Applicability of a Three-Layer Model for the Dynamic Analysis of Ballasted Railway Tracks

vibration

Article Applicability of a Three-Layer Model for the Dynamic Analysis of Ballasted Railway Tracks

André F. S. Rodrigues 1 and Zuzana Dimitrovová 2,3,*

1 Track Systems and Development, Infrastructure, Banedanmark, DK-2450 Copenhagen, Denmark; [email protected] 2 Departamento de Engenharia Civil, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal 3 IDMEC, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal * Correspondence: [email protected]

Abstract: In this paper, the three-layer model of ballasted railway track with discrete supports is analyzed to access its applicability. The model is referred as the discrete support model and abbreviated by DSM. For calibration, a 3D ﬁnite element (FE) model is created and validated by experiments. Formulas available in the literature are analyzed and new formulas for identifying parameters of the DSM are derived and validated over the range of typical track properties. These formulas are determined by ﬁtting the results of the DSM to the 3D FE model using metaheuristic optimization. In addition, the range of applicability of the DSM is established. The new formulas are presented as a simple computational engineering tool, allowing one to calculate all the data needed for the DSM by adopting the geometrical and basic mechanical properties of the track. It is demonstrated that the currently available formulas have to be adapted to include inertial effects of the dynamically activated part of the foundation and that the contribution of the shear stiffness, being determined by ballast and foundation properties, is essential. Based on this conclusion, all similar models that

Citation: Rodrigues, A.F.S.; neglect the shear resistance of the model and inertial properties of the foundation are unable to Dimitrovová, Z. Applicability of a reproduce the deﬂection shape of the rail in a general way. Three-Layer Model for the Dynamic Analysis of Ballasted Railway Tracks. Keywords: ballasted track; structural vibrations; ﬁnite element method; discrete support model; Vibration 2021, 4, 151–174. https:// metaheuristic optimization; numerical calibration doi.org/10.3390/vibration4010013

Academic Editor: Hitoshi Tsunashima 1. Introduction Received: 1 January 2021 Numerical models of the railway track are fundamental tools for the study of their Accepted: 17 February 2021 Published: 22 February 2021 dynamic behavior, with implications for the safety and comfort of rail transport. The importance of these models has increased alongside the speed of the railway vehicles and

Publisher’s Note: MDPI stays neutral capacity of the network over the last decades. with regard to jurisdictional claims in The use of 3D FE models is common practice, but reduced models are still relevant, published maps and institutional afﬁl- due to simplicity of implementation and results interpretation, and low computational cost. iations. Despite the wide use of the reduced models, there are still relatively few studies about their overall applicability and how to select appropriate parameters based on the properties of the railway track. Most works determine the necessary parameters in a way to match the response of the model to experimental measurements from the track. This means that the range of applicability of the methods developed and the conclusions reached in these Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. works are always, to some extent, limited. This article is an open access article This paper provides a detailed analysis of the viability and applicability of the three- distributed under the terms and layer discrete support model [1], for simplicity referred to as the discrete support model conditions of the Creative Commons and abbreviated by DSM, expanding on the work presented in [2] and analyzing the Attribution (CC BY) license (https:// formulas proposed in [3]. The DSM is a 2D model composed of a beam, representing the creativecommons.org/licenses/by/ rails, connected to sets of springs, dampers and masses that represent the support elements 4.0/). of the track: the fastening system, sleepers, ballast, and subgrade. The DSM has signiﬁcant

Vibration 2021, 4, 151–174. https://doi.org/10.3390/vibration4010013 https://www.mdpi.com/journal/vibration Vibration 2021, 4 152

advantages over the more detailed 3D FE model, which presents a high computational cost, large number of results to analyze, the need for special boundary conditions, necessity for high level of discretization, etc. Given its computational efficiency, the DSM and other, even simpler models, are often used to study complex phenomena that arise in different railway transportation problems. Examples of published works dealing with specific railway problems include: (i) modelling the vehicle–track dynamic interaction on plain line [4–13] but also in the case when the line is passing over bridge [14] or crossing a transition zone [15,16]; (ii) assessing and diagnosing the deterioration of track components [17–22], including switches and crossings [23–25], rail wear/corrugation [26–29] and hanging sleepers [30,31]; (iii) studying other problems related to wheel–rail interaction, including the effect of wheel flats [32–37] and rolling noise [38–42]. Published works with the aim of defining the properties of reduced models with discrete supports consider the ballast pyramid model referred to in [43,44] and the Saller assumption from 1932 which can be consulted in [45]. The pyramid model is further detailed in [4], under the name stress cone model, where it is used not only for the definition of the vertical ballast stiffness, but also for the identification of the oscillating ballast mass. Further improvements include the superposition of the cones between adjacent sleepers, i.e., in the longitudinal direction [3]. Nevertheless, these works have two notable limitations: (i) they do not propose theoretical expressions for all the elements of the model, (ii) they each validate their models and theoretical expressions by comparison with experimental measurements of a single railway track, and therefore cannot prove that they are widely applicable to a range of different track properties. The two limitations are particularly serious when considered together; since some parameters of the models are obtained by fitting its results to experimental measurements for a single set of track parameters, it is not possible to unequivocally determine if the other parameters are intrinsic to the physical phenomena, or if the good approximation is obtained simply by virtue of the fitting process. Therefore, the present work aims to provide a detailed analysis of the viability and applicability of the DSM, to derive formulas for its components that are still missing and to enhance formulas that are already available. This is achieved by fitting the DSM steady- state vertical displacement of the rail under moving force to reference solutions that cover a wide range of representative values of the track. The reference solutions are obtained by a linear elastic 3D FE model, which is validated by the experimental measurements published by Paixão [46]. The use of a linear elastic model is appropriate because the focus is on the track’s short-term dynamic behavior. Analyzing the deflection shape of the rail is also adequate, because obtaining a good approximation of the steady-state rail displacement is a necessary prerequisite for studying more complex phenomena. The DSM is fitted to the reference solutions using genetic algorithms, first individually for each combination of track parameters, to evaluate its ability to approximate the results of the 3D FE model and to define proportionality relationships between the properties of the track and those of the DSM. After that, optimization is conducted simultaneously for various combinations of track parameters. Based on these results, theoretical expressions for all parameters of the DSM are derived as functions of the geometry and mechanical properties of the track. It is shown that the currently available formulas have to be adapted in a way to include inertial effects of the dynamically activated part of the foundation and that the contribution of the shear stiffness which must account not only for ballast but also for foundation, is essential. This is in agreement with [47,48], where it is shown that if the shear stiffness of the model is neglected, then the critical velocity of a uniformly moving force tends to zero with the dynamically activated mass of the model, which is not physical. With this conclusion, all similar models that neglect the shear resistance of the model and inertial properties of the foundation are unable to reproduce the deflection shape of the rail in a general way. In addition to these new conclusions, the stress cone formulas are improved by incorporating the superposition of the cones in the lateral direction. All formulas are presented as a simple computational engineering tool, allowing the reader to Vibration 2020, 3 FOR PEER REVIEW 2 Vibration 2021, 4 153 allowing the reader to calculate all the data needed for the DSM simply by adopting the geometrical and basic mechanical properties of the track constituents. calculateThis allpaper the datais organized needed forin the following DSM simply way: by Section adopting 2 describes the geometrical the DSM and and basic re- mechanicalviews the relevant properties literature of the track, Section constituents. 3 presents the 3D FE model and its validation, Sec- tion This4 is dedicated paper is organized to the calibration in the following of the DSM way: based Section on2 describes the results the of DSM the 3D and FE reviews model theand relevant derivation literature, of the theoretical Section3 presents expressions the 3Dand FE Section model 5 andsummarizes its validation, the main Section conclu-4 issions. dedicated to the calibration of the DSM based on the results of the 3D FE model and derivation of the theoretical expressions and Section5 summarizes the main conclusions. 2. DSM 2. DSMRegarding the evolution of the reduced models, it is worthwhile to mention that dis- creteRegarding supports were the evolutionintroduced of in the [49 reduced] to remove models, the limitations it is worthwhile of the Winkler to mention and thatPas- discreteternak models, supports which were consist introduced of a beam in [49 continuously] to remove the supported limitations by an of theelastic Winkler or viscoe- and Pasternaklastic foundation. models, Newton which consist and Clark of a [5 beam0] were continuously the first to supporteduse the abbreviation by an elastic DSM or vis-and coelasticto consider foundation. the sleepers Newton as discrete and Clark mass [50] wereelements the first, which to use separate the abbreviationd the stiffness DSM and and todamping consider introduced the sleepers by as the discrete fastening mass system elements, from which the one separated due to the the ballast stiffness bed and and damp- sub- inggrade, introduced creating bytwo the layers. fastening In [5 system1], another from set the of one masses due to that the represent ballast bed the and ballast subgrade, were creatingintroduced two, layers. and therefore In [51], another, the stiffness set of masses and damping that represent of the the ballast ballast were and of introduced, the sub- andgrade/foundation therefore, the stiffnessgot also separated and damping, creating of the three ballast layers. and ofZhai the and subgrade/foundation Sun [4] then intro- gotduced also springs separated, and dampers creating threeconnecting layers. consecutive Zhai and Sunballast [4] masses then introduced to model the springs shear and be- dampershavior of connecting the ballast, consecutive which already ballast cover masses all the to DSM model components the shear behavior, as depicted of the in ballast, Figure which1. already cover all the DSM components, as depicted in Figure1.

FigureFigure 1.1. DSM,DSM, basedbased onon [[4].4].

