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Rail. Eng. Science (2022) 30(2):162–182 https://doi.org/10.1007/s40534-021-00265-8 On dynamic analysis method for large-scale train–track–substructure interaction 1,2 Lei Xu Received: 13 August 2021 / Revised: 9 November 2021 / Accepted: 9 November 2021 / Published online: 18 March 2022 The Author(s) 2022 Abstract Train–track–substructure dynamic interaction is excitation difference of different track irregularity an extension of the vehicle–track coupled dynamics. It spectrums. contributes to evaluate dynamic interaction and perfor- mance between train–track system and its substructures. Keywords Train Track dynamic interaction Railway For the ﬁrst time, this work devotes to presenting engi- substructures Finite elements Dynamics system neering practical methods for modeling and solving such Iterative solution Tunnel Bridge large-scale train–track–substructure interaction sys- tems from a uniﬁed viewpoint. In this study, a train con- sists of several multi-rigid-body vehicles, and the track is modeled by various ﬁnite elements. The track length needs only satisfy the length of a train plus boundary length at 1 Introduction two sides, despite how long the train moves on the track. The substructures and their interaction matrices to the 1.1 Research background upper track are established as independent modules, with no need for additionally building the track structures above In a rail transit system, the main body includes the train and substructures, and accordingly saving computational cost. its upper pantograph catenary system of power supply and Track–substructure local coordinates are deﬁned to assist lower rail infrastructural system of supporting and guid- the conﬁrming of the overlapped portions between the ance. From aspects of civil engineering, one of the mostly train–track system and the substructural system to effec- concerned issue is analyzing the dynamic interaction tively combine the cyclic calculation and iterative solution between the train and its infrastructural system including procedures. The advancement of this model lies in its the tracks and substructures, e.g., the tunnel and bridge, convenience, efﬁciency and accuracy in continuously since train–track–substructure (TTS) interaction is associ- considering the vibration participation of multi-types of ated with very important railway engineering problems substructures against the moving of a train on the track. such as the evaluation of train running safety and riding Numerical examples have shown the effectiveness of this comfort [1–3], prediction of structural fatigue damage and method; besides, inﬂuence of substructures on train–track vibration noise  and revealing the mechanism of wheel– dynamic behaviors is illustrated accompanied by clarifying rail contact [5, 6]. To evaluate the TTS interaction, computer modeling of the TTS system seems to be the most economic and & Lei Xu practical way except the basic experimental work. As a firstname.lastname@example.org typically large and complex system, the train, the tracks, School of Civil Engineering, Central South University, and the substructures are expected to be elaborated Changsha 410075, China sophisticatedly to accurately depict the system’s dynamic National Engineering Laboratory for High-Speed Railway behaviors, but sometimes making a compromise to Construction, Changsha 410075, China 123 On dynamic analysis method for large-scale train–track–substructure interaction 163 efﬁciency. Generally, two categories of analytical and system, vertically layered and tying the train and the sub- numerical methods are adopted to the TTS dynamics, but structures as an integrated system. Zhai and Cai  well the analytical method can be merely suitable for very recognized this matter and established a train–track–bridge simpliﬁed structures with moving constant, pulsating or interaction model as an extension of vehicle–track coupled continuous forces with constant speed [7, 8], and accord- dynamics . Extensively, Zhai et al. [25, 26] developed ingly, the most research nowadays concentrates on a framework to conduct the train–track–bridge dynamic numerical methods. interaction analysis, and also, validations from experiments onsite were performed. Using this modeling framework, 1.2 State-of-the-art review the running safety and ride comfort of trains passing through bridges can be evaluated. Lou  presented an In conventional research, an emphasis is mainly put on integrated vertical vehicle–track–bridge model based on train–bridge interaction where the detailed conﬁguration of hypotheses of vehicle rigid body and track–bridge Ber- track structure is neglected, for more than one hundred noulli–Euler beams; besides, the wheels were assumed to years. Yang and Yau  presented a vehicle–bridge ele- be always in contact with the upper beam. To develop a ment to accurately and efﬁciently modeling and analyzing simple-to-implement algorithm for analyzing train–track– the vehicle–bridge interaction. Xia and Zhang [10, 11] substructure interaction, Fedorova and Sivaselvan  established representative vehicle–bridge interaction mod- applied kinematic constraints to couple the separately built els: one is built in global equations with time-dependent bridge and train together and the wheel–rail contact sepa- coefﬁcients and the other is solved by an inter-system ration was also formulated as linear complementary prob- iteration method. Iterative methods has also been applied lem. Using a special wheel–rail interaction element , by scholars in a series of work, such as Feriani et al.  Liu and Gu  presented a modiﬁed substructure method for solving vertical vehicle–bridge interaction, Wang et al. for numerical simulation of vehicle–track–bridge systems.  for vehicle–track systems based on the prediction of In the ﬁnite element framework, Zeng et al.  applied wheel–rail forces, Zhu et al.  for multi-time-step the energy variation method to formulate matrix equations solution of layered train–track–bridge systems, Melo et al. of motion for a 3D train–slab track–bridge interaction  for the nonlinear behavior of the track–deck interface system. Galvı ´n et al. , Bucinskas and Andersen  did and Xu et al.  for presenting a multi-time-step solution interesting work on considering the vibration participation for vehicle–track related dynamics. Except for iterative of soil in train passages by introducing boundary element procedures, researchers have also done interesting work on method. coupled solutions, i.e., the entire dynamics system are Except the bridge substructures, tunnel has gradually directly obtained by solving integrated dynamic equations been considered in train–track interaction process. of motion globally. Datta et al.  presented a non-iter- Degrande et al. [34, 