applied sciences
Article DEM Analysis of Track Ballast for Track Ballast–Wheel Interaction Simulation
Nam-Hyoung Lim 1 , Kyoung-Ju Kim 2, Hyun-Ung Bae 2 and Seungjun Kim 3,*
1 Department of Civil Engineering, Chungnam National University, Daejeon 34134, Korea 2 Road Kinematics Co., Ltd., Cheonan 31094, Korea 3 School of Civil, Environmental, and Architectural Engineering, Korea University, Seoul 02841, Korea * Correspondence: [email protected]; Tel.: +82-2-3290-4868
Received: 3 January 2020; Accepted: 13 April 2020; Published: 15 April 2020
Abstract: This study aims to suggest a rational analysis method for a track ballast–wheel interaction that could be further developed to model the interaction in a train-derailment event, based on the discrete-element method (DEM). Track ballast is filled with gravel to form the trackbed. Although finite-element analysis (FEA) is widely applied in structural analysis, track ballast cannot be analyzed using conventional FEA because this approach does not allow separation of elements that share nodes. The DEM has been developed to analyze the dynamic behavior of separable objects, assuming that the objects are rigid. Therefore, track ballast can be modeled as separable rigid pieces of gravel, and its dynamic behavior can be analyzed using a rational contact model. In this study, a rational numerical strategy for track ballast–wheel interaction was investigated using the DEM approach. The suggested analysis method was validated through comparison with the experimental results of a drop test. In addition, case studies were conducted to investigate the effects of the contact-model parameters on the simulation result.
Keywords: track ballast; gravel; DEM; explicit dynamics; contact model; drop test
1. Introduction Derailment of a rapidly moving train may lead to fatal consequences, thus such accidents should be prevented through effective means, such as derailment-prevention devices and structures. In Europe and in Japan, Korea, and several other countries, effective derailment-prevention methodologies have been studied and developed, such as derailment-containment walls [1–4]. In addition to the fundamental prevention concepts, these studies have suggested design methods and provisions for the walls. Recently, a new type of structural system for derailment-containment prevention (DCP), which is constructed between the rails, has been developed in Korea [4], as shown in Figure1. The concrete structural member is located between rails to restrict the horizontal movement of derailed train. To design the structural member and to validate its performance as a DCP system, the structural responses of the member under contact, and the impact forces induced by a derailed train, should be investigated by rational procedures. Through simulations and experiments, the performance of the newly proposed system has been verified, and an appropriate design procedure for this system has been suggested [1,4]. However, the prevention performance has been investigated only for concrete track ballast, even though in several cases, the track ballast can be constructed as a gravel track bed. This indicates that, apart from concrete ballast, the DCP systems should be additionally examined for train derailment over a gravel ballast.
Appl. Sci. 2020, 10, 2717; doi:10.3390/app10082717 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, x FOR PEER REVIEW 2 of 20
Unit: mm Derailed bogie
Appl. Sci. 2020, 10, 2717 Rail 2 of 19 Appl. Sci. 2020, 10, x FOR PEER REVIEW DCP 2 of 20
170 Unit: mm 500 Derailed bogie
Figure 1. Overview of the derailment-containment prevention (DCP)Rail wall developed in Korea [1]. DCP
Through simulations and experiments,170 the performance of the newly proposed system has been verified, and an appropriate design procedure for 500this system has been suggested [1,4]. However, the prevention performance has been investigated only for concrete track ballast, even though in several cases, the track ballast can be constructed as a gravel track bed. This indicates that, apart from concrete ballast, the DCP systems should be additionally examined for train derailment over a gravel FigureFigure 1. 1. OverviewOverview of of the the derailment-containment derailment-containment preven preventiontion (DCP) (DCP) wall wall developed developed in in Korea Korea [1]. [1]. ballast. ThroughAs mentioned simulations before, and in several experiments, cases, thethe perfor track manceballast ofcomprises the newly pieces proposed of gravel system of hasirregular been verified,shapes, which and an form appropriate the track design bed, asprocedure shown in for Figure this system 22.. WhenWhen has aa been traintrain suggested isis derailed,derailed, [1,4]. thethe However, wheelwheel ofof thethethe preventionderailed train performance will contact has the been ballast, investigated and the inte interaction onlyraction for concrete between track the gravel ballast, pieces even thoughand the incontacted several cases,wheel thewill track aaffectffect ballast the subsequent can be constructed behavior of as the a derailedgrdeavelrailed track train. bed. Therefore, This indicates for designing, that, apart verifying, verifying, from concreteand optimizing ballast, rational the DCP DCP systems systems should with be gravel additi balla ballastonallyst examinedsystems, this for interaction train derailment should over be analyzed a gravel ballast.to investigate the behavior of the derailed train after it contacts thethe ballast.ballast. As mentioned before, in several cases, the track ballast comprises pieces of gravel of irregular shapes, which form the track bed, as shown in Figure 2. When a train is derailed, the wheel of the derailed train will contact the ballast, and the interaction between the gravel pieces and the contacted wheel will affect the subsequent behavior of the derailed train. Therefore, for designing, verifying, and optimizing rational DCP systems with gravel ballast systems, this interaction should be analyzed to investigate the behavior of the derailed train after it contacts the ballast.
(a) (b)
Figure 2. Conventional track ballast (ballast for rapid trains in Korea): ( aa)) track track ballast; ( (b)) pieces of gravel for the ballast.