TheThe railrail isis modelledmodelled asas aa TimoshenkoTimoshenko beam,beam, for which the bending ( EIEI)) andand shearshear stiffnessstiffness ((GAGA∗*),), andand thethe massmass perper unitunit length (mm)) are are well-known. well-known. Likewise, Likewise, thethe fasteningfastening system stiffness and damping coefficient (K and C ) can be determined experimentally. system stiffness and damping coefficient (padKpad andpad Cpad ) can be determined experimen- Thetally. sleepers The sleepers experience experience very littlevery little deformation deformation during during the normalthe normal use use of the of the track, track, so they are modelled as rigid elements that only contribute with their mass (Msleep), which is so they are modelled as rigid elements that only contribute with their mass ( Msleep ), which well-defined.is well-defined. The The remaining remaining unknown unknown parameters parameters are are the the vertical vertical stiffness stiffness and and damping damp- coefficienting coefficient of the of ballastthe ballast (Kb and ( K C andb), the C vertical), the vertical stiffness stiffness and damping and damping coefficient coefficient of the subgrade (K and C ), the shear stiffnessb andb damping coefficient of the ballast (K and C ) of the subgradef ( Kf and C ), the shear stiffness and damping coefficient of thew ballastw ( and the dynamicallyf activatedf ballast mass (M ). K and C ) and the dynamically activated ballastb mass ( M ). w In [3,4],w formulas for calculation of the vertical stiffnessb of the ballast are proposed asIn an[3,4], adaptation formulas offor the calculation stress cone of (pyramid)the verticalmethod, stiffness whichof the ballast assumes are that proposed the stresses as an areadaptation transmitted of the from stress the cone effective (pyramid) load-bearing method, area which under assumes the sleepers that the to stresses the ballast, are trans- and thenmitted dissipate from the at effective a constant load angle,-bearing as shown area under in Figure the 2sleepersa. The effective to the ba load-bearingllast, and then area dis- undersipate theat a sleepers constant is angle, determined as shown in [ 45in], Figure based 2a). on experimental The effective observationsload-bearing andarea simple under geometricalthe sleepers considerations,is determined in requiring [45], based only on the experimental sleeper’s dimensions observations and and the tracksimple gauge. geo- Then,metrical the considerations, vertical stiffness requiringKb can be only determined the sleeper’s as a function dimensio ofns a distribution and the track angle gauge.αb,

theThen, Young the vertical modulus stiffness of the ballast can and be itsdetermined depth. In as [43 a], function a reduction of a factordistribution for the angle vertical  b stiffness, the Young due modulus to superposition of the ballast of stress and conesits depth. of 0.5 Inwas [43], proposed a reduction based factor on for experimental the vertical observations.stiffness due to In superposition [3], superposition of stress of the cones adjacent of 0.5 stress was cones proposed in the based longitudinal on experimental direction is introduced exactly. The cone volume such deﬁned is then used for the estimation of the

Vibration 2020, 3 FOR PEER REVIEW 2 Vibration 2021, 4 154 observations. In [3], superposition of the adjacent stress cones in the longitudinal direction is introduced exactly. The cone volume such defined is then used for the estimation of the dynamically activated ballast mass, M . The area of the base of the stress cone is further dynamically activated ballast mass, Mb. Theb area of the base of the stress cone is further usedused to to calculate calculate the the foundation foundation vertical vertical stiffness, stiffness,K f ,K butf , but for for this, this, the the experimentally experimentally determineddetermined subgrade subgrade reaction reaction modulus modulus must must be be known. known.

(a) (b)

Figure 2. Stress cone load distribution in the ballast, adapted from [3]: (a) without superposition; (b) with superposition. Figure 2. Stress cone load distribution in the ballast, adapted from [3]: (a) without superposition; (b) with superposition.

No functional dependence is proposed for all damping coefficients (Cb, Cf and Cw), No functional dependence is proposed for all damping coefficients ( C , C and C for the shear stiffness of the ballast (Kw) and, in fact, also for the foundationb stiffnessf K f . w There is also no justification for the value of the distribution angle α . It is therefore the ), for the shear stiffness of the ballast ( Kw ) and, in fact, also for theb foundation stiffness objective of the present work to propose general formulas for all components of the DSM . There is also no justification for the value of the distribution angle  . It is therefore without the necessity to call on experimental data. Alongside this, the range ofb applicability of the DSMobjective is obtained. of the present Contrary work to the to arrangementpropose general given formulas in Figure for1, itall is components proven that theof the dynamicallyDSM without activated the necessity part of theto call foundation on experimental must be data. added Alongside to the ballast this, massthe ra andnge that of ap- theplicability shear stiffness of the of DSM the ballast is obtained. must be Contrary increased to by the the arrangement contribution given from thein foundationFigure 1, it is soils.proven If this that is notthe respected,dynamically then activated the DSM part is unableof the foundation to reproduce must the be deflection added to shape the ballast of themass rail inand a generalthat the way. shear stiffness of the ballast must be increased by the contribution from the foundation soils. If this is not respected, then the DSM is unable to reproduce the de- 3. Three-Dimensionalflection shape of the Modelrail in a general way. To obtain the reference results necessary to calibrate the DSM, a 3D FE model of the railway3. Three track-Dimensional is developed. Model This model has linear elastic behavior and can be used to obtain static andTo dynamicobtain the vertical reference displacements results necessary in the rail.to calibrate It is implemented the DSM, a using3D FE the model ANSYS of the parametricrailway track design is languagedeveloped. [51 This], so model the model has linear can be elastic generated behavior in a fullyand can automatic be used way to ob- fortain each static set of and parameters. dynamic vertical displacements in the rail. It is implemented using the AN- SYSStatic, parametric modal anddesign transient language analyses [51], so are the used model to definecan be thegenerated necessary in a discretization fully automatic andway which for each type set of boundaryof parameters. conditions (BCs) are suitable to the problem in study. The static andStatic, modal modal analyses and transient are necessary analyses to validateare used the to define stiffness the component necessary ofdiscretization the BCs. Theand 3D which FE model type isof validated boundary by conditions modelling (BCs) an existing are suitable railway to the track problem for which in study. the rail The displacementsstatic and modal due to analy a trainses passageare necessary are available to validate in [46 the]. stiffness component of the BCs. TheThe 3D sleepers, FE model ballast is validated and subgrade by modelling are modelled an existing by brick railway and wedge track for solid which elements, the rail anddisplacements beam elements due are to used a train for passage the rail andare available discrete spring–damper in [46]. elements in the three orthogonalThe directionssleepers, ballast simulate and the subgrade fastening are system. modelled The by Timoshenko brick and wedge beam has solid the elements, cross- sectionand beam properties elements of a are 60E1 used rail for profile the rail [52 and] and discrete shear factorspring of–damper 0.4 as in element [5]. Thes in sleepers the three areorthogonal spaced by directions 0.6 m and simulate their geometry the fastening is based system. on the The DW Timoshenko post-tensioned beam mono-block has the cross- concretesection sleepers properties used of bya 60E1 the Portugueserail profile [52] railway and manager,shear factor I.P. of S.A. 0.4 [as53 in,54 [5].], but The altered sleepers to achieve a regular mesh (the following parameters in m are used: length 2.6, bottom are spaced by 0.6m and their geometry is based on the DW post-tensioned mono-block width 0.3, top width 0.2 and height 0.22). The moment of inertia is approximately the concrete sleepers used by the Portuguese railway manager, I.P. S.A. [53,54], but altered to same as the real section, and the change in volume is corrected by applying a factor to the achieve a regular mesh (the following parameters in m are used: length 2.6, bottom width mass of the material of 0.86. Another simplification is that no loss of contact is possible 0.3, top width 0.2 and height 0.22). The moment of inertia is approximately the same as between distinct materials, however, in the absence of sudden changes in stiffness or other the real section, and the change in volume is corrected by applying a factor to the mass of irregularities, no loss of contact occurs anyway. Symmetry along the vertical plane parallel

Vibration 2020, 3 FOR PEER REVIEW 2 Vibration 2021, 4 155 the material of 0.86. Another simplification is that no loss of contact is possible between distinct materials, however, in the absence of sudden changes in stiffness or other irregularities, no loss of contact occurs anyway. Symmetry along the vertical plane parallel to tothe the rails rails (the (the xy-xyplane)-plane) is assumed, is assumed, reducing reducing the size the of size the of model the model to half, to as half, seen as in seen Figure in Figure3. The 3shoulder. The shoulder width of width 0.5m ofand 0.5 slope m and 1:2 slopeare assumed 1:2 are for assumed the ballast for thelayer. ballast Two layer.thick- Twonesses thicknesses will be used will of be 0.3 used and of0.6m, 0.3 leading and 0.6 to m, the leading bottom to width the bottom of 4.8 widthand 6m. of The 4.8 andsub- 6grade m. The is discretized subgradeis using discretized a regular using orthogonal a regular mesh. orthogonal The regularity mesh. of The the regularity mesh simplifies of the meshthe process simpliﬁes of implementing the process of implementingBCs and avoids BCs spurious and avoids reflections spurious between reﬂections elements between of elementsdifferent ofsizes different [55]. Preliminary sizes [55]. Preliminarytests have also tests shown have that also t shownhe BCs thatperform the BCs worse perform at the worseinterface at thebetween interface different between element different sizes. element Convergence sizes. Convergencestudies for static, studies dynamic for static, and dynamictransient andanalyses transient led to analyses the choice led toof thethechoice mesh: ofrail the (x mesh:) 0.05m; rail longitudinal (x) 0.05 m; longitudinaldirection (x) direction0.075m, lateral (x) 0.075 (z) m,and lateral vertical (z) direction and vertical except direction for sleepers except (y for) 0.1m, sleepers along (y )sleeper 0.1 m, height along sleeper(y) 0.055m. height (y) 0.055 m.

FigureFigure 3.3. 3D3D FEFE modelmodel ofof thethe railwayrailway track.track.