35] devoted to predicting the tunnel– ative solution method in a time marching pattern using soil dynamic responses from underground railways, where dynamic ‘‘mortar elements’’ for vehicle–bridge interaction ﬁnite element–boundary element hybrid modeling method analysis. Dimitrakopoulos and Zeng  presented a gen- is applied. Ignoring the inﬂuence of train system and sub- eric scheme for interaction between a train and the curved stituting with a series of moving loads, Forrest et al.  bridge elaborated in 3D space. Based on an arbitrary and Di et al.  presented an analytical Pipe-in-Pipe Lagrangian–Eulerian approach, a moving mesh strategy model to analyze the dynamic performance of underground was developed by Greco and Lonetti  to analyze the tunnels, also a 2.5D model in . Besides, some work vehicle–bridge interaction subject to moving load appli- regarded the train–track and the tunnel system as two cations. Besides, Antolı ´n et al.  considered four wheel– systems and using a two-step method; that is, the train rail contact models in the vehicle–bridge interaction to test loads derived from the vehicle–track coupled model were the applicability of various models. Zhu et al.  estab- treated as the excitation of the tunnel and soil models, such lished a linear complementarity method for analyzing as the work in [39–41]. Zhou et al.  presented a vertical vehicle–bridge dynamic interaction, where the wheel–rail dynamic model for metro train–track–tunnel–soil interac- contact and separation scenarios are considered. Through tion using a semi-analytical approach, and the sub-systems establishment of a set of second-order ordinary differential were coupled by wheel–rail forces and rail fastener forces. equations, Stoura et al.  presented a dynamic parti- Further, Zhou and its cooperators such as Di and He et al. tioning method to solve the vehicle–bridge interaction [43, 44] had conducted extensive work on the establish- problem. ment of vehicle–track–tunnel–soil to predict the train-in- The work displayed above mainly aims at train–bridge duced vibrations. By a so-called 2.5D ﬁnite element– dynamics, and track structures are not considered in detail. boundary element hybrid method, Jin et al.  and It is known that the track is also a very important sub- Ghangale et al.  analyzed the train-induced ground Rail. Eng. Science (2022) 30(2):162–182 164 L. Xu vibrations and energy ﬂow radiation using numerical method to especially construct the tunnel with rings and model for track/tunnel/soil system, though the train is segments, straight joined and stagger joined . But simpliﬁed as moving loads or by two-step dealing method. totally different from the previously coupled modeling Zhu et al.  combined the pseudo-excitation method and methods, such as the one presented in , where the the 2.5D ﬁnite element-perfectly matched layers method to train–track–substructure system must be established in build a model for subway train-induced ground vibration advance, in this work, the substructures participate in the prediction. Recently, Xu and Zhai  established an computation only when there exists overlapped computa- entirely coupled model for train–track–tunnel interaction as tional portion between the train–track system and sub- matrix formulations, where the entire system is solved structural system. In this manner, both the computational simultaneously without iteration. mode and the efﬁciency has been promoted. As observed from the above state-of-the-art review, train– The rest of this paper is organized as follows: track–substructure dynamic interaction has obtained signif- (1) In Sect. 2, a brief introduction on the modeling of the icant progress in the last decades. As to the large-scale train– TTS dynamic interaction is presented. track interaction, methods seem to be matured gradually (2) In Sect. 3, the method for achieving large-scale TTS [49–51]. However, two main deﬁciencies still exist in the dynamic solution is elucidated. train–track–substructure dynamic simulation as follows: (3) In Sect. 4, numerical examples are presented to (1) The periodicity of the geometric and mechanical illustrate the practicality of this model and performing properties of track structures is widely adopted to be extensive work. capable of introducing highly efﬁcient methods such (4) In Sect. 5, conclusions are drawn from the numerical as Green function method and 2.5D ﬁnite element. studies. However, the track–substructure coupled system shows non-periodicity in most realistic cases for high-speed slab track system, because of the non- 2 Modeling of the train–track–substructure consistency and non-integer times of the slab length, dynamic interaction the bridge span and the tunnel ring length. Therefore, a more robust method based on ﬁnite elements still In this work, the subgrade is substituted by a Winkler needs to be developed. foundation represented by spring–dashpot elements, and (2) The typical substructures of bridge and tunnel are accordingly, the substructures indicate mainly the bridge generally treated as independent structural systems and tunnel systems. that participated in the train–track interaction instead In [48, 54], the train–track–bridge and train–track–tun- of continuously being considered in the train moving nel coupled systems are, respectively, established, as process, and consequently, the inﬂuence of multi- shown in Fig. 1, where the vehicle is modeled as a multi- substructural inﬂuence on train running performance rigid-body system, and the track and substructures are cannot be evaluated properly. modeled by various ﬁnite elements such as bar, beam, thin- plate, solid and iso-parametric elements. 1.3 Goal of this work In this work, a uniﬁed model is developed, in which typical bridge and tunnel substructures are integrated into To meet the above referred two issues, this work is devoted the vehicle–track system. As shown in Fig. 2, the track to developing a versatile modeling and computational structure represented by dotted lines are not required to be method to achieve large-scale train–track–substructure built; moreover, three coordinate systems are deﬁned, dynamic interaction in a 3D ﬁnite element framework, but namely global coordinate system O X Y Z , 0 0 0 0 this work is not involved in far-ﬁeld vibration of the soil moving coordinate system O X Y Z and local m m m m below or around the substructures. track–substructure coordinate system O X Y Z . l l l l The proposed method compiled in a uniﬁed computer Firstly, track model is established at the left side with a program is a far more step founded on a series of previous minimum length of l þ 2l , and generally set as an train b work, i.e., cyclic calculation method to solve inﬁnite length integral multiple of a slab length, and then the bridge, moving of a