For the modeling and simulation of the track ballast, the finite-element method (FEM) has been widely used because of its reliability and convenience. By using the conventional FEM approach, the ballast is modeled( asa) shell or solid elements, with material properties(b) equivalent to those of gravel layers in the two-or three-dimensional domain, respectively. Even when appropriate material Figure 2. Conventional track ballast (ballast for rapid trains in Korea): (a) track ballast; (b) pieces of models are applied, there is a limitation regarding the interaction between the gravel pieces in contact. gravel for the ballast. The gravel pieces can be in contact or separate from each other. However, if the gravel layer is Appl. Sci. 2020, 10, 2717 3 of 19 modeled using the conventional FEM, the interaction characteristics cannot be effectively considered because this approach does not allow the separation of elements that share nodes and exhibit a very large deformation. To rationally design the DCP, the post-behavior of the derailed train should be analyzed first to evaluate the required physical quantities, including the impact load on the DCP structural members due to the derailed train. When the train is derailed over gravel ballast, it can be assumed that the interaction between the ballast and train, which is induced by the contact between the gravel and the wheel of the bogie, directly affects the subsequent behavior of the derailed train. Therefore, rational numerical modeling and simulation methods for the ballast should be studied to perform the post-behavioral analysis of a train derailed over gravel ballast. As mentioned earlier, the track ballast has been modeled based on FEM approaches. Paderno [5] studied the long-term settlement characteristics and the effect of the tamping process on the dynamic behavior of the ballast, using FEM approaches. In that study, the ballast was modeled as shell elements, and the equivalent material properties, such as internal friction angle, dilation angle, and cohesion, which had been evaluated through a simple experiment, were applied using the Mohr–Coulomb plastic model. Although this approach could be applied in ballast modeling for relatively small-deformation problems, it still has limitations with respect to directly considering the interactions between the pieces of gravel in contact, especially the interlocking effect between these pieces. Zhou et al. [6] studied methodologies for analyzing the ultimate lateral pressure of penetrated pile in undrained clay based on FEM. Although the approach can be applied for relatively large deformable soil layers, separation of the gravels cannot be simulated by the approach. Ahmadi and Eskandari [7,8] suggested the vibration analysis method of a rigid circular disk embedded in a transversely isotropic solid. Eskandrai et al. [9] studied the closed-form solution for lateral translation of an inextensible circular membrane embedded in a transversely isotropic half-space. However, there are still limitations for considering the time-varying change of the geotechnical properties of the ballast due to the applied forces. To overcome these limitations, other approaches have been studied based on the discrete-element method (DEM). Using DEM, each piece of gravel can be modeled as an individual object with efficient calculation; therefore, the interaction between the gravel elements in the ballast, including slip, separation, and re-attachment, can be analyzed. Zhou et al. [10] performed a DEM-based analytical study to simulate the effect of the tamping process on the compactness of track ballast. Mahmoud et al. [11] studied a simulation method for the permanent settlement of track ballast due to cyclic loading, based on the DEM approach. Kim et al. [12] investigated the influencing factors in ballast settlement using a DEM simulation. Furthermore, Pi et al. [13] studied the relation between the geogrid rib size and the particle size distribution of ballast materials using a DEM simulation. When DEM approaches are used, the modeling method used for the gravel, including the shape and initial positions of the gravel pieces, is important. Thakur et al. [14] studied a modeling method for arbitrarily shaped gravel pieces using a clump of circles in 2D for DEM simulation. Mollon and Zaho [15] studied the generation and particle-packing method of sand in a DEM simulation. They suggested a method to determine the number and initial positions of the particles to satisfy the pre-defined size distribution, shape, and density of the sand layer. Campello and Cassares [16] also studied a method to rapidly generate a particle layer using the DEM approach. In recent studies, the individual particles have been modeled as sphere-type rigid bodies or clumps consisting of spheres, to ensure efficient calculation. Nezami et al. [17] adopted a DEM simulation approach using specifically shaped objects with fast tracking for contact between the objects, using their in-house simulation code, DBLOCK3D. As alternatives to the conventional FEM, DEM approaches are being used to model and simulate ballast. If a rational modeling and simulation method for track ballast is used, the simulation method for analyzing the behavior of the derailed train can be investigated considering the interaction between the ballast and the derailed bogie. For the simulation, DEM approaches can be effectively used. This study aims to propose a method for modeling track ballast filled with gravel pieces and simulating a ballast–wheel collision. Because the pieces of gravel can be separated and re-attached, Appl. Sci. 2020, 10, 2717 4 of 19 conventional FE methods, which do not allow the separation of the elements, exhibit limitations regarding the modeling of gravel-filled ballast. To overcome this limitation, DEM-based modeling and simulation methods were investigated in this study. Using a DEM approach, the individual pieces of gravel can be modeled; hence, the interactions between the pieces of gravel in contact and between the gravel and other objects can be simulated. For a rational simulation of the ballast–wheel interaction, appropriate modeling and simulation strategies should be adopted considering the characteristics of the gravel used for the track ballast. To consider the interlocking of the gravel elements in contact, a clump-type rigid object is mainly used for each piece of gravel. Whereas the gravel pieces are modeled based on DEM, the other objects that contact and clash with the gravel layer are modeled based on FEM to calculate and consider the energy absorption of the deformable bodies. Thus, a DEM–FEM combined simulation method is adopted in this research. An experiment was conducted to examine the behavior of a freely dropped object that clashes with the gravel layer. To validate the simulation method, the motions of the dropped weight, obtained using both the experiment and the simulation, were directly compared. In addition, case studies were conducted to investigate the effects of the parameters of the gravel–gravel and gravel–object contact models on the simulation results for the ballast–wheel interaction.
2. Theoretical Background for Ballast Modeling and Analysis Several researchers have modeled ballast using the FEM with shell and solid elements for two- and three-dimensional analyses, respectively. In such studies, the equivalent soil properties, such as elastic modulus, Poisson’s ratio, internal friction angle, dilation angle, and cohesion, should be defined using appropriate material models such as the Mohr–Coulomb model. Thus, the equivalent material properties should be evaluated for modeling track ballast filled with irregular pieces of gravel. Although the FEM has been widely used in numerical simulations of various structural and geotechnical engineering problems, it has significant limitations regarding the modeling and simulation of the behaviors of a track ballast filled with gravel, because of the fundamental assumptions of the method. Using the FEM approach, the interlocking and separation of the gravel pieces cannot be directly considered because the conventional FEM does not allow the separation of elements attached to each other. Specifically, the collision between the ballast and the wheel of a derailed train cannot be rationally simulated using conventional FE approaches because of this limitation. Thus, the ballast should be modeled using another approach. The DEM is one of the alternative solutions for modeling track ballast. Using the DEM approach, each piece of gravel can be modeled as an individual object that does not share nodes. Thus, all the pieces of gravel in contact can be detached and re-attached after their positions are updated. Therefore, the limitation of the FE approach can be overcome using the DEM. Based on the advantages of the DEM approach, an appropriate simulation strategy for modeling gravel ballast and simulating ballast–wheel collisions is suggested in this study.
