Three-Dimensional Rail-Bridge Coupling Element of Unequal Lengths for Analyzing Train-Track-Bridge Interaction System

Abstract

A three-dimensional rail-bridge coupling element of unequal lengths in which the length of the rail element is shorter than that of the bridge element is presented in this paper to investigate the spatial dynamic responses of a train-track-bridge interaction system. Formulation of stiffness and damping matrices for the fastener, ballast, and bearing, as well as the three-dimensional equations of motion in matrix form for a train-track-bridge interaction system using the proposed element are derived in detail using the energy principle. The accuracy of the proposed three-dimensional rail-bridge coupling element is verified using the existing two-dimensional element. Three examples of a seven-span continuous beam bridge are shown: the first investigates the influence of the efficiency and accuracy of the lengths of the rail and bridge elements on the spatial dynamic responses of the train-track-bridge interaction system, and the other two illustrate the influence of two types of track models and two types of wheel-rail interaction models on the dynamic responses of the system. Results show that (1) the proposed rail-bridge coupling element is not only able to help conserve calculation time, but it also gives satisfactory results when investigating the spatial dynamic responses of a train-track-bridge interaction system; (2) the double-layer track model is more accurate in comparison with the single-layer track model, particularly in relation to vibrations of bridge and rail; and (3) the no-jump wheel-rail interaction model is generally reliable and efficient in predicting the dynamic responses of a train-track-bridge interaction system.

Keywords:
Three-dimensional rail-bridge; coupling element; unequal length; bridge; track; finite element method

1 INTRODUCTION

A considerable amount of research has been conducted on the dynamic responses of railway bridge/track structures subjected to a moving train (Sun and Dhanasekar, 2002; Liu et al., 2009; Lu et al., 2009; Wang et al., 2010; Kim, 2011; Zakeri et al., 2014; Lei and Wang, 2014; Xu et al., 2015). Such research has been conducted particularly in the past three decades and mostly in relation to the rapid development of high-speed railways worldwide. However, due to the massive volume of work conducted, it is difficult to have a complete count of the number of studies and it is only possible to cite a few of those that are most relevant here.

The dynamic response of structures in relation to moving vehicles has been studied by previous researchers by modeling a moving vehicle as a moving load, moving mass, or a moving sprung mass with consideration of suspension (Ayre et al., 1950; Frýba, 1972; Chu et al., 1979; Wu and Dai, 1987; Chatterjee et al., 1994; Ichikawa et al., 2000). More sophisticated models that also consider the vertical dynamic interaction between the moving train and structures have also been implemented by a large number of researchers in recent years. For example, Zhai and Sun (1994) developed a new and detailed model to investigate the vertical interaction between a vehicle and the track in which the vehicle was modeled as a multi-body system with 10 degrees of freedom (DOFs), the track as an infinite Euler beam, and the wheel-rail interaction as a Hertzian nonlinear contact spring. In addition, Yang et al. (1999) derived a vehicle-bridge interaction element by considering a vehicle as a rigid beam supported by two suspension units and a bridge as beam elements, and Cheng et al. (2001) proposed a bridge-track-vehicle element in which the vehicles were modeled as mass-spring-damper systems, the rails as an upper beam element, and the bridge deck as a lower beam element. Furthermore, Lei and Noda (2002) developed a dynamic computational model for a vehicle and track coupling system using the finite element method (FEM), in which the vehicle-track coupling dynamic responses were analyzed in time and frequency domains due to the random irregularity of the track vertical profile. Thereafter, Wu and Yang (2003) investigated the vertical dynamic responses of a vehicle-rails-bridge interaction system using a condensation technique, which included the steady-state response and riding comfort of the train as well as the impact response of the rails and bridges. Based on the principle of a stationary value of total potential energy of dynamic system, Zeng (2003), Lou (2005), and Lou and Zeng (2005) derived equations of motion in a matrix form for three types of vehicle-track-bridge vertical interaction elements, in which the rails and the bridge deck were represented by an elastic Bernoulli-Euler upper beam with finite length and a simply supported Bernoulli-Euler lower beam, respectively. A later study by Lou (2007) investigated the vertical dynamic responses of a train-track-bridge interaction (TTBI) system using FEM, and by discretizing the slab track subsystem into track elements that flow with the moving vehicle, Lei and Wang (2014) developed a new approach with finite elements in a moving frame of reference to investigate the dynamic behavior of the train and slab track coupling system.