train on ﬁnite length track , multi-scale tunnel and their interaction to upper tracks will be ﬁnite element coupling method to achieve versatile inter- constructed. action matrix establishment between the track and various To characterize the interaction between sub-systems, substructures , multi-time-step iteration method to vehicle–track coupled dynamics developed by Zhai [2, 24] realize the track–substructure interaction at arbitrary is introduced, and the train, track and substructure can be moments or positions , the substructural modeling assembled by coupled matrix formulations as Rail. Eng. Science (2022) 30(2):162–182 On dynamic analysis method for large-scale train–track–substructure interaction 165 8 9 2 3 € direction and angles around the X-, Y- and Z-axes, and each > > M 00 < = VV wheelset has ﬁve DOFs, i.e., displacements in the lateral 4 5 € 0 M 0 X TT > > and vertical directions, and angles around the X-, Y- and Z- : ; 00 M SS S axes. 8 9 2 3 The dynamic equations of motion for the train can be > V > C C 0 < = VV VT assembled as 4 5 _ þ C C C TV TT TS T > > : ; 0 C C € _ ST SS _ M X þ C X þ K X ¼F : ð2Þ VV V VV V VV V V 8 9 2 3 > V > K K 0 < = VV VT The detail formulations for the train mass, damping, and 4 5 þ K K K X TV TT TS T stiffness matrix, i.e., M , C and K , have been VV VV VV > > : ; 0 K K ST SS illustrated in . 2 3 6 7 2.2 Track model ¼ 4 F 5; ð1Þ The track is modeled as a ballastless track system, con- sisting of the rail by Bernoulli–Euler beams, track slab by where M, C and K denote the mass, damping and stiffness _ € thin-plate elements in general, and the supporting layer by matrixes, respectively; F is the force vector; X, X and X solid elements generally regarded as base plate or base- denote the displacement, velocity and acceleration ment, or equivalently regarded as a mortar layer. The response vectors, respectively; the subscripts ‘‘V’’, ‘‘T’’ interaction between track layers is connected by spring– and ‘‘S’’ denote the train, track and substructure sub-system dashpot elements. respectively; the matrixes with subscripts ‘‘VV’’, ‘‘TT’’ and The dynamic equations of motion for the tracks can be ‘‘SS’’ denote the self-matrices of the train, track and sub- assembled as structure sub-system, respectively, and the ones with sub- scripts ‘‘VT’’, ‘‘TV’’, ‘‘TS’’ and ‘‘ST’’ denote the € _ M X þ C X þ K X ¼F : ð3Þ TT T TT T TT T T interaction matrices between sub-systems. For the matrix formulations, one can refer to . 2.1 Train model 2.3 Bridge model The train includes several identical vehicles regarded as The bridge, as a simply supported type, is modeled as an multi-rigid-body systems. Each vehicle consists of one car assemblage of girders by thin-plate elements and piles by body, two bogie frames and four wheelsets. The car body bar elements with extensive lateral motion and rotation. and the bogie frames have six degrees of freedom (DOFs), i.e., displacements in the longitudinal, lateral, vertical (a) (b) Tunnel–soil Car body (k , c ) ts,y interaction element ts,y , (k c ) ,z ,z ts ts Tunnel structure Bogie frame Rail Wheelset Track slab Support layer ksy sy Girder Pier ca cz Fig. 1 Two representative train–track–substructure dynamic system: a train–track–bridge interaction system; b train–track–tunnel interaction system Rail. Eng. Science (2022) 30(2):162–182 b 166 L. Xu Fig. 2 Train–track–substructure dynamic interaction model: a conﬁguration for the train–track–substructure system; b substructure 1: tunnel; c substructure 2: bridge The dynamic equations of motion for the bridge can be 2.5 General methods for the coupling of sub-systems assembled as  To form uniﬁed train–track interaction system and to pre- € _ M X þ C X þ K X ¼ F ; ð4Þ bb b bb b bb b b pare for track–substructure interaction, the wheel–rail where M , C and K are, respectively, the mass, coupling matrices and the interaction matrices between the bb bb bb damping and stiffness matrices of the bridge, respectively; track and substructures are required. X and F are, respectively, the displacement and force b b vectors of the bridge. For formulations of these matrixes, 2.5.1 Wheel–rail coupling matrices one can refer to . Derived from wheel–rail dynamics coupling method  2.4 Tunnel model and energy variation principle, the wheel–rail coupling matrices can be formulated as  8 9 As to the tunnel, which possesses particularity in spatial 2 3 > > K K 0 < = ww wI conﬁguration and components as a ring structure, eight- 4 5 K K K X Iw II Ir I node iso-parametric element is applied to model tunnel > > 0 K K rI rr segments and the segmental joints and ring joints are r 8 9 2 3 regarded as spring–dashpot elements. > w > C C 0 < = ww wI The dynamic equations of motion for the tunnel can be 4 5 þ C C C X Iw II Ir > > : ; assembled as 0 C C rI rr 2 3 € _ M X þ C X þ K X ¼F ; ð5Þ tt t tt t tt t t 0 6 7 ¼ 0 ; ð6Þ 4 5 where M , C and K are, respectively, the mass, damping tt tt tt and stiffness matrices of the tunnel, respectively; X and F 0 t t are, respectively, the displacement and force vectors of the where the subscripts ‘‘w’’, ‘‘I’’ and ‘‘r’’ denote the wheel, tunnel. For formulations of these matrixes, one can refer to track irregularity and rail, respectively. . Rail. Eng. Science (2022) 30(2):162–182 Computational region X X 0 0 O X x l l L 4 l l r t t 1 t b b Substructure 1 Substructure 2 Winkler Z Z Soil layer unnel structure ls Girder Bridge substructure Pier l On dynamic analysis method for large-scale train–track–substructure interaction 167 From Eq. (6), it is known that track irregularities have x_ ¼ a x þ a x been treated virtual DOFs in the wheel–rail interaction, and nþ1 0 nþ1 i niþ1 i¼1 accordingly the force vector in Eq. (2) and partial force ð10Þ vector in Eq. (3) can be obtained as € _ _ x ¼ a x þ a x : nþ1 0 nþ1 i niþ1 i¼1 F ¼K X C X þ G V wI I wI I V ; ð7Þ 10 15 1 1 F ¼K X C X with a ¼ , a ¼ , a ¼ , a ¼ ; where x, x_ and T rI I rI I 0 1 2 3 6Dt 6Dt Dt 6Dt x € denote the displacement, velocity and acceleration vector where G is the gravitational force vector of the train. of the dynamic system; Dt is the time step size; n denotes the n-th time step. 2.5.2 Interaction matrices between the track By introducing Eq. (10) into Eq. (1), it can be derived that and substructures M a x þ a G þ G þ CðÞ a x þ Gþ Kx nþ1 0 0 1 0 nþ1 0 nþ1 In the track–substructure iterative procedures, the track– ¼ FðÞ ðn þ 1ÞDt substructure interaction force must be transferred from one ð11Þ to another. The interaction matrices between the track and with substructure are therefore demanded. Using the multi-scale 3 3 X X ﬁnite element coupling strategy , we have G ¼ a x ; G ¼ a x_ ; 0 i niþ1 1 i niþ1 () () "# i¼1 i¼1 X 0 0 K T 0 C T TS TS þ ¼ : ð8Þ K 0 C 0 _ where F is the force vector. ST X ST 0 S X From Eq. (11), the displacement responses of related