2.1. DEM Analysis In a DEM analysis, the individual objects are modeled as rigid bodies. The simplest way to model gravel is to use spherical rigid bodies. For modeling gravel pieces as simple rigid bodies, an appropriate contact model should be used to consider the elasticity of the actual object. Hence, after modeling the rigid bodies using the contact model between them, the analysis can be conducted. The analysis is performed based on Newton’s second law. Hence, the increments in the velocity and displacement of the individual bodies due to the applied loads are determined by calculating their acceleration. Figure3 shows the general procedure of the DEM analysis based on an explicit structural dynamic analysis. Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 of 20 Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 of 20 acceleration. Figure 3 shows the general procedure of the DEM analysis based on an explicit structural dynamic analysis. Appl.acceleration. Sci. 2020, 10 Figure, 2717 3 shows the general procedure of the DEM analysis based on an explicit5 of 19 structural dynamic analysis. Define Initial positions and velocities Define Initial positions and velocities Detect collision Detect collision Calculate contact forces and torques Update all the Calculate contact forces and torques physicalUpdate quantities all the for Calculate accelerations by Newton’s 2nd law physicaltn+1 quantities=tn+∆t for Calculate accelerations by Newton’s 2nd law tn+1=tn+∆t Calculate new velocities and positions Calculate new velocities and positions End
End Figure 3. General procedure of discrete-element (DEM) analysis based on explicit structural FigureFigure 3. 3.General General procedure procedure of of discrete-elementdynamic analysis. (DEM) (DEM) analysis analysis based based on onexplicit explicit structural structural dynamic analysis. dynamic analysis. At the preparation stage, the initial positions and velocities of the rigid bodies are defined. BecauseAtAt thetheall preparationpreparationthe objects stage,are considered the initial as positions individual an andd rigidvelocities bodies, of the initial rigid overlapping bodies are betweendefined. defined. BecausecontactedBecause allall objects the the objects objectsis not are allowed. are considered considered Therefore, as individual as ifindivi the initial rigiddual bodies,positionrigid bodies, initial of each overlapping initial object overlapping is defined, between the contactedbetween initial objectsanalysiscontacted is should notobjects allowed. be is conductednot Therefore,allowed. to Therefore,determine if the initial ifthe the position a initialppropriate ofposition each position object of each iswithou defined,objectt anyis thedefined, overlapping. initial the analysis initial To shouldavoidanalysis the be should conductedinitial beanalysis conducted to determine and todefine determine the the appropriate appropriate the appropriate position initial without positionpositions any withou of overlapping. thet anyobjects, overlapping. Toa numerical avoid theTo initialmethodavoid analysisthe such initial as and particle analysis define packing theand appropriate define can be the used. initialappropriate The positions details initial of of particle the positions objects, packing aof numerical theare describedobjects, method a innumerical the such next as particlesection.method packingsuch as particle can be used. packing The can details be used. of particle The de packingtails of particle are described packing in are the described next section. in the next section.Once the initial conditions are defined,defined, including the initial positions, velocities, and boundary conditionsOnce the of theinitial domain, conditions the dynamic are defined, analysis including cancan be the conducted initial positions, to obtain velocities, thethe dynamicdynamic and structuralstructuralboundary responsesconditions ofof allthe thethe domain, objects,objects, the includingincl dynamicuding the analysis rigidrigid bodiesbodiescan be andandconducted deformabledeformab to obtainle structuralstructural the dynamic members, structural in the model.responses Again,Again, of all the thethe DEM DEMobjects, analysis analysis incl isuding suitable is suitablethe for rigid dynamic forbodies dynamic problems and deformab problems becausele the structuralbecause displacement the members, displacement increments in the areincrementsmodel. calculated Again, are by calculatedthe integrating DEM analysisby the integrating acceleration is suitable the increments. acce forleration dynamic As increments. shown problems in Figure Asbecause shown2, the collisionthe in Figuredisplacement between 2, the individualcollisionincrements between bodies are calculated isindividual verified by to bodies determineintegrating is verified whether the acce to the determineleration contact increments. force whether and torquethe As contact shown should forcein be Figure calculated and torque2, the or not.shouldcollision If the be between forcecalculated and individual torque or not. of If thebodies the object force is verified areand not torque calculated,to determine of the object the whether acceleration are not the calculated, incrementcontact force the of acceleration theand object torque is calculatedincrementshould be usingcalculatedof the theobject force or isnot. andcalculated If momentthe force using determined and the torque force atof and the the previousmoment object are determined time not step. calculated, If at the the force the previous acceleration and torque time havestep.increment toIf the be calculated, offorce the and object torque the is forces calculated have are to calculated be using calculated the based fo,rce the on and forces the pre-definedmoment are calculated determined contact based model. at on the the Various previous pre-defined contact time modelscontactstep. If themodel. exist force for Various DEMand torque analysis. contact have models Among to be existcalculated these, for the DEM, Hertz-based the analysis. forces are modelAmong calculated was these, used based the in Hertz-based on this the study pre-defined because model thewascontact model used model. in can this beVarious study effectively because contact applied the models model for exist considering can for be DEMeffectively contact analysis. applied behaviors Among for considering these, of elastic the bodiesHertz-based contact modelled behaviors model as rigidofwas elastic used spherical bodies in this bodies. studymodelled because The as applied rigid the modelspherical contact can model bodies. be effectively is The shown applied applied in Figure contact for4. considering model is shown contact in behaviors Figure 4. of elastic bodies modelled as rigid spherical bodies. The applied contact model is shown in Figure 4.
Ri : radius of the rigid spherical object i : displacement along the reference axes Ri , : radius of the rigid spherical object i K C t t : rotation Rigid spherical , : displacement along the reference axes K C : spring constant and damping coefficient for t t object 2 , : rotation
UsingUsing thethe contactcontact model,model, thethe elasticelastic behaviorbehavior of of thethe objectobject cancan bebe takentaken intointo account,account, eveneven thoughthough thethe objects objects are are modeled modeled as as rigid rigid bodies bodies for for numerical numerica simplicity.l simplicity. Thus, Thus, the the sti stiffnessffness for for normal normal contact contact is estimatedis estimated considering considering the elasticthe elastic modulus modulus andthe and size the of size the objectsof the objects in contact. in contact. In addition, In addition, the friction the betweenfriction between the objects the in objects contact in is contact considered is considered as a tangential as a tangential contact force. contact Once force. the friction Once the coe ffifrictioncient forcoefficient the objects for inthe contact objects is in defined, contact theis defined, friction the force friction can be force calculated can be calculated because the because normal the force normal has beenforce calculated. has been Incalculated. this simulation, In this a simple simulation, linear frictiona simple model linear was applied.friction Consequently,model was applied. all the forceConsequently, components all requiredthe force forcomponents defining therequired motion fo equationr defining of the each motion object, equation including of inertiaeach object, and dampingincluding forces, inertia were and calculateddamping forces, and considered. were calcul Theated normal and considered. contact force TheFn normalbetween contact two spherical force particlesbetween intwo contact spherical can be particles calculated in contact using the can Hertz be calculated solution, asusing shown the inHertz Equation solution, (1): as shown in Equation (1): p 4 3 Fn = E4∗ √R δ , (1) F 3= E*√R δ3, (1) n 3 2 2 R R 1 v 1 v 1 2 1 − 1 2 wherewhereR == ;; ∗ == + − ;; E i is is the the Young’s Young’s modulus modulus of of particle particlei; iv; i is the is the Poisson’s Poisson’s ratio ratio of R 1 + R2 E ∗ E 1 E 2 particleof particlei; and i; andδ is the is normal the normal overlap overlap between between two sphericaltwo spherical particles particles in contact. in contact. AsAs shownshown inin EquationEquation (1),(1), evaluationevaluation of δ is is very very important important because because it it determines determines the thecontact contact force F . In the simulation, δ was evaluated as the approach distance between remote points on the force Fn n. In the simulation, was evaluated as the approach distance between remote points on the contactingcontacting spheresspheres in in every every time time increment increment during during explicit explicit dynamic dynamic analysis. analysis. F µ TheThe tangent tangent contact contact force force t can can be be calculated calculated using using the the pre-defined pre-defined friction friction coe fficoefficientcient , while μ, while the FD FD C dampingthe damping forces forcesn and andt can be can calculated be calculated using using the pre-defined the pre-defined damping damping coeffi coefficientscients n and C andt, respectively. , respectively. AsAs mentionedmentioned previously,previously, aa rationalrational definitiondefinition ofof thethe initialinitial positionposition ofof thethe objectsobjects isis important.important. ParticleParticle packingpacking isis oneone ofof thethe usefuluseful methodsmethods forfor modelingmodeling aa layerlayer filledfilled with with rigid rigid particles. particles. InIn thisthis study,study, a a random random particle-packing particle-packing methodmethod waswas primarilyprimarily usedused toto modelmodel thethe tracktrack ballastballast filledfilled withwith irregularlyirregularly sized sized gravel gravel pieces. pieces. FigureFigure5 5presents presents the the general general procedure procedure of of the the packing packing method. method.