TheThe subgradesubgrade depth depth can can be be defined defined as theas the depth depth at which at which a significantly a significantly rigid rigid substrate sub- isstrate found, is found, or as a or reasonable as a reasonable depth depth after whichafter which the deformations the deformations due todue surface to surface loads loads are negligibleare negligible (the (the so-called so-called active active depth depth of theof the soil soil [56 [56,57]).,57]). A A great great variety variety of of such such values values cancan bebe foundfound inin thethe literature:literature: somesome adoptadopt valuesvalues ofof thethe orderorder ofof 33 mm [[58,59],58,59], whilewhile othersothers considerconsider muchmuch higherhigher values,values, from 10 m [[60,61]60,61] toto 5050 mm [[6262––6464]] andand asas highhigh asas 7070 mm [[65].65]. Given this variability, five different depths of the subgrade (hs) are studied: 3, 6 and 9 m Given this variability, five different depths of the subgrade ( hs ) are studied: 3, 6 and 9 m (relatively shallow subgrade), 25 m (an average depth) and 50 m (a high depth). Not (relatively shallow subgrade), 25 m (an average depth) and 50 m (a high depth). Not all all the depths are considered for all the different analyses, as will be discussed in the the depths are considered for all the different analyses, as will be discussed in the relevant relevant section. section. All materials are assumed to have linear elastic behavior, since the purpose of the All materials are assumed to have linear elastic behavior, since the purpose of the model is to analyze short-term behavior due to vehicle passage. Besides the elastic com- model is to analyze short-term behavior due to vehicle passage. Besides the elastic components, there are discrete linear viscous dampers in the fastening system. No material ponents, there are discrete linear viscous dampers in the fastening system. No material damping was considered in the ballast and subgrade, to avoid damping the dynamic response.damping was The materialconsidered properties in the ballast of the and rail subgrade, and sleepers to avoid are well damping defined, the since dynamic they are re- manufacturedsponse. The material components properties that followof the therail relevantand sleepers standards are well [53 ,defined,54]. The since properties they are of themanufactured ballast and components subgrade, on that the follow other hand, the relevant show great standards variability [53,5 across4]. The the properties literature, of particularlythe ballast and the subgrade, Young modulus on the other (see Rodrigues hand, show [66 great]), which variability generally across has the the literature, greatest influenceparticularly on thethe short-termYoung modulus response (see of theRodrigues track. Three [66]), values which of generally the Young has modulus the greatest were choseninfluence for o then the ballast short and-term subgrade, response which of the cover track. the Three relevant values range of found the Young in the literaturemodulus whilewere excludingchosen for extreme the ballast values and that subgrade, are not representative which cover the of typical relevant railway range tracks found in in good the condition.literature while Table1 excluding summarizes extreme these values,values thethat most are not typical representative values of Poisson’s of typical ratio railway and mass density and the properties of the sleepers and rail.

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Vibration 2021, 4 156 tracks in good condition. Table 1 summarizes these values, the most typical values of Pois- son’s ratio and mass density and the properties of the sleepers and rail.

TableTable 1. 1.Elastic Elastic material material properties properties for for the the three-dimensional three-dimensional railway railway model. model.

PropertyProperty RailRail SleeperSleeper BallastBallast SubgradeSubgrade 3 3 Young’sYoung modulus’s modulus [MPa] [MPa] 210·10210∙103 3838∙10·103 50,100,15050,100,150 50,100,15050,100,150 Poisson’sPoisson ratio’s ratio 0.30.3 0.20.2 0.250.25 0.350.35 DensityDensity [kg/m [kg/m3] 3] 7850 7850 206420641 1 17501750 19001900 1 Modiﬁed1 Modified by by correction correction factor. factor.

TheThe range range considered considered for for Young’s Young’s modulus modulus of theof the ballast ballast may may be considered be considered large, large, but itbut was it justiﬁedwas justified by extensive by extensive literature literature search search and by and summarizing by summarizing values values from morefrom thanmore 40than sources. 40 sources. The histogram The histogram for ballast for ballast Young’s Young’s modulus, modulus, Poisson’s Poisson’s ratio ratio and densityand density are presentedare presented in Figure in Figure4. 4.

(a)

(b)

(c)

FigureFigure 4. 4.Summary Summary of of ballast ballast properties properties over over the the literature: literature: (a )(a Young’s) Young’s modulus, modulus, (b )(b Poisson’s) Poisson’s ratio, (cratio) density., (c) density.

ForFor the the properties properties of of the the fastening fastening system, system, the the Vossloh Vossloh Zw687a Zw687a EVA EVA rail rail pad pad with with directdirect fastening fastening is is used, used, to to match match IP IP S.A. S.A. [ 54[5].4]. In In agreement agreement with with [ 67[67],], the the static static stiffness stiffness of 1300 MN/m and 100 MN/m, dynamic stiffness of 3550 MN/m and 280 MN/m and damping coefficient of 36 kNs/m and 10 kNs/m were used, with the first value reflecting

Vibration 2020, 3 FOR PEER REVIEW 2

of 1300MN/m and 100MN/m, dynamic stiffness of 3550MN/m and 280MN/m and damp- Vibration 2021, 4 ing coefficient of 36kNs/m and 10kNs/m were used, with the first value reflecting the ver-157 tical and the second value the lateral and longitudinal directions, respectively (see [66] for more details). the verticalIn regard and to the the secondelastic part value of thethe BCs, lateral the and normal longitudinal stiffness of directions, 4/Gr and respectively tangential (seestiffness [66] for of more2/Gr details). is used at the lateral surfaces, as derived for spherical waves in [68,69], becauseIn regard they were to the found elastic to part work of better the BCs, than the the normal ones derived stiffness for of 4theG/ cylindricalr and tangential waves stiffnessthat were of used2G/r inis used[70]. The at the bottom lateral bo surfaces,undary asis considered derived for either spherical fixed waves or Eh inoed [/68, 69and], becauseGh/ are they used were for foundthe normal to work and better tangential than the stiffness, ones derived respectively, for the as cylindrical in [70]. Here, waves G that were used in [70]. The bottom boundary is considered either fixed or Eoed/h and G/h stands for the shear modulus, r for the distance from the source, Eoed is the oedometric are used for the normal and tangential stiffness, respectively, as in [70]. Here, G stands modulus and h is the depth of the foundation not being modelled. At all (non-fixed) for the shear modulus, r for the distance from the source, Eoed is the oedometric modulus boundaries damping coefficients of c for normal and c for tangential direction, as and h is the depth of the foundation notp being modelled. Ats all (non-fixed) boundaries damping coefficients of ρc for normal and ρc for tangential direction, as in [68–71] are in [68–71] are used. Here, p is the density ands cp , cs , are the velocity of propagation of used. Here, ρ is the density and c , c , are the velocity of propagation of pressure and shear pressure and shear waves, respectively.p s waves, respectively. The model was tested for different combinations of properties and showed good con- The model was tested for different combinations of properties and showed good vergence for the mesh specified for static displacement, the 10 first natural frequencies, convergence for the mesh specified for static displacement, the 10 first natural frequencies, the transient response to a pulse load and to a moving load at velocities of 50 and 100 m/s. the transient response to a pulse load and to a moving load at velocities of 50 and 100 m/s. Finally, the model was validated by comparison with results published in [46], where the Finally, the model was validated by comparison with results published in [46], where the measured vertical vibrations of a railway line in Portugal due to the passage of a train at measured vertical vibrations of a railway line in Portugal due to the passage of a train at 220 km/h are compared to the results of a 2D FE model. 66kN per wheel is used in the 220 km/h are compared to the results of a 2D FE model. 66 kN per wheel is used in the front bogie. By adapting the properties of the 3D FE model developed for the purpose of front bogie. By adapting the properties of the 3D FE model developed for the purpose of this paper, the vertical displacements seen in Figure 5 were obtained. this paper, the vertical displacements seen in Figure5 were obtained.

(a)

(b)

FigureFigure 5. 5.Vertical Vertical displacement displacement of of the the rail: rail: comparison comparison between between experimental experimental measurements, measurements, ﬁtted fitted numericalnumerical resultsresults (2D) from [4 [466]] and and numerical numerical results results (3D) (3D) obtained obtained here; here; (a) full (a) fullvehicle; vehicle; (b) first (b) ﬁrstbogie. bogie.

It can be concluded that the 3D FE model provides a good approximation to the experimental results, particularly when taking into account the variability of the track properties and the random nature of the dynamic response for high frequencies due to the short-wave irregularities of the rails and wheels. These results also conﬁrm that the

Vibration 2021, 4 158

simplifications introduced in the model do not significantly impact its applicability, namely, the assumption of linear elasticity and the use of moving forces. The input data for the 3D FE model are taken from [46]. Besides the typical properties related to the rail, rail pads and sleepers, there are other materials characterized by its thickness, Youngs’s modulus, density and Poisson’s ratio: the ballast layer (0.3 m, 130 MPa, 1530 kg/m3, 0.2), sub-ballast layer (0.3 m, 200 MPa, 1935 kg/m3, 0.3), capping layer (0.2 m, 2820 MPa, 1935 kg/m3, 0.3), embankment soils (~7 m, 80 MPa, 2040 kg/m3, 0.3) and natural foundation (~3.5 m, 300 MPa, 2040 kg/m3, 0.3). The depth of the embankment soils and the stiffness of the natural foundation in relation to that of the former were deemed high enough that the latter can be considered as a rigid substrate, therefore subgrade was implemented with (7 m, 80 MPa, 2040 kg/m3, 0.3) because the capping layer, having low thickness, does not influence the equivalent stiffness much. As far as the ballast layer, as the generic model used for optimization has only one layer, (0.6 m, 150 MPa, 1530 kg/m3, 0.2) was implemented, where 150 MPa corresponds to an equivalent modulus obtained by considering the two layers characterized by equivalent springs connected in series.

4. Optimization The optimization is performed using genetic algorithms (GA), which, being a heuristic method, provides no guarantee that the solution found is the global optimum. Therefore, several independent runs are necessary to guarantee the stability of the results obtained. In what follows, only stable results will be shown for each set of optimization runs, for the sake of brevity. For all optimization runs, the following parameters are used: the population consists of 100 individuals, the number of generations is 20, the mutation rate is 1% and an elite of two individuals is kept between generations (see [72] for definitions). The objective function Φ is equal to the L2-norm of the difference between the rail displacements resulting from the DSM and from the 3D FE model, divided by the L2-norm of the displacement obtained by the 3D FE model. Therefore, Φ can theoretically reach zero when both deflection shapes are identical. The philosophy behind the optimization is the following: firstly, dependence on basic mechanical properties is determined, identifying the proportionality factors; secondly, mechanistic expressions are used to specify these proportionality factors in terms of some geometric or similar specifications of the model. This is very important because the solution of the optimization problem is not unique, i.e., several combinations of parameters can lead to practically identical rail deflection shapes, but for some of those solutions, the parameters have no physical meaning. Expressing the proportionality factors in terms of some geometric parameters impose natural limits on the design space and consequently guarantee the obtainment of physically admissible values. Besides this, the optimization is divided in two steps. First, the stiffness parameters characterizing the DSM are determined from the static analysis (v = 0 m/s). Then, a dynamic analysis (v = {50, 100} m/s) is used to determine damping and mass parameters. In addition, it is also important to distinguish the individual and combined optimizations. For the individual optimization, a unique set of the discrete values of the parameters is selected and the objective function Φ is used as specified above. For the combined optimization, some parameters are unique (fixed) while others assume all possible combinations. Then the objective function is defined as Φ = maxΦi, where i identifies the particular i combination. As specified in previous section, the discrete values selected for variation are:

• The depth of the ballast (hb = {0.3, 0.6} m); • The Young modulus of the ballast (Eb = {50, 150, 300} MPa); • The depth of the subgrade (hs = {3, 6, 9, 25, 50} m); • The Young modulus of the subgrade (Es = {50, 100, 150} MPa); • The load velocity (v = {0, 50, 100} m/s). Since the analysis is linear and velocities are mostly subcritical, the smooth evolution of the response on all parameters can be assumed allowing for interpolation. For the ﬁxed Vibration 2021, 4 159

values of the Poisson ratio and density, the lowest velocity of propagation of Rayleigh waves that is considered is approximately 92.3 m/s in the subgrade, which means that the load velocity of v = 100 m/s is supercritical in this case.