With acquisition of Eq. (8), the interaction forces to the DOFs of this dynamic system can be calculated by track and substructure system can be, respectively, hi 0 0 ~ ~ x ðÞ U ¼ K U F U ð12Þ obtained by nþ1 nþ1 nþ1 ~ _ F ¼K X C X T TS S TS S with : ð9Þ F ¼K X C X S ST T ST T 0 0 0 0 0 0 0 0 K U ;U ¼a M U ;U þa C U ;U þ K U ;U ; nþ1 0 Finally, the force vector acting can be assembled by ~ 0 0 0 0 F ¼ F þF . T T T ~ F U ¼F U a M U ; U G ðÞ U nþ1 nþ1 0 0 0 0 0 0 M U ; U G ðÞ U C U ; U G ðÞ U ; 1 0 3 Method for achieving large-scale train–track– where U and U represent the global DOFs of the dynamic sys- substructure dynamic interaction tem time-dependently following the moving train and the corre- sponding local DOFs at the stiffness, damping and mass matrices. To achieve the solution for such a large-scale dynamic sys- With acquisition of x ðÞ U , the velocity and acceler- tem, especially when long-length calculation and multi-type nþ1 ation vector can be consequently obtained by substructure are considered, practical computation tactics are required. Here a computational method for achieving x_ ðÞ U ¼ðÞ 10x ðÞ U 15x ðÞ U þ 6x ðÞ U x ðÞ U < nþ1 nþ1 n n1 n2 6Dt TTS interaction is elaborated, which mainly includes the time-domain direct integration method, cyclic calculation € _ _ _ _ x ðÞ U ¼ðÞ 10x ðÞ U 15x ðÞ U þ 6x ðÞ U x ðÞ U nþ1 nþ1 n n1 n2 6Dt method and iterative procedures for TTS interaction. ð13Þ 3.1 Park integration method From Eq. (13), it is known that the Park method can be only started with the responses of the previous three steps To solve this time-dependent nonlinear dynamic system, which are obtained by Wilson-h method. The Park method Park method  is selected, in which the integral schemes can be assembled by an operator form as follows: are expressed as ðÞ x € ; x_ ; x¼P M; C; K; F ; x ; x_ ; U; U ; nþ1 nþ1 nþ1 nþ1 n n2 n n2 ð14Þ Rail. Eng. Science (2022) 30(2):162–182 168 L. Xu m m U ¼ðÞ n t N þ 1, U ¼ðÞ t t þ 1 N , 0 1 c 2 1 c c;1 c;2 where the subscript ‘‘n n 2’’ denotes time steps n, n 0 0 with t ¼ x = L þ l þ 1, t ¼ x = L þ l þ 1, 1 and n 2, respectively. 1 4 t 2 1 t t t where the symbol ‘‘:’’ denotes an operator of the left number to right number with increment of 1; t and t 3.2 Mapping relation of the degrees of freedom 1 2 denote the track slab number with respect to the with respect to various coordinate systems positions of x and x , respectively; N denotes the total 4 1 c number of DOFs for a baseplate; n denotes the initial An improved cyclic calculation method has been devel- 0 baseplate against the start position of the substructure. oped in , and its essence is achieving the DOFs map- • When L \x \L is satisﬁed, ping between the global coordinate system x;1 4 x;2 O X Y Z l l l l l l O X Y Z and the moving coordinate system, U ¼ U : U ; 0 0 0 0 c c;1 c;2 O X Y Z . With the participation of the sub- l l m m m m U ¼ðÞ t n N þ 1, U ¼ðÞ t n þ 1 N ; 1 0 c 2 0 c c;1 c;2 structures, local track–substructure coordinate system O O X Y Z m m m m m m m U ¼ U : U ; U ¼1; c c;1 c;2 c;1 X Y Z is further established. l l l U ¼ðÞ t t þ 1 N : To ascertain DOFs of the sub-systems participating in the 2 1 c c;2 numerical integration scheme. The position of a train on the track–substructure systems should be pre-determined. For a For conditions x L : 1 x;2 train with vehicle numbers more than 2, the positions of the ﬁrst and last wheelset of the vehicles in a train at the global • When x \L is satisﬁed, 4 x;1 O X Y Z l l l coordinate O X Y Z are obtained by l l l l 0 0 0 0 U ¼ U : U , U ¼1, c c;1 c;2 c;1 x ¼ VT þð2l þ 2l þ l þ 2ðl þ l Þði 2Þþ l ði 2ÞÞ 1;i t;1 c;1 mt t;2 c;2 tt l U ¼ðþ n n 1ÞN , ; 1 0 c c;2 x ¼ VT ð2l þ 2l þ l þ 2ðl þ l Þði 1Þþ l ði 2ÞÞ 4;i t;1 c;1 mt t;2 c;2 tt O X Y Z m m m m m m U ¼ U : U , c c;1 c;2 ð15Þ U ¼ðÞ n t N þ 1, 0 1 c c;1 where V is the train speed; T is the running time; l and m t;1 U ¼ðÞ n t þ 1 N ; 1 1 c c;2 l are the semi-longitudinal distance between wheelsets in c;1 where n denotes the end baseplate number against the a bogie and between bogies for a motor vehicle, and l and t;2 end position of the substructure. l are the semi-longitudinal distance between wheelsets in c;2 • When L x \L is satisﬁed, a bogie and between bogies for a trailer; l is the distance x;1 4 x;2 mt O X Y Z l l l l l l l between the last wheelset of the motor and the ﬁrst U ¼ U : U , U ¼ðÞ t n N þ 1, 1 0 c c c;1 c;2 c;1 wheelset of the trailer and l is the distance between the l tt U ¼ðÞ n n þ 1 N , 1 0 c c;2 last and the ﬁrst wheelset between trailers; symbol ‘‘i’’ O X Y Z m m m m m m m U ¼ U : U , U ¼1, c c;1 c;2 c;1 denotes the i-th vehicle. U ¼ðÞ n t þ 1 N : From Eq. (15), the start and end positions of the com- 1 1 c c;2 putational region in O X Y Z can be conﬁrmed 0 0 0 0 c The calculation period after running through the as x ¼ x þ l , x ¼ x l , where n indicates the 1 1;i¼1 b 4 4;i¼n b substructure. total number of vehicles in a train. Obviously x [ x . 1 4 When x L is satisﬁed, only simulation for train– 1 x;2 In O X Y Z , the computational region is 0 0 0 0 track dynamic interaction is performed. constantly moving along the running of the train. We O X Y Z 0 0 0 0 0 0 In O X Y Z , U ¼ U : U 0 0 0 0 c c;1 c;2 denote by L and L the longitudinal initial and end x,1 x,2 0 0 where U ¼ðÞ t 1 N þ 1, U ¼t N . 1 c 2 c c;1 c;2 coordinates of the substructure, respectively. a The calculation period before running through the substructure: 3.3 Iterative procedures for this large-scale When x L , only simulation for train–track 1 x;1 dynamic system dynamic interaction is performed. b The calculation period running through the substructure: The non-iterative procedures for solving the train–track dynamic equations of motion have been presented in , For conditions L \x \L : x;1 1 x;2 here not presented for brevity. • When x L is satisﬁed, 4 x;1 In the iterative procedures, steps below can be followed: O X Y Z l l l l l l l O X Y Z U ¼ U : U ; U ¼1, 0 0 0 0 c c;1 c;2 c;1 Step 1 Set X U as convergence index. U ¼ðÞ t n þ 1 N , 2 0 c O X Y Z c;2 0 0 0 0 8 If X U e, where e¼ 10 , go to step 5; or O X Y Z m m m m m m U ¼ U : U , c c;1 c;2 go to step 2. Rail. Eng. Science (2022) 30(2):162–182 On dynamic analysis method for large-scale train–track–substructure interaction 169 0 0 0 Step 2 Calculate the train–track system responses. O X Y Z l l l l F ¼C n þ 1 : n þ N ; U cS gp c d d c The train–track system DOFs in the O X Y Z _ 0 0 0 0 O X Y Z and O X Y Z coordinate 0 0 0 0 m m m m X U j OmXmYmZm ; c U systems are, respectively, represented as U and U . The TT TT 0 0 0 O X Y Z l l l l force vector for the train–track system includes the wheel– F ¼K n þ 1 : n þ N ; U cS gp k d d c rail force vector F and the boundary force exerted by the O X Y Z 0 0 0 0 X U j O X Y Z ; m m m m substructure, that is, c U O X Y Z O X Y Z m m m m m m m m F n þ U ¼ F n þ U O X Y Z 0 0 0 0 TT d 0 d c c where U j O X Y Z denotes the DoF vector of m m m m c U F F F m c k supporting layer chosen from coordinate of ð16Þ O X Y Z . 