Define the space with a boundary ↓ Assume the initial position of arbitrary particles (following a normal-distribution random process) ↓ Create tetrahedral meshes at every position using Delaunay triangulation ↓ Create spheres with an initial radius ↓ Find the maximum radius of each particle that allows no overlap between particles in contact ↓ Determine the initial position and radius of each particle
FigureFigure 5.5. GeneralGeneral procedureprocedure forfor closeclose packingpacking ofof particles.particles.
2.2.2.2. KoreanKorean RegulationsRegulations forfor SizeSize DistributionDistribution ofof BallastBallast GravelGravel InIn Korea,Korea, thethe sizesize distributiondistribution ofof thethe gravelgravel forfor tracktrack ballastballast isis determineddetermined accordingaccording toto thethe specificationsspecifications forfor railroadrailroad equipmentequipment [[18].18]. Table1 1 shows shows the the details details of of the the size size distribution distribution of of the the gravelgravel forfor tracktrack ballastballast presentedpresented by by the the regulation. regulation.
Table 1. Size distribution of the gravel for track ballast used in Korea [18]. Table 1. Size distribution of the gravel for track ballast used in Korea [18]. Size (mm) 10 22.4 31.5 40 50 63 Size (mm) 10 22.4 31.5 40 50 63 % passed by sieve analysis - 0–5 5–35 30–65 60–100 100 % passed by sieve analysis - 0–5 5–35 30–65 60–100 100 Appl. Sci. 2020, 10, 2717 7 of 19
The procedure shown in Figure5 is followed using the given number of particles. Thus, this procedure can be performed to determine the maximum radius of each particle for Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 20 the given boundary with the given number of particles. To achieve the required size distribution for the gravelThe layer, procedure the procedure shown shown in Figure in Figure 5 is 5followed should beusin revised.g the given First, thenumber number of particles. of particles Thus, should this beprocedure treated as can one ofbe the performed variables toto bedetermine determined the via ma theximum optimization radius process.of each Therefore,particle for the the variables given toboundary be found with are the the number given number of particles of particles. and the sizeTo ac andhieve initial the locationrequired of size each distribution particle. The for boundarythe gravel islayer, considered the procedure as the dimensions shown in ofFigure the target 5 should layer be to revised. be formed. First, The the object number functions of particles to be minimized should be aretreated defined as one by calculatingof the variables the overlappingto be determined length via and the voidoptimization of the layer. process. Figure Therefore,6 shows the variables revised processto be found for the are close the number packing. of particles and the size and initial location of each particle. The boundary is consideredThe revised as the procedure dimensions canbe of successfullythe target layer performed to be formed. based The on object the genetic functions algorithm, to be minimized which is aare powerful defined optimization by calculating algorithm. the overlapping Using this length procedure, and void close of packing the layer. of the Figure particles 6 shows can be the obtained revised consideringprocess for the givenclose packing. size distribution.
Define the space with a boundary ↓ Generate the particles satisfying the pre-defined size distribution (*the number of particles is assumed first) ↓
*Perform the sequence following optimization algorithms Arrange the locations of all the generated particles ↓ Calculate the overlapping length and void of the current layer ↓ Verify the result ↓ Determine the initial position and radius of each particle
FigureFigure 6. 6.Revised Revised general general procedure procedure for for close close packing packing of of the the particles. particles.
3. StrategyThe revised and Validation procedure of can Numerical be successfully Implementation performed based on the genetic algorithm, which is a powerful optimization algorithm. Using this procedure, close packing of the particles can be 3.1. Simulation Strategy for Ballast–Wheel Collision obtained considering the given size distribution. Ballast–wheel collisions were analyzed to trace the path of a derailed train and investigate the3. Strategy effects of and the Validation ballast on theof Numerical energy absorption Implementation under various conditions, including the ballast condition and kinetic characteristics of the derailed train. To investigate the behaviors, the change in3.1. the Simulation position, Strategy forces, and for Ballast–Wheel energies of the Collision clashing objects should be simulated in the time domain. Therefore,Ballast–wheel a DEM–FEM collisions combined were simulationanalyzed to should trace the be conducted,path of a derailed wherein train the ballastand investigate is modeled the usingeffects a DEMof the approach, ballast on and the the energy colliding absorption object is modeled under usingvarious aFEM conditions, approach. including Figure7 showsthe ballast the procedurecondition forand the kinetic simulation characteristics of the ballast–wheel of the derailed collision. train. To investigate the behaviors, the change in the position, forces, and energies of the clashing objects should be simulated in the time domain. Therefore, a DEM–FEM combined simulation should be conducted, wherein the ballast is modeled using a DEM approach, and the colliding object is modeled using a FEM approach. Figure 7 shows the procedure for the simulation of the ballast–wheel collision.
Appl. Sci. 2020, 10, 2717 8 of 19 Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 20 Appl. Sci. 2020, 10, x FOR PEER REVIEW 8 of 20
Model Modelthe ballast the ballastfilled with filled gravel with piecesgravel considpieces eringconsid theering given the size given distribution size distribution (using DEM)(using DEM) ↓ ↓ Model Modelthe object the that object will that clas willh with clas hthe with ballast the ballast(using FEM)(using FEM) ↓ ↓ Define Definethe analysis the analysis protocol, protocol, including including initial time initial st ep,time total step, time, total load time, sequence, load sequence, and initial and and initial boundary and boundary conditionsconditions
Define Definethe contact the contact model betweenmodel between the gravel the piecesgravel andpieces gravel–wheel and gravel–wheel ↓ ↓ ConductConduct an explicit an explicit dynamic dynamic analysis analysis to calculate to calculate the motion, the motion, force, and force, energies and energies of all the of objectsall the objectsin the in the time domaintime domain Figure 7. Simulation procedure for ballast–wheel collision. FigureFigure 7. 7.Simulation Simulation procedure procedure for for ballast–wheel ballast–wheel collision. collision.
3.2. Validation3.2.3.2. ValidationValidation To validateToTo validatevalidate the numerical the numerical numerical strategy, strategy, strategy, a drop a test adrop drop was test conducted. test was was conducted. conducted. As shown As shown in As Figure shown in Figure 8, a in1000 8, Figure a× 10008, × 1300 ×a 700 1000 mm 1300(width 700× breadth mm (width × height)breadth steel box waheight)s filled steel with box gravel was pieces filled withup to gravela height pieces of up 1300 × ×700 mm× (width × breadth × height) steel× box was filled with gravel pieces up to a height of 600.0 tomm.600.0 a height The mm. sizes of The 600.0 of sizes all mm. the of allgravel The the sizes gravelpieces of all pieceswere the gravel31.5–50.0 were pieces31.5–50.0 mm, were which mm, 31.5–50.0 whichis suitable mm,is suitable for which track for is ballast. suitabletrack ballast. The for track The 656-kgballast.656-kg weight Theweight was 656-kg freely was weightdroppedfreely dropped was from freely heights from dropped heights of 0.5 from andof 0.5 1.0 heights and m. 1.0During of m. 0.5 During the and drop 1.0 the m. test, drop During the test, motion thethe dropmotion of test, of the weightthethe motion weight was measured ofwas the measured weight in the was timein the measured domain. time domain. inFurthe the timeFurthermore, domain.rmore, the depth Furthermore,the depthof penetration of penetration the depth into the ofinto penetrationgravel the gravel layer intowaslayer themeasured. was gravel measured. layer To prevent was To measured.prevent a local a effect local To prevent dueeffect to due adeformation local to edeformationffect due of the to deformationbox,of the the box, wall the of of thewall the box, boxof the the was box wall was of sufficientlythesufficiently box stiffened. was sustiffened.ffi cientlyThe test The sti ffwas ened.test conducted was The testconducted was twice conducted twforice each for twice dropeach for height.drop each dropheight. During height. During the During test, the the thetest, test, the accelerationtheacceleration acceleration of the weightof the of the weight was weight measured was was measured measured via vision via via -postprocessingvision vision-postprocessing-postprocessing in the timein the in domain, the time time domain, and domain, then, and andthe then, then, the velocitythevelocity and velocity displacement and and displacement displacement were evaluated were were evaluated evaluated via numerical via via numerical numerical integration integration integration of the measuredof ofthe the measured measured acceleration. acceleration. acceleration.
(a) (a) (b) (b) Figure 8. Cont. Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 20 Appl. Sci. 2020, 10, 2717 9 of 19 Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 20
(c) (d) (c) (d) Figure 8. Drop test to simulate the collision between the dropped weight and gravel layer: (a) Front Figure 8. Drop test to simulate the collision between the dropped weight and gravel layer: (a) Front view of the drop test; (b) gravel layer; (c) weight penetration; (d) gravel layer after penetration. Figureview of 8. theDrop drop test test; to simulate (b) gravel the layer; collision (c) weight between penetration; the dropped (d) gravel weight layer and aftergravel penetration. layer: (a) Front view of the drop test; (b) gravel layer; (c) weight penetration; (d) gravel layer after penetration. FigureFigure9 shows 9 shows the numericalthe numerical model model for simulating for simula theting drop-testthe drop-test with with the samethe same geometric geometric conditions.Figureconditions. 9 As shows shownAs shown the in numerical the in figure,the figure, model the gravelthe forgravel layersimula layer wasting was modeled the modeled drop-test based based onwith DEM, on the DEM, whilesame while thegeometric weightthe weight conditions.waswas modeled modeled As usingshown using FEM. in FEM.the In figure, this In this simulation, the simulation, gravel the layer gravelthe was gravel was modeled was modeled modeled based as simpleon as DEM,simple rigid while rigid sphere thesphere weight clumps. clumps. wasThus, modeledThus, there there was using was a limitation FEM. a limitation In this regarding simulation, regarding the interlocking thethe interlockigravel was engffect modeled effect between between as thesimple arbitrarilythe rigidarbitrarily sphere shaped shaped clumps. pieces pieces Thus,of gravel.of there gravel. This was This coulda limitation could be one be regardingone of the of sourcesthe sourcesthe interlocki of the of ditheffngerence difference effect between between between the the resultsthe arbitrarily results from from theshaped experimentthe experimentpieces ofand gravel.and the test.the This test. Following could Following be one the procedureofthe the procedure sources shown ofshown the in Figuredifference in Figure7, explicit between7, explicit dynamic the dynamic results analyses analysesfrom were the wereexperiment conducted, conducted, andand theand the test. motionthe Followingmotion of the of dropped thethe dropped procedure weight weight shown was was obtained. in obtained.Figure In 7, addition, explicitIn addition, dynamic after after the simulation,analysesthe simulation, were the conducted, penetrationthe penetration anddepth thedepth andmotion and the ofthe motion the motion dropped of theof the droppedweight dropped was weight obtained.weight were were In observed addition, observed for after for comparison comparison the simulation, with with the the the penetration resultresult fromfrom the depththe dropdrop and test.the motionThe The DEM–FEM DEM–FEM of the dropped simulation simulation weight was was were conducted conducted observed using usingfor ABAQUScomparison ABAQUS V6.17 with V6.17 [19]. the [19 Theresult]. Theparticle from particle thepacking droppacking wastest. wasperformedThe performedDEM–FEM using usingsimulation a MATLAB-based a MATLAB-based was conducted in-house in-house using code ABAQUS, code, following following V6.17 the [19].procedure the The procedure particle shown shown packing in Figure in 5. wasFigure performedAfter5. Aftermodeling, modeling,using thea MATLAB-based explicit the explicit dynamic dynamic in-house analyses analyses code were, following wereconducted. conducted. the Theprocedure initial The initial timeshown timestep in wasFigure step set was 5. as 1.0 Afterset asE modeling,− 1.05 s, E and5 s, the andthe total explicit the totalsimulation dynamic simulation time analyses timewas 1.5 was were s. 1.5 After conducted. s. After simulation simulation The ofinitial the of drop thetime drop test,step test, thewas vertical the set vertical as 1.0motion − Emotion−5 s,and and and final the final penetrationtotal penetration simulation depth depth time of the ofwas the weight 1.5 weight s. After were were simulationevaluated evaluated andof andthe compared compareddrop test, with the with thosevertical those obtained obtainedmotion from andfrom finalthe the experiment. penetration experiment. depth of the weight were evaluated and compared with those obtained from the experiment.
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(a) Figure 9. Cont. (b) Appl. Sci. 2020, 10, 2717 10 of 19 Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 20
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FigureFigure 9. DEM–FEM 9. DEM–FEM simulation simulation of the of dropthe drop tests: tests: (a) dropping(a) dropping the weight;the weight; (b) penetration( b) penetration after after (c) (d) collision;collision; (c) weight (c) weight stopped; stopped; (d) interrupted (d) interrupted gravel gravel layer layer due todue penetration. to penetration. Figure 9. DEM–FEM simulation of the drop tests: (a) dropping the weight; (b) penetration after FiguresFigures 10collision; and 10 and11 (c) showweight 11 show stopped; the comparisonthe (d )comparison interrupted between gravel between layer the due motions theto penetration. motions of the of weights,the weights, obtained obtained from from the the dropdrop test andtest and the simulation.the simulation. In the In simulation,the simulation, diff erentdiffere frictionnt friction coeffi coefficientscients for thefor gravelthe gravel pieces pieces in in contact were appliedFigures 10 to and verify 11 show the the eff comparisonect of friction between between the motions the of gravel the weights, pieces. obtained Although from the the curves contactdrop were test appliedand the simulation. to verify In the the effectsimulation, of fricti differeonnt betweenfriction coefficients the gravel for the pieces. gravel Althoughpieces in the curves are notare perfectlynotcontact perfectly were fitted, appliedfitted, the to suggestedthe verify suggested the effect method ofmethod fricti canon between becan adopted be theadopted gravel for pieces.thefor simulationthe Although simulation the of curves ballast–wheel of ballast–wheel interactionsinteractionsare because not becauseperfectly the tendenciesfitted,the tendencies the suggested of the of motionmethodthe motion can of thebe of adopted weightsthe weights for arethe similar.simulationare similar. From of ballast–wheelFrom the curves,the curves, we can we can interactions because the tendencies of the motion of the weights are similar. From the curves, we can selectselect the timethe time for thefor firstthe first contact contact between between the droppedthe dropped weight weight and and the gravelthe gravel layer, layer, following following which, which, the velocitiesselect and the time changes for the in first displacement contact between of the the dro weightpped weight can and be the directly gravel layer, compared. following As which, shown in the the velocitiesthe velocities and and changes changes in in displacementdisplacement of theof theweight weight can be candirectly be compared.directly compared. As shown in Asthe shown in the figures,figures, thefigures, curvesthe curves theof curves the of velocityofthe the velocity velocity and and and displacement displaceme displacement of of ntthe the of weights weightsthe weights obtained obtained obtainedfrom the from tests from the and teststhethe tests and and the the simulationsimulationsimulation are similar,are similar,are similar, including including including the magnitudes thethe magnitudes magnitudes and and theand time time the required time required required for full for penetration. full for penetration. full penetration.
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Figure 10.FigureComparison 10. Comparison between between the motions the motions of the of dropped the dropped weight, weight, obtained obtained from from tests tests and and simulations simulations (dropped from 0.5 m height): (a) acceleration; (b) velocity; (c) displacement. (dropped from 0.5 m height): (a) acceleration; (b) velocity; (c) displacement.
The displacement curves obtained from the postprocessing of the test, shown in Figure9c, reveal significant rebounding after full penetration. However, this is not the actual situation; it originated from the cumulative error of the numerical integration. To verify further, the penetration depth after the test was compared with that after the simulation. As shown in Table2, the penetration depths measured from the tests and simulations were very similar. The simulation results clearly show that the penetration depth is affected by the applied friction coefficient between the pieces of gravel. Friction is one of the influencing factors in the interlocking between the pieces of gravel in contact. It can be expected that large friction induces extensive interlocking between the gravels. The physical mechanism is clearly expressed in the simulation results. The test results are not well-matched with each other because tamping was not performed after creating the layer. If tamping had been conducted, the layers would have had very similar properties. Appl. Sci. 2020, 10, 2717 12 of 19 Appl. Sci. 2020, 10, x FOR PEER REVIEW 12 of 20
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Figure 11. Comparison between the motions of the dropped weight, obtained from tests and Figure 11. Comparison between the motions of the dropped weight, obtained from tests and simulations simulations (dropped from 1.0 m height): (a) acceleration; (b) velocity; (c) displacement. (dropped from 1.0 m height): (a) acceleration; (b) velocity; (c) displacement. The displacement curves obtained from the postprocessing of the test, shown in Figure 9c, reveal significant rebounding after full penetration. However, this is not the actual situation; it originated from the cumulative error of the numerical integration. To verify further, the penetration depth after the test was compared with that after the simulation. As shown in Table 2, the penetration depths measured from the tests and simulations were very similar. The simulation results clearly show that the penetration depth is affected by the applied friction coefficient between the pieces of gravel. Friction is one of the influencing factors in the interlocking between the pieces of gravel in contact. It can be expected that large friction induces extensive interlocking between the gravels. The physical mechanism is clearly expressed in the simulation results. The test results are not well-matched with each other because tamping was not performed after creating the layer. If tamping had been conducted, the layers would have had very similar properties.
Table 2. Comparison between penetration depths.
Test Simulation Test-1 Test-2 Average Friction = 0.5 Friction = 0.7 0.5-m drop height 0.33 0.24 0.28 0.35 0.26 1.0-m drop height 0.46 0.35 0.41 0.47 0.38
4. Case Study: Effects of Contact Model Parameters on Simulation Results
4.1. Effect of Friction between Gravel Pieces In this section, the effect of the friction between the pieces of gravel on the simulation results is described in detail. In the case study, a friction coefficient of 0.4–0.7 was applied in the contact model of the gravel and, then, the motion of the penetrating weight and the friction-energy absorption were numerically compared to investigate the interlocking due to the applied friction coefficients. Appl. Sci. 2020, 10, 2717 13 of 19
Table 2. Comparison between penetration depths.
Test Simulation Test-1 Test-2 Average Friction = 0.5 Friction = 0.7 0.5-m drop height 0.33 0.24 0.28 0.35 0.26 1.0-m drop height 0.46 0.35 0.41 0.47 0.38
4. Case Study: Effects of Contact Model Parameters on Simulation Results
4.1. Effect of Friction between Gravel Pieces In this section, the effect of the friction between the pieces of gravel on the simulation results is described in detail. In the case study, a friction coefficient of 0.4–0.7 was applied in the contact model of the gravel and, then, the motion of the penetrating weight and the friction-energy absorption were numerically compared to investigate the interlocking due to the applied friction coefficients. As shown in Figure 12, the friction between the pieces of gravel significantly affects the motion of the weight after the impact. The equivalent geotechnical properties of the gravel layer are determined by the effect of interlocking between the contacted gravel pieces. Based on the simulation results, it can be concluded that the considered friction coefficient between the pieces of gravel governs the geotechnical properties of the gravel ballast, including the rigidity of the layer. Figure 13 shows the energy dissipation due to friction in the different gravel layers with different friction coefficients between the gravel pieces. As shown in the figure, the energy dissipation due to friction decreasesAppl. as Sci. the2020, 10 friction, x FOR PEER REVIEW coe fficient increases. This clearly proves14 of that20 the layer would be stiffened owing to theAs shown considerable in Figure 12, the interlockingfriction between the pieces effect of gravel induced significantly when affects athe larger motion friction coefficient of the weight after the impact. The equivalent geotechnical properties of the gravel layer are between the pieces ofdetermined gravel by isthe applied.effect of interlocking Thus, between the th frictione contacted gravel coeffi pieces.cient Based between on the simulation the gravel pieces is one results, it can be concluded that the considered friction coefficient between the pieces of gravel of the most influentialgoverns factors the geotechnical for the properties simulation. of the gravel ballast, including the rigidity of the layer.
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Figure 12. Cont. Appl. Sci. 2020, 10, x FOR PEER REVIEW 15 of 20
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0.4 (c) Displacement (m) Figure 12. Vertical motion of weights clashing with gravel layers for different friction coefficients between the gravel pieces 0.2(friction coefficient between a Friction gravel coefficient=0.4 piece and the weight = 0.6): (a) Friction coefficient=0.5 acceleration; (b) velocity; (c) displacement. Friction coefficient=0.6 Friction coefficient=0.7 0.0 Figure 13 shows the energy 0.0dissipation 0.3 due 0.6 to fricti 0.9on in the 1.2 different 1.5 gravel layers with different Time (s) friction coefficients between the gravel pieces. As shown in the figure, the energy dissipation due to (c) friction decreases as the friction coefficient increases. This clearly proves that the layer would be Figure 12. Vertical motion of weights clashing with gravel layers for different friction coefficients Figure 12. Vertical motion of weights clashing with gravel layers for different friction coefficients stiffened owing tobetween the considerablethe gravel pieces (friction interlocking coefficient between effect a gravelinduced piece and when the weight a larger= 0.6): (a )friction coefficient betweenbetween the pieces the gravelacceleration; of gravel pieces (b) velocity;is (friction applied. (c) displacement. coe Thus,fficient the between friction acoefficient gravel piece between and the the weight gravel= pieces0.6): (a is) one acceleration; (b) velocity; (c) displacement. of the most influentialFigure factors 13 shows thefor energy the simulation. dissipation due to friction in the different gravel layers with different friction coefficients between the gravel pieces. As shown in the figure, the energy dissipation due to friction decreases as the friction coefficient increases. This clearly proves that the layer would be stiffened owing6000 to the considerable interlocking effect induced when a larger friction coefficient between the pieces of gravel is applied. Thus, the friction coefficient between the gravel pieces is one of the most5500 influential factors for the simulation.
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Frictionenergy dissipation (J) 1500 1500 Friction coefficient=0.4 1000 Friction Frictioncoefficient=0.5 coefficient=0.4 1000 500 Friction coefficient=0.6 Friction Frictioncoefficient=0.7 coefficient=0.5 0 Friction coefficient=0.6 500 0.0 0.3 0.6 0.9 1.2 1.5 Time (s) Friction coefficient=0.7. 0 0.0 0.3 0.6(a) 0.9 1.2 1.5 Time (s) . (a)
Figure 13. Cont. Appl. Sci. 2020,, 10,, x 2717 FOR PEER REVIEW 1615 of of 20 19
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Figure 13. ComparisonComparison of of friction friction energy energy dissipation dissipation wi withth different different friction coecoefficientsfficients between (b) pieces of gravel: ( a)) variation; ( b) change of maximum friction dissipation owing to the considered friction coefficient. coeFigurefficient. 13. Comparison of friction energy dissipation with different friction coefficients between pieces of gravel: (a) variation; (b) change of maximum friction dissipation owing to the considered friction coefficient. 4.2. Effect Effect of Friction between Gravel and Dropped Weight In addition 4.2. Effect to of the Friction gravel between pieces Gravel in and contact, Dropped the Weight friction between the gravel and the object may be one of the influentialinfluentialIn addition factors to the gravel for the pieces simulation in contact, theresult result. friction. To To between investigate the gravel the the and effect eff theect objectof this may factor be on the one of the influential factors for the simulation result. To investigate the effect of this factor on the result, a case study considering various friction co coeefficientsfficients between the gravel and the weight was conducted.result, For a the case parametric study considering study, various a friction friction coe coefficientsfficient between of 0.4–0.7 the gravel was appliedand the weight to the was DEM–FEM conducted.conducted. For the parametricFor the parametric study, study, a friction a friction coefficient coefficient of of 0.4–0.7 0.4–0.7 was wasapplied applied to the DEM–FEM to the DEM–FEM model, with model, a friction with a frictioncoefficient coefficient coefficient of 0.5 of between 0.5 between the the gravel pieces. pieces. Figure Figure 14 shows 14 showsshows the vertical the the motionvertical vertical motion of the weights of the dropped weights dropped fromfrom afr om height a height of 0.5of 0.5 m. m.
2.0 2.0
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(a) Appl. Sci. 2020, 10, 2717 16 of 19 Appl. Sci. 2020, 10, x FOR PEER REVIEW 17 of 20
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Figure 14.FigureVertical 14. Vertical motion motion of weights of weights clashing clashing withwith gravel gravel layers layers for different for diff frictionerent frictioncoefficients coe fficients between the gravel and the weight (friction coefficient between gravel pieces = 0.5): (a) acceleration; between the gravel and the weight (friction coefficient between gravel pieces = 0.5): (a) acceleration; (b) velocity; (c) displacement. (b) velocity; (c) displacement. According to the simulation results, the friction between the gravel and the weight mainly affects Accordingthe vertical to the motion simulation of the penetrating results, theweight. friction Although between the fluctuation the gravel patterns and of the the weight acceleration mainly affects the verticalare motion different of for the different penetrating friction weight.coefficients, Although the vertical the velocity fluctuation and vertical patterns displacement of the do acceleration not are differ significantly. Specifically, the time-series vertical displacement and penetration depth appear different for different friction coefficients, the vertical velocity and vertical displacement do not differ to be affected by the friction coefficient when coefficients smaller than 0.5 are applied. This indicates significantly.that the Specifically, dynamic behavior the time-series of the weight verticalused for the displacement drop test is mainly and affected penetration by the characteristics depth appear to be affected byof thethe frictiongravel layer. coe Asffi cientshown whenin Figure coe 15,ffi thecients friction-energy smaller than dissipation 0.5 are converges applied. to Thisa constant indicates that the dynamic behavior of the weight used for the drop test is mainly affected by the characteristics of the gravel layer. As shown in Figure 15, the friction-energy dissipation converges to a constant value as the considered friction coefficient increases. According to the simulation, the friction between the disk-type steel object and the gravel for the track ballast should be larger than 0.5. Appl. Sci. 2020, 10, x FOR PEER REVIEW 18 of 20
value as the considered friction coefficient increases. According to the simulation, the friction Appl. Sci. 2020 10 between the, disk-type, 2717 steel object and the gravel for the track ballast should be larger than 0.5.17 of 19
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0 0.4 0.5 0.6 0.7 Friction coefficient between gravels Figure 15.15. Comparison of friction-energy dissipation for different different friction coecoefficientsfficients between the gravel and the weight. 5. Conclusions 5. Conclusions In this study, a rational modeling method and an analysis for gravel-filled track ballast were In this study, a rational modeling method and an analysis for gravel-filled track ballast were investigated to simulate track-derailed train interaction. Because of the characteristics of the track investigated to simulate track-derailed train interaction. Because of the characteristics of the track ballast, a DEM-based modeling and analysis strategy was used for modeling the ballast. Once the ballast, a DEM-based modeling and analysis strategy was used for modeling the ballast. Once the track ballast was modeled with numerous rigid particles, the interaction between the track ballast and track ballast was modeled with numerous rigid particles, the interaction between the track ballast objects of any shape could be simulated. and objects of any shape could be simulated. • BasedBased on theon DEMthe DEM approach, approach, each pieceeach piece of gravel of gravel is modeled is modeled as a rigid as bodya rigid to body ensure toan ensure efficient an • simulation.efficient Tosimulation. consider theTo elasticityconsider ofthe the elasticity gravel and of the frictiongravel and effects the between friction theeffects gravel between pieces in contact,the gravel an pieces appropriate in contact, contact an modelappropriate should contact be applied. model Inshould this study,be applied. the Hertz–Mindlin In this study, contactthe Hertz–Mindlin model was applied contact to considermodel was the applied normal to and consider tangential the normal contact and forces tangential and the frictioncontact force.forces The and nonlinear the friction normal force. and The tangential nonlinear sti ffnonessesrmal and were tangential estimated stiffnesses using the were equation estimated of the contactusing model the equation considering of the the contact size, elastic model modulus, considering and Poisson’sthe size, elastic ratio of modulus, the gravel. and Apart Poisson’s from theratio contact of betweenthe gravel. the piecesApart offrom gravel, the acontact contact between model between the pieces the gravelof gravel, and contacteda contact objectsmodel wasbetween also used the along gravel with and the contacted Hertz–Mindlin objects was model. also used along with the Hertz–Mindlin model. • To validate the simulation method, a drop test was conducted, and the experiment was To validate the simulation method, a drop test was conducted, and the experiment was reproduced • reproduced via a DEM-based simulation. The comparison results, including vertical via a DEM-based simulation. The comparison results, including vertical acceleration, velocity, acceleration, velocity, displacement, and the penetration depth of the freely dropped displacement, and the penetration depth of the freely dropped weights, exhibit very good weights, exhibit very good agreement. Therefore, the feasibility and rationality of the agreement. Therefore, the feasibility and rationality of the simulation method was verified. simulation method was verified. The effects of the applied friction coefficient on the interaction between the gravel layer and • • The effects of the applied friction coefficient on the interaction between the gravel layer and the clashing object were studied. According to the results, the friction coefficient between the the clashing object were studied. According to the results, the friction coefficient between the contacted gravel pieces significantly affects the geotechnical properties of the layer filled with contacted gravel pieces significantly affects the geotechnical properties of the layer filled with gravel. The coefficient governs the interlocking effect in the gravel layer and, thus, this coefficient gravel. The coefficient governs the interlocking effect in the gravel layer and, thus, this also affects the rigidity of the layer. coefficient also affects the rigidity of the layer. The friction coefficient between the gravel and the object does not significantly affect the interaction • • The friction coefficient between the gravel and the object does not significantly affect the between the layer and the clashing object. According to the simulation, the results converge interaction between the layer and the clashing object. According to the simulation, the results when a coefficient over 0.5 is applied. In addition, based on both the experiment and simulation, converge when a coefficient over 0.5 is applied. In addition, based on both the experiment a coefficient of 0.5 for the track ballast–steel wheel contact is appropriate. and simulation, a coefficient of 0.5 for the track ballast–steel wheel contact is appropriate. Although the simulation results agree well with the test results, limitations still exist to be improved for simulation of derailed train-track ballast interaction as below: Appl. Sci. 2020, 10, 2717 18 of 19
The effect of the shapes of different pieces of gravel could not be considered because simple rigid • sphere clumps were used. To simulate derailed train-track ballast interaction, the collision cases which induce the shearing • motion of the gravel of track as well as vertical motion should be further validated. For real track ballasts tamping effect should be taken into account. • Further study is needed to find the rational range of friction coefficients between irregular gravels • for real ballast track.
Author Contributions: Conceptualization, S.K., N.-H.L. and K.-J.K.; methodology, S.K. and H.-U.B.; software, S.K.; validation, S.K. and H.-U.B.; formal analysis, S.K. and H.-U.B.; investigation, S.K. and H.-U.B.; resources, N.-H.L.; data curation, K.-J.K. and H.U.B.; writing—original draft preparation, S.K. and N.-H.L.; writing—review and editing, S.K. and N.-H.L.; visualization, H.-U.B.; supervision, S.K.; project administration, N.-H.L.; funding acquisition, N.-H.L. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the Railway Technology Research Program, funded by the Ministry of Land, Infrastructure, and Transport of the Korean government, grant number 17RTRP-B122273-02. Conflicts of Interest: The authors declare no conflict of interest.
References
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