Numerical Simulation of Vertical Random Vibration of Train-Slab Track-Bridge Interaction System by PEM

Abstract

The paper describes the numerical simulation of the vertical random vibration of train-slab track-bridge interaction system by means of finite element method and pseudoexcitation method. Each vehicle is modeled as four-wheelset mass-spring-damper system with two-layer suspension systems. The rail, slab, and bridge girder are modeled by three-layer elastic Bernoulli-Euler beams connected with each other by spring and damper elements. The equations of motion for the entire system are derived according to energy principle. By regarding rail irregularity as a series of multipoint, different-phase random excitations, the random load vectors of the equations of motion are obtained by pseudoexcitation method. Taking a nine-span simply supported beam bridge traveled by a train consisting of 8 vehicles as an example, the vertical random vibration responses of the system are investigated. Firstly, the suitable number of discrete frequencies of rail irregularity is obtained by numerical experimentations. Secondly, the reliability and efficiency of pseudoexcitation method are verified through comparison with Monte Carlo method. Thirdly, the random vibration characteristics of train-slab track-bridge interaction system are analyzed by pseudoexcitation method. Finally, applying the 3σ rule for Gaussian stochastic process, the maximum responses of train-slab track-bridge interaction system with respect to various train speeds are studied.

1. Introduction

The dynamic analysis of railway track or bridge subjected to a moving train has long been a subject of great interest in the field of civil engineering [1–10]. As is well known [11–19], rail irregularity, with random nature, is one of the most important factors that can amplify the vibration responses of train-track/bridge interaction system. The dynamic responses in most of the previous researches are usually calculated by using only one time-history sample of rail irregularity. From the point of view of random vibration, those results can merely represent one of a series of possible random samples due to the fact that each time-history sample generated from the PSD function of rail irregularity is random.

With the increasing speed and axle load of train, the random vibration of train-track/bridge interaction system has received more and more attention and is usually analyzed by means of Monte Carlo method (MCM) [12, 16]. In this approach, the time-history samples of dynamic responses are calculated by using a series of samples of rail irregularity firstly, and then the statistical characteristics of responses are obtained according to the maximum values of each sample of dynamic responses. In order to ensure the reliability of the simulation, tens, hundreds, or even thousands of samples must be calculated, which may result in intolerable computer time, especially for the sophisticated train-slab track-bridge interaction system.

Pseudoexcitation method (PEM), originally developed from the field of bridge seismic resistance [20], has been applied to analyze efficiently and accurately the linear time-independent or time-dependent systems with stationary or nonstationary random excitations [21–25]. Lu et al. [23] studied the stationary random vibration responses of vehicle-ballasted track interaction system by PEM in which a time-independent vehicle-track interaction model was adopted. By adopting a train-bridge interaction element with the track structure omitted [26], PEM was applied to research the nonstationary random vibration of time-dependent train-bridge interaction system [24].

With the increasing use of ballastless track on bridge in modern high-speed railway, this paper investigates the vertical random vibration responses of slab ballastless track and bridge subjected to a moving train based on finite element method (FEM) and PEM. The train, slab track, and bridge are regarded as an integrated system. Each moving vehicle is modeled as four-wheelset mass-spring-damper system with two-layer suspension systems possessing 6 degrees of freedom (DOFs) [17]. The rail, slab, and bridge girder are modeled by three-layer elastic Bernoulli-Euler beams [17, 19, 27]. The elasticity and damping properties of the fastener and bridge bearing are represented by discrete springs and dampers. Simultaneously, the elasticity and damping properties of the concrete asphalt layer (CA layer) are represented by continuous springs and dampers [27, 28]. The equations of motion of finite element form for the entire system are derived by means of energy principle [28]. The effects of rail irregularity are regarded as a series of multipoint, different-phase random excitations by taking time lags between the wheels into account. The random excitations between rail and wheels caused by rail irregularity are then transformed into a series of deterministic pseudoharmonic excitation vectors by PEM, so that the random vibration responses of train, slab track, and bridge can be obtained by step-by-step integration method such as Newmark-β method [29] or Wilson- method [30]. Taking a nine-span simply supported beam bridge traveled by a train consisting of 2 motor cars and 6 trailer cars as an example, the suitable number of discrete frequencies of rail irregularity for calculating the random vibration responses of the system is obtained by numerical experimentations, the reliability and efficiency of the random vibration responses obtained by PEM are verified through comparison with those by MCM; the random vibration characteristics of train-slab track-bridge interaction system are analyzed by PEM, and finally the maximum responses of train-slab track-bridge interaction system with respect to various train moving speeds are studied by applying the 3σ rule for Gaussian stochastic process.

2. Models of Train, Slab Track, and Bridge

2.1. Model of Train

Figure 1 shows a train consisting of a series of four-wheelset vehicles moving with constant speed on a slab track structure resting on a series of simply supported beams to model a railway bridge and the two approach subgrades. The train consists of the front and rear motor cars numbered 1 and 2, respectively, and trailer cars numbered from right to left. It is assumed that the downward vertical displacements and clockwise direction rotation of train are taken as positive and that they are measured with reference to their respective static equilibrium positions before moving upon the track concerned.

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(b)

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Figure 1 

Model of train-slab track-bridge interaction system. (a) A train moving upon a slab track resting on a multispan simply supported beam bridge. (b) The th trailer car moving upon the th span simply supported beam.

Each trailer car in the train is modeled as a mass-spring-damper system consisting of one carbody, two bogies, four wheelsets, and two-stage suspensions. As shown in Figure 1, 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 a mass and a moment of inertia about the transverse horizontal axis through its center of gravity. Similarly, each bogie is considered as a rigid body having a mass and a moment of inertia about the transverse horizontal axis through its center of gravity. Each wheelset has a mass . The spring and shock absorber in the primary suspension for each wheelset are characterized by spring stiffness and damping coefficient , respectively. Likewise, the secondary suspension between carbody and each bogie is characterized by spring stiffness and damping coefficient . As the carbody is assumed to be rigid, the motions of the th trailer carbody may be described by the vertical displacement and rotation at its center of gravity, where the subscript denotes the trailer car number. Similarly, the motions of the front bogie of the th trailer car may be described by the vertical displacement and rotation at its center of gravity; the motions of the rear bogie of the th trailer car may be described by the vertical displacement and rotation at its center of gravity. The motions of the four wheelsets from right to left of the th trailer car may be described by the vertical displacements , , , and , respectively. Therefore, the total number of DOFs for each trailer car is ten. However, it is assumed that each wheelset of all vehicles is always in full contact with the rail in this paper; that is, the vertical displacement of each wheelset is constrained by the displacement of the rail. Consequently, the independent DOFs for each trailer car become six.

Each motor car in the train is also modeled as a mass-spring-damper system consisting of one carbody, two bogies, four wheelsets, and two-stage suspensions. The carbody has a mass and a moment of inertia . Each bogie has a mass and a moment of inertia . Each wheelset has a mass . The primary suspension is characterized by spring stiffness and damping coefficient . Likewise, the secondary suspension is characterized by spring stiffness and damping coefficient . The independent DOFs for each motor car include, too, the carbody vertical displacement and rotation , the front bogie vertical displacement and rotation , and the rear bogie vertical displacement and rotation , where the subscript denotes the motor car number.

2.2. Models of Slab Track and Bridge

As shown in Figure 1, the rail, slab, and bridge girder are all modeled as elastic Bernoulli-Euler beam. The two rails are effectively treated as one in the subsequent analysis. On the basis of FEM, the rail, slab, and bridge girder are all divided into a series of beam elements of equal length . The elasticity and damping properties of the fastener are represented by discrete massless springs with stiffness and dampers with damping coefficient . The elasticity and damping properties of the CA layer underlying the slab are represented by continuous massless springs with stiffness and dampers with damping coefficient . Simultaneously, the elasticity and damping properties of the bridge bearing are represented by massless springs with stiffness and dampers with damping coefficient . It is assumed that the damping of the rail and slab is neglected [19, 31], and the bridge girder has linear viscous damping [15, 32]. In addition, by neglecting axial deformations of the rail, slab, and bridge girder, each node of the rail, slab, and bridge girder has two DOFs, that is, vertical displacement and rotation about an axis normal to the plane of paper. The cubic Hermitian interpolation polynomials are used as shape functions of the rail, slab, and bridge girder elements.

It is assumed also that the downward deflections of rail, slab, and bridge are taken as positive and that they are measured with reference to their respective vertical static equilibrium positions. Let denote the initial vertical irregularities of rail and be measured with reference to smooth profile of rail; that is, , if the top surface of rail is smooth. Likewise, it is considered positive in the downward direction.

3. Equations of Motion for a Train-Slab Track-Bridge Interaction System

By using the energy principle, such as the principle of a stationary value of total potential energy of a dynamic system [28], one can derive the equations of motion written in submatrix for the train-slab track-bridge interaction system as where the subscripts “,” “,” “,” and “” denote the train, rail, slab, and bridge girder, respectively. The displacement vector, the matrices of mass, stiffness, and damping, as well as the load vector for the train, rail, slab, and bridge girder, are explained as follows.

3.1. Displacement Vectors

The total train displacement vector with order    can be written as where the superscript “” denotes the transpose of the vector and () and () are the displacement vectors of the th trailer car and the th motor car, respectively, which can be expressed as

The displacement vector of rail with order can be written as where denotes the total number of DOFs of rail.

The displacement vector with order for a series of continuously supported beams to model slab can be written as where () denotes the displacement vector of the th slab, denotes the total number of slab, and denotes the total number of DOFs of all slab. with order and can be expressed as

The displacement vector with order for a series of simply supported beams to model the bridge can be written as where () denotes the displacement vector of the th bridge girder, denotes the total number of bridge girder, and denotes the total number of DOFs of all bridge girder. with order and can be expressed as

3.2. Matrices for Train

The matrices of train are marked with the subscript “.” The mass matrix of train, with order () × (), can be written as where and with order denote the mass matrices of the th trailer car and th motor car, respectively, and can be expressed as

The stiffness matrix of train, with order () × (), can be written as where and with order , denoting the stiffness matrices of the th trailer car and th motor car, respectively, can be expressed as in which denotes half of longitudinal distance between the centers of gravity of trailer car’s front and rear bogies, denotes half of longitudinal distance between the centers of gravity of motor car’s front and rear bogies, denotes half of trailer car’s bogie axle base, and denotes half of motor car’s bogie axle base.