In addition, with the exception of work that has been restricted to mainly analyzing the two-dimensional (2D) dynamic responses of a train-track/bridge interaction system, a great volume of research has also dealt with three-dimensional (3D) aspects of the system. For example, Zhai et al. (1996) presented a new vertical and lateral coupling model of vehicle-track interaction, and investigated the safety limits against derailment due to track twist and the combined alignment and cross-level irregularities. In addition, Xia et al. (2000) studied the dynamic responses of the bridge-train system, and the derailment and the offload factors related to the running safety of the train, using a 3D finite element model to represent the bridge. Furthermore, Wu et al. (2001) developed a vehicle-rail-bridge interaction model to analyze the 3D dynamic interaction between moving trains and the railway bridge, and Dinh et al. (2009) developed a formulation for 3D dynamic interactions between a bridge and a high-speed train using wheel-rail interfaces, where the bridge eccentricities and deck displacement due to torsion were accounted for in bridge deck modeling. Papers have also been written addressing the dynamic interaction between the track/bridge and the moving train, and some monographs have focused on this subject. For example, Song et al. (2003), Kwasniewski et al. (2006), Nguyen et al. (2009), Lei and Zhang (2011), Xin and Gao (2011), and Zhai et al. (2013) proposed a theory and method for dealing with the dynamic problem of the vehicle-track/bridge interaction system, respectively.

In the aforementioned works, most researchers have established the track-bridge interaction model using FEM, in which a rail-bridge coupling element of equal lengths (i.e., with the length of the rail element equal to that of the bridge element) is adopted. When the length of the bridge increases, the DOFs of the track-bridge interaction system also increase, and thus making a dynamic analysis of a track-bridge interaction system is a relatively time consuming process when using a rail-bridge coupling element of equal lengths. Therefore, the aim of this paper is to present a 3D rail-bridge coupling element of unequal lengths, in which sleepers are considered and where the length of the bridge element is longer than that of the rail element, to investigate the spatial dynamic responses of a TTBI system under the action of track irregularities. This paper can therefore be regarded as an extension of the theory presented by Lou et al. (2012), in which a 2D (vertical) rail-bridge coupling element of unequal lengths was proposed to analyze the vertical dynamic responses of a TTBI system. However, the possibility of considering the lateral responses of a TTBI system in the current work allows for a more realistic analyses.

In this study, a seven-span continuous beam bridge is used as an example, the influences of the lengths of the bridge and rail elements, two types of track models, and two types of wheel-rail interaction models on the efficiency and accuracy for calculating the spatial dynamic responses of the TTBI system excited by track irregularities are carried out, based on which some conclusions are drawn.

2 A 3D RAIL-BRIDGE COUPLING ELEMENT OF UNEQUAL LENGTHS

2.1 Model

A typical 3D rail-bridge coupling element of unequal lengths is shown in Figure 1 in which the length of the rail element is shorter than that of the bridge element (the corresponding physical parameters in Figure 1 are defined in Table 1). In the present study, the spatial dynamic behavior is studied while the axial deformations of the rail and bridge are ignored. Using a case of a double-track bridge, the dynamic responses of only one track is investigated and the other track is considered to be the dead load of the bridge, because the flexural rigidity of the bridge is usually thousands of times greater than that of the rails (or even tens of thousands). The proposed 3D coupling element ultimately consists of several rail elements of equal lengths (including the left and right rail), a bridge element, a few sleepers, a series of fasteners, and a series of discrete ballasts. It can also include a bearing that connects a pier node at a supporting point of the bridge. The rails, bridges, and piers are modeled as uniform Bernoulli-Euler beams, while each sleeper is modeled as a rigid body, and the lateral and vertical elasticity and damping properties of the fastener, ballast, and bearing are modeled using discrete massless springs and dampers. The mass of the ballast is also added to the dead load of the bridge. As the longitudinal vibrations are neglected, each node in the rail and bridge elements has five DOFs, i.e., a lateral displacement along the y-axis, a vertical displacement along the z-axis, and three rotations about the x-, y-, and z-axes. Each sleeper and each node in the pier element has three DOFs, i.e., a lateral displacement along the y-axis, a vertical displacement along the z-axis, and a rotation about the x-axis. The positive directions of these DOFs accord with those of the co-ordinate, as shown in Figure 1. In addition, it is assumed that the length of the bridge element (LBE) is an integer number of times of the length of the rail element (LRE).