4.1. Identiﬁcation of Dependencies on Mechanical Properties 4.1.1. Stiffness Parameters (Static Solution), Individual Optimization

The parameters to be determined (compare with Figure1) are Kb, Kw and K f . First, the individual optimizations are performed, to establish some general trends. Each of the 18 possible combinations of Eb, Es and hb are assumed separately, while hs is kept at a constant value of 6 m. The following observations were made: • Φ is small for all combinations (1%–4%), so the DSM can adequately approximate the 3D FE model solution for a static load. • The optimum value of K f varies linearly with Es and has the greatest influence on Φ. • The optimum value of Kw varies significantly with both Eb and Es (and therefore with the respective shear moduli, Gb and Gs), with the latter having a stronger influence, thus the resulting optimum value of Kw can be suitably approximated using a linear combination of Gb and Gs. • Above a certain limit, the value of Kb does not have a significant impact on Φ, and so the optimum values are spurious, with no clear dependence on Eb, Es or hb.

Although no clear conclusions can be drawn as regarding Kb, further analyses will show that the formula proposed in [3] provides a good approximation. Therefore, taking also into account the observations above, the following relations are proposed and will be used in the combined optimization:

Kb = fK,bEb oed K f = fK,sEs (1) Kw = fK,w,bGb + fK,w,sGs

oed where Es is the oedometric modulus of the subgrade. The proportionality factors ( fK,b, fK,s, fK,w,b, fK,w,s) are in meters.

4.1.2. Stiffness Parameters (Static Solution), Combined Optimization

The static solution is optimized for all possible combinations of Eb and Es simultaneously, while hb is considered unique and hs are grouped in two sets: a typical range hs = {6, 25, 50} m and a shallow range hs = {3, 6, 9} m. Some results can be presented for all three depths from the set, some have to be separated, as shown in Table2. For the former set, the reference results of the 3D FE model were obtained using the elastic BCs at the bottom, so that only a fraction of the subgrade depth had to be modelled, while for the latter, the full subgrade depth was modelled with a ﬁxed bottom boundary. The results of four proportionality coefﬁcients from Equation (1) are summarized in Table2.

Table 2. Optimum values for the static case in combined optimizations.

hs = {6,25,50} m hs = {3,6,9} m

Parameter hb = 0.3 m Parameter hb = 0.3 m Parameter hb = 0.3 m

fK,b [m] 1.911 1.483 fK,b [m] 1.956 1.349 fK,s,6 [m] 0.215 0.221 fK,s,3 [m] 0.329 0.346 fK,s,25 [m] 0.151 0.137 fK,s,6 [m] 0.209 0.225 fK,s,50 [m] 0.151 0.140 fK,s,9 [m] 0.173 0.183 fK,w,b [m] 0.250 0.722 fK,w,b [m] 0.109 0.499 fK,w,s,6 [m] 3.674 3.751 fK,w,s,3 [m] 2.807 3.856 fK,w,s,25 [m] 4.920 6.398 fK,w,s,6 [m] 4.175 5.202 fK,w,s,50 [m] 5.033 6.906 fK,w,s,9 [m] 5.078 6.872 Φ 2.4%–5.7% 3.4%–5.5% Φ 1.7%–4.9% 1.7%–6.5% Vibration 2021, 4 160

Some qualitative observations are listed below:

• fK,b decreases as hb increases, but not as much as for a purely inversely proportional relation, which supports the stress cone theory; • fK,s does not change signiﬁcantly with hb, which suggests that superposition of the stress cones occurs at a relatively shallow depth (see Figure2); • fK,s decreases when hs increases, but is stays nearly constant from 25 to 50 m, which suggests that the relation between K f and hs is asymptotic in nature; • fK,w,b increases with hb at a rate higher than direct proportionality; • fK,w,s increases with hs, particularly in the shallower range, but not as much from 25 to 50, again suggesting an asymptotic relation; • fK,w,s also increases with hb, in some cases very signiﬁcantly, while in others not so much; • fK,b and fK,s,6 are very similar for the two different optimization sets; • Despite differences in fK,w,b and fK,w,s,6 between the two sets, the optimum value of Kw is similar.

The differences observed between the two sets for hs = 6 m are because for the second set, the complete subgrade depth was modelled instead of using elastic BCs at the bottom, which introduces some differences in the results. The impact on Φ, however, is not signiﬁcant.

4.1.3. Damping and Mass Parameters (Dynamic Solution), Individual Optimization

Admitting Kb, Kw and K f obtained for the static case, the remaining parameters according to Figure1 are: Cb, Cw, Cf and Mb. They are optimized for the dynamic case with a load velocity v = {50, 100} m/s. For simplicity, only a subgrade depth of hs = 6 m is assumed, using the viscoelastic BCs discussed in the previous section. The following observations were made for the individual optimizations: • For v = 50 m/s, Φ ranges from 8% to 13%, while for v = 100 m/s, from 8% to 18%; • The worst results for v = 100 m/s occur when Es = 50 MPa. When these combinations are omitted, the maximum value of Φ is 13%, the same that was observed for v = 50 m/s; • The effect of Cb and Cw on Φ is negligible, and the optimum values are close to zero; • Cf has the most effect on Φ, and its optimum value increases with Es, but not with Eb; • Mb shows significant variation in its optimum value (2.3–5.0 tons). The average value is higher for hb = 0.6 m than for hb = 0.3 m, in agreement with [3]; • The effect of Mb on Φ is noticeable, but relatively small in comparison with Cf . Using its average value for all combinations of Eb and Es does not significantly affect the solution. The higher value of Φ for the soft subgrade when v = 100 m/s is because in this case, the velocity is supercritical, as already mentioned. As the load velocity gets closer to the critical velocity, the displacements are significantly amplified, especially in the upward direction. If the load velocity is higher than the critical one, then there is a wider region affected by the displacements and the maximum downward value shifts back gradually from the point of load application. If the two models being compared have different critical velocities, then this limits the validity of the comparison. Obviously, the wave propagation in DSM is somehow limited leading to differences between the two models that cannot be removed by improved optimization. In addition, the omission of horizontal displacements prevents the development of Rayleigh waves in the DSM. Given the results above, the dampers Cb and Cw were set to zero for combined optimizations. It should be noted that this would not be the case if material damping was used. Since only radiation/geometric damping is considered, the following relation was adapted from [73]: p 3.4 Gsρs Cf = fC,rad,s (2) π(1 − νs) Vibration 2021, 4 161

where fC,rad,s has unit of square meters. As for the mass, Mb, although it showed variation with the properties of the foundation, it was preliminarily assumed that the oscillating mass is proportional to the mass density of the ballast, as suggested in [3]: Mb = fM,bρb where fM,b has a unit of cubic meters.

4.1.4. Damping and Mass Parameters (Dynamic Solution), Combined Optimization The dynamic solutions for v = 50 m/s and v = 100 m/s are optimized for all possible combinations of Eb and Es simultaneously, but hb is kept ﬁxed and only hs = 6 m is assumed. The results are summarized in Table3, including the resulting value of Mb.

Table 3. Optimum values for the dynamic case in combined optimizations.

v = 50 m/s v = 100 m/s v = 50 m/s v = 100 m/s v = 50 m/s v = 100 m/s v = 50 m/s v = 100 m/s

fC,rad,s [m] 0.387 0.418 fC,rad,s [m] 0.353 0.436 3 3 fM,b [m ] 1.876 2.521 fM,b [m ] 1.411 1.734 Mb [t] 3.283 4.412 Mb [t] 2.469 3.035 Φ 4.6%–8.2% 5.2%–11.5% Φ 7.0%–18.5% 8.5%–19.0%

Some qualitative observations:

• fC,rad,s increases with hb in a similar way to fK,s. This suggests that the area of the base of the stress cone inﬂuences the rate of radiation damping, which agrees with [73]; 3 • fM increases by 0.645 and 0.323 m with the increase in hb for v = 50 m/s and v = 100 m/s, respectively; • fC,rad,s for v = 100 m/s is slightly lower than for v = 50 m/s for hb = 0.3 m, but slightly higher for hb = 0.6 m. These differences are likely due to the stochasticity of the optimization method; • fM for v = 100 m/s is 25%–31% lower than for v = 50 m/s.

Due to the high values of Mb obtained by the optimization, it is clear that the dependence on foundation properties is important and it is thus necessary to increase the oscillating mass by contribution from the foundation, as preliminarily indicated in the previous section: M = Mb + Ms (3)

where Ms is the dynamically activated foundation mass. Then, M will be used in place of Mb in the DSM description from Figure1. This means that the assumptions made in [ 3] are qualitatively correct, but for the shear spring (Kw) as well as for the dynamically activated mass (M), besides the ballast contribution, foundation contribution must also be added, which is the main new contribution of this paper. In general, it can be concluded that the DSM provides a very good approximation to the 3D FE model, when the velocity of the load is not too close to the critical. Nevertheless, the approximation for velocities close to the critical can still be considered as reasonable, as can be seen in Figure6.