The damping matrix of train with order () × () can be obtained by simply replacing in the corresponding stiffness matrix by .

3.3. Matrices for Rail, Slab, and Bridge

The matrices of rail are marked with the subscript “.” The mass matrix of rail, with order , can be written as with where , with order , represents the overall mass matrix of the rail itself, represents rail mass per unit length, represents the total number of rail elements, represents the local coordinate measured from the left node of a beam element, and , with order , represents the shape function matrix for the th rail element. It should be noted that each element is zero in except for those corresponding to the four DOFs of the two nodes of the th rail element. , with order , represents the overall mass matrix induced by all the wheel masses of trailer cars; , with order , represents the shape function matrix for the rail element, evaluated at the position of the th wheelset of the th trailer car; , , , and , respectively, represent the distances between the 1st, 2nd, 3rd, and 4th wheelsets of the th trailer car and the left node of the rail element upon which the wheelset is acting. It should be noted that each element is zero in except those corresponding to the four DOFs of the two nodes of the rail element upon which the th wheelset of the th trailer car is acting. is time dependent as the th wheelset of the th trailer car moves from one position to another within one rail element. As the th wheelset of the th trailer car moves to the next rail element, will shift in position corresponding to the DOFs of the rail element where the th wheelset of the th trailer car is positioned. Similarly, , with order , represents the overall mass matrix induced by all the wheel masses of motor cars; , with order , represents the shape function matrix for the rail element, evaluated at the position of the th wheelset of the th motor car; , , , and , respectively, represent the distances between each wheelset of the th motor car and the left node of the corresponding rail element upon which the wheelset is acting. It should be noted too for that, apart from those elements corresponding to the four DOFs of the two nodes of the rail element upon which the th wheelset of the th motor car is acting, all other elements are zero. is also time dependent.

The stiffness matrix of rail, with order , can be similarly expressed in terms of the overall stiffness matrix of the rail itself, overall stiffness matrix induced by all trailer cars, overall stiffness matrix induced by all motor cars, and overall stiffness matrix induced by the stiffness of all fasteners as with where denotes Young’s modulus of the rail and denotes the constant moment of inertia of the rail cross section and the prime differentiation with respect to local coordinate . In the formulation of , the shape function matrix () of order for the rail element is evaluated at the position of the th fastener; denotes the distance between the th fastener and the left node of the rail element containing the th fastener, and denotes the total number of fastener. It should be noted for that, apart from those elements corresponding to four DOFs of the two nodes of the rail element containing the th fastener, all other elements are zero.

With omission of the damping of the rail itself, the damping matrix of rail, with order , can be written in terms of the overall damping matrix induced by all trailer cars, overall damping matrix induced by all motor cars, and overall damping matrix induced by the damping of all fasteners as with

The matrices of slab, with order , are marked with the subscript “” and can be written in terms of the matrices of the slab itself, the matrices induced by fastener on the slab, and the matrices induced by CA layer underlying the slab. The matrices for bridge, with order , are marked with the subscript “” and can be written in terms of the matrices of the bridge girder itself, the matrices induced by CA layer lying on the bridge girder, and the matrices induced by bearing at the support of bridge girder. The matrices of slab and bridge are not derived in detail here but can be obtained by following the similar procedure for derivation of rail matrices.

3.4. Matrices for Train-Rail-Slab-Bridge Interaction

The matrices for train-rail interaction are marked with subscript “” or “.” The stiffness matrices of order , of order , of order , and of order induced by the interaction between the train and rail can be written, respectively, as with in which and represent the stiffness matrices induced by the interaction between the th wheelset of the th trailer car and rail, and and are the corresponding damping matrices. , , , and consist of zero row vectors except for those corresponding to the two DOFs of front bogie of the th trailer car, while , , , and consist of similar column vectors. Accordingly, , , , , , , , and are formed by row or column vectors where the only nonzero elements correspond to the two DOFs of the rear bogie.

The matrices induced by the interaction between motor car and rail can be worked out similarly.

The matrices for rail-slab-bridge interaction can be also derived by following the similar procedure for derivation of train-rail interaction matrices. The matrices for rail-slab interaction, induced by the matrices of fastener between rail and slab, are marked with subscript “” or “.” The matrices for slab-bridge interaction, induced by the matrices of CA layer between slab and bridge girder, are marked with subscript “” or “.”

3.5. Load Vector for Train, Rail, Slab, and Bridge

The load vector of train with order can be written as where the load vector of the th trailer car and and the load vector of the th motor car and with order can be written, respectively, as in which denotes the first derivative of rail irregularity .