Figure 1:
Typical 3D rail-bridge coupling element of unequal lengths: (a) frontal view, (b) left side view, and (c) top view.

Table 1:
Major parameters of track and bridge.

2.2 Formulation of Stiffness and Damping Matrices of Fastener

For both the vertical and lateral discrete springs and dampers representing a fastener, one end point connects with an element of the left or right rail, while the other end point connects with a sleeper, as shown in Figures 1 and 2. Taking as an example the vertical discrete spring modeling with a left fastener connecting the ith left rail element and a sleeper (Figure 2 (a) and (b)), the upper end point has a dependent DOF depending on the vertical displacement, z_Lr , and rotation, ϑ_xLr, about the x-axis of the ith left rail element, while the lower end point also has a dependent DOF depending on the vertical displacement, z_s , and rotation, ϑ_xs , about the x-axis of the corresponding sleeper. The elastic strain energy of the vertical spring, П^{LZ, e}_fas , can then be expressed as

Figure 2:
Sleeper attached to the ith left rail element by fastener:frontal view, (b) left side view, (c) top view.

(1)

with

(2)

where ξ_rs denotes the longitudinal distance between the left node of the ith left rail element and the discrete spring, and k^{LZ, e} _fas denotes the stiffness matrix of the vertical discrete spring for a left fastener.

Similarly, one end point of the lateral spring for a left fastener connecting the ith left rail element and a sleeper has a dependent DOF depending on the lateral displacement, y_Lr , and rotation, ϑ_xLr , about the x-axis of the ith left rail element, while the other end point has an independent DOF, i.e., the lateral displacement, y _s , of the corresponding sleeper (Figure 2 (b) and (c)). The stiffness matrix of the lateral discrete spring for a left fastener, k^{LY, e} _fas , can then be expressed as

(3)

with

N_ry,1 =1-3(ξ_rs/l_re )² + 2(ξ_rs/l_re )³ N_ry,2 = ξ_rs [1-2(ξ_rs/l_re )+ ξ_rs/l_re )²]

N_ry,3 = 3(ξ_rs/l_re )² - 2(ξ_rs/l_re )³ N_ry,4 = ξ_rs [(ξ_rs/l_re )² - (ξ_rs/l_re )]

The vertical and lateral damping matrices, c^{LZ, e} _fas and c^{LY, e} _fas , of the discrete damper for a left fastener can be obtained simply by replacing "k_rsz " and "k_rsy " in the corresponding stiffness matrices, k^{LZ, e} _fas and k^{LY, e} _fas , using "c_rsz " and "c_rsy ", respectively.

The vertical stiffness and damping matrices for a right fastener, k^{RZ, e} _fas and c^{RZ, e} _fas , as well as the lateral stiffness and damping matrices, k^{RY, e} _fas and c^{RY, e} _fas , can be obtained by following a procedure similar to that given above.

2.3 Formulation of Stiffness and Damping Matrices of Ballast

For both the vertical and lateral discrete spring and damper representing a ballast, one end point connects to a sleeper, while the other end point connects to a bridge element, as shown in Figures 1 and 3. If we use as an example the vertical discrete spring in modeling a ballast connecting a sleeper, and the ith bridge element (Figure 3 (a) and (b)), the upper end point has a dependent DOF that depends on the vertical displacement, z_s , and the rotation, ϑ_xs , about the x-axis of the sleeper, while the lower end point also has a dependent DOF depending on the vertical displacement, z_b , and the rotation, ϑ_xb , about the x-axis of the ith bridge element. The stiffness matrix of the vertical discrete spring for a ballast, k^Z,e _bal , can then be expressed as follows,

(4)

with

Nsbz ,1 = 1-3(ξsb/lbe )2 + 2(ξsb/lbe )3 Nsbz ,2 = ξsb [-1+2(ξsb/lbe )- (ξsb/lbe )2

_^Nsbz ,₃ = 3(ξ_sb/l_be )² - