.5E − 02 8E − 05 2E − 02 1E − 03 6E − 05 1.5E − 02 4E − 05 1E − 02 5E − 04 2E − 05 5E − 03 0E +00 0E +00 0E +00 0 50 100 150 200 250 0 50 100 150 200 250 0 50 100 150 200 250 Time (mm/s) Time (mm/s) Time (mm/s) Conceret 1 Conceret 1 Conceret 1 Conceret 2 Conceret 2 Conceret 2 (a) (b) (c) 1.6E − 04 2.5E − 03 4.5E − 02 4E − 02 1.4E − 04 2E − 03 3.5E − 02 1.2E − 04 3E − 02 1E − 04 1.5E − 03 2.5E − 02 8E − 05 2E − 02 6E − 05 1E − 03 1.5E − 02 4E − 05 1E − 02 5E − 04 2E − 05 5E − 03 0E +00 0E +00 0E +00 0 50 100 150 200 250 0 50 100 150 200 250 0 50 100 150 200 250 Time (mm/s) Time (mm/s) Time (mm/s) Rubber 1 Rubber 1 Rubber 1 Rubber 2 Rubber 2 Rubber 2 (d) (e) (f ) 2.5E − 03 4.5E − 02 1.6E − 04 4E − 02 1.4E − 04 2E − 03 3.5E − 02 1.2E − 04 3E − 02 1E − 04 1.5E − 03 2.5E − 02 8E − 05 2E − 02 1E − 03 6E − 05 1.5E − 02 4E − 05 1E − 02 5E − 04 2E − 05 5E − 03 0E +00 0E +00 0E +00 0 50 100 150 200 250 0 50 100 150 200 250 0 50 100 150 200 250 Time (mm/s) Time (mm/s) Time (mm/s) Polyurethane 1 Polyurethane 1 Polyurethane 1 Polyurethane 2 Polyurethane 2 Polyurethane 2 (g) (h) (i) Figure 11: Trend of displacement, velocity, and acceleration in two nodes located before and after the barrier. vertical to horizontal particle motions were mostly counter 6. Conclusions clockwise; this could indicate the presence of Rayleigh waves The results of the proposed elaborations at the finite elements [11]. have been finalized to the assessment of the incidence of the barrier on the vibration state induced from the passage 5. Results of a high speed locomotive. The response of three different materials for the barrier was confronted and the following The graphs of the outcomes of the various analyses are shown conclusions can be made: in Figures 9 and 10. To estimate the effectiveness of the barrier we have esti- (i) concrete seems to provide a better reduction of mated the dynamic characteristics (speed and acceleration) the vibration. In spite of the greater density of the before and after the discontinuity, in correspondence of an material it involves an increase of the reflection eventual foundation plan of a building placed at a distance of phenomena and a consequent increase of the phe- 1.5 meters from the barrier. nomenon at the top side of the barrier (Figures 9 and As a general result of the analyses, as illustrated in Figures 10); 9 and 11, for the considered loads, it is concluded that, when (ii) polyurethane and rubber chip materials seem to the load is travelling with the speed of 200 Km/h, concrete respond in a similar way to the solicitations; however barriersprovide amoreefficient vibration shielding than the their damping contribution, in the analyzed geomet- rubber and polyurethane barriers. ric configuration, do not contribute meaningfully. Figure 11 shows, for the three kinds of barrier, the course of the displacement, velocity, and acceleration in time. The results obtained were given with the use of different Theresults giveninFigures 9 and 11 supply useful materials for the barrier, in identical geometrical conditions, information for the making of barriers, especially when it in size and position of the proposed damping barriers. A is required to respect regulation’s limit for vibration, as follow up to the presented study, requiring further analyses, expressed in terms of displacements and velocities (i.e., is in course, to estimate the influence of the position and of Italian regulation UNI 9916/2220). the thickness of the barriers on their dampening ability. Displacement Displacement Displacement Velocity (m/s) Velocity (m/s) Velocity (m/s) 2 2 2 Acceleration (m/s ) Acceleration (m/s ) Acceleration (m/s ) Advances in Acoustics and Vibration 7 References [1] S. Francois, L. Pyl, H. R. Masoumi, and G. Degrande, “The influence of dynamic soil-structure interaction on traffic induced vibrations in buildings,” Soil Dynamics and Earthquake Engineering, vol. 27, no. 7, pp. 655–674, 2007. [2] X. Sheng, C. J. C. Jones, and M. Petyt, “Ground vibration generated by a load moving along a railway track,” Journal of Sound and Vibration, vol. 228, no. 1, pp. 129–156, 1999. [3] G. Lombaert, G. Degrande, and D. Clouteau, “Numeri- cal modelling of free field traffic-induced vibrations,” Soil Dynamics and Earthquake Engineering, vol. 19, no. 7, pp. 473– 488, 2000. [4] P. Fiala, G. Degrande, and F. Augusztinovicz, “Numerical modelling of ground-borne noise and vibration in buildings due to surface rail traffic,” Journal of Sound and Vibration, vol. 301, no. 3–5, pp. 718–738, 2007. [5] W. Hubert, K. Friedrich, G. Pflanz, and G. Schmid, “Frequency- and time-domain BEM analysis of rigid track on a half-space with vibration barriers,” Meccanica, vol. 36, no. 4, pp. 421–436, 2001. [6] L. Andersen and S. R. K. Nielsen, “Reduction of ground vibration by means of barriers or soil improvement along a railway track,” Soil Dynamics and Earthquake Engineering, vol. 25, no. 7–10, pp. 701–716, 2005. [7] J.P.D.Hartog, Mechanical Vibrations, Dover, Oxford, UK; Oxford University Press, Oxford, UK, 1975. [8] R.W.Clough andJ.Penzien, Dynamics of Structures, McGraw- Hill, Singapore, 1985. [9] E. Boschi and M. Dragoni, Sismologia, UTET, Torino, Italy, [10] K. F. Graff, Wave Motion in Elastic Solids, Dover, Oxford, UK, [11] L. Hall, “Simulations and analyses of train-induced ground vibrations in finite element models,” Soil Dynamics and Earthquake Engineering, vol. 23, no. 5, pp. 403–413, 2003. 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