4.2. Identiﬁcation of Dependencies on Geometrical and Other Properties 4.2.1. Ballast Vertical Stiffness The stress cone model assumes that stresses are transmitted from the effective load- bearing area under the sleeper and then dissipate at a constant angle, αb. This assumption approximates the static stress distribution of the 3D FE model, as shown in Figure7. Vibration 2020, 3 FOR PEER REVIEW 2

MMM=+bs (3)

where Ms is the dynamically activated foundation mass. Then, M will be used in place Vibration 2020, 3 FOR PEER REVIEW 2 of Mb in the DSM description from Figure 1. This means that the assumptions made in

[3] are Duequalitatively to the high correct, values ofbut M forb obtainedthe shear by spring the optimization, ( Kw ) as well it is as clear for thatthe dynamicallythe de- activatedpendence mass on foundation ( M ), besides properties the ballast is important contribution and it ,is foundation thus necessary contribution to increase must the also beoscillating added, which mass is by the contribution main new from contribution the foundation, of this aspaper. preliminar ily indicated in the previousIn general, section: it can be concluded that the DSM provides a very good approximation to the 3D FE model, when the velocity of the load is not too close to the critical. Nevertheless, MMM=+bs (3) the approximation for velocities close to the critical can still be considered as reasonable,

as wherecan be Mseens is in the Figure dynamically 6. activated foundation mass. Then, M will be used in place

of Mb in the DSM description from Figure 1. This means that the assumptions made in

[3] are qualitatively correct, but for the shear spring ( Kw ) as well as for the dynamically activated mass ( M ), besides the ballast contribution, foundation contribution must also be added, which is the main new contribution of this paper. In general, it can be concluded that the DSM provides a very good approximation to the 3D FE model, when the velocity of the load is not too close to the critical. Nevertheless, Vibration 2021, 4 162 the approximation for velocities close to the critical can still be considered as reasonable, as can be seen in Figure 6.

Figure 6. Normalized vertical displacement of the rail for the 3D FE model and the DSM for

v = 100m/s , hb = 0.6m , hs = 6m and different combinations of Eb and Es ; xF is the point of load application.

4.2. Identification of Dependencies on Geometrical and Other Properties

4.2.1. Ballast Vertical Stiffness FigureFigureThe 6.6. stress NormalizedNormalized cone verticalmodel vertical displacement assumes displacement that of ofthe stresses the rail rail for fortheare the3D transmitted FE 3D model FE model and from the and DSM the the effectivefor DSM for load-

bearingvv==100100m/s aream/s,, underhhb b== 0.60.6m them,, hsleepershs==66mm and and different then different dissipate combinations combinations at a constant of ofE bEandb and Eangle,s ; xEFs ;is x theF bis. point Thisthe point of assumption load of approximatesapplication.load application. the static stress distribution of the 3D FE model, as shown in Figure 7. Vibration 2020, 3 FOR PEER REVIEW 2 4.2. Identification of Dependencies on Geometrical and Other Properties 4.2.1. Ballast Vertical Stiffness effective load-bearing area of the sleeper. According to [45], the effective length of the sleeper loadedThe stress under cone each model rail assumes is given that by: stresses are transmitted from the effective load- bearing area under the sleeper and then dissipate at a constant angle,  b . This assumption =− approximates the static lstresse l sleep distribution l g of the 3D FE model, as shown in Figure 7. (4)

l where lsleep is the length of the sleeper and g is the track gauge. The total depth of the ballast is h , while h and h are the height at which adjacent FigureFigure 7. 7. StaticStatic vertical stress stress in in the the 3D 3D FEb modelFE model of thex of railway the railway track.z Blue:track. maximum Blue: max negativeimum stressnegative stress cones overlap in the longitudinal and transversal directions, respectively: stressin the in ballast; the ballast; red: 0% red: to 10% 0% ofto the10% maximum of the max negativeimum stress; negative grey: stress; values grey: out of values range; outhb = of0.6 range;m. . In [3], the superposition of the stress cones in the longitudinal direction is considered. = − = − Here, thehx superpositionmin(( l s l b of) the( 2 tan cones b) in , h both b) , h directions x h b h x (i.e., also in the transversal direction

betweenIn [3] rails,, the superposition as shown in Figure of the8) stress is taken cone intos in account. the longitudinal Figure9 also direction illustrates is considered. the (5) h=min l − l 2 tan , h , h = h − h Here,effectiveFigure the 7.z superposition load-bearingStatic vertical(( g stress area e) of( in ofthe the the cones 3D sleeper. b )FE inmodel b) both According z of thedirections b railway z to [ 45track. (i.e.,], the Blue: also effective max in theimum length transversal negative of the direc- tionsleeperstress between in loaded the ballast; rails, under red:as eachshown 0% to rail 10% in is Figureof given the max by: 8)imum is taken negative into stress; account. grey: Figure values out 9 also of range; illustrates the where ls is. the spacing between sleepers. Thus, the dimensions of the cone-base are: le = lsleep − lg (4) lx=+min( l b 2 h x tan b , l s ) In [3], the superposition of the stress cone s in the longitudinal direction is considered. (6) whereHere, lthesleep superpositionis the length ofof thethe sleepercones in and bothlg isdirections the track (i.e., gauge. also in the transversal direc- lz= l e +tan b( h b + h z ) tion between rails, as shown in Figure 8) is taken into account. Figure 9 also illustrates the

Figure 8. Superposition of the stress cones in the transversal direction. Figure 8. Superposition of the stress cones in the transversal direction.

Figure 9. Geometry of the stress cone with superposition in both directions.

The vertical stiffness of the ballast is the inverse of the integral of the virtual strain over its depth due to a vertical load at the top. Due to superposition, it is necessary to combine three different sections:

fK, b=1 ( f b ,1 + f b ,2 + f b ,3 ) , Kb= E b( f b,1 + f b ,2 + f b ,3 ) (7)

 lb l e+ 2 tan b min( h x , h z ) =− fb,1 ln 2( l b l e) tan b (8a) l l+ 2 tan min h , h e b b( x z )

Vibration 2020, 3 FOR PEER REVIEW 2

effective load-bearing area of the sleeper. According to [45], the effective length of the sleeper loaded under each rail is given by:

le=− l sleep l g (4)

where lsleep is the length of the sleeper and lg is the track gauge.

The total depth of the ballast is hb , while hx and hz are the height at which adjacent stress cones overlap in the longitudinal and transversal directions, respectively:

hx=min(( l s − l b) ( 2 tan b) , h b) , h x = h b − h x (5) = − = − hzmin(( l g l e) ( 2 tan b) , h b) , h z h b h z

where ls is the spacing between sleepers. Thus, the dimensions of the cone-base are:

lx=+min( l b 2 h x tan b , l s ) (6) lz= l e +tan b( h b + h z )

Vibration 2021, 4 163

Figure 8. Superposition of the stress cones in the transversal direction.

FigureFigure 9.9.Geometry Geometry ofof thethe stressstresscone conewith with superposition superposition in in both both directions. directions.

TheThe totalvertical depth stiffness of the of ballast the ballast is hb, whileis the hinversex and h ofz are the the integral height of at the which virtual adjacent strain stressover its cones depth overlap due to in thea ve longitudinalrtical load at and the transversal top. Due to directions, superposition, respectively: it is necessary to combine three different sections: h = min((l − l )/(2 tan α ), h ), h = h − h x s b b b x b x (5) fK, b=1 ( f b ,1 + f b ,2 + f b ,3 ), Kb= E b( f b,1 + f b ,2 + f b ,3 ) (7) hz = min lg − le /(2 tan αb), hb , hz = hb − hz

where ls is the spacing between sleepers. Thus, the dimensions of the cone-base are: lb l e+ 2 tan b min( h x , h z ) =− fb,1 ln 2( l b l e) tan b (8a) l l+ 2 tan minlx h= , hmin(lb + 2hx tan αb, ls) e b b( x z ) (6) lz = le + tan αb(hb + hz)

fK,b = 1/( fb,1 + fb,2 + fb,3), Kb = Eb/( fb,1 + fb,2 + fb,3) (7) lb[le + 2 tan αbmin(hx, hz)] fb,1 = ln /[2(lb − le) tan αb] (8a) le[lb + 2 tan αbmin(hx, hz)]

" 1 1 #  ( l +hx tan α ){le+tan α [hz+ (lg−le)]} ln 2 b b b 2  1 l +h l + h + 1 l −l  ( 2 b z tan αb){ e tan αb[ x 2 ( g e)]} = , hz < hx fb,2 tan α [(2le−l +(lg−le) tan α )] (8b)  b b b  le+2hz tan αb  ln /(2ls tan α ), hz ≥ hx le+2hx tan αb b

 h 1 i   le + tan αb hb + 2 lg − le  fb,3 = ln /(ls tan αb) (8c) h 1 i  le + tan αb max(hx, hz) + 2 lg − le 

4.2.2. Ballast Shear Stiffness

The shear stiffness of the ballast, here designated as Kw,b does not come directly from the stress cone assumption, and [3] does not provide a way to estimate it. It is proposed here to use classical expressions from beam theory for shear and bending stiffness, adapted to a representative ballast area. Given that the vertical stiffness of the ballast is calculated based on the stress cone distribution, it is reasonable to assume that the representative ballast area, Ab, is the cross-sectional area of the stress cone, as depicted in Figure 10:

2 2 Ab = hb + 2hbhz − hz tan αb/2 + lehb (9) Vibration 2020, 3 FOR PEER REVIEW 2

 11l+ htan l + tan  h + l − l  ( 22b x b) e b z( g e )  ln  ( 11l+ htan) l + tan  h + l − l  22b z b e b x( g e )   , hh f =  zx (8b) b,2 tan 2l− l + l − l tan  b( e b( g e) b )    lhe+ 2 z tan b ln( 2l tan ) , h h  +  s b z x  lhe2 x tan b

l+tan  h +1 l − l e b b2 ( g e ) flb,3 = ln( s tan b ) (8c) l+tan  max h , h +1 l − l e b( x z) 2 ( g e )

4.2.2. Ballast Shear Stiffness

The shear stiffness of the ballast, here designated as Kwb, does not come directly from the stress cone assumption, and [3] does not provide a way to estimate it. It is proposed here to use classical expressions from beam theory for shear and bending stiffness, adapted to a representative ballast area. Given that the vertical stiffness of the ballast is calculated based on the stress cone distribution, it is reasonable to assume that the repre-

sentative ballast area, A b , is the cross-sectional area of the stress cone, as depicted in Vibration 2021, 4 Figure 10: 164

22 Ab=( h b +2 h b h z − h z) tan b 2 + l e h b (9)

Figure 10. Cross-section of the stress cone in the transversal direction. Figure 10. Cross-section of the stress cone in the transversal direction. The shear stiffness Kw,b is then equal to: The shear stiffness is then equal to: − ∗ −1 −1! 1 3 2 2 G A 12−1 E I 5 h l + 4lelz + l −−11b b b b ∗ 3 2 2 b e z Kw,b = *  +  , Ahb =ll++A4b,llIb = (10) GAEIb b12 b b 3 * 5 b ( e e z z ) + KA=  +ls   ,lsAI =, = 6 36(le lz) (10) wb, l  l3  b6 b b 36 ll+ s  s   ( e z ) 4.2.3. Subgrade Stiffness In [3], it is assumed that the reaction modulus of the subgrade is known from ex- 4.2.3. Subgrade Stiffness perimental measurements, and therefore the vertical stiffness of the subgrade, K f , can be calculatedIn [3], by it is multiplying assumed that this the value reaction by the modulus area of of the the base subgrade of stress is coneknown of from the ballast. exper-

Inimental this paper, measurements a general expression, and therefore will the be proposed vertical stiffness based on of the the Vlasov subgrade, model K [f74, can] that be prescribescalculated evolution by multiplying of the verticalthis value displacement by the area alongof the thebase active of stres depths cone of of the the soil, ballast.hs: In this paper, a general expression will be proposed based on the Vlasov model [74] that sinh(γ(hs − y)) prescribes evolution of the verticalf (y displacement) = along the active depth of the soil, h(11): sinh(γhs) s

where γ is a parameter in m−1 that describes the rate at which the vertical displacement in the foundation decreases with depth. When γ = 0, the displacement decreases linearly with depth and the vertical stress in the soil is constant. Both the subgrade vertical reaction modulus, Ks, and the shear reaction modulus, Ks,P, are calculated by minimizing the total strain energy due to a uniform surface load ([74]):

hs 2 R oed d f oed sinh(γhs) cosh(γhs)+γhs Ks = Es 2 dy = Es γ 2 dy 2sinh (γhs) 0 (12) hs R 2 sinh(γhs) cosh(γhs)−γhs K = Gs f (y)dy = Gs s,P 2γsinh2(γh ) 0 s

These values, when multiplied by the area of the base of stress cone of the ballast (A f = lxlz), give the subgrade stiffness, K f , and the discrete rotational stiffness due to shear, which can then be used to calculate the shear stiffness of the subgrade, Kw,s:

K f = Ks A f 2 (13) Kw,s = Ks,P A f /ls

Figure 11 shows the variation of Ks and Ks,P with depth for different values of γ. Except for γ = 0, both values become nearly constant after a certain depth of the model, which is lower when γ is higher. This is in agreement with the results of the 3D FE model, for which the static displacement for hs = 25 m and hs = 50 m was very similar, as were the numerical optimal values of the DSM (Table2). Applying the principles above to a substructure with two distinct layers shows that, as long as the bottom layer is signiﬁcantly thicker than the top layer, the total vertical reaction modulus can be approximated as a series of springs, while the shear reaction modulus can be approximated as parallel rotation springs. This is consistent with the fact that the shear stiffness is proportional with the Vibration 2020, 3 FOR PEER REVIEW 2

sinh( (hys − )) fy( ) = (11) sinh( hs )

where  is a parameter in m-1 that describes the rate at which the vertical displacement in the foundation decreases with depth. When  = 0 , the displacement decreases linearly with depth and the vertical stress in the soil is constant. Both the subgrade vertical reaction

modulus, K s ,, and the shear reaction modulus, KsP, , are calculated by minimizing the total strain energy due to a uniform surface load ([74]):

h s df2 sinh(h) cosh(  h) +  h K== Eoed dy E oed s s s s s22 s 0 dy2sinh ( hs ) h (12) s sinh(h) cosh(  h) −  h K== G f2 y dy G s s s s, P s( ) s 2 0 2 sinh ( hs )

These values, when multiplied by the area of the base of stress cone of the ballast (

Af= l x l z ), give the subgrade stiffness, K f , and the discrete rotational stiffness due to shear, which can then be used to calculate the shear stiffness of the subgrade, K : Vibration 2021, 4 ws, 165 KKA= f s f 2 (13) Kw,, s= K s P A f l s layer’s depth, and agrees with the numerical ﬁnding that the shear stiffness element of the Figure 11 shows the variation of and with depth for different values of . DSM, Kw, can be approximated as a linear combination of Gb and Gb.

(a)

(b)

Figure 11. (a) Ks and (b) K as a function of hs for different values of γ.  Figure 11. (a) Ks and (b) s,KP sP, as a function of hs for different values of . 4.2.4. Dynamically Activated Mass Except for  = 0 , both values become nearly constant after a certain depth of the In [3], the ballast mass is calculated from the volume of the stress cone, which is here adaptedmodel, which to consider is lower superposition when is in higher. both directions: This is in agreement with the results of the 3D

FE model, for which the static displacement for hs = 25m and hs = 50m was very simi- = + + = ( + + ) lar, as were thefM ,bnumericalfM,b,1 optimalfM,b,2 valuesfM,b,3, ofM theb DSMfM,b ,1(TablefM ,2b,2). ApplyingfM,b,3 ρ bthe principles(14) 4 f = tan2 α min(h , h )3 + (l + l ) tan α min(h , h )2 + l l min(h , h ) (15a) M,b,1 3 b x z b e b x z b e x z  (h − h )l l + 2 tan2 α h2 + h h + h2 + 3l − l (h + h )  x z b e 3 b x x z z g e x z 1 fM,b,2 = + 2 tan αb lb lg − le + hx + hz + 2le(hx + hz) , hz < hx (15b)  ls(hz − hx)(le + tan αb(hx + hz)), hz ≥ hx 1 f = l (h − max(h , h ))2l + tan α l − l + h + max(h , h ) (15c) M,b,3 2 s b x z e b g e b x z The other contribution speciﬁed in Equation (3) is based on [75] where vibrating mass of the soil under foundation is analyzed. Using these formulas and taking into account that the only half the track is being modelled, the following deﬁnition of the subgrade mass is proposed: 3/2 2ρs 2lxlz Ms = (16) 1 − νs π

4.2.5. Subgrade Damping For the foundation damping, the formula proposed in [73] speciﬁes:

fC,rad,s = cZ A f (17) Vibration 2021, 4 166

where A f is the area of the foundation, in this case, the area of the base of the stress cone, and cz is the rate of absorption, when cz = 1, in theory, full absorption of incident waves occurs.

5. Fitting the Decisive Parameters to the Numerical Results The expressions proposed in Equations (1), (3), (7), (8), (10), (12)–(17) define all the parameters of the DSM from the known properties of the track, with the exception of the angle of stress distribution in the ballast, αb, the rate of displacement decrease with depth in the subgrade, γ, and the rate of absorption due to radiation damping, cz. To determine the values for these parameters, the expressions above are fitted to the numerical results obtained in the previous sections. Since there are only a few discrete points to fit, a simple norm of the relative difference between the theoretical and numerical results is used. The first parameter to optimize is αb, which must fit both ballast depths, hb = 0.3 m ◦ and hb = 0.6 m. The resulting optimum value for the first set in Table2 is αb = 49.8 , and ◦ produces an error of ±9%, while for the second set in Table2 it is αb = 47.6 with an error of ±4%. The values are shown in Table4.

Table 4. Optimized value of αb, fK,b [m] and respective error, optimum values from Table2.

◦ ◦ h Opt. Val. αb = 49.8 Opt. Val. αb = 47.6 b Error Error [m] fK,b fK,b fK,b fK,b 0.3 1.911 2.086 +9% 1.956 2.029 +4% 0.6 1.483 1.344 −9% 1.349 1.302 −4%

Using these values of αb, it is possible to deﬁne the area of the base of the stress cone and proceed to the optimization of γ and fK,s. The optimum value for the ﬁrst set in Table2 −1 is γ = 0.268 m , with an error of fK,s from −16% to +18%, and for the second set in Table2 −1 it is γ = 0.324 m , with an error of fK,s from −11% to +15%. Then, it is possible to use the above determined values of αb and γ to compute the shear stiffness of the ballast and the subgrade. Table5 shows a summary of these results.

Table 5. Value of fK,w,b [m] and respective error, optimum values from Table2.

◦ ◦ h Opt. Val. αb = 49.8 Opt. Val. αb = 47.6 b Error Error [m] fK,w,b fK,w,b fK,w,b fK,w,b 0.3 0.250 0.226 −9% 0.109 0.222 +104% 0.6 0.722 0.949 +31% 0.499 0.930 +86%

It is seen that the error is quite high, nevertheless, this is not very important, because the main shear contribution comes from the foundation. For the subgrade shear stiffness, using the optimized values of αb and γ, the error of fK,w,s ranges from −9% to +34% for the ﬁrst set in Table2 and from −29% to −14% for the second set. Both fK,s and fK,w,s are plotted as a function of hs in Figure 12, for a rough average of the optimum values of γ = 0.3 m−1. It can be concluded that the approximation is quite reasonable, especially taking into account that the optimum values of γ were obtained only from optimizing of fK,s. The value of Φ obtained for the static solution using these expressions ranges from 3.3% to 14.8% for the ﬁrst optimization set in Table2 and from 3.3% to 13.8% for the second set. Vertical displacements for the representative load of 40 kN are compared in Figure 13. It can be concluded that the agreement between DSM with the proposed expressions and the 3D FE model is excellent. Vibration 2020, 3 FOR PEER REVIEW 2

produces an error of 9% , while for the second set in Table 2 it is b = 47.6 ° with an error of 4% . The values are shown in Table 4.

Table 4. Optimized value of b ,, fKb, [m] and respective error, optimum values from Table 2.

hb Opt. val. b = 49.8 ° Opt. val. ° Error Error m  fKb, 0.3 1.911 2.086 +9% 1.956 2.029 +4% 0.6 1.483 1.344 −9% 1.349 1.302 -4%

Using these values of  b , it is possible to define the area of the base of the stress cone

and proceed to the optimization of  and fKs, . The optimum value for the first set in Table 2 is  = 0. 268m−1 , with an error of from -16% to +18%, and for the second set in Table 2 it is  = 0. 324m−1 , with an error of from -11% to +15%. Then, it is possible to use the above determined values of and to compute the shear stiffness of the ballast and the subgrade. Table 5 shows a summary of these results.

Table 5. Value of fK,, w b [m] and respective error, optimum values from Table 2.

Opt. val. ° Opt. val. ° Error Error fK,, w b 0.3 0.250 0.226 −9% 0.109 0.222 +104% 0.6 0.722 0.949 +31% 0.499 0.930 +86%

It is seen that the error is quite high, nevertheless, this is not very important, because the main shear contribution comes from the foundation. For the subgrade shear stiffness,

using the optimized values of and , the error of fK,, w s ranges from -9% to +34% for the first set in Table 2 and from -29% to -14% for the second set. Both and f are Vibration 2021, 4 K,, w s 167

plotted as a function of hs in Figure 12, for a rough average of the optimum values of  = 0. 3m−1 .

Vibration 2020, 3 FOR PEER REVIEW 2

set. Vertical displacements for the representative load of 40kN are compared in Figure 13. (a) (b) It can be concluded that the agreement between DSM with the proposed expressions and = FigureFigure 12.12.Optimum Optimum values values fromthe from 3D Table Table FE2 model(crosses) 2 (crosses) is and excellent. and proposed proposed expression expression (blue (blue line) whenline) whenhb 0.3hb m= :(0.3ma) fK: ,(sa;() b)fKsf, K;, w(,bs.)

fK,, w s .

It can be concluded that the approximation is quite reasonable, especially taking into account that the optimum values of were obtained only from optimizing of . The value of  obtained for the static solution using these expressions ranges from 3.3% to 14.8% for the first optimization set in Table 2 and from 3.3% to 13.8% for the second

= FigureFigure 13. 13.Vertical Vertical displacement displacement of of the the rail rail for for the the 3D 3D FE FE model model and and the the DSM DSM for hfobr= h0.6b m0.6m, hs =, 6 m

andhs = different6m and combinations different combinations of Eb and Eofs. Eb and Es .

Further,Further, thethe samesame comparisoncomparison mustmust bebe carriedcarriedout out for for the the dynamic dynamic case. case. UsingUsingthe the optimum value of α the error in the proposed expression can be calculated. The results optimum value of b the error in the proposed expression can be calculated. The results are summarized in Tableb 6. are summarized in Table 6. Table 6. Values of the mass and respective error, optimum values from Table3. Table 6. Values of the mass and respective error, optimum values from Table 3. Opt. Val. ◦ Opt. Val. ◦ h αb = 49.8 αb = 49.8 h b Opt.v = val. 50 m/s Error vOpt.= 100 val. m/s Error b[m] b = 49.8 ° ° v = 50m/sMMMM Error v = 100m/s Error m 0.3M 3.283 3.273 −3% 2.469 3.273 +33% 0.30.6 3.283 4.412 4.9003.273 +11%-3% 2.469 3.035 4.9003.273 +61%+33% 0.6 4.412 4.900 +11% 3.035 4.900 +61% It is clear that the proposed equations provide a good approximation for the lower velocity,It is but clear not that for the the proposed high velocity, equations for which provide the a optimal good approximation mass is 20% to for 30% the lower, lower likelyvelocity, because but not the for latter the ishigh close velocity, to the for critical which velocity the optimal of the mass track is for 20% a soft to 30% subgrade. lower, Sincelikelythe because proposed the latter equations is close do to the not critical account velocity for this of effect,the track they for work a soft wellsubgrade. for low Since to moderatethe proposed velocities, equations but do not not for account high velocities, for this effect, for which they work the dynamic well for ampliﬁcationlow to moderate is morevelocities, pronounced. but not for high velocities, for which the dynamic amplification is more pronounced.As for the radiation damping, the numerical optimal values in Table3 are used to deduceAs the for optimum the radiation value damping, of cz. The resultsthe numerical are summarized optimal invalues Table 7in. ItTable can be3 concludedare used to

thatdeducecz = the0.4 optimumis a good value recommendation of cz . The results for railway are summarized models with in Table properties 7. It can inside be con- the range studied here. cluded that cz = 0.4 is a good recommendation for railway models with properties inside the range studied here.

Table 7. Optimum values of cz , based on Table 3.

 hb m 0.3 0.393 0.358 0.6 0.347 0.361

The value of Φ obtained for the steady-state dynamic solution using these expressions ranges from 8.1% to 13.1% for and from 8.3% to 43.6% for .

The higher errors are always observed for the soft subgrade ( Es = 50MPa ), particularly for , as shown in Figure 14, due to the aforementioned dynamic amplification factor due to the critical velocity.

Vibration 2021, 4 168

Table 7. Optimum values of cz, based on Table3.

hb[m] v = 50 m/s v = 100 m/s 0.3 0.393 0.358 0.6 0.347 0.361

The value of Φ obtained for the steady-state dynamic solution using these expressions ranges from 8.1% to 13.1% for v = 50 m/s and from 8.3% to 43.6% for v = 100 m/s. The higher errors are always observed for the soft subgrade (Es = 50 MPa), particularly for Vibration 2020, 3 FOR PEER REVIEW v = 100 m/s, as shown in Figure 14, due to the aforementioned dynamic ampliﬁcation2 Vibration 2020, 3 FOR PEER REVIEW factor due to the critical velocity. 2

(a) (b) (a) (b) Figure 14. Normalized vertical displacement of the rail for the 3D FE model and the DSM, v = 100m/s ; hb = 0.3m (a) Figure 14. Normalized vertical displacement of the rail for the 3D FE model and the DSM,v = v = 100m/sh ;= h = 0.3ma (a) stiff ballastNormalized and subgrade; vertical (b) soft displacement ballast and of subgrade the rail for ( x theF is 3D the FE point model of load and theapplication DSM, ). 100 m/s; b 0.3b m ( ) stiff ballast and subgrade; (b) soft ballast and subgrade (x is the point of load application). stiff ballast and subgrade; (b) soft ballast and subgradeF ( xF is the point of load application). It is of note that the maximum upward and downward displacements for the worst It is of note that the maximum upward and downward displacements for the worst case Itare is relativelyof note that close the tomaximum the ones upward obtained and on downwardthe 3D FE displacementsmodel, so the DSMfor the and worst the case are relatively close to the ones obtained on the 3D FE model, so the DSM and the caseproposed are relatively expressions close provide to the aones reasonable obtained approximation on the 3D FE to model, the rail so displacement the DSM and even the proposed expressions provide a reasonable approximation to the rail displacement even proposedwhen the loadexpressions is moving provide with thea reasonable velocity is approximation close to the velocity to the of rail propagation displacement of elastic even when the load is moving with the velocity is close to the velocity of propagation of elastic whenwaves the in theload soil. is moving with the velocity is close to the velocity of propagation of elastic waves in the soil. waveFinallys in the, thesoil. results of the DSM are compared with the experimental results published Finally, the results of the DSM are compared with the experimental results−1 published in [46Finally], as was, the the results case forof the the DSM 3D FE are model, compared using with  the=◦ 50experimental°,  = 0.3m−1 results and published c = 0.4 . in [46], as was the case for the 3D FE model, using αb = b50 , γ = 0.3 m and cz = 0.4z . The  =  = −1 = inverticalThe [4 6vertical], as displacement was displacement the case obtained for obtainedthe 3D in theFE in DSMmodel, the DSM due using to due the to passageb the50 passage°, of the 0.3mof full the train full and and train c az singleand0.4 a. Thesinglebogie vertical atbogie 220 displacement km/hat 220 iskm/h presented is obtained presented in Figure in in the Figure 15 DSM. 1 due5. to the passage of the full train and a single bogie at 220 km/h is presented in Figure 15.

(a) (b) (a) (b) Figure 15.15. VerticalVertical displacementdisplacement ofof thethe rail:rail: comparisoncomparison between experimental measurements, the 3D FE model and the FigureDSM using 15. Vertical the mechanistic displacement expressions;expressions; of the rail: ((aa)) fullfullcomparison vehicle;vehicle; ( (bbetweenb)) ﬁrstfirst bogie.bogie. experimental measurements, the 3D FE model and the DSM using the mechanistic expressions; (a) full vehicle; (b) first bogie. It can be concluded that the results of the DSM are very close to the ones of the 3D FE modelIt can andbe concluded provide a thatgood t heapproximation results of the of DSM the experimentalare very close results. to the onesIn fact, of thethe 3Dre- FEsults model are in and better provide agreement a good than approximation the ones of theof the 2D experimental model presented results. in [46], In fact, where the there- sultsmultibody are in vehiclebetter agreement model was than considered, the ones ofjustifying the 2D model the assumption presented in of [46],moving where forces the multibodyadopted here vehicle. model was considered, justifying the assumption of moving forces adopted here. 6. Comparison with Other Works 6. ComparisonAs most of with the literatureOther Works does not present formulas for establishing the model data, as a referenceAs most of for the comparison literature does with notother present works formulas the model for presentedestablishing in the[3] ismodel chosen data,. In as a reference for comparison with other works the model presented in [3] is chosen. In [3], values of Kb and Mb are calculated using b = 35 ° and subgrade stiffness is calcu-

[3], values of Kb and Mb are calculated using b = 35 ° and subgrade stiffness is calculated using the given subgrade modulus K s . Therefore, the comparison is performed in

lated using the given subgrade modulus K s . Therefore, the comparison is performed in

