# Random Vibration Analysis of Train Moving over Slab Track on Bridge under Track Irregularities and Earthquakes by Pseudoexcitation Method

Authors: Zhiping Zeng; Kunteng Zhu; Xianfeng He; Wentao Xu; Lingkun Chen; Ping Lou
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

This paper investigates the random vibration and the dynamic reliability of operation stability of train moving over slab track on bridge under track irregularities and earthquakes by the pseudoexcitation method (PEM). Each vehicle is modeled by multibody dynamics. The track and bridge is simulated by a rail-slab-girder-pier interaction finite element model. The coupling equations of motion are established based on the wheel-rail interaction relationship. The random excitations of the track irregularities and seismic accelerations are transformed into a series of deterministic pseudoexcitations by PEM. The time-dependent power spectral densities (PSDs) of the random vibration of the system are obtained by step-by-step integration method, and the corresponding dynamic reliability is estimated based on the first-passage failure criterion. A case study is then presented in which a high-speed train moves over a slab track resting on a simply supported girder bridge. The PSD characteristics of the random vibration of the bridge and train are analyzed, the influence of the wheel-rail-bridge interaction models on the random vibration of the bridge and train is discussed, and furthermore the influence of train speed, earthquake intensity, and pier height on the dynamic reliability of train operation stability is studied.


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## Body

## 1. Introduction

High-speed railway has become one of the most important forms of public transportation in many countries [1–3]. Meanwhile, bridges have been widely used as the supporting structures for high-speed railway [4, 5]; for example, bridges account for approximately 75% of the length of the Taipei-Kaohsiung high-speed railway in Taiwan [6]. This considerably increases the probability of earthquakes taking place when trains are crossing bridges. As for railway bridges, it is possible that the bridge itself may remain safe during an earthquake but may not be safe enough for the train to move over it due to excessive vibration of the sustaining bridge [7]. Evidently, the dynamic response characteristic and the stability of the moving trains over bridges shaken by earthquakes have become a subject of great concern for the railway engineers [8–11], especially in those countries that are earthquake-prone [12–14]. In the study by Miura [15], the emphasis was placed on the earthquake-induced displacement of track and structures, as well as the damage of trains caused by earthquake excitations, rather than on the dynamic stability of trains during an earthquake. Miyamoto et al. [16] analyzed analytically the operation safety of railway vehicles under the action of earthquakes using a three-dimensional simplified vehicle model, where sine waves were used as the input excitation and the vehicle was assumed to remain stationary on the track. Yang et al. [17] investigated the dynamic stability of the train, initially static or traveling over bridges shaken by earthquakes by a three-dimensional model for the train, track, and bridge. In their works, the maximum allowable speeds for the train to run safely under four specific seismic accelerations were evaluated on some threshold values. Matsumoto et al. [18] developed a vehicle/structure dynamic interaction analysis program, known as DIASTARS, to study the operation safety of railway vehicles subjected to earthquake motion. Zhang et al. [19] proposed a method to simulate the dynamic responses of the vehicle-bridge interaction system under multisupport seismic excitation, by which the histories of the dynamic responses were obtained for a high-speed train traversing a steel truss cable-stayed bridge with different seismic intensities and different train speeds. In these pioneering works, most researchers either neglected the track system or took only the conventional ballasted track into consideration. Since the track system is a flexible medium vibrating with the train and bridge, it can affect the extent of interaction between the two subsystems, especially for the train moving in the high-speed range. To the knowledge of the authors, rather few researches have conducted on the random vibrations and the stability of trains running over ballastless track on bridges shaken by earthquake. It follows that the index values computed for the evaluation of the random vibrations and the stability of trains may not be accurate enough for modern high-speed and urban railways, in which the ballastless tracks, for example, slab track, are widely used [20–25]. On the other hand, the track irregularity and seismic motion are usually only treated as one or few time-history samples to compute the dynamic responses of train-track/bridge interaction system in most of the previous literatures. In fact, these results can only be regarded as the particular cases of a sequence of possible outcomes because of the randomness of track irregularity and seismic motion. Therefore it is of great importance to evaluate the random vibrations and the stability of the moving trains on a random vibration basis in order to ensure the reliability of the simulation. Unfortunately, the conventional approach for random vibration analysis [26] is computationally inefficient, especially for the sophisticated train-slab track-bridge interaction (TSTBI) system. Therefore more efficient and accurate algorithm should be employed to analyze the random vibrations of the TSTBI system as well as the stability of the moving trains, such as PEM [6, 27–29]. Zhang et al. [6] have investigated the random vibrations for the train-bridge system subjected to horizontal earthquake by PEM. However, the track was assumed to be attached firmly to the girder, and the wheel displacements were assumed to be fully constrained by the girder displacements and track irregularities; that is, the inherent relative displacements between the wheel, the track, and the girder were not taken into consideration.In present paper, a three-dimensional TSTBI model is constructed based on the assumptions made for modeling such a system. The equations of motion for the major components of the model, that is, the vehicle, rail, slab, girder, and pier, are formulated by means of the finite element method (FEM) and energy principle [30], and, using these, the equations of motion for the entire TSTBI system are assembled. The track irregularities are regarded as a series of uniformly modulated, multipoint, different-phase random excitations by taking the time lags between the wheels into account, while the earthquakes are assumed as a series of uniformly modulated, nonstationary, evolutionary random excitations. Thus, the random excitations caused by the track irregularities and earthquakes are transformed into a series of deterministic pseudoharmonic excitation vectors according to PEM and the wheel-rail interaction relationship, so that the time-dependent PSDs of the random vibration responses of the system excited by track irregularities and earthquakes can be obtained by step-by-step integration method such as Wilson-θ method. A high-speed train consisting of eight vehicles moving over a slab track resting on a fifteen-span simply supported girder bridge is presented. Firstly, the PSD characteristics of the random dynamic responses of the bridge and train are investigated. Secondly, the influence of the wheel-rail-bridge interaction models on the random vibration characteristic of the bridge and train are studied. Thirdly, the influence of the train speed, the earthquake intensity, and the pier height on the dynamic reliability of train operation stability is analyzed. Finally, some useful conclusions are drawn.


## 2. Three-Dimensional Models of Train, Slab Track, and Bridge

Figure1 depicts a train consisting of a series of four-axle vehicles moving with a constant speed v on a slab track resting on a simply supported girder bridge shaken by the track irregularities and earthquakes. For the sake of simplicity, it is assumed that no inelastic deformation occurs on each subsystem during earthquakes.Figure 1
Three-dimensional model for the TSTBI system. (a) Frontal view. (b) Left side view (thejth trailer car, track, and bridge). (c) Top view (half of the jth trailer car without track and bridge).
(a)
(b)
(c)


### 2.1. Model of Train

The train consists of the rear and front motor cars numbered 1 and 2, respectively, andNv trailer cars numbered 1,2,…,Nv from left to right.Each trailer car in the train is modeled as a mass-spring-damper system consisting of one carbody, two bogies, four wheelsets, and a two-stage suspension system. As shown in Figure1, the carbody rests on the front and rear bogies, each of which in turn is supported by two wheelsets. The carbody is modeled as a rigid body with mass mc and three moments of inertia Icx, Icy, and Icz about the x-axis, y-axis, and z-axis through its center of gravity. Similarly, each bogie is considered as a rigid body with mass mt and three moments of inertia Itx, Ity, and Itz. Each wheelset is considered as a rigid body with mass mw and two moments of inertia Iwx and Iwz. The secondary suspension between the carbody and each bogie is characterized by a three-dimensional spring-damper system with stiffness ksx, ksy, and ksz and damping coefficients csx, csy, and csz, respectively. Likewise the springs and shock absorbers in the primary suspension for each wheelset are characterized by kpx, kpy, and kpz and cpx, cpy, and cpz, respectively. By neglecting of the longitudinal motions, the motions of the jth trailer carbody with respect to its center of gravity may be described by the lateral displacement ycj, the vertical displacement zcj, the rolling displacement θcj, the pitching displacement φcj, and the yawing displacement ψcj, where the subscript j denotes the trailer car number. Similarly, the motions of both the rear and front bogies of the jth trailer car may be also described, respectively, by yt1j, zt1j, θt1j, φt1j, and ψt1j and yt2j, zt2j, θt2j, φt2j, and ψt2j. The motion from left to right of the hth (h = 1–4) wheelset of the jth trailer car may be described by ywhj, zwhj, θwhj, and ψwhj, respectively. Therefore, the total number of DOFs for each trailer car is 31. However, it is assumed that no jump occurs between each wheel of all vehicles and the rail in this paper; that is, the vertical and rolling displacements of each wheelset are constrained by the displacements of the rail. Consequently, the independent DOFs for each trailer car become 23.Similarly, each motor carbody has massmJc and three moments of inertia IJcx, IJcy, and IJcz. Each bogie has mass mJt and three moments of inertia Itx, Ity, and Itz. Each wheelset has mass mJw and two moments of inertia IJwx and IJwz. The secondary suspension is characterized by springs with stiffness kJsx, kJsy, and kJsz and dampers with damping coefficients cJsx, cJsy, and cJsz. The primary suspension is characterized by springs with stiffness kJpx, kJpy, and kJpz and dampers with damping coefficients cJpx, cJpy, and cJpz. The DOFs of the carbody are denoted as yJci, zJci, θJci, φJci, and ψJci. The DOFs of the rear bogie are denoted as yJt1i, zJt1i, θJt1i, φJt1i, and ψJt1i. The DOFs of the front bogie are denoted as yJt2i, zJt2i, θJt2i, φJt2i, and ψJt2i. The DOFs of the four wheelsets are denoted as yJw1i–yJw4i and ψJw1i–ψJw4i. Herein the subscript i denotes the motor car number.


### 2.2. Models of Slab Track and Bridge

As shown in Figure1, the rail, slab, girder, and pier are all modeled as elastic Bernoulli-Euler beams. On the basis of FEM, the rail, slab, girder, and pier are all divided into a series of beam elements. The lateral elasticity and damping properties of the fastener are represented by discrete massless springs with stiffness krsy and dampers with damping coefficient crsy. The vertical elasticity and damping properties of the fastener are represented by krsz and crsz. The lateral elasticity and damping properties of the cement asphalt mortar (CAM) beneath the slab are represented by continuous massless springs with stiffness k-sby and dampers with damping coefficient c-sby. The vertical elasticity and damping properties of CAM are represented by k-sbz and c-sbz. The elasticity and damping properties of the bridge bearing are represented by massless springs with stiffness kbby, kbbz, and kbbθ and dampers with damping coefficients cbby, cbbz, and cbbθ. In addition, the damping property of the beams is assumed to be of the Rayleigh type [17]. By neglecting the longitudinal motion, each node of the rail, slab, and girder has five DOFs, that is, lateral displacement, vertical displacement, and rotations about the x-axis, y-axis, and z-axis. Each node of the pier has three DOFs, that is, lateral displacement, vertical displacement, and rotation about the x-axis.


## 2.1. Model of Train

The train consists of the rear and front motor cars numbered 1 and 2, respectively, andNv trailer cars numbered 1,2,…,Nv from left to right.Each trailer car in the train is modeled as a mass-spring-damper system consisting of one carbody, two bogies, four wheelsets, and a two-stage suspension system. As shown in Figure1, the carbody rests on the front and rear bogies, each of which in turn is supported by two wheelsets. The carbody is modeled as a rigid body with mass mc and three moments of inertia Icx, Icy, and Icz about the x-axis, y-axis, and z-axis through its center of gravity. Similarly, each bogie is considered as a rigid body with mass mt and three moments of inertia Itx, Ity, and Itz. Each wheelset is considered as a rigid body with mass mw and two moments of inertia Iwx and Iwz. The secondary suspension between the carbody and each bogie is characterized by a three-dimensional spring-damper system with stiffness ksx, ksy, and ksz and damping coefficients csx, csy, and csz, respectively. Likewise the springs and shock absorbers in the primary suspension for each wheelset are characterized by kpx, kpy, and kpz and cpx, cpy, and cpz, respectively. By neglecting of the longitudinal motions, the motions of the jth trailer carbody with respect to its center of gravity may be described by the lateral displacement ycj, the vertical displacement zcj, the rolling displacement θcj, the pitching displacement φcj, and the yawing displacement ψcj, where the subscript j denotes the trailer car number. Similarly, the motions of both the rear and front bogies of the jth trailer car may be also described, respectively, by yt1j, zt1j, θt1j, φt1j, and ψt1j and yt2j, zt2j, θt2j, φt2j, and ψt2j. The motion from left to right of the hth (h = 1–4) wheelset of the jth trailer car may be described by ywhj, zwhj, θwhj, and ψwhj, respectively. Therefore, the total number of DOFs for each trailer car is 31. However, it is assumed that no jump occurs between each wheel of all vehicles and the rail in this paper; that is, the vertical and rolling displacements of each wheelset are constrained by the displacements of the rail. Consequently, the independent DOFs for each trailer car become 23.Similarly, each motor carbody has massmJc and three moments of inertia IJcx, IJcy, and IJcz. Each bogie has mass mJt and three moments of inertia Itx, Ity, and Itz. Each wheelset has mass mJw and two moments of inertia IJwx and IJwz. The secondary suspension is characterized by springs with stiffness kJsx, kJsy, and kJsz and dampers with damping coefficients cJsx, cJsy, and cJsz. The primary suspension is characterized by springs with stiffness kJpx, kJpy, and kJpz and dampers with damping coefficients cJpx, cJpy, and cJpz. The DOFs of the carbody are denoted as yJci, zJci, θJci, φJci, and ψJci. The DOFs of the rear bogie are denoted as yJt1i, zJt1i, θJt1i, φJt1i, and ψJt1i. The DOFs of the front bogie are denoted as yJt2i, zJt2i, θJt2i, φJt2i, and ψJt2i. The DOFs of the four wheelsets are denoted as yJw1i–yJw4i and ψJw1i–ψJw4i. Herein the subscript i denotes the motor car number.


## 2.2. Models of Slab Track and Bridge

As shown in Figure1, the rail, slab, girder, and pier are all modeled as elastic Bernoulli-Euler beams. On the basis of FEM, the rail, slab, girder, and pier are all divided into a series of beam elements. The lateral elasticity and damping properties of the fastener are represented by discrete massless springs with stiffness krsy and dampers with damping coefficient crsy. The vertical elasticity and damping properties of the fastener are represented by krsz and crsz. The lateral elasticity and damping properties of the cement asphalt mortar (CAM) beneath the slab are represented by continuous massless springs with stiffness k-sby and dampers with damping coefficient c-sby. The vertical elasticity and damping properties of CAM are represented by k-sbz and c-sbz. The elasticity and damping properties of the bridge bearing are represented by massless springs with stiffness kbby, kbbz, and kbbθ and dampers with damping coefficients cbby, cbbz, and cbbθ. In addition, the damping property of the beams is assumed to be of the Rayleigh type [17]. By neglecting the longitudinal motion, each node of the rail, slab, and girder has five DOFs, that is, lateral displacement, vertical displacement, and rotations about the x-axis, y-axis, and z-axis. Each node of the pier has three DOFs, that is, lateral displacement, vertical displacement, and rotation about the x-axis.


## 3. Equations of Motion for the TSTBI System

By using the energy principle [30], one can derive the three-dimensional equations of motion written in submatrix form for the TSTBI system as(1)Mtt00000Mrr00000Mss00000Mbb00000MppX¨tX¨rX¨sX¨bX¨p+CttCtr000CrtCrrCrs000CsrCssCsb000CbsCbbCbp000CpbCppX˙tX˙rX˙sX˙bX˙p+KttKtr000KrtKrrKrs000KsrKssKsb000KbsKbbKbp000KpbKppXtXrXsXbXp=FtFrFsFbFp,where the subscripts “t,” “r,” “s,” “b,” and “p” denote the train, rail, slab, girder, and pier, respectively. The displacement vector, mass matrix, stiffness matrix, damping matrix, and the load vector of the train, rail, slab, girder, and pier are explained as follows briefly, and the detailed derivation can refer to [30, 31].


### 3.1. Displacement Vectors for Tran, Rail, Slab, Girder, and Pier

The total train displacement vectorXt of order Tdof×1(Tdof=23×Nv+23×2) can be written as(2)Xt=XJ1Xv1Xv2⋯XvNvXJ2T,where the superscript “T” denotes the transpose of the matrix and Xvj(j=1,2,…,Nv) and XJi(i=1,2) the displacement vectors of the jth trailer car and the ith motor car, respectively. These can be expressed as(3)Xvj=ycjzcjθcjφcjψcjyt1jzt1jθt1jφt1jψt1jyt2jzt2jθt2jφt2jψt2jyw1jψw1jyw2jψw2jyw3jψw3jyw4jψw4j,XJi=yJcizJciθJciφJciψJciyJt1izJt1iθJt1iφJt1iψJt1iyJt2izJt2iθJt2iφJt2iψJt2iyJw1iψJw1iyJw2iψJw2iyJw3iψJw3iyJw4iψJw4i.The displacement vector of the railXr of order 2N-r×1, comprising the displacement vectors XLr of the left rail and XRr of the right rail, both of order N-r×1, can be written as(4)Xr=XLrXRrT,XLr=XRr=qr1qr2⋯qrN-r,where N-r denotes the total number of DOFs of each rail.Similarly, the displacement vector of the slabXs of order N-s×1, the displacement vector of the girder Xb of order N-b×1, and the displacement vector of the pier Xp of order N-p×1 can be written, respectively, as(5)Xs=Xs1Xs2⋯XsNsT,(6)Xb=Xb1Xb2⋯XbNbT,(7)Xp=Xp1Xp2⋯XpNpT,where Xsi  (i=1,2,…,Ns) denotes the displacement vector of the ith slab, Ns the total number of slabs, and N-s the total number of DOFs of all slabs; Xbi (i=1,2,…,Nb) denotes the displacement vector of the ith girder, Nb the total number of girders, and N-b the total number of DOFs of all girders; and Xpi (i=1,2,…,Np) denotes the displacement vector of the ith pier, Np the total number of piers, and N-p the total number of DOFs of all piers.


### 3.2. Matrices for Train, Rail, Slab, Girder, and Pier

The matrices of the train are marked with the subscript “tt.” The mass matrix Mtt, the stiffness matrix Ktt, and the damping matrix Ctt of the train, all of order (23×Nv+23×2)×(23×Nv+23×2), can be written, respectively, as(8)Mtt=diag⁡MJ1Mv1Mv2⋯MvNvMJ2,(9)Ktt=diag⁡KJ1Kv1Kv2⋯KvNvKJ2,(10)Ctt=diag⁡CJ1Cv1Cv2⋯CvNvCJ2,where Mvj, Kvj, and Cvj, all of order 23 × 23, denote the mass, stiffness, and damping matrices of the jth trailer car, respectively; MJi, KJi, and CJi, all of order 23 × 23, denote the mass, stiffness, and damping matrices of the ith motor car, respectively.The matrices of the rail are marked with the subscript “rr.” The mass matrix Mrr of the rail with order 2N-r×2N-r, comprising mass matrices MLrr of the left rail and MRrr of the right rail, both of order N-r×N-r, can be written as(11)Mrr=diag⁡MLrrMRrr.The stiffness matrixKrr of the rail of order 2N-r×2N-r, which is composed of the stiffness matrices of the left and right rails KLrr and KRrr, both of order N-r×N-r, and the left rail-right rail interaction stiffness matrices KLrRr and KRrLr of order N-r×N-r, can be written as(12)Krr=KLrrKLrRrKRrLrKRrr,where the stiffness matrices KLrRr and KRrLr are induced by the train’s weight acting upon the rails by the wheelsets.The damping matrixCrr of the rail of order 2N-r×2N-r, consisting of the damping matrices of the left and right rails CLrr and CRrr, respectively, both of order N-r×N-r, and can be written as(13)Crr=diag⁡CLrrCRrr.The matrices of the slab, girder, and pier, marked with the subscripts “ss,” “bb,” and “pp,” respectively, are not given here but can be derived by following a procedure similar to that given above.


### 3.3. Matrices for Train-Rail-Slab-Girder-Pier Interaction

The matrices for the train-rail interaction, marked with subscripts “tr” or “rt,” consist of the train-left rail interaction matrices marked with subscript “tLr,” and the train-right rail interaction matrices marked with subscript “tRr.” The stiffness matrices Ktr and Krt of order Tdof×2N-r and damping matrices Ctr and Crt of order Tdof×2N-r, describing the train-rail interaction, can be written according to the vertical and lateral wheel-rail interaction relationships [32, 33] as(14)Ktr=KtLrKtRrTdof×2Nr,Ctr=CtLrCtRrTdof×2Nr,Krt=KtrT,Crt=CtrT.The matrices for the rail-slab interaction, marked with subscripts “rs” or “sr,” are induced by the stiffness and damping of the fastener between the rail and slab. The matrices for the slab-girder interaction, marked with subscripts “sb” or “bs,” are induced by the stiffness and damping of the CAM between the slab and girder. The matrices for the girder-pier interaction, marked with subscripts “bp” or “pb,” are induced by the stiffness and damping of the bearing between the girder and pier. All of them can be derived similarly.


### 3.4. Load Vectors for Train, Rail, Slab, Girder, and Pier

The load vectorFt of the train of order Tdof×1 can be written as(15)Ft=Ft1V+Ft2V+Ft1C+Ft2C+Ft1A+Ft2A+Ft2Gwith(16)Ft1V=FJ11VFv11VFv21V⋯FvNv1VFJ21VT,Ft1C=FJ11CFv11CFv21C⋯FvNv1CFJ21CT,Ft1A=FJ11AFv11AFv21A⋯FvNv1AFJ21AT,Ft2V=FJ12VFv12VFv22V⋯FvNv2VFJ22VT,Ft2C=FJ12CFv12CFv22C⋯FvNv2CFJ22CT,Ft2A=FJ12AFv12AFv22A⋯FvNv2AFJ22AT,Ft2G=FJ12GFv12GFv22G⋯FvNv2GFJ22GT,where Ft1V, Ft1C, and Ft1A represent the load vectors caused by the track elevation, cross level, and alignment irregularities, respectively; Ft2V, Ft2C, Ft2A, and Ft2G represent the load vectors caused by the velocity of the track elevation, cross level, alignment, and gauge irregularities, respectively.F
v
j
1
V, Fvj1C, Fvj1A, Fvj2V, Fvj2C, Fvj2A, and Fvj2G for the jth trailer car can be written, respectively, as (17)Fvj1V=∑h=14kpzrxjhVRvhV,Fvj1C=∑h=14kpzrxjhCRvhC,Fvj1A=∑h=14Waxleλb0rxjhARvhA,Fvj2V=∑h=14cpzr˙xjhVRvhV,Fvj2C=∑h=14cpzr˙xjhCRvhC,Fvj2A=∑h=14fLjh22+fRjh22r˙xjhARvhA,Fvj2G=∑h=1412fLjh22-fRjh22r˙xjhGRvhA,Rv1V=000000202Lt00000000000000,Rv2V=00000020-2Lt00000000000000,Rv3V=00000000000202Lt000000000,Rv4V=0000000000020-2Lt000000000,Rv1C=Rv2C=0000000b2000000000000000,Rv3C=Rv4C=000000000000b20000000000,Rv1A=00000000000000010000000,Rv2A=00000000000000000100000,Rv3A=00000000000000000001000,Rv4A=00000000000000000000010,where r(xjhV), r(xjhC), r(xjhA), and r(xjhG) denote the track elevation, cross level, alignment, and gauge irregularities, respectively, at the hth wheel-rail contact point of the jth trailer car; r˙(·) is the first derivative of track irregularity r(·); b0 is half of the transverse distance between the contact points of the wheel and rail; b2 is half of the transverse distance between the vertical primary suspension systems (Figure 1(b)); Lt is half of the bogie axle base; Waxle is the axle weight; λ is the slope of the wheel tread; and fLjh22 and fRjh22 (h = 1–4) are the lateral creepage coefficients between the hth wheelset of the jth trailer car and the left and right rails, respectively. The lateral creepage coefficient is a function of the wheel-rail normal contact force and the wheel-rail curvature radius at the contact point. In order to make the calculation easier, the coefficient is considered as a constant for certain vehicle by taking the wheel-rail normal contact force as the static wheel weight and assuming that the wheel-rail contact point lies in a cone surface for the wheel and in a cylindrical surface for the rail [33].F
J
i
1
V, FJi1C, Fji1A, FJi2V, FJi2C, FJi2A, and FJi2G for the ith motor car can be obtained similarly.The load vector of the railFr of order 2N-r×1 can be written as(18)Fr=FrLFrR,FrL=FrL0+FrL1+FrL2+FrL3+FrL4+FrL5+FrL6+FrL7+FrL8+FrL9+FrL10,FrR=FrR0+FrR1+FrR2+FrR3+FrR4+FrR5+FrR6+FrR7+FrR8+FrR9+FrR10,where FrL0, FrL1, FrL2, FrL3, and FrL4 represent the load vectors of each wheelset acting upon the left rail caused by the train’s weight, the track elevation irregularity, the cross level irregularity, the alignment irregularity, and the gauge irregularity, respectively; FrL5, FrL6, FrL7, and FrL8 represent the load vectors of each wheelset acting upon the left rail caused by the velocity of track elevation, cross level, alignment, and gauge irregularities, respectively; FrL9 and FrL10 represent the load vectors of each wheelset acting upon the left rail caused by the acceleration of track elevation and cross level irregularities, respectively. Similarly, FrR0–FrR10 represent the load vectors of each wheelset acting upon the right rail.F
r
L
0
–
F
r
L
10 can be written, respectively, as (19)FrL0=-∑j=1Nv∑h=1412gmw+12mt+14mcNzjhT-∑i=12∑h=1412gmJw+12mJt+14mJcNzJihT,FrL1=-∑j=1Nv∑h=14kpzrxjhVNzjhT-∑i=12∑h=14kJpzrxJihVNzJihT,FrL2=-∑j=1Nv∑h=1412kpzrxjhCNzjhT-∑i=12∑h=1412kJpzrxJihCNzJihT,FrL3=-∑j=1Nv∑h=14Waxleλb0rxjhANzjhT-∑i=12∑h=14WJaxleλb0rxJihANzJihT,FrL4=-∑j=1Nv∑h=14Waxleλ2b0rxjhGNzjhT-∑i=12∑h=14WJaxleλ2b0rxJihGNzJihT,FrL5=-∑j=1Nv∑h=14cpzr˙xjhVNzjhT-∑i=12∑h=14cJpzr˙xJihVNzJihT,FrL6=-∑j=1Nv∑h=1412cpzr˙xjhCNzjhT-∑i=12∑h=1412cJpzr˙xJihCNzJihT,FrL7=-∑j=1Nv∑h=14fLjh22r˙xjhANyjhT-∑i=12∑h=14fLJih22r˙xJihANyJihT,FrL8=-∑j=1Nv∑h=1412fLjh22r˙xjhGNyjhT-∑i=12∑h=1412fLJih22r˙xJihGNyJihT,FrL9=-∑j=1Nv∑h=1412mwr¨xjhVNzjhT-∑i=12∑h=1412mJwr¨xJihVNzJihT,FrL10=-∑j=1Nv∑h=1414mwr¨xjhCNzjhT-∑i=12∑h=1414mJwr¨xJihCNzJihT,where Nzjh and NzJih, both of order 1×N-r, are the time-dependent shape function vectors in the xz plane for the rail element evaluated at the position of the hth wheelset of the jth trailer car and the ith motor car, respectively; Nyjh and NyJih, both of order 1×N-r, are the time-dependent shape function vectors in the xy plane for the rail element evaluated at the position of the hth wheelset of the jth trailer car and the ith motor car, respectively; g is the acceleration of gravity; WJaxle is the axle weight of the motor car; r¨(·) is the second derivative of track irregularity; and fLJih22  (h = 1–4) is the lateral creepage coefficient between the hth wheelset of the ith motor car and the left rail.F
r
R
0
–
F
r
R
10 can be derived similarly.Each element for the load vector of the slabFs of order N-s×1 is zero.The load vector of the girderFb of order N-b×1 can be written as(20)Fb=Fb1Fb2⋯FbNbT,Fbi=fb1fb2⋯fbnbifbnbi+1,fbi=mbiagymbiagz000,where nbi denotes the number of elements of the ith girder, mbi the mass of node of the ith girder, and ayg and azg the lateral horizontal and vertical seismic accelerations, respectively.Similarly the load vector of the pierFp of order N-p×1 can be written as(21)Fp=Fp1Fp2⋯FpNpT,Fpi=fp1fp2⋯fpnpifpnpi+1,fpi=mpiagympiagz0,where npi denotes the number of element of the ith pier and mpi the mass of node of the ith pier.Let the following definitions be true:(22)FGRTt=0Tdof×1FrL0FrR00N-s×10N-b×10N-p×1,FRND1t=Ft1VFrL1FrR10N-s×10N-b×10N-p×1,FRND2t=Ft1CFrL2FrR20N-s×10N-b×10N-p×1,FRND3t=Ft1AFrL3FrR30N-s×10N-b×10N-p×1,FRND4t=0Tdof×1FrL4FrR40N-s×10N-b×10N-p×1,FRND5t=Ft2VFrL5FrR50N-s×10N-b×10N-p×1,FRND6t=Ft2CFrL6FrR60N-s×10N-b×10N-p×1,FRND7t=Ft2AFrL7FrR70N-s×10N-b×10N-p×1,FRND8t=Ft2GFrL8FrR80N-s×10N-b×10N-p×1,FRND9t=0Tdof×1FrL9FrR90N-s×10N-b×10N-p×1,FRND10t=0Tdof×1FrL10FrR100N-s×10N-b×10N-p×1,FRND11t=0Tdof×10Nr×10Nr×10N-s×1FbFp.Then, the load vector of the TSTBI system simultaneously excited by track irregularities and earthquakesF(t) can be expressed as(23)Ft=FtFrLFrRFsFbFp=FGRTt+∑i=111FRNDit.


## 3.1. Displacement Vectors for Tran, Rail, Slab, Girder, and Pier

The total train displacement vectorXt of order Tdof×1(Tdof=23×Nv+23×2) can be written as(2)Xt=XJ1Xv1Xv2⋯XvNvXJ2T,where the superscript “T” denotes the transpose of the matrix and Xvj(j=1,2,…,Nv) and XJi(i=1,2) the displacement vectors of the jth trailer car and the ith motor car, respectively. These can be expressed as(3)Xvj=ycjzcjθcjφcjψcjyt1jzt1jθt1jφt1jψt1jyt2jzt2jθt2jφt2jψt2jyw1jψw1jyw2jψw2jyw3jψw3jyw4jψw4j,XJi=yJcizJciθJciφJciψJciyJt1izJt1iθJt1iφJt1iψJt1iyJt2izJt2iθJt2iφJt2iψJt2iyJw1iψJw1iyJw2iψJw2iyJw3iψJw3iyJw4iψJw4i.The displacement vector of the railXr of order 2N-r×1, comprising the displacement vectors XLr of the left rail and XRr of the right rail, both of order N-r×1, can be written as(4)Xr=XLrXRrT,XLr=XRr=qr1qr2⋯qrN-r,where N-r denotes the total number of DOFs of each rail.Similarly, the displacement vector of the slabXs of order N-s×1, the displacement vector of the girder Xb of order N-b×1, and the displacement vector of the pier Xp of order N-p×1 can be written, respectively, as(5)Xs=Xs1Xs2⋯XsNsT,(6)Xb=Xb1Xb2⋯XbNbT,(7)Xp=Xp1Xp2⋯XpNpT,where Xsi  (i=1,2,…,Ns) denotes the displacement vector of the ith slab, Ns the total number of slabs, and N-s the total number of DOFs of all slabs; Xbi (i=1,2,…,Nb) denotes the displacement vector of the ith girder, Nb the total number of girders, and N-b the total number of DOFs of all girders; and Xpi (i=1,2,…,Np) denotes the displacement vector of the ith pier, Np the total number of piers, and N-p the total number of DOFs of all piers.


## 3.2. Matrices for Train, Rail, Slab, Girder, and Pier

The matrices of the train are marked with the subscript “tt.” The mass matrix Mtt, the stiffness matrix Ktt, and the damping matrix Ctt of the train, all of order (23×Nv+23×2)×(23×Nv+23×2), can be written, respectively, as(8)Mtt=diag⁡MJ1Mv1Mv2⋯MvNvMJ2,(9)Ktt=diag⁡KJ1Kv1Kv2⋯KvNvKJ2,(10)Ctt=diag⁡CJ1Cv1Cv2⋯CvNvCJ2,where Mvj, Kvj, and Cvj, all of order 23 × 23, denote the mass, stiffness, and damping matrices of the jth trailer car, respectively; MJi, KJi, and CJi, all of order 23 × 23, denote the mass, stiffness, and damping matrices of the ith motor car, respectively.The matrices of the rail are marked with the subscript “rr.” The mass matrix Mrr of the rail with order 2N-r×2N-r, comprising mass matrices MLrr of the left rail and MRrr of the right rail, both of order N-r×N-r, can be written as(11)Mrr=diag⁡MLrrMRrr.The stiffness matrixKrr of the rail of order 2N-r×2N-r, which is composed of the stiffness matrices of the left and right rails KLrr and KRrr, both of order N-r×N-r, and the left rail-right rail interaction stiffness matrices KLrRr and KRrLr of order N-r×N-r, can be written as(12)Krr=KLrrKLrRrKRrLrKRrr,where the stiffness matrices KLrRr and KRrLr are induced by the train’s weight acting upon the rails by the wheelsets.The damping matrixCrr of the rail of order 2N-r×2N-r, consisting of the damping matrices of the left and right rails CLrr and CRrr, respectively, both of order N-r×N-r, and can be written as(13)Crr=diag⁡CLrrCRrr.The matrices of the slab, girder, and pier, marked with the subscripts “ss,” “bb,” and “pp,” respectively, are not given here but can be derived by following a procedure similar to that given above.


## 3.3. Matrices for Train-Rail-Slab-Girder-Pier Interaction

The matrices for the train-rail interaction, marked with subscripts “tr” or “rt,” consist of the train-left rail interaction matrices marked with subscript “tLr,” and the train-right rail interaction matrices marked with subscript “tRr.” The stiffness matrices Ktr and Krt of order Tdof×2N-r and damping matrices Ctr and Crt of order Tdof×2N-r, describing the train-rail interaction, can be written according to the vertical and lateral wheel-rail interaction relationships [32, 33] as(14)Ktr=KtLrKtRrTdof×2Nr,Ctr=CtLrCtRrTdof×2Nr,Krt=KtrT,Crt=CtrT.The matrices for the rail-slab interaction, marked with subscripts “rs” or “sr,” are induced by the stiffness and damping of the fastener between the rail and slab. The matrices for the slab-girder interaction, marked with subscripts “sb” or “bs,” are induced by the stiffness and damping of the CAM between the slab and girder. The matrices for the girder-pier interaction, marked with subscripts “bp” or “pb,” are induced by the stiffness and damping of the bearing between the girder and pier. All of them can be derived similarly.


## 3.4. Load Vectors for Train, Rail, Slab, Girder, and Pier

The load vectorFt of the train of order Tdof×1 can be written as(15)Ft=Ft1V+Ft2V+Ft1C+Ft2C+Ft1A+Ft2A+Ft2Gwith(16)Ft1V=FJ11VFv11VFv21V⋯FvNv1VFJ21VT,Ft1C=FJ11CFv11CFv21C⋯FvNv1CFJ21CT,Ft1A=FJ11AFv11AFv21A⋯FvNv1AFJ21AT,Ft2V=FJ12VFv12VFv22V⋯FvNv2VFJ22VT,Ft2C=FJ12CFv12CFv22C⋯FvNv2CFJ22CT,Ft2A=FJ12AFv12AFv22A⋯FvNv2AFJ22AT,Ft2G=FJ12GFv12GFv22G⋯FvNv2GFJ22GT,where Ft1V, Ft1C, and Ft1A represent the load vectors caused by the track elevation, cross level, and alignment irregularities, respectively; Ft2V, Ft2C, Ft2A, and Ft2G represent the load vectors caused by the velocity of the track elevation, cross level, alignment, and gauge irregularities, respectively.F
v
j
1
V, Fvj1C, Fvj1A, Fvj2V, Fvj2C, Fvj2A, and Fvj2G for the jth trailer car can be written, respectively, as (17)Fvj1V=∑h=14kpzrxjhVRvhV,Fvj1C=∑h=14kpzrxjhCRvhC,Fvj1A=∑h=14Waxleλb0rxjhARvhA,Fvj2V=∑h=14cpzr˙xjhVRvhV,Fvj2C=∑h=14cpzr˙xjhCRvhC,Fvj2A=∑h=14fLjh22+fRjh22r˙xjhARvhA,Fvj2G=∑h=1412fLjh22-fRjh22r˙xjhGRvhA,Rv1V=000000202Lt00000000000000,Rv2V=00000020-2Lt00000000000000,Rv3V=00000000000202Lt000000000,Rv4V=0000000000020-2Lt000000000,Rv1C=Rv2C=0000000b2000000000000000,Rv3C=Rv4C=000000000000b20000000000,Rv1A=00000000000000010000000,Rv2A=00000000000000000100000,Rv3A=00000000000000000001000,Rv4A=00000000000000000000010,where r(xjhV), r(xjhC), r(xjhA), and r(xjhG) denote the track elevation, cross level, alignment, and gauge irregularities, respectively, at the hth wheel-rail contact point of the jth trailer car; r˙(·) is the first derivative of track irregularity r(·); b0 is half of the transverse distance between the contact points of the wheel and rail; b2 is half of the transverse distance between the vertical primary suspension systems (Figure 1(b)); Lt is half of the bogie axle base; Waxle is the axle weight; λ is the slope of the wheel tread; and fLjh22 and fRjh22 (h = 1–4) are the lateral creepage coefficients between the hth wheelset of the jth trailer car and the left and right rails, respectively. The lateral creepage coefficient is a function of the wheel-rail normal contact force and the wheel-rail curvature radius at the contact point. In order to make the calculation easier, the coefficient is considered as a constant for certain vehicle by taking the wheel-rail normal contact force as the static wheel weight and assuming that the wheel-rail contact point lies in a cone surface for the wheel and in a cylindrical surface for the rail [33].F
J
i
1
V, FJi1C, Fji1A, FJi2V, FJi2C, FJi2A, and FJi2G for the ith motor car can be obtained similarly.The load vector of the railFr of order 2N-r×1 can be written as(18)Fr=FrLFrR,FrL=FrL0+FrL1+FrL2+FrL3+FrL4+FrL5+FrL6+FrL7+FrL8+FrL9+FrL10,FrR=FrR0+FrR1+FrR2+FrR3+FrR4+FrR5+FrR6+FrR7+FrR8+FrR9+FrR10,where FrL0, FrL1, FrL2, FrL3, and FrL4 represent the load vectors of each wheelset acting upon the left rail caused by the train’s weight, the track elevation irregularity, the cross level irregularity, the alignment irregularity, and the gauge irregularity, respectively; FrL5, FrL6, FrL7, and FrL8 represent the load vectors of each wheelset acting upon the left rail caused by the velocity of track elevation, cross level, alignment, and gauge irregularities, respectively; FrL9 and FrL10 represent the load vectors of each wheelset acting upon the left rail caused by the acceleration of track elevation and cross level irregularities, respectively. Similarly, FrR0–FrR10 represent the load vectors of each wheelset acting upon the right rail.F
r
L
0
–
F
r
L
10 can be written, respectively, as (19)FrL0=-∑j=1Nv∑h=1412gmw+12mt+14mcNzjhT-∑i=12∑h=1412gmJw+12mJt+14mJcNzJihT,FrL1=-∑j=1Nv∑h=14kpzrxjhVNzjhT-∑i=12∑h=14kJpzrxJihVNzJihT,FrL2=-∑j=1Nv∑h=1412kpzrxjhCNzjhT-∑i=12∑h=1412kJpzrxJihCNzJihT,FrL3=-∑j=1Nv∑h=14Waxleλb0rxjhANzjhT-∑i=12∑h=14WJaxleλb0rxJihANzJihT,FrL4=-∑j=1Nv∑h=14Waxleλ2b0rxjhGNzjhT-∑i=12∑h=14WJaxleλ2b0rxJihGNzJihT,FrL5=-∑j=1Nv∑h=14cpzr˙xjhVNzjhT-∑i=12∑h=14cJpzr˙xJihVNzJihT,FrL6=-∑j=1Nv∑h=1412cpzr˙xjhCNzjhT-∑i=12∑h=1412cJpzr˙xJihCNzJihT,FrL7=-∑j=1Nv∑h=14fLjh22r˙xjhANyjhT-∑i=12∑h=14fLJih22r˙xJihANyJihT,FrL8=-∑j=1Nv∑h=1412fLjh22r˙xjhGNyjhT-∑i=12∑h=1412fLJih22r˙xJihGNyJihT,FrL9=-∑j=1Nv∑h=1412mwr¨xjhVNzjhT-∑i=12∑h=1412mJwr¨xJihVNzJihT,FrL10=-∑j=1Nv∑h=1414mwr¨xjhCNzjhT-∑i=12∑h=1414mJwr¨xJihCNzJihT,where Nzjh and NzJih, both of order 1×N-r, are the time-dependent shape function vectors in the xz plane for the rail element evaluated at the position of the hth wheelset of the jth trailer car and the ith motor car, respectively; Nyjh and NyJih, both of order 1×N-r, are the time-dependent shape function vectors in the xy plane for the rail element evaluated at the position of the hth wheelset of the jth trailer car and the ith motor car, respectively; g is the acceleration of gravity; WJaxle is the axle weight of the motor car; r¨(·) is the second derivative of track irregularity; and fLJih22  (h = 1–4) is the lateral creepage coefficient between the hth wheelset of the ith motor car and the left rail.F
r
R
0
–
F
r
R
10 can be derived similarly.Each element for the load vector of the slabFs of order N-s×1 is zero.The load vector of the girderFb of order N-b×1 can be written as(20)Fb=Fb1Fb2⋯FbNbT,Fbi=fb1fb2⋯fbnbifbnbi+1,fbi=mbiagymbiagz000,where nbi denotes the number of elements of the ith girder, mbi the mass of node of the ith girder, and ayg and azg the lateral horizontal and vertical seismic accelerations, respectively.Similarly the load vector of the pierFp of order N-p×1 can be written as(21)Fp=Fp1Fp2⋯FpNpT,Fpi=fp1fp2⋯fpnpifpnpi+1,fpi=mpiagympiagz0,where npi denotes the number of element of the ith pier and mpi the mass of node of the ith pier.Let the following definitions be true:(22)FGRTt=0Tdof×1FrL0FrR00N-s×10N-b×10N-p×1,FRND1t=Ft1VFrL1FrR10N-s×10N-b×10N-p×1,FRND2t=Ft1CFrL2FrR20N-s×10N-b×10N-p×1,FRND3t=Ft1AFrL3FrR30N-s×10N-b×10N-p×1,FRND4t=0Tdof×1FrL4FrR40N-s×10N-b×10N-p×1,FRND5t=Ft2VFrL5FrR50N-s×10N-b×10N-p×1,FRND6t=Ft2CFrL6FrR60N-s×10N-b×10N-p×1,FRND7t=Ft2AFrL7FrR70N-s×10N-b×10N-p×1,FRND8t=Ft2GFrL8FrR80N-s×10N-b×10N-p×1,FRND9t=0Tdof×1FrL9FrR90N-s×10N-b×10N-p×1,FRND10t=0Tdof×1FrL10FrR100N-s×10N-b×10N-p×1,FRND11t=0Tdof×10Nr×10Nr×10N-s×1FbFp.Then, the load vector of the TSTBI system simultaneously excited by track irregularities and earthquakesF(t) can be expressed as(23)Ft=FtFrLFrRFsFbFp=FGRTt+∑i=111FRNDit.


## 4. Random Vibration Analysis of the TSTBI System under the Actions of Track Irregularities and Earthquakes by PEM

### 4.1. Pseudoexcitation for the System

PEM is a highly efficient and accurate algorithm for random vibration analysis, which has been established and successfully used for many time-dependent systems subjected to different kinds of random excitations, for example, earthquakes [27] and wind gusts [34].For any zero-mean-valued random system, the relationship between the PSDSFF of excitation FRND and the PSD SUU of response U can be expressed as(24)SUU=H2ωSFF,where H(ω) denotes the frequency response function.The basic principle of PEM is the following. Given the PSD functionSFF and defining the pseudoexcitation F~RND=SFFeiωt, one then can obtain, according to (24),(25)U~=SFFHωeiωtwith(26)F~RND∗F~RND=SFFe-iωtSFFeiωt=SFF,U~∗U~=SUUH∗ωe-iωtSUUHωeiωt=H∗ωSFFHω=SUU,where U~ denotes the pseudoresponse and the superscript “∗” the complex conjugate.Letx be the distance from the left-hand starting point of the TSTBI model to the wheel-rail contact point at time t. There then exists a transformation from the PSD Srr(Ω) of r(x) in the space domain to the PSD Srr(ω) of r(t) in the time domain, according to x=vt:(27)Srrω=SrrΩv,where ω (rad/s) denotes the time frequency, Ω (rad/m) the spatial frequency, and v (m/s) the train speed. Obviously, ω=Ωv.As the duration of an earthquake is usually comparable with the time it takes for a train to pass over a large bridge, it is reasonable to represent the earthquakes by the nonstationary random process as follows:(28)agy=gtay,agz=gtaz,where ay and az denote the zero-mean-valued, stationary, lateral, and vertical seismic accelerations with the PSDs of Sayay(ω) and Sazaz(ω), respectively, and g(t) a specified slowly varying modulation function.According to PEM and (23), the pseudoexcitation F~RND(ω,t) induced by FRND(t) can be written as(29)F~RNDω,t=∑i=111F~RNDiω,twith(30)F~RND1ω,t=F~t1Vω,tF~rL1ω,tF~rR1ω,t0N-s×10N-b×10N-p×1,F~RND2ω,t=F~t1Cω,tF~rL2ω,tF~rR2ω,t0N-s×10N-b×10N-p×1,F~RND3ω,t=F~t1Aω,tF~rL3ω,tF~rR3ω,t0N-s×10N-b×10N-p×1,F~RND4ω,t=0Tdof×1F~rL4ω,tF~rR4ω,t0N-s×10N-b×10N-p×1,F~RND5ω,t=F~t2Vω,tF~rL5ω,tF~rR5ω,t0N-s×10N-b×10N-p×1,F~RND6ω,t=F~t2Cω,tF~rL6ω,tF~rR6ω,t0N-s×10N-b×10N-p×1,F~RND7ω,t=F~t2Aω,tF~rL7ω,tF~rR7ω,t0N-s×10N-b×10N-p×1,F~RND8ω,t=F~t2Gω,tF~rL8ω,tF~rR8ω,t0N-s×10N-b×10N-p×1,F~RND9ω,t=0Tdof×1F~rL9ω,tF~rR9ω,t0N-s×10N-b×10N-p×1,F~RND10ω,t=0Tdof×1F~rL10ω,tF~rR10ω,t0N-s×10N-b×10N-p×1,F~RND11ω,t=0Tdof×10Nr×10Nr×10N-s×1F~bω,tF~pω,t.


### 4.2. Calculation of Random Responses of the System

Based on (1) and (23), the equations of motion for the TSTBI system can be expressed as(31)MtU¨+CtU˙+KtU=FGRTt+FRNDt,where M(t), C(t), and K(t) denote the mass, damping, and stiffness matrices, respectively, of the total TSTBI system; U, U˙, and U¨ denote the displacement, velocity, and acceleration vectors, respectively, of the system.The response of the system caused by the train’s weightFGRT(t) is deterministic and can be obtained easily in the time domain using a step-by-step integration method. While, the pseudoresponse U~(ω,t) of the system caused by the pseudoexcitation F~RND(ω,t) can be obtained by solving the following equation in both the frequency and time domains:(32)MtU¨~+CtU˙~+KtU~=F~RNDω,t.According to PEM, the time-dependent PSDSUU(ω,t) of arbitrary response U~(ω,t) of the system can be written as(33)SUUω,t=U~∗ω,tU~Tω,t.Then, one can estimate the global maximum valuemax⁡(U(t)), the minimum value min⁡(U(t)) and the dynamic reliability p of the random responses by the first-passage failure criterion according to the following equations [6, 35, 36]:(34a)max⁡Ut=max⁡U-t+Eη+κσησ-U,(34b)min⁡Ut=min⁡U-t-Eη+κσησ-U,(35)pb,-b=exp⁡∫0T-υtexp⁡-b22σU2t,where η is a dimensionless parameter known as the peak factor, κ is a safety factor, E(η) and ση are the expectation value and the standard deviation of η, respectively, σ-U is the equivalent stationary standard deviation of random response U, υ is the zero-passage rate, and b is the allowable safety limit.


### 4.3. Solution Procedures

The flowchart for efficiently analyzing the random vibrations of the TSTBI system under the actions of track irregularities and seismic accelerations is shown in Figure2.Figure 2
Flowchart for analyzing the random vibrations of the TSTBI system.


## 4.1. Pseudoexcitation for the System

PEM is a highly efficient and accurate algorithm for random vibration analysis, which has been established and successfully used for many time-dependent systems subjected to different kinds of random excitations, for example, earthquakes [27] and wind gusts [34].For any zero-mean-valued random system, the relationship between the PSDSFF of excitation FRND and the PSD SUU of response U can be expressed as(24)SUU=H2ωSFF,where H(ω) denotes the frequency response function.The basic principle of PEM is the following. Given the PSD functionSFF and defining the pseudoexcitation F~RND=SFFeiωt, one then can obtain, according to (24),(25)U~=SFFHωeiωtwith(26)F~RND∗F~RND=SFFe-iωtSFFeiωt=SFF,U~∗U~=SUUH∗ωe-iωtSUUHωeiωt=H∗ωSFFHω=SUU,where U~ denotes the pseudoresponse and the superscript “∗” the complex conjugate.Letx be the distance from the left-hand starting point of the TSTBI model to the wheel-rail contact point at time t. There then exists a transformation from the PSD Srr(Ω) of r(x) in the space domain to the PSD Srr(ω) of r(t) in the time domain, according to x=vt:(27)Srrω=SrrΩv,where ω (rad/s) denotes the time frequency, Ω (rad/m) the spatial frequency, and v (m/s) the train speed. Obviously, ω=Ωv.As the duration of an earthquake is usually comparable with the time it takes for a train to pass over a large bridge, it is reasonable to represent the earthquakes by the nonstationary random process as follows:(28)agy=gtay,agz=gtaz,where ay and az denote the zero-mean-valued, stationary, lateral, and vertical seismic accelerations with the PSDs of Sayay(ω) and Sazaz(ω), respectively, and g(t) a specified slowly varying modulation function.According to PEM and (23), the pseudoexcitation F~RND(ω,t) induced by FRND(t) can be written as(29)F~RNDω,t=∑i=111F~RNDiω,twith(30)F~RND1ω,t=F~t1Vω,tF~rL1ω,tF~rR1ω,t0N-s×10N-b×10N-p×1,F~RND2ω,t=F~t1Cω,tF~rL2ω,tF~rR2ω,t0N-s×10N-b×10N-p×1,F~RND3ω,t=F~t1Aω,tF~rL3ω,tF~rR3ω,t0N-s×10N-b×10N-p×1,F~RND4ω,t=0Tdof×1F~rL4ω,tF~rR4ω,t0N-s×10N-b×10N-p×1,F~RND5ω,t=F~t2Vω,tF~rL5ω,tF~rR5ω,t0N-s×10N-b×10N-p×1,F~RND6ω,t=F~t2Cω,tF~rL6ω,tF~rR6ω,t0N-s×10N-b×10N-p×1,F~RND7ω,t=F~t2Aω,tF~rL7ω,tF~rR7ω,t0N-s×10N-b×10N-p×1,F~RND8ω,t=F~t2Gω,tF~rL8ω,tF~rR8ω,t0N-s×10N-b×10N-p×1,F~RND9ω,t=0Tdof×1F~rL9ω,tF~rR9ω,t0N-s×10N-b×10N-p×1,F~RND10ω,t=0Tdof×1F~rL10ω,tF~rR10ω,t0N-s×10N-b×10N-p×1,F~RND11ω,t=0Tdof×10Nr×10Nr×10N-s×1F~bω,tF~pω,t.


## 4.2. Calculation of Random Responses of the System

Based on (1) and (23), the equations of motion for the TSTBI system can be expressed as(31)MtU¨+CtU˙+KtU=FGRTt+FRNDt,where M(t), C(t), and K(t) denote the mass, damping, and stiffness matrices, respectively, of the total TSTBI system; U, U˙, and U¨ denote the displacement, velocity, and acceleration vectors, respectively, of the system.The response of the system caused by the train’s weightFGRT(t) is deterministic and can be obtained easily in the time domain using a step-by-step integration method. While, the pseudoresponse U~(ω,t) of the system caused by the pseudoexcitation F~RND(ω,t) can be obtained by solving the following equation in both the frequency and time domains:(32)MtU¨~+CtU˙~+KtU~=F~RNDω,t.According to PEM, the time-dependent PSDSUU(ω,t) of arbitrary response U~(ω,t) of the system can be written as(33)SUUω,t=U~∗ω,tU~Tω,t.Then, one can estimate the global maximum valuemax⁡(U(t)), the minimum value min⁡(U(t)) and the dynamic reliability p of the random responses by the first-passage failure criterion according to the following equations [6, 35, 36]:(34a)max⁡Ut=max⁡U-t+Eη+κσησ-U,(34b)min⁡Ut=min⁡U-t-Eη+κσησ-U,(35)pb,-b=exp⁡∫0T-υtexp⁡-b22σU2t,where η is a dimensionless parameter known as the peak factor, κ is a safety factor, E(η) and ση are the expectation value and the standard deviation of η, respectively, σ-U is the equivalent stationary standard deviation of random response U, υ is the zero-passage rate, and b is the allowable safety limit.


## 4.3. Solution Procedures

The flowchart for efficiently analyzing the random vibrations of the TSTBI system under the actions of track irregularities and seismic accelerations is shown in Figure2.Figure 2
Flowchart for analyzing the random vibrations of the TSTBI system.


## 5. Case Study

### 5.1. Properties of the TSTBI System

A fifteen-span simply supported boxing girder high-speed railway bridge, with the span length of 32 m and the pier height of 20 m, is considered as shown in Figure1. The central part of the slab track is supported on the bridge, while the left and right parts of the track are supported on the subgrades adjacent to the bridge. The lengths of element of the rail, slab, and girder are all equal to the fastener spacing of 0.625 m, while the length of the pier element equals 1.0 m. The high-speed train comprises one front motor car, six trailer cars, and one rear motor car moving at a constant velocity v. The fundamental data of the motor car and trailer car can refer to [37]. The major fundamental data of the slab track and bridge are listed in Table 1. The PSDs of the German high-speed track spectrum of low irregularity [38] are adopted. In addition, the PSDs of the lateral horizontal and vertical seismic accelerations and the uniform modulation function (Figure 3) are taken as follows [6, 35]:(36a)Sayayω=ωgy4+4ζgy2ωgy2ω2ωgy2-ω22+4ζgy2ωgy2ω2·ω4ω2-ωfy22+4ζfy2ωfy2ω2S0y,(36b)Sazazω=ωgz4+4ζgz2ωgz2ω2ωgz2-ω22+4ζgz2ωgz2ω2·ω4ω2-ωfz22+4ζfz2ωfz2ω2S0z,(37)gt=ttb20≤t<tb1tb≤t<tcexp⁡ct-tct≥tc,where S0y and S0z denote the spectral intensity factor, ζgy and ζgz denote the damping ratio of the site, ωgy and ωgz denote the predominant frequency of the site, ζfy, ζfz, ωfy, and ωfz denote the filter parameters, c denotes the attenuation coefficient, and tb and tc denote the times at the start and end, respectively, of the stationary main shock. Additionally, ζgz = ζgy, ωgz=1.58ωgy, ζfy=ζgy, ωfy=0.1ωgy–0.2ωgy, ζfz=ζgz, ωfz=0.1ωgz–0.2ωgz, and S0z=0.218S0y. The parameters for the earthquakes with different intensities and site types are listed in Table 2. It is assumed that the DBGA is 0.15 g and the bridge site is Type II. The spatial frequency of the track irregularity PSDs ranges from 0.004 × 2π to 1 × 2π rad/m, while the frequency of the seismic acceleration PSDs lies in the range of 0–40 Hz. It is worth noting that the extreme state for the maximum responses of the bridge and each moving vehicle may not occur at the same time when the track irregularities and earthquakes are considered simultaneously, and therefore cannot be easily identified. This makes it rather difficult to determine how the earthquakes’ starting times affect the results. Here, the earthquakes are assumed to start exactly at the instant when the train enters the bridge [17].Table 1
Major parameters for slab track and bridge.
Notation
Item
Unit
Value
Slab track
Er
Young’s modulus of rail
N/m2
2.06 × 1011
Iry
Flexural moment of inertia abouty-axis of cross section of rail
m4
3.217 × 10−5
Irz
Flexural moment of inertia aboutz-axis of cross section of rail
m4
5.24 × 10−6
m-r
Mass per unit length of rail
kg/m
60.64
Es
Young’s modulus of slab
N/m2
3.6 × 1010
Isy
Flexural moment of inertia of cross section of slab
m4
1.4 × 10−3
Isz
Flexural moment of inertia of cross section of slab
m4
0.219
m-s
Mass per unit length of slab
kg/m
1.2 × 103
krsy
Lateral stiffness of fastener
N/m
3.0 × 107
krsz
Vertical stiffness of fastener
N/m
5.0 × 107
k-sby
Lateral stiffness of CAM per unit length
N/m2
1.5 × 109
k-sbz
Vertical stiffness of CAM per unit length
N/m2
1.5 × 109
Girder
Eb
Young’s modulus of girder
N/m2
3.45 × 1010
Iby
Flexural moment of inertia abouty-axis of cross section of girder
m4
12.744
Ibz
Flexural moment of inertia aboutz-axis of cross section of girder
m4
96.435
m-b
Mass per unit length of girder
kg/m
2.972 × 104
ωb1
Fundamental frequency of girder
Hz
5.58
ωb2
Second natural frequency of girder
Hz
13.26
Pier
Ep
Young’s modulus of pier
N/m2
3.15 × 1010
Ipx
Flexural moment of inertia aboutx-axis of cross section of pier
m4
73.23
m-p
Mass per unit length of pier
kg/m
5.2 × 104Table 2
Parameters of earthquake for different site conditions.
Parameter
Unit
Site classification
Type I
Type II
Type III
ω
g
y
rad/s
25.13
17.95
13.96
ζ
g
y
—
0.64
0.72
0.80
t
b
s
0.5
0.8
1.2
t
c
s
5.5
7.0
9.0
c
—
0.45
0.35
0.25
Duration of acting time of earthquake
s
6.69
8.41
10.92
S
0
y for DBGA = 0.15 g∗∗
cm2/s3
275
364
440
S
0
y for DBGA = 0.30 g∗∗
cm2/s3
1099
1456
1759
S
0
y for DBGA = 0.40 g∗∗
cm2/s3
1780
2359
2849
∗
∗DBGA is the design basic acceleration of ground motion.Figure 3
Plots of modulation functiong(t).


### 5.2. Analysis of the PSD Characteristic of the Random Vibration Responses of Bridge and Train

The random vibration responses of the bridge and train calculated by PEM with train speed of 300 km/h are still used.The PSDs of the vertical and lateral acceleration of the girder midpoint to the passage of the train shaken by track irregularities and earthquakes are plotted in Figure4. It can be seen from Figure 4(a) that there exists only one dominant vibration frequency (DVF) of 5.34 Hz, which is very close to the vertical fundamental frequency of the girder of 5.58 Hz. Similarly, only one DVF of 3.07 Hz can be found from Figure 4(b), which is very close to the lateral fundamental frequency of the total bridge (including the girder and pier) of 3.11 Hz.Figure 4
PSD of acceleration of bridge girder midpoint. (a) Vertical. (b) Lateral.
(a)
(b)Figure5 exhibit the PSDs of the vertical and lateral accelerations of the trailer carbody. As can be seen, the PSDs change violently with the vibration frequency, implying the great influence of track irregularities and earthquakes on the dynamic response of the carbody. Of interest is that there exist two DVFs with DVF1 = 0.83 Hz and DVF2 = 5.19 Hz in Figure 5(a), which are approximately equal to the vertical fundamental frequency of the trailer carbody and that of the girder, respectively. Similarly, two DVFs with DVF1 = 1.49 Hz and DVF2 = 2.98 Hz can be seen easily from Figure 5(b), which are also close to the lateral fundamental frequency of the trailer carbody and of the total bridge, respectively. Obviously, the vertical DVF1 and the lateral DVF1 are mainly excited by track irregularities, while the vertical DVF2 and the lateral DVF2 are mainly induced by the bridge vibration under the shake of earthquakes. On the other hand, the vertical and lateral PSDs excited by track irregularities vary only slightly with the time, which indicates that the influence of the bridge vibration on the carbody vibration is insignificant if there is no earthquake, because of the comparatively high mass and flexural rigidity of the bridge. However, the vertical PSD induced by earthquakes varies periodically with the vehicle passing through different girder (Figure 5(a)), while similar trend cannot be found for the lateral PSD induced by earthquakes (Figure 5(b)).Figure 5
PSD of acceleration of trailer carbody. (a) Vertical. (b) Lateral.
(a)
(b)A similar phenomenon for the vertical and lateral accelerations of the motor carbody can be also observed. Herein it is not discussed in detail to save the length of the paper.


### 5.3. Influence of Train Speed on the Random Vibration Characteristic of Bridge and Train

In reality, the train may move over the bridge at various speeds during earthquakes. There exists a need to investigate the random vibration characteristic of the train moving over the bridge under various train speeds, as they may be different. The train is assumed to move over the bridge with a constant speed varying from 150 to 420 km/h at 10 km/h intervals. The other parameters are the same as listed in Section5.1. Furthermore, two calculation cases (Table 3) are computed for comparison. In case B, the relative displacements between wheel, track, and girder are neglected [6]. The maximum values of the PSD and the DVF of the bridge and train with respect to train speed are shown in Figures 6–10 and Tables 4 and 5.Table 3
Calculation cases for studying the influence of train speed.
Calculation case
A
B
Calculation model
TSTBI model
Train-bridge interaction model [6]
Excitations
Track irregularity+ earthquake
Track irregularity+ earthquakeTable 4
Influence of train speed on PSDs of bridge midpoint acceleration.
Train speed(km/h)
Vertical
Lateral
DVF (Hz)
Peak value (m2/s3)
DVF (Hz)
Peak value (m2/s3)
Case A
Case B
Case A
Case B
Case A
Case B
Case A
Case B
150
5.34
5.34
1.50E − 01
1.58E − 01
2.98
3.07
2.01E − 01
2.17E − 01
170
5.27
5.42
1.53E − 01
1.61E − 01
3.03
3.11
2.20E − 01
2.24E − 01
190
5.27
5.42
1.59E − 01
1.67E − 01
3.03
3.11
2.26E − 01
2.30E − 01
210
5.36
5.36
1.50E − 01
1.63E − 01
2.99
3.08
2.19E − 01
2.34E − 01
230
5.26
5.40
1.63E − 01
1.72E − 01
3.02
3.10
2.30E − 01
2.33E − 01
250
5.26
5.41
1.66E − 01
1.80E − 01
3.02
3.10
2.33E − 01
2.35E − 01
270
5.37
5.37
1.53E − 01
1.66E − 01
2.90
3.08
2.25E − 01
2.42E − 01
290
5.31
5.46
1.80E − 01
1.72E − 01
3.01
3.05
2.35E − 01
2.31E − 01
310
5.31
5.37
1.57E − 01
1.70E − 01
3.01
3.08
2.25E − 01
2.44E − 01
330
5.26
5.41
1.68E − 01
1.84E − 01
3.01
3.10
2.36E − 01
2.41E − 01
350
5.28
5.42
1.77E − 01
1.85E − 01
3.03
3.11
2.38E − 01
2.37E − 01
370
5.28
5.28
1.76E − 01
1.83E − 01
3.03
3.11
2.37E − 01
2.36E − 01
390
5.26
5.41
1.70E − 01
1.84E − 01
3.02
3.11
2.34E − 01
2.39E − 01
410
5.12
5.12
1.62E − 01
1.70E − 01
3.02
3.08
2.27E − 01
2.41E − 01Table 5
Influence of train speed on PSDs of motor carbody acceleration.
Train speed (km/h)
Vertical
Lateral
DVF1(Hz)
Peak value(m2/s3)
DVF2(Hz)
Peak value(m2/s3)
DVF1(Hz)
Peak value(m2/s3)
DVF2(Hz)
Peak value(m2/s3)
150
Case A
0.96
3.69E − 03
5.20
1.59E − 03
1.07
1.59E − 03
2.98
1.03E − 02
Case B
0.96
3.75E − 03
5.49
1.55E − 03
—
—
3.07
1.12E − 02
170
Case A
0.92
4.44E − 03
5.27
1.92E − 03
1.21
1.92E − 03
3.11
8.21E − 03
Case B
0.92
4.53E − 03
5.42
1.93E − 03
—
—
3.11
1.12E − 02
190
Case A
0.94
5.22E − 03
5.27
2.05E − 03
1.03
2.05E − 03
3.20
9.24E − 03
Case B
0.94
5.32E − 03
5.27
2.13E − 03
—
—
3.11
1.09E − 02
210
Case A
0.96
5.70E − 03
5.36
2.12E − 03
1.10
2.12E − 03
3.16
8.28E − 03
Case B
0.96
5.81E − 03
5.36
2.08E − 03
—
—
3.08
1.08E − 02
230
Case A
0.94
6.23E − 03
5.11
1.65E − 03
1.24
1.65E − 03
3.02
1.00E − 02
Case B
0.94
6.35E − 03
5.56
1.75E − 03
—
—
3.10
1.08E − 02
250
Case A
0.92
6.62E − 03
5.11
1.52E − 03
1.37
1.52E − 03
2.94
8.63E − 03
Case B
0.92
6.75E − 03
5.56
1.61E − 03
—
—
3.10
1.12E − 02
270
Case A
0.91
7.21E − 03
5.37
1.87E − 03
0.96
1.87E − 03
3.08
9.61E − 03
Case B
0.91
7.34E − 03
5.52
1.93E − 03
—
—
3.08
1.12E − 02
290
Case A
0.92
7.61E − 03
5.31
2.80E − 03
1.03
2.80E − 03
3.13
8.92E − 03
Case B
0.92
7.74E − 03
5.46
2.89E − 03
—
—
3.13
1.10E − 02
310
Case A
0.94
7.89E − 03
5.22
1.95E − 03
1.07
1.95E − 03
3.13
9.57E − 03
Case B
0.94
8.03E − 03
5.22
2.00E − 03
—
—
3.08
1.11E − 02
330
Case A
0.94
8.17E − 03
5.41
2.10E − 03
1.14
2.10E − 03
3.10
1.05E − 02
Case B
0.92
8.31E − 03
5.41
2.21E − 03
—
—
3.10
1.13E − 02
350
Case A
0.92
8.39E − 03
5.13
2.13E − 03
1.21
2.13E − 03
3.11
8.15E − 03
Case B
0.92
8.54E − 03
5.28
2.07E − 03
—
—
3.11
1.14E − 02
370
Case A
0.92
8.58E − 03
5.43
2.26E − 03
1.35
2.26E − 03
2.95
8.59E − 03
Case B
0.92
8.72E − 03
5.43
2.38E − 03
—
—
3.11
1.15E − 02
390
Case A
0.92
8.76E − 03
4.98
1.74E − 03
1.14
1.74E − 03
3.03
1.07E − 02
Case B
0.92
8.91E − 03
4.98
1.76E − 03
—
—
3.11
1.14E − 02
410
Case A
0.89
8.97E − 03
4.98
3.06E − 03
1.08
3.06E − 03
2.94
1.03E − 02
Case B
0.87
9.12E − 03
5.12
3.09E − 03
—
—
2.94
1.12E − 02Figure 6
Girder midpoint acceleration with respect to train speed. (a) DVF. (b) Maximum value of PSD.
(a)
(b)Figure 7
Motor carbody vertical acceleration with respect to train speed. (a) DVF. (b) Maximum value of PSD.
(a)
(b)Figure 8
Motor carbody lateral acceleration with respect to train speed. (a) DVF. (b) Maximum value of PSD.
(a)
(b)Figure 9
Trailer carbody vertical acceleration with respect to train speed. (a) DVF. (b) Maximum value of PSD.
(a)
(b)Figure 10
Trailer carbody lateral acceleration with respect to train speed. (a) DVF. (b) Maximum value of PSD.
(a)
(b)Some conclusions can be drawn from Figure6 and Table 4: (1) both the maximum values of the PSD and the DVF of the girder vary slightly as the train speed increases in both cases A and B, which indicates that the vertical and lateral vibrations of the girder are induced basically by earthquakes and not by the train and the track irregularities. (2) The maximum values of the PSD in case B tend to be larger than those in case A because the effects of the constraint of the track are not taken into account in case B, but the degree of increase is below 10%.From Figures7–10 and Table 5, the following observations can be made easily: (1) the DVF1 and DVF2 for both the vertical and lateral accelerations of the carbody change slightly around the corresponding fundamental frequencies of the train and bridge with the rising of train speed in both cases A and B. (2) For the carbody vertical acceleration, the maximum values of the PSD of DVF1 increase as the train speed increases in both cases A and B. (3) The maximum values of the PSD of DVF1 for the carbody lateral acceleration do not show a trend of monotonic increase for higher train speeds in case A. This trend can be also found from the maximum values of the PSD of DVF2 for both the vertical and lateral accelerations of the carbody in case A. (4) For the carbody vertical acceleration, the maximum values of the PSD of DVF1 are generally bigger than those of DVF2 in both cases A and B, implying that the influence of the track irregularities in the vertical direction is more obvious than that of earthquakes. However the opposite is the case for the carbody lateral acceleration. (5) Unlike in case A, there exists only one DVF for the lateral accelerations of the carbody in case B, which generally coincides with the lateral fundamental frequency of the bridge. Meanwhile, the peak values of the PSD of the carbody acceleration in case B are generally larger than those in case A, but the difference between the lateral accelerations of the carbody is much larger than that for the vertical accelerations. For the motor car, the lateral and vertical accelerations of case B are larger than those of case A by up to 36% and 6%, respectively. One reason behind these is that the influence of earthquakes on the carbody lateral accelerations is greater than that of track irregularities. Another reason is that the inherent relative displacements between the wheel and track in the lateral direction are probably much larger than that in the vertical direction in case A.


### 5.4. Analysis of Dynamic Reliability of Train Operation Stability

As was mentioned in [7], the train operation stability over bridges under the action of earthquakes is an important subject for the railway engineers because the moving trains may not be safe enough due to the excessive vibrations of bridges. The dynamic reliability of train operation stability is investigated in this paper according to (35), in which “b” is equal to the allowable safety limit of each evaluation index. The standards used for evaluation of train operation stability are listed in Table 6 [38].Table 6
Evaluation standards for train operation stability.
Evaluation indices
Unit
Allowable limit
Carbody vertical acceleration
m/s2
1.30
Carbody lateral acceleration
m/s2
1.00
Lateral wheelset force for motor car
kN
53.83 (= 0.85 × (10 +Pw/3))∗∗∗
Lateral wheelset force for trailer car
kN
49.87 (= 0.85 × (10 +Pw/3))∗∗∗
Wheel load decrement ratio
—
0.65
∗
∗
∗
P
w denotes the static wheelset load, which is equal to 160.0 kN and 146.0 kN for motor car and trailer car in this paper, respectively.Three numerical examples are applied to analyze the influence of the train speed, the earthquake intensity, and the pier height on the dynamic reliability of train operation stability, respectively. To investigate the influence of the train speed on the dynamic reliability of train operation stability over bridge, the train is also assumed to move over the bridge with a constant speed varying from 150 to 420 km/h at 10 km/h intervals, with the same track irregularities and earthquakes as listed in Section5.1. The estimated dynamic reliability of train operation stability with respect to train speed is plotted in Figures 11–14. To investigate the influence of the earthquake intensity on the dynamic reliability of train operation stability over bridge, seven lateral seismic accelerations varying from 0.0 g to 0.3 g are used as the input excitations, respectively. For each case, the train is assumed to pass through the bridge at a speed of 300 km/h. The other parameters are as for Section 5.1. The estimated dynamic reliability of train operation stability with respect to seismic acceleration is shown in Figures 15–18. To investigate the influence of the pier height on the dynamic reliability of train operation stability over bridge, the analysis is performed by applying the height of pier from 10 m to 40 m at 2 m intervals and train running at a speed of 300 km/h, with other parameters same as those listed in Section 5.1. Figures 19–22 show the estimated dynamic reliability of train operation stability with respect to pier height.Figure 11
Dynamic reliability for carbody vertical acceleration with respect to train speed.Figure 12
Dynamic reliability for carbody lateral acceleration with respect to train speed.Figure 13
Dynamic reliability for lateral wheelset force with respect to train speed.Figure 14
Dynamic reliability for wheel load decrement ratio with respect to train speed.Figure 15
Dynamic reliability for carbody vertical acceleration with respect to seismic acceleration.Figure 16
Dynamic reliability for carbody lateral acceleration with respect to seismic acceleration.Figure 17
Dynamic reliability for lateral wheelset force with respect to seismic acceleration.Figure 18
Dynamic reliability for wheel load decrement ratio with respect to seismic acceleration.Figure 19
Dynamic reliability for carbody vertical acceleration with respect to pier height.Figure 20
Dynamic reliability for carbody lateral acceleration with respect to pier height.Figure 21
Dynamic reliability for lateral wheelset force with respect to pier height.Figure 22
Dynamic reliability for wheel load decrement ratio with respect to pier height.The following observations can be drawn from Figures11–14: (1) In general, the dynamic reliability decrease with the increase of train speed. (2) The dynamic reliability for the vertical acceleration of both the motor carbody and the trailer carbody as well as for the wheel load decrement ratio of the motor car change relatively slowly. (3) The dynamic reliability for the carbody vertical acceleration and the lateral wheelset force reduces drastically as the train speed exceeds 300 km/h. (4) The dynamic reliability for the motor carbody acceleration decreases faster than that of the trailer carbody as the train speed increases. (5) The dynamic reliability for the wheel load decrement of the trailer car drops rapidly as the train speed exceeds 350 km/h.The following are conclusions made from Figures15–18: (1) as is expected, the dynamic reliability decreases generally with the increase of earthquake intensity. (2) The dynamic reliability related to the carbody lateral acceleration drops more rapidly than that related to the carbody vertical acceleration. (3) The dynamic reliability for the lateral acceleration of the motor carbody and the trailer carbody begins to decrease rapidly as the seismic acceleration reaches 0.10 g and 0.15 g, respectively. (4) The dynamic reliability for the lateral wheelset force of both the motor car and the trailer car begins to drop quickly as the seismic acceleration reaches 0.20 g.Some conclusions can be drawn from Figures19–22: (1) in general, the dynamic reliability also decreases with the increase of pier height. (2) Compared with the very slight change of the dynamic reliability related to the carbody vertical acceleration, the dynamic reliability related to the carbody lateral acceleration drops remarkably with the increasing of pier height. (3) The dynamic reliability for the lateral acceleration of the motor carbody and the trailer carbody begins to drop rapidly as the pier height reaches 16 m and 22 m, respectively. (4) The dynamic reliability for the lateral wheelset force of the trailer car begins to decrease quickly as the pier height reaches 30 m.


## 5.1. Properties of the TSTBI System

A fifteen-span simply supported boxing girder high-speed railway bridge, with the span length of 32 m and the pier height of 20 m, is considered as shown in Figure1. The central part of the slab track is supported on the bridge, while the left and right parts of the track are supported on the subgrades adjacent to the bridge. The lengths of element of the rail, slab, and girder are all equal to the fastener spacing of 0.625 m, while the length of the pier element equals 1.0 m. The high-speed train comprises one front motor car, six trailer cars, and one rear motor car moving at a constant velocity v. The fundamental data of the motor car and trailer car can refer to [37]. The major fundamental data of the slab track and bridge are listed in Table 1. The PSDs of the German high-speed track spectrum of low irregularity [38] are adopted. In addition, the PSDs of the lateral horizontal and vertical seismic accelerations and the uniform modulation function (Figure 3) are taken as follows [6, 35]:(36a)Sayayω=ωgy4+4ζgy2ωgy2ω2ωgy2-ω22+4ζgy2ωgy2ω2·ω4ω2-ωfy22+4ζfy2ωfy2ω2S0y,(36b)Sazazω=ωgz4+4ζgz2ωgz2ω2ωgz2-ω22+4ζgz2ωgz2ω2·ω4ω2-ωfz22+4ζfz2ωfz2ω2S0z,(37)gt=ttb20≤t<tb1tb≤t<tcexp⁡ct-tct≥tc,where S0y and S0z denote the spectral intensity factor, ζgy and ζgz denote the damping ratio of the site, ωgy and ωgz denote the predominant frequency of the site, ζfy, ζfz, ωfy, and ωfz denote the filter parameters, c denotes the attenuation coefficient, and tb and tc denote the times at the start and end, respectively, of the stationary main shock. Additionally, ζgz = ζgy, ωgz=1.58ωgy, ζfy=ζgy, ωfy=0.1ωgy–0.2ωgy, ζfz=ζgz, ωfz=0.1ωgz–0.2ωgz, and S0z=0.218S0y. The parameters for the earthquakes with different intensities and site types are listed in Table 2. It is assumed that the DBGA is 0.15 g and the bridge site is Type II. The spatial frequency of the track irregularity PSDs ranges from 0.004 × 2π to 1 × 2π rad/m, while the frequency of the seismic acceleration PSDs lies in the range of 0–40 Hz. It is worth noting that the extreme state for the maximum responses of the bridge and each moving vehicle may not occur at the same time when the track irregularities and earthquakes are considered simultaneously, and therefore cannot be easily identified. This makes it rather difficult to determine how the earthquakes’ starting times affect the results. Here, the earthquakes are assumed to start exactly at the instant when the train enters the bridge [17].Table 1
Major parameters for slab track and bridge.
Notation
Item
Unit
Value
Slab track
Er
Young’s modulus of rail
N/m2
2.06 × 1011
Iry
Flexural moment of inertia abouty-axis of cross section of rail
m4
3.217 × 10−5
Irz
Flexural moment of inertia aboutz-axis of cross section of rail
m4
5.24 × 10−6
m-r
Mass per unit length of rail
kg/m
60.64
Es
Young’s modulus of slab
N/m2
3.6 × 1010
Isy
Flexural moment of inertia of cross section of slab
m4
1.4 × 10−3
Isz
Flexural moment of inertia of cross section of slab
m4
0.219
m-s
Mass per unit length of slab
kg/m
1.2 × 103
krsy
Lateral stiffness of fastener
N/m
3.0 × 107
krsz
Vertical stiffness of fastener
N/m
5.0 × 107
k-sby
Lateral stiffness of CAM per unit length
N/m2
1.5 × 109
k-sbz
Vertical stiffness of CAM per unit length
N/m2
1.5 × 109
Girder
Eb
Young’s modulus of girder
N/m2
3.45 × 1010
Iby
Flexural moment of inertia abouty-axis of cross section of girder
m4
12.744
Ibz
Flexural moment of inertia aboutz-axis of cross section of girder
m4
96.435
m-b
Mass per unit length of girder
kg/m
2.972 × 104
ωb1
Fundamental frequency of girder
Hz
5.58
ωb2
Second natural frequency of girder
Hz
13.26
Pier
Ep
Young’s modulus of pier
N/m2
3.15 × 1010
Ipx
Flexural moment of inertia aboutx-axis of cross section of pier
m4
73.23
m-p
Mass per unit length of pier
kg/m
5.2 × 104Table 2
Parameters of earthquake for different site conditions.
Parameter
Unit
Site classification
Type I
Type II
Type III
ω
g
y
rad/s
25.13
17.95
13.96
ζ
g
y
—
0.64
0.72
0.80
t
b
s
0.5
0.8
1.2
t
c
s
5.5
7.0
9.0
c
—
0.45
0.35
0.25
Duration of acting time of earthquake
s
6.69
8.41
10.92
S
0
y for DBGA = 0.15 g∗∗
cm2/s3
275
364
440
S
0
y for DBGA = 0.30 g∗∗
cm2/s3
1099
1456
1759
S
0
y for DBGA = 0.40 g∗∗
cm2/s3
1780
2359
2849
∗
∗DBGA is the design basic acceleration of ground motion.Figure 3
Plots of modulation functiong(t).


## 5.2. Analysis of the PSD Characteristic of the Random Vibration Responses of Bridge and Train

The random vibration responses of the bridge and train calculated by PEM with train speed of 300 km/h are still used.The PSDs of the vertical and lateral acceleration of the girder midpoint to the passage of the train shaken by track irregularities and earthquakes are plotted in Figure4. It can be seen from Figure 4(a) that there exists only one dominant vibration frequency (DVF) of 5.34 Hz, which is very close to the vertical fundamental frequency of the girder of 5.58 Hz. Similarly, only one DVF of 3.07 Hz can be found from Figure 4(b), which is very close to the lateral fundamental frequency of the total bridge (including the girder and pier) of 3.11 Hz.Figure 4
PSD of acceleration of bridge girder midpoint. (a) Vertical. (b) Lateral.
(a)
(b)Figure5 exhibit the PSDs of the vertical and lateral accelerations of the trailer carbody. As can be seen, the PSDs change violently with the vibration frequency, implying the great influence of track irregularities and earthquakes on the dynamic response of the carbody. Of interest is that there exist two DVFs with DVF1 = 0.83 Hz and DVF2 = 5.19 Hz in Figure 5(a), which are approximately equal to the vertical fundamental frequency of the trailer carbody and that of the girder, respectively. Similarly, two DVFs with DVF1 = 1.49 Hz and DVF2 = 2.98 Hz can be seen easily from Figure 5(b), which are also close to the lateral fundamental frequency of the trailer carbody and of the total bridge, respectively. Obviously, the vertical DVF1 and the lateral DVF1 are mainly excited by track irregularities, while the vertical DVF2 and the lateral DVF2 are mainly induced by the bridge vibration under the shake of earthquakes. On the other hand, the vertical and lateral PSDs excited by track irregularities vary only slightly with the time, which indicates that the influence of the bridge vibration on the carbody vibration is insignificant if there is no earthquake, because of the comparatively high mass and flexural rigidity of the bridge. However, the vertical PSD induced by earthquakes varies periodically with the vehicle passing through different girder (Figure 5(a)), while similar trend cannot be found for the lateral PSD induced by earthquakes (Figure 5(b)).Figure 5
PSD of acceleration of trailer carbody. (a) Vertical. (b) Lateral.
(a)
(b)A similar phenomenon for the vertical and lateral accelerations of the motor carbody can be also observed. Herein it is not discussed in detail to save the length of the paper.