0 0 0 0 Consequently, the TTS system responses by updating the with substructural system solution can be obtained by 0 0 O X Y Z l l l l F ¼ M U ; n þ 1 : n þ N m cS gp c d d ðÞ x € ; x_ ; x¼P M ; C ; K ; F ; x ; x_ ; U ; U ; nþ1 nþ1 nþ1 cS cS cS S n n2 n n2 S X U þ U þ U þ 1 : U þ U þ U þ N ; r t c r t c gp ð19Þ 0 0 O X Y Z l l l l F ¼ C U ; n þ 1 : n þ N c cS gp in which U ¼U þ U þ U þ 1 : U þ U þ U þ N , c d d S r t c r t c gp 0 0 0 and U ¼n þ 1 : n þ N . gp S d d X U þ U þ U þ 1 : U þ U þ U þ N ; r t c r t c gp 0 0 Step 4 Calculate the maximum absolute value O X Y Z l l l l F ¼K U ; n þ 1 : n þ N k cS gp c d d O X Y Z O X Y Z 0 0 0 0 0 0 0 0 X = max X U X U . error n1 n c c X U þ U þ U þ 1 : U þ U þ U þ N ; r t c r t c gp If X e is satisﬁed, go to Step 5; or go to Step 2. error where n and n are, respectively, the total number of Step 5 Jump out of the iterative loop and update the DOFs of the rail and track slab in the O X Y Z m m m m displacement and velocity response vector in the previous and O X Y Z coordinate systems; U , U and U l l l l r t c three steps, preparing for the next Park integration, namely denote the total number of DOFs of the rail, track slab and 8 8 x ¼x x_ ¼x_ > > n2 n1 n2 n1 < < support layer in O X Y Z coordinate system, 0 0 0 0 x ¼x ; and x_ ¼x_ : ð20Þ n1 n n1 n respectively; N is the total number of the substructural gp > > : : DOFs; M , C and K denote the mass, damping and x ¼x x_ ¼x_ cS cS cS n nþ1 n nþ1 stiffness matrices of the supporting layer–substructure Step 6 Perform non-iterative computation illustrated in coupling system, respectively. , or go to step 1 to conduct the next iteration solution. Then TTS system responses with train–track system In summary, the train–track–substructure interaction can solution update can be obtained by following Eq. (14)as be modeled and solved by the following framework, as shown in Fig. 3. ðÞ x € ; x_ ; x¼P M ; C ; K ; F ; x ; x_ ; U ; U : nþ1 nþ1 nþ1 TT TT TT TT n n2 n n2 TT TT ð17Þ 4 Numerical examples Step 3 Calculate the substructural system response. The force vector of the substructural system excited by Three examples are presented to show the accuracy, efﬁ- the supporting layer is ciency, and engineering practicality of this model in ana- 0 0 0 F ¼F F F ð18Þ m m m lyzing the TTS dynamic interactions. The ﬁrst one is to validate the effectiveness of the computational method, and with the second one is to illustrate the inﬂuence of substructures 0 0 0 O X Y Z l l l l F ¼M n þ 1 : n þ N ; U on the TTS system dynamic performance, and the last one cS gp m d d c is to show the inﬂuence of track irregularities on TTS O X Y Z € 0 0 0 0 X U j OmXmYmZm ; c U dynamic interactions. The train consists of three identical vehicles with a running speed of 300 km/h, the moving length of each time step is 0.1 m. The train parameters, track structures, and Rail. Eng. Science (2022) 30(2):162–182 170 L. Xu tunnel and bridge substructural parameters are shown in the substructure portions, remarkable response ﬂuctuation Tables 4, 5, 6 and 7 in Appendix. is noticed, and the deteriorating length coincides with the The conﬁguration of the TTS system (Fig. 4) in the geometry conﬁguration of the substructures. examples is illustrated as below: In a computer with Inter(R) Core (TM) i7-10700 K CPU e ¼ 0, l ¼ 35 m, l ¼ 141:24 m, L ¼ 99:42 m, @ 3.80 GHz, the time consumed by this model and the b s0 t l ¼ 31:18 m, and L ¼ 94:93 m representing a three-span combined model is, respectively, 2987 and 13,833 s. d b simply-supported bridge. Obviously this model possesses higher computational efﬁciency. Moreover, though coincident results have been 4.1 Validation of the proposed model obtained between this model and the combined model, the modeling complex and large computer storage are highly To validate the accuracy and efﬁciency of this model, a required in the combined TTS model; mostly important, combined TTS system based on the model in [48, 54]is long-length computation subject to various substructures constructed as the veriﬁcation model, in which an entire simultaneously cannot be conveniently achieved in previ- model following the TTS system conﬁguration is estab- ous models. lished. Using models of iteration and coupling, Figs. 5 and 6 present comparisons on the car body accelerations and 4.2 Clariﬁcation of the inﬂuence of substructures wheel–rail forces, and the time-dependent rail vibrations on train–track responses beneath the ﬁrst moving wheelset with and without track irregularity excitations, from which it can be observed that To clarify the inﬂuence of substructures on vehicle–track all system responses derived by this model and the com- dynamic performance, substructural conditions are set as. bined model agree rather well with each other, and gen- C : no substructures are considered. erally the difference is smaller than 0.01%, which makes C : only the tunnel substructure is considered. no difference in engineering practices. C : only the bridge substructure is considered. From the response curve of dynamic indices without C : both tunnel and bridge substructures are considered. track irregularity excitations (Fig. 5), it can be clearly seen For comparative analysis, the stiffness under the sup- that response periodicity appears due to the periodic porting layer is uniﬁed as the same along the longitudinal structural geometry characteristics of the rail, track slab, abscissa. The normal foundations are consolidated, and the tunnel segment and bridge girder span, etc. Besides, it can substructures are elastic ﬂexible bodies. The equivalent be observed from Fig. 4 that when a train moves through supporting stiffness in normal foundation regions is Trai n Trac k Substructure Car body Bogie fram es Wheelset Rail Track slab Support layer Br idge Tunnel Secondar y Prim ary Rail pads Filling l ayer Spring–dasphot elements suspension suspension Wheel–rail c oupli ng matri ces Trac k–substructure coupling matrices Solution method Cycl ic c alculati on method Adaptive iteration method Suitable scenarios Br idge and tunnel substruct ures Br idge and tunnel substruct ures are not involved in t he are involved in the computational boundaries computational boundaries Trai n–track interact ion analysis subj ect to various substructures time dependently Fig. 3 Modeling and analysis framework for train–track–substructure dynamic interaction Rail. Eng. Science (2022) 30(2):162–182 On dynamic analysis method for large-scale train–track–substructure interaction 171 Fig. 4 Conﬁguration of the TTS system (a) (b) (c) (d) Fig. 5 Comparisons on system responses without considering track irregularity excitations: a car body vertical acceleration; b wheel–rail vertical force; c rail vertical displacement; d rail vertical acceleration therefore larger than those in the substructural regions. performing maximum deviations of 7% and 1.65%. In Under this computational condition, the time responses of Fig. 9c, power spectral density (PSD) distributions are pre- car body vertical and lateral displacement are shown in sented against different track infrastructural conditions, for Figs. 7 and 8, respectively, from which when a train runs more clearly analyzing the inﬂuence brought out by the par- into the tunnel portion, both the car body vertical and lat- ticipation of the substructures. Set C ¼ PSD PSD , C C i i 1 eral displacement are increased, by about 0.328 mm and i ¼ 2; 3; 4. It can be seen from Fig. 9d that the inﬂuence of 0.085 mm, respectively, and substantially on the bridge substructures concentrates mainly on frequencies lower than portion, the car body displacements are also larger than 50 Hz. In the frequency range of 24.4–46.7 Hz, the car body those in normal foundations. vertical acceleration is decreased by the participation of Moreover, Fig. 9 further presents the response differences substructures, and the car body vibration is mostly deterio- of car body vertical acceleration against various track struc- rated in frequencies lower than 4.8 Hz. The main frequencies tural conditions. To separate the response difference, the with ampliﬁed car body accelerations exist around 1.22 and results of C are set as the reference values. As illustrated in 2.03 Hz for the C and C , respectively. 1 2 3 Figure 10 illustrates the related results of wheel–rail Fig. 9a, set C ¼ R R , i ¼ 2; 3; 4, and R denotes time- C C i i 1 vertical forces. It can be seen from Fig. 10b that the wheel– domain responses; signiﬁcant variations can be observed in rail force difference can reach 9.77 and 2.12 kN in the the tunnel and bridge sections, with the maximum differences -3 -3 tunnel and bridge sections respectively, namely causing of 5.596 9 10 g and 1.316 9 10 g respectively, that is, Rail. Eng. Science (2022) 30(2):162–182 g 172 L. Xu (a) (b) (c) (d) Fig. 6 Comparisons on system responses considering track irregularity excitations: a car body vertical acceleration; b wheel–rail vertical force; c rail vertical displacement; d rail vertical acceleration (a) (b) Fig. 7 Inﬂuence on car body vertical displacement: a under various track conditions; b response difference between C and C 1 4 maximum deviations of 4.83% and 1.01% compared to the sections due to the stiffness softening in supporting the maximum values of C conditions. Besides, it can be seen track structures. Besides, the track slab displacement, when from Fig. 10d that the inﬂuence of the substructures is the train moves through the tunnel, is larger than those focused on the frequency range of 12.21–64.29 Hz. moving through the bridge. As illustrated in Fig. 11c, d, the As to the track dynamic performance, set track slab displacement responses at frequencies lower than 38.25 Hz vibration as an example, the track slab vertical displace- are signiﬁcantly increased by the vibration participation of ment and acceleration are, respectively, shown in Figs. 11 the substructures. A notable frequency 15.46 Hz is and 12. Like the rail displacement curve shown in Fig. 5, observed, which corresponds to the wavelength of a track the track slab displacement shown in Fig. 11a also per- slab length. As to the slab acceleration, it can be seen from forms an amplitude ampliﬁcation tendency at substructural Fig. 12b that the slab acceleration is increased by about Rail. Eng. Science (2022) 30(2):162–182 g On dynamic analysis method for large-scale train–track–substructure interaction 173 (a) (b) Fig. 8 Inﬂuence on car body lateral displacement: a under various track conditions; b response difference between C and C 1 4 (a) (b) (c) (d) Fig. 9 Inﬂuence on car body vertical acceleration: a car body vertical acceleration for C condition; b the time-domain response difference between C and C ; c PSD against various conditions; d PSD difference between conditions C , C , C and C 1 4 2 3 4 1 22.57% with a maximum difference value of 6.95 m/s . 4.3 Inﬂuence of track irregularities on TTS Moreover, the slab acceleration is increased at a wide dynamic performance frequency range of less than 208 Hz, especially, the slab acceleration is greatly increased at the frequency range of To show the effectiveness of this model in revealing sub- 41–71 Hz, and increased at the characteristic low-fre- structural dynamics and illustrate the inﬂuence of track irregularities on TTS dynamic performance, two types of quency range caused by the tendency item of slab acceleration. track irregularity spectrum are adopted as excitations for comparisons: one is the China high-speed spectrum, and Rail. Eng. Science (2022) 30(2):162–182 PSD (g /Hz) PSD difference (g /Hz) 174 L. Xu (a) (b) (c) (d) Fig. 10 Inﬂuence on wheel–rail vertical forces: a wheel–rail vertical force for C condition; b the time-domain response difference between C 1 1 and C ; c PSD against various conditions; d PSD difference the other is the German high-speed low-disturbance spec- Set the time-domain track irregularities transformed trum; the detail expressions are as follows. from above two track irregularity spectrums as system excitations, as shown in Fig. 13. Figure 14 presents the • China high-speed spectrum : comparisons of car body lateral and vertical accelerations k under these two different excitation conditions, from which SðfÞ¼ Af ; ð12Þ the car body accelerations excited by the China spectrum where S denotes the power spectral density; f denotes the are far smaller than those by German spectrum, e.g., the spatial frequency; A and k denote the coefﬁcients as shown maximum car body lateral and vertical accelerations are, in Tables 1 and 2. respectively, 0.0025g and 0.0129g for the China spectrum, and 0.0081g and 0.0383g for German spectrum. Besides, • German high-speed low-disturbance spectrum : the car body acceleration by China spectrum is smaller than A X v that by German spectrum at almost all frequencies. > c S ðXÞ¼ ðvertical profileÞ > v > 2 2 2 2 > ðX þ X ÞðX þ X Þ Figure 15 further presents the comparisons on wheel–rail r c < 2 A X forces. The maximum wheel–rail lateral and vertical forces S ðXÞ¼ (alignment) ; 2 2 2 2 are, respectively, 8.68 and 136.63 kN with respect to the China ðX þ X ÞðX þ X Þ r c 2 2 spectrum, and 10.04 and 150.94 kN for the German spectrum, A X X > c > S ðXÞ¼ ðcross-levelÞ 2 2 2 2 2 2 namely the maximum wheel–rail forces excited by the China b ðX þ X ÞðX þ X ÞðX þ X Þ r c c c s spectrum are also smaller than those excited by German ð13Þ spectrum. However, it is noted that the wheel–rail forces where S , S and S denote the power spectral density of v a x excited by the China spectrum are larger than those by the vertical proﬁle irregularity, alignment irregularity and German spectrum at frequencies above 101.7 and 83.03 Hz. cross-level irregularity, respectively; X denotes the spatial Apart from the comparison on typical train dynamics, wavenumber, in rad/m; truncated wavenumbers X ¼ c inﬂuence of track irregularities on track–substructural 0:8246 rad/m and X ¼ 0:0206 rad/m; and for low-distur- system responses can be evaluated. From the results, the bance spectrum, coefﬁcients A ¼ 4:032 10 m rad, maximum rail vertical acceleration subject to the excitation 2 2 A ¼ 2:119 10 m rad, and b ¼ 0:75 m. of China spectrum is 193.05 m/s , but only 75.34 m/s for Rail. Eng. Science (2022) 30(2):162–182 On dynamic analysis method for large-scale train–track–substructure interaction 175 (a) (b) (c) (d) Fig. 11 Inﬂuence on track slab vertical displacement: a slab displacement against various conditions; b time-domain response difference; c PSD against various conditions; d PSD difference (a) (b) (c) (d) Fig. 12 Inﬂuence on track slab vertical acceleration: a slab acceleration against various conditions; b time-domain response difference; c PSD against various conditions; d PSD difference Rail. Eng. Science (2022) 30(2):162–182 176 L. Xu Table 1 The coefﬁcients for the ﬁtting formula of the track irregularity spectrum Item The 1st section The 2nd section The 3rd section The 4th section –5 –3 –4 –4 A (9 10 ) kA (9 10 ) kA (9 10 ) kA(9 10 ) k Gauge 5497.8 0.8282 5.0701 1.9037 1.8778 4.5948 – – Cross-level 361.48 1.7278 43.685 1.0461 45.867 2.0939 – – Alignment 395.13 1.8670 11.047 1.5354 7.5633 2.8171 – – Vertical proﬁle 1.0544 3.3891 3.5588 1.9271 197.84 1.3643 3.9488 3.4516 Table 2 The spatial frequency and corresponding wavelength of sectional points for the spectrum Item The 1st–2nd section The 2nd–3rd section The 3rd–4th section -1 -1 -1 Frequency (m ) Wavelength (m) Frequency (m ) concluavelength (m) Frequency (m ) Wavelength (m) Gauge 0.1090 9.2 0.2938 3.4 – – Cross-level 0.0258 38.8 0.1163 8.6 – – Alignment 0.0450 22.2 0.1234 8.1 – – Vertical proﬁle 0.0187 53.5 0.0474 21.1 0.1533 6.5 (a) (b) Fig. 13 Time-domain track irregularities: a vertical track irregularity; b lateral track irregularity German spectrum. The reason lies in the high sensitivity of the German spectrum. It can be observed from Fig. 16 that the the rail beam to high-frequency excitations. As displayed in differences of tunnel lateral acceleration shown in Fig. 17 Fig. 15, German spectrum possesses better status in high- with respect to these two track spectrums mainly exist in frequency track irregularities compared to the China frequencies larger than 101.7 Hz, where the tunnel lateral spectrum, and consequently, the rail acceleration against acceleration excited by China high-speed spectrum is sig- China spectrum at the high-frequency domain is signiﬁ- niﬁcantly larger than that excited by the German low-distur- cantly larger than that of the German spectrum. bance spectrum, and it causes the larger values of tunnel As to the track slab, tunnel segment and bridge girder lateral acceleration excited by China spectrum. accelerations, the maximum lateral and vertical accelerations are listed in Table 3. It can be seen from Table 3 that the vertical accelerations of the track slab, tunnel, and bridge with 5 Conclusions respect to China high-speed spectrum are smaller than those with respect to German low-disturbance spectrum. It is found An original method for solving large-scale train–track– that the lateral accelerations of the track slab and tunnel under substructure dynamic interaction has been presented in this China spectrum excitations are obviously larger than those of paper for the ﬁrst time. The numerical model is based on a Rail. Eng. Science (2022) 30(2):162–182 On dynamic analysis method for large-scale train–track–substructure interaction 177 (a) (b) -0 (c) (d) Fig. 14 Comparisons on car body accelerations: a lateral acceleration at time-domain; b PSD of lateral acceleration; c vertical acceleration at time-domain; d PSD of vertical acceleration (a) (b) (c) (d) Fig. 15 Comparisons on wheel–rail forces: a lateral acceleration at time-domain; b PSD of lateral acceleration; c vertical acceleration at time- domain; d PSD of vertical acceleration Rail. Eng. Science (2022) 30(2):162–182 g 178 L. Xu Table 3 Maximum track and substructural acceleration Track Component China high-speed spectrum German low-disturbance spectrum 2 2 2 2 Lateral (m/s ) Vertical (m/s ) Lateral (m/s ) Vertical (m/s ) Track slab 1.01 4.89 0.66 7.24 Tunnel 1.30 3.73 0.60 6.20 Bridge 0.39 0.62 0.59 0.81 (a) (b) Fig. 16 Comparisons on rail vertical acceleration: a time-domain responses; b PSD responses (a) (b) Fig. 17 Comparisons on tunnel lateral acceleration: a time-domain responses; b PSD responses coupled train–track model and substructural ﬁnite element of multiple substructures, such as tunnel and bridge, in the models and solved in the time domain. In a robust and cost- train–track interaction process at arbitrary time or position effective manner, substructures are modeled as indepen- by DOF mapping relation and iterative solution. dent matrices including their interaction matrices to the Numerical applications show that this model is equally upper track. The train–track system is coupled as an entire accurate to the train–track–substructure coupled solution, one by matrix equations, only demanding a rather limited but with higher efﬁciency and less computer storage con- modeling length, namely satisfying being larger than the sumption. Numerical studies have shown that the sub- total length of a train and two times the boundary length, structures have relatively slight inﬂuence on train dynamic and then, the inﬁnite length of a train moving on tracks can behaviors, e.g., the car body vertical acceleration and be achieved by a cyclic calculation method. An ingenious wheel–rail vertical force are increased by 7% and 4.83%, tactics is developed to consider the vibration participation respectively. However, the track structure vibrations are Rail. Eng. Science (2022) 30(2):162–182 On dynamic analysis method for large-scale train–track–substructure interaction 179 Open Access This article is licensed under a Creative Commons signiﬁcantly inﬂuenced no matter on response amplitudes Attribution 4.0 International License, which permits use, sharing, or on response frequencies, besides, the train–track system adaptation, distribution and reproduction in any medium or format, as responses are generally ampliﬁed at the low-frequency long as you give appropriate credit to the original author(s) and the domain, but the sensitive frequencies are different for source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this different sub-systems, depending on the parametric prop- article are included in the article’s Creative Commons licence, unless erties of the substructures. Moreover, the inﬂuence of track indicated otherwise in a credit line to the material. If material is not irregularities on train–track–substructure system perfor- included in the article’s Creative Commons licence and your intended mance has also been clariﬁed. It has proved that the high use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright superiority of China high-speed spectrum in controlling holder. To view a copy of this licence, visit http://creativecommons. middle–long wavelength irregularities but being worse in org/licenses/by/4.0/. short-wavelength irregularities compared to the German low-disturbance spectrum. Acknowledgments This work was supported by the National Natural Appendix Science Foundation of China (Grant Nos. 52008404), and the National Natural Science Foundation of Hunan Province (Grant No. 2021JJ30850). See Tables 4–7. Table 4 Main parameters of the vehicle Notation Parameter Value M Car body mass (kg) 48,000 M Bogie mass (kg) 3200 M Wheelset mass (kg) 2400 I Mass moment of inertia of car body about X-axis (kgm ) 115,000 cx I Mass moment of inertia of car body about Y-axis (kgm ) 2,300,000 cy I Mass moment of inertia of car body about Z-axis (kgm ) 2,300,000 cz I Mass moment of inertia of bogie about X-axis (kgm ) 3200 tx I Mass moment of inertia of bogie about Y-axis (kgm ) 7200 ty I Mass moment of inertia of bogie about Z-axis (kgm ) 6800 tz I Mass moment of inertia of wheelset about X-axis (kgm ) 1200 wx I Mass moment of inertia of wheelset about Y-axis (kgm ) 200 wy I Mass moment of inertia of wheelset about Z-axis (kgm ) 1200 wz k Stiffness coefﬁcient of primary suspension along X-axis (MN/m) 9 px k Stiffness coefﬁcient of primary suspension along Y-axis (MN/m) 3 py k Stiffness coefﬁcient of primary suspension along Z-axis (MN/m) 1.04 pz k Stiffness coefﬁcient of secondary suspension along X-axis (MN/m) 0.24 sx k Stiffness coefﬁcient of secondary suspension along Y-axis (MN/m) 0.24 sy k Stiffness coefﬁcient of secondary suspension along Z-axis (MN/m) 0.40 sz c Damping coefﬁcient of primary suspension along Z-axis (kNs/m) 45 pz c Damping coefﬁcient of secondary suspension along Y-axis (kNs/m) 3 sy c Damping coefﬁcient of secondary suspension along Z-axis (kNs/m) 98 sz L Semi-longitudinal distance between bogies (m) 7.85 L Semi-longitudinal distance between wheelsets in bogie (m) 1.25 Y Semi-lateral distance for the primary suspension system (m) 0.988 Y Semi-lateral distance for the secondary suspension system (m) 0.988 H Vertical distance between car body centroid to the upper plane of the secondary suspension (m) 1.112 cb H Vertical distance between bogie centroid to the lower plane of the secondary suspension (m) -0.081 bt H Vertical distance between the centroids of the bogie and the wheelset (m) 0.14 tw R Wheel radius (m) 0.46 Rail. Eng. Science (2022) 30(2):162–182 180 L. Xu Table 5 Main parameters of track structures Notation Parameter Value 2 11 E Elastic modulus of the rail (N/m ) 2.059 9 10 q Mass of the rail per unit length (kg/m) 60.64 4 –5 I Rail second moment of the area about the Y-axis (m ) 3.215 9 10 ry 4 –6 I Rail second moment of the area about the Z-axis (m ) 5.24 9 10 rz k Fastener stiffness in the vertical direction (N/m) 3 9 10 rz k Fastener stiffness in the lateral direction (N/m) 3.5 9 10 ry c Fastener damping in the vertical direction (Ns/m) 6 9 10 rz c Fastener damping in the lateral direction (Ns/m) 6 9 10 ry 2 10 E Elastic modulus of the track slab (N/m ) 3.564 9 10 l Poisson ratio of the concrete slab 0.2 m Mass per unit volume of the slab (kg/m ) 2500 H Height of the slab (m) 0.19 L Length of the slab (m) 5.35 W Width of the slab (m) 2.5 E Elastic modulus of the supporting layer (Pa) 3.45 9 10 tz H Height of the supporting layer (m) 0.09 3 12 k Vertical contact area stiffness between the track slab and supporting layer (N/m ) 2.3266 9 10 ts;z 3 12 k Lateral contact area stiffness between the track slab and supporting layer (N/m ) 1.4125 9 10 ts;y 3 8 k Vertical contact area stiffness between the supporting layer and substructure (N/m ) 3.5 9 10 ss;z 3 9 k Lateral contact area stiffness between the supporting layer and substructure (N/m ) 1.0 9 10 ss;y 3 4 c Vertical contact area damping between the track slab and supporting layer (Ns/m)5 9 10 ts;z 3 4 c Lateral contact area damping between the track slab and supporting layer (Ns/m)5 9 10 ts;y Table 6 Main parameters of the tunnel Notation Parameter Values 2 10 E Elastic modulus of the tunnel segment (N/m ) 3.25 9 10 l Poisson ratio of the tunnel segment 0.3 l Length of a tunnel segment (m) 1.5 d Spacing between adjacent two tunnel segments (m) 0.02 ðr ; r Þ Radius of inner and outer circle of the tunnel wall (m) (3, 3.2) 1 2 n Number of tunnel segment in a tunnel ring 6 8 9 9 (k ; k ; k ) Longitudinal, lateral, and vertical interaction stiffness coefﬁcients between tunnel segments (N/ (10 , 3.8 9 10 , 1.33 9 10 ) xx yy zz m ) 4 4 4 (c ; c ; c ) Longitudinal, lateral, and vertical interaction damping coefﬁcients between tunnel segments (10 , 3.8 9 10 , 1.33 9 10 ) xx yy zz (Ns/m ) 0 0 0 7 7 Longitudinal, lateral, and vertical interaction stiffness coefﬁcients between the tunnel wall and (1.7 9 10 , 4.0 9 10 , (k ; k ; k ) xx yy zz 3 7 soil (N/m ) 1.7 9 10 ) 5 5 5 (c ; c ; c ) Longitudinal, lateral, and vertical interaction damping coefﬁcients between tunnel segments (10 ,10 ,10 ) xx yy zz (Ns/m ) Rail. Eng. Science (2022) 30(2):162–182 On dynamic analysis method for large-scale train–track–substructure interaction 181 Table 7 Main parameters of the bridge girder and pier Notation Parameter Value 2 10 E Elastic modulus of the bridge girder (N/m ) 3.45 9 10 l Poisson ratio of the girder 0.2 m Mass per unit volume of the girder (kg/m ) 2549 H Height of the pier (m) 5 A Cross-sectional area of the pier (m ) 26.72 2 10 E Elastic modulus of the bridge pier (N/m ) 7.5 9 10 10 10 (k ; k ) Lateral and vertical interaction stiffness coefﬁcients between the pier and soil (N/m) (1.5 9 10 ,1 9 10 ) ps;y ps;z 4 4 (c ; c ) Lateral and vertical interaction damping coefﬁcients between the pier and soil (Ns /m) (1.5 9 10 ,1 9 10 ) ps;y ps;z 16. Xu L, Li Z, Zhao Y, Yu Z, Wang K (2020) Modelling of vehicle– References track related dynamics: a development of multi-ﬁnite-element coupling method and multi-time-step solution method. Veh Syst 1. 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Railway Engineering Science – Springer Journals
Published: Jun 1, 2022
Keywords: Train; Track dynamic interaction; Railway substructures; Finite elements; Dynamics system; Iterative solution; Tunnel; Bridge
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