Vibration 2020, 3 FOR PEER REVIEW 2 Vibration 2021, 4 169 the following way: the data kept in both models are related to the rail: EI = 6.62MNm2 , m = 60.64kg/m ; to the sleepers: M = 125.5kg , l = 0.545m , l = 0.273m , l = 0.95m ; It can be concluded that the results ofsleep the DSM are verys close to the onesb of the 3D FEe model and provide a good approximation of the experimental results. In fact, the results are to the rail pads: Kpad = 65MN/m , Cpad = 75kNs/m and to the ballast: Eb = 110MPa , in better agreement than the ones of the 2D model presented in [46], where the multibody  = 3 hvehicleb = 0.45m model, wasb considered,1800kg/m justifying. Ballast the vertical assumption and shear of moving damping forces adoptedare neglected here. as in the final optimization steps. 6. Comparison with Other Works There is no indication for E in [3]. Thus, K = 90MPa/m given in [3] is used with As most of the literature does nots present formulas fors establishing the model data, as a reference for comparison-1 with other works the model presented in [3] is chosen. In [3], the optimized  = 0. 3m and hs = 6m , to give Es = 295.7MPa . Table 8 summarizes the values of K and M are calculated using α = 35◦ and subgrade stiffness is calculated differences bin both bmodels. b using the given subgrade modulus Ks. Therefore, the comparison is performed in the following way: the data kept in both models are related to the rail: EI = 6.62 MNm2, Table 8. Model data summary. m = 60.64 kg/m; to the sleepers: Msleep = 125.5 kg, ls = 0.545 m, lb = 0.273 m, le = 0.95 m; to the rail pads: Kpad = 65 MN/m, Cpad = 75 kNs/m and to the ballast: Eb = 110 MPa, Parameter 3 Formulas in this paper [3] hb = 0.45 m, ρb = 1800 kg/m . Ballast vertical and shear damping are neglected as in the ﬁnal optimizationKwb, [MN/m] steps. 27.3 78.4 * There is no indication for Es in [3]. Thus, Ks = 90 MPa/m given in [3] is used with Kws, [MN/m] −1 500.9 --- the optimized γ = 0.3 m and hs = 6 m, to give Es = 295.7 MPa. Table8 summarizes the

differencesKw in[MN/m] both models. 528.2

Table 8. ModelMb data [kg] summary. 595.7 531.4

MParameters [kg] Formulas in This3033.6 Paper [3] ---

MKw, b[kg][MN/m] 27.33629.3 78.4 * K [MN/m] 500.9 — K [MN/m]w,s b Kw [MN/m] 528.2 168.27 137.75 Mb [kg] 595.7 531.4 K f [MN/m] 88.8 77.5 Ms [kg] 3033.6 — M [kg] 3629.3 C f [kNs/m] 308 31.15 * Kb [MN/m] 168.27 137.75 * Means that Kthef [MN/m] value was used without justification. 88.8 77.5 Cf [kNs/m] 308 31.15 * * MeansIt is that necessary the value was to used highlight without justiﬁcation. that values for and M , given in Table 8 for the model

developedIt is necessary in this topaper, highlight are thatvery values high fordueKw toand theM ,contribution given in Table f8ro form the the model foundation, as justifieddeveloped by inoptimizations. this paper, are In very fact high, it would due to be the more contribution adequate from to theinclude foundation, them in as a separate foundationjustiﬁed by optimizations.layer. In fact, it would be more adequate to include them in a separate foundation layer. The representative force of 40kN is used for results comparison, as in previous sec- The representative force of 40 kN is used for results comparison, as in previous tionssections.. Static Static displacement displacement is is depicted depicted in in Figure Figure 16 .16.

Figure 16. Static vertical displacement of the rail. Comparison of the model proposed in this paper, Figurethe model 16. Static from [ 3v]ertical and the displacement 3D FE model of the rail. Comparison of the model proposed in this paper, the model from [3] and the 3D FE model

It is seen that the high shear stiffness of the model attenuates the upward displacements which are not present in the 3D FE model. The deflection shape agreement is better for the model presented in this paper. Further, the stabilized steady-state deflection shape is compared to the moving load in Figure 17.

Vibration 2021, 4 170

Vibration 2020, 3 FOR PEER REVIEW 2 It is seen that the high shear stiffness of the model attenuates the upward displacements which are not present in the 3D FE model. The deﬂection shape agreement is better for the model presented in this paper. Further, the stabilized steady-state deﬂection shape Vibration 2020, 3 FOR PEER REVIEW is compared to the moving load in Figure 17. 2

Figure 17. Steady-state vertical displacement of the rail: v = 100m/s . Comparison of the model proposed in this paper, the model from [3] and the 3D FE model.

FigureFinally, 17. Steady-state the maximum vertical displacement displacement of the values rail: v = in100 them/s absolute. Comparison value of the are model extracted as a Figure 17. Steady-state vertical displacement of the rail: v = 100m/s . Comparison of the model functionproposed inof this velocity paper, the. In model Figure from 18 [3], andthe the maximum 3D FE model. value over the structure is plotted to- proposedgether with in this the paper, value the in modelthe load from position. [3] and the The 3D v FEalues model are. normalized by static displace- Finally, the maximum displacement values in the absolute value are extracted as a ment. The typical fact can be observed that by getting closer to the critical velocity, the functionFinally, of velocity. the m Inaximum Figure 18 displacement, the maximum values value over in the structureabsolute is value plotted are together extracted as a maximumwith the value downward in the load displacement position. The values does are not normalized occur at bythe static load displacement. position. It The is confirmed function of velocity. In Figure 18, the maximum value over the structure is plotted to- thattypical the factincrease can be in observed oscillating that mass by getting as suggested closer to thein this critical paper velocity, makes the the maximum prediction of the gether with the value in the load position. The values are normalized by static displace- criticaldownward velocity displacement more reasonable, does not occur bearing at the in load mind position. the value It is conﬁrmed of E . This that concludes the the ment.increase The in typical oscillating fact mass can as be suggested observed in thisthat paper by getting makes the closer prediction to the ofs critical the critical velocity, the maximumcomparison.velocity more downward reasonable, displacement bearing in mind does the value not occur of Es. Thisat the concludes load position. the comparison. It is confirmed that the increase in oscillating mass as suggested in this paper makes the prediction of the

critical velocity more reasonable, bearing in mind the value of Es . This concludes the comparison.

Figure 18. Maximum downward normalized displacement and normalized displacement under the Figure 18. Maximum downward normalized displacement and normalized displacement under force as a function of velocity. Comparison of the model proposed in this paper and in [3]. the force as a function of velocity. Comparison of the model proposed in this paper and in [3]. It was shown that the new formulas led to much better agreement with the 3D

FE model,It was notshown only inthat regard the new to the formulas deﬂection led shape, to much but also better for the agreement prediction ofwith the the 3D FE Figuremodelcritical 18., not velocity. Maximum only in regarddownward to the normalized deflection displacement shape, but andalso normalized for the prediction displacement of the under critical the force as a function of velocity. Comparison of the model proposed in this paper and in [3]. velocity.7. Conclusions ItThe was present shown work that establishes the new the formulas validity andled theto much range ofbetter applicability agreement of the with DSM. the 3D FE 7. Conclusions modelIt proposes, not only general in regard formulas to for the identifying deflection all shape, DSM properties but also withoutfor the prediction the necessity of of the critical calling on experimental results. The formulas proposed are simple and straight-forward to velocity.The present work establishes the validity and the range of applicability of the DSM. Ituse proposes and avoid general the necessity formulas of numerical for identifying calibration. all DSM properties without the necessity of 7.calling Conclusions on experimental results. The formulas proposed are simple and straight-forward to use and avoid the necessity of numerical calibration. The present work establishes the validity and the range of applicability of the DSM. It was shown that the DSM shows a good approximation to the 3D FE model for the It proposes general formulas for identifying all DSM properties without the necessity of whole range of parameters considered—ballast and subgrade depth and the Young mod- calling on experimental results. The formulas proposed are simple and straight-forward uli. However, since the DSM does not model the elastic wave propagation in the soil, it is to use and avoid the necessity of numerical calibration. less reliable when the load moves at a speed close to the critical velocity. Good agreement It was shown that the DSM shows a good approximation to the 3D FE model for the was observed up to a speed of 75% of the Rayleigh wave velocity in the subgrade (8%– whole range of parameters considered—ballast and subgrade depth and the Young mod- 13% overall error), while at a speed very close to the Rayleigh waves, the error became uli. However, since the DSM does not model the elastic wave propagation in the soil, it is more significant (18%). less reliable when the load moves at a speed close to the critical velocity. Good agreement was observed up to a speed of 75% of the Rayleigh wave velocity in the subgrade (8%– 13% overall error), while at a speed very close to the Rayleigh waves, the error became more significant (18%).

Vibration 2021, 4 171

It was shown that the DSM shows a good approximation to the 3D FE model for the whole range of parameters considered—ballast and subgrade depth and the Young moduli. However, since the DSM does not model the elastic wave propagation in the soil, it is less reliable when the load moves at a speed close to the critical velocity. Good agreement was observed up to a speed of 75% of the Rayleigh wave velocity in the subgrade (8%–13% overall error), while at a speed very close to the Rayleigh waves, the error became more signiﬁcant (18%). All DSM components can be determined from basic mechanical and geometrical properties, except for three fundamental values, for which the following values were found to provide a good approximation: ◦ • Angle of stress distribution in the ballast: αb = 50 ; • Rate of displacement reduction with depth in the subgrade: γ = 0.3 m−1; • Rate of absorption for the radiation damping of the subgrade: cz = 0.4. Regarding the formulas available in the literature, it was concluded that the oscillating mass of the model cannot represent only the dynamically activated ballast mass, but the contribution from the foundation must be added. The same conclusion was taken regarding the shear stiffness of the model: the value determined for the ballast has to be superposed with the contribution coming from the foundation. The shear stiffness of the model is essential for achieving a good agreement. This means that reduced models that are focusing mainly on the vertical dynamic equilibrium and/or are neglecting the dynamically activated mass in the foundation, cannot represent the rail deﬂection shape accurately within the range of typical track properties even in the subcritical velocity range.

Author Contributions: This research article is based on Ph.D. thesis of the first author, A.F.S.R., su- pervised by the second author, Z.D. Conceptualization, Z.D. and A.F.S.R.; methodology, A.F.S.R. and Z.D.; software, A.F.S.R.; validation, A.F.S.R.; writing—original draft preparation, A.F.S.R.; writing— review and editing, Z.D.; supervision, Z.D. Both authors have read and agreed to the published version of the manuscript. Funding: This research was funded in the form of a PhD grant, SFRH/BD/75375/2010 and supported by project UIDB/50022/2020. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Not applicable. Acknowledgments: This work was supported by the Portuguese Foundation for Science and Tech- nology (FCT), through IDMEC, under LAETA, project UIDB/50022/2020. Conflicts of Interest: The authors declare no conflict of interest.

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