## 5.3. Influence of Train Speed on the Random Vibration Characteristic of Bridge and Train

In reality, the train may move over the bridge at various speeds during earthquakes. There exists a need to investigate the random vibration characteristic of the train moving over the bridge under various train speeds, as they may be different. The train is assumed to move over the bridge with a constant speed varying from 150 to 420 km/h at 10 km/h intervals. The other parameters are the same as listed in Section5.1. Furthermore, two calculation cases (Table 3) are computed for comparison. In case B, the relative displacements between wheel, track, and girder are neglected [6]. The maximum values of the PSD and the DVF of the bridge and train with respect to train speed are shown in Figures 6–10 and Tables 4 and 5.Table 3
Calculation cases for studying the influence of train speed.
Calculation case
A
B
Calculation model
TSTBI model
Train-bridge interaction model [6]
Excitations
Track irregularity+ earthquake
Track irregularity+ earthquakeTable 4
Influence of train speed on PSDs of bridge midpoint acceleration.
Train speed(km/h)
Vertical
Lateral
DVF (Hz)
Peak value (m2/s3)
DVF (Hz)
Peak value (m2/s3)
Case A
Case B
Case A
Case B
Case A
Case B
Case A
Case B
150
5.34
5.34
1.50E − 01
1.58E − 01
2.98
3.07
2.01E − 01
2.17E − 01
170
5.27
5.42
1.53E − 01
1.61E − 01
3.03
3.11
2.20E − 01
2.24E − 01
190
5.27
5.42
1.59E − 01
1.67E − 01
3.03
3.11
2.26E − 01
2.30E − 01
210
5.36
5.36
1.50E − 01
1.63E − 01
2.99
3.08
2.19E − 01
2.34E − 01
230
5.26
5.40
1.63E − 01
1.72E − 01
3.02
3.10
2.30E − 01
2.33E − 01
250
5.26
5.41
1.66E − 01
1.80E − 01
3.02
3.10
2.33E − 01
2.35E − 01
270
5.37
5.37
1.53E − 01
1.66E − 01
2.90
3.08
2.25E − 01
2.42E − 01
290
5.31
5.46
1.80E − 01
1.72E − 01
3.01
3.05
2.35E − 01
2.31E − 01
310
5.31
5.37
1.57E − 01
1.70E − 01
3.01
3.08
2.25E − 01
2.44E − 01
330
5.26
5.41
1.68E − 01
1.84E − 01
3.01
3.10
2.36E − 01
2.41E − 01
350
5.28
5.42
1.77E − 01
1.85E − 01
3.03
3.11
2.38E − 01
2.37E − 01
370
5.28
5.28
1.76E − 01
1.83E − 01
3.03
3.11
2.37E − 01
2.36E − 01
390
5.26
5.41
1.70E − 01
1.84E − 01
3.02
3.11
2.34E − 01
2.39E − 01
410
5.12
5.12
1.62E − 01
1.70E − 01
3.02
3.08
2.27E − 01
2.41E − 01Table 5
Influence of train speed on PSDs of motor carbody acceleration.
Train speed (km/h)
Vertical
Lateral
DVF1(Hz)
Peak value(m2/s3)
DVF2(Hz)
Peak value(m2/s3)
DVF1(Hz)
Peak value(m2/s3)
DVF2(Hz)
Peak value(m2/s3)
150
Case A
0.96
3.69E − 03
5.20
1.59E − 03
1.07
1.59E − 03
2.98
1.03E − 02
Case B
0.96
3.75E − 03
5.49
1.55E − 03
—
—
3.07
1.12E − 02
170
Case A
0.92
4.44E − 03
5.27
1.92E − 03
1.21
1.92E − 03
3.11
8.21E − 03
Case B
0.92
4.53E − 03
5.42
1.93E − 03
—
—
3.11
1.12E − 02
190
Case A
0.94
5.22E − 03
5.27
2.05E − 03
1.03
2.05E − 03
3.20
9.24E − 03
Case B
0.94
5.32E − 03
5.27
2.13E − 03
—
—
3.11
1.09E − 02
210
Case A
0.96
5.70E − 03
5.36
2.12E − 03
1.10
2.12E − 03
3.16
8.28E − 03
Case B
0.96
5.81E − 03
5.36
2.08E − 03
—
—
3.08
1.08E − 02
230
Case A
0.94
6.23E − 03
5.11
1.65E − 03
1.24
1.65E − 03
3.02
1.00E − 02
Case B
0.94
6.35E − 03
5.56
1.75E − 03
—
—
3.10
1.08E − 02
250
Case A
0.92
6.62E − 03
5.11
1.52E − 03
1.37
1.52E − 03
2.94
8.63E − 03
Case B
0.92
6.75E − 03
5.56
1.61E − 03
—
—
3.10
1.12E − 02
270
Case A
0.91
7.21E − 03
5.37
1.87E − 03
0.96
1.87E − 03
3.08
9.61E − 03
Case B
0.91
7.34E − 03
5.52
1.93E − 03
—
—
3.08
1.12E − 02
290
Case A
0.92
7.61E − 03
5.31
2.80E − 03
1.03
2.80E − 03
3.13
8.92E − 03
Case B
0.92
7.74E − 03
5.46
2.89E − 03
—
—
3.13
1.10E − 02
310
Case A
0.94
7.89E − 03
5.22
1.95E − 03
1.07
1.95E − 03
3.13
9.57E − 03
Case B
0.94
8.03E − 03
5.22
2.00E − 03
—
—
3.08
1.11E − 02
330
Case A
0.94
8.17E − 03
5.41
2.10E − 03
1.14
2.10E − 03
3.10
1.05E − 02
Case B
0.92
8.31E − 03
5.41
2.21E − 03
—
—
3.10
1.13E − 02
350
Case A
0.92
8.39E − 03
5.13
2.13E − 03
1.21
2.13E − 03
3.11
8.15E − 03
Case B
0.92
8.54E − 03
5.28
2.07E − 03
—
—
3.11
1.14E − 02
370
Case A
0.92
8.58E − 03
5.43
2.26E − 03
1.35
2.26E − 03
2.95
8.59E − 03
Case B
0.92
8.72E − 03
5.43
2.38E − 03
—
—
3.11
1.15E − 02
390
Case A
0.92
8.76E − 03
4.98
1.74E − 03
1.14
1.74E − 03
3.03
1.07E − 02
Case B
0.92
8.91E − 03
4.98
1.76E − 03
—
—
3.11
1.14E − 02
410
Case A
0.89
8.97E − 03
4.98
3.06E − 03
1.08
3.06E − 03
2.94
1.03E − 02
Case B
0.87
9.12E − 03
5.12
3.09E − 03
—
—
2.94
1.12E − 02Figure 6
Girder midpoint acceleration with respect to train speed. (a) DVF. (b) Maximum value of PSD.
(a)
(b)Figure 7
Motor carbody vertical acceleration with respect to train speed. (a) DVF. (b) Maximum value of PSD.
(a)
(b)Figure 8
Motor carbody lateral acceleration with respect to train speed. (a) DVF. (b) Maximum value of PSD.
(a)
(b)Figure 9
Trailer carbody vertical acceleration with respect to train speed. (a) DVF. (b) Maximum value of PSD.
(a)
(b)Figure 10
Trailer carbody lateral acceleration with respect to train speed. (a) DVF. (b) Maximum value of PSD.
(a)
(b)Some conclusions can be drawn from Figure6 and Table 4: (1) both the maximum values of the PSD and the DVF of the girder vary slightly as the train speed increases in both cases A and B, which indicates that the vertical and lateral vibrations of the girder are induced basically by earthquakes and not by the train and the track irregularities. (2) The maximum values of the PSD in case B tend to be larger than those in case A because the effects of the constraint of the track are not taken into account in case B, but the degree of increase is below 10%.From Figures7–10 and Table 5, the following observations can be made easily: (1) the DVF1 and DVF2 for both the vertical and lateral accelerations of the carbody change slightly around the corresponding fundamental frequencies of the train and bridge with the rising of train speed in both cases A and B. (2) For the carbody vertical acceleration, the maximum values of the PSD of DVF1 increase as the train speed increases in both cases A and B. (3) The maximum values of the PSD of DVF1 for the carbody lateral acceleration do not show a trend of monotonic increase for higher train speeds in case A. This trend can be also found from the maximum values of the PSD of DVF2 for both the vertical and lateral accelerations of the carbody in case A. (4) For the carbody vertical acceleration, the maximum values of the PSD of DVF1 are generally bigger than those of DVF2 in both cases A and B, implying that the influence of the track irregularities in the vertical direction is more obvious than that of earthquakes. However the opposite is the case for the carbody lateral acceleration. (5) Unlike in case A, there exists only one DVF for the lateral accelerations of the carbody in case B, which generally coincides with the lateral fundamental frequency of the bridge. Meanwhile, the peak values of the PSD of the carbody acceleration in case B are generally larger than those in case A, but the difference between the lateral accelerations of the carbody is much larger than that for the vertical accelerations. For the motor car, the lateral and vertical accelerations of case B are larger than those of case A by up to 36% and 6%, respectively. One reason behind these is that the influence of earthquakes on the carbody lateral accelerations is greater than that of track irregularities. Another reason is that the inherent relative displacements between the wheel and track in the lateral direction are probably much larger than that in the vertical direction in case A.


## 5.4. Analysis of Dynamic Reliability of Train Operation Stability

As was mentioned in [7], the train operation stability over bridges under the action of earthquakes is an important subject for the railway engineers because the moving trains may not be safe enough due to the excessive vibrations of bridges. The dynamic reliability of train operation stability is investigated in this paper according to (35), in which “b” is equal to the allowable safety limit of each evaluation index. The standards used for evaluation of train operation stability are listed in Table 6 [38].Table 6
Evaluation standards for train operation stability.
Evaluation indices
Unit
Allowable limit
Carbody vertical acceleration
m/s2
1.30
Carbody lateral acceleration
m/s2
1.00
Lateral wheelset force for motor car
kN
53.83 (= 0.85 × (10 +Pw/3))∗∗∗
Lateral wheelset force for trailer car
kN
49.87 (= 0.85 × (10 +Pw/3))∗∗∗
Wheel load decrement ratio
—
0.65
∗
∗
∗
P
w denotes the static wheelset load, which is equal to 160.0 kN and 146.0 kN for motor car and trailer car in this paper, respectively.Three numerical examples are applied to analyze the influence of the train speed, the earthquake intensity, and the pier height on the dynamic reliability of train operation stability, respectively. To investigate the influence of the train speed on the dynamic reliability of train operation stability over bridge, the train is also assumed to move over the bridge with a constant speed varying from 150 to 420 km/h at 10 km/h intervals, with the same track irregularities and earthquakes as listed in Section5.1. The estimated dynamic reliability of train operation stability with respect to train speed is plotted in Figures 11–14. To investigate the influence of the earthquake intensity on the dynamic reliability of train operation stability over bridge, seven lateral seismic accelerations varying from 0.0 g to 0.3 g are used as the input excitations, respectively. For each case, the train is assumed to pass through the bridge at a speed of 300 km/h. The other parameters are as for Section 5.1. The estimated dynamic reliability of train operation stability with respect to seismic acceleration is shown in Figures 15–18. To investigate the influence of the pier height on the dynamic reliability of train operation stability over bridge, the analysis is performed by applying the height of pier from 10 m to 40 m at 2 m intervals and train running at a speed of 300 km/h, with other parameters same as those listed in Section 5.1. Figures 19–22 show the estimated dynamic reliability of train operation stability with respect to pier height.Figure 11
Dynamic reliability for carbody vertical acceleration with respect to train speed.Figure 12
Dynamic reliability for carbody lateral acceleration with respect to train speed.Figure 13
Dynamic reliability for lateral wheelset force with respect to train speed.Figure 14
Dynamic reliability for wheel load decrement ratio with respect to train speed.Figure 15
Dynamic reliability for carbody vertical acceleration with respect to seismic acceleration.Figure 16
Dynamic reliability for carbody lateral acceleration with respect to seismic acceleration.Figure 17
Dynamic reliability for lateral wheelset force with respect to seismic acceleration.Figure 18
Dynamic reliability for wheel load decrement ratio with respect to seismic acceleration.Figure 19
Dynamic reliability for carbody vertical acceleration with respect to pier height.Figure 20
Dynamic reliability for carbody lateral acceleration with respect to pier height.Figure 21
Dynamic reliability for lateral wheelset force with respect to pier height.Figure 22
Dynamic reliability for wheel load decrement ratio with respect to pier height.The following observations can be drawn from Figures11–14: (1) In general, the dynamic reliability decrease with the increase of train speed. (2) The dynamic reliability for the vertical acceleration of both the motor carbody and the trailer carbody as well as for the wheel load decrement ratio of the motor car change relatively slowly. (3) The dynamic reliability for the carbody vertical acceleration and the lateral wheelset force reduces drastically as the train speed exceeds 300 km/h. (4) The dynamic reliability for the motor carbody acceleration decreases faster than that of the trailer carbody as the train speed increases. (5) The dynamic reliability for the wheel load decrement of the trailer car drops rapidly as the train speed exceeds 350 km/h.The following are conclusions made from Figures15–18: (1) as is expected, the dynamic reliability decreases generally with the increase of earthquake intensity. (2) The dynamic reliability related to the carbody lateral acceleration drops more rapidly than that related to the carbody vertical acceleration. (3) The dynamic reliability for the lateral acceleration of the motor carbody and the trailer carbody begins to decrease rapidly as the seismic acceleration reaches 0.10 g and 0.15 g, respectively. (4) The dynamic reliability for the lateral wheelset force of both the motor car and the trailer car begins to drop quickly as the seismic acceleration reaches 0.20 g.Some conclusions can be drawn from Figures19–22: (1) in general, the dynamic reliability also decreases with the increase of pier height. (2) Compared with the very slight change of the dynamic reliability related to the carbody vertical acceleration, the dynamic reliability related to the carbody lateral acceleration drops remarkably with the increasing of pier height. (3) The dynamic reliability for the lateral acceleration of the motor carbody and the trailer carbody begins to drop rapidly as the pier height reaches 16 m and 22 m, respectively. (4) The dynamic reliability for the lateral wheelset force of the trailer car begins to decrease quickly as the pier height reaches 30 m.


## 6. Summary

In this study, the three-dimensional vibration model of the TSTBI system is firstly established by FEM, the equations of motion for the entire system are then derived by means of energy principle. The excitations caused by random track irregularities and seismic accelerations are transformed into a series of deterministic pseudoharmonic excitation vectors via PEM. Taking a high-speed train traveling over a fifteen-span simply supported girder bridge as an example, the random vibration characteristic of the coupled system and the dynamic reliability of train operation stability are investigated. From the numerical results obtained in this work, the following conclusions can be drawn:(1)
The time-dependent PSDs of the random vibration responses of the TSTBI system simultaneously excited by track irregularities and earthquakes can be obtained easily by PEM, which may be useful for unveiling the mechanism underlying the phenomena of resonance and cancellation of the system, as well as for the mitigation of vibrations of the system.(2)
The neglecting of the relative displacements between the wheel, track, and bridge may lead to an increase of the peak values of the PSD of the TSTBI system. Furthermore, the extent of the amplification for the lateral accelerations of the carbody is much larger than that for the vertical accelerations of the carbody.(3)
PEM may serve as a useful aid to the railway engineers for optimizing the design of dynamic parameters of the vehicle, track, and bridge to ensure the dynamic stability of a train traveling over a bridge under the actions of track irregularities and earthquakes on the basis of random vibration.



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*Source: 436567-2015-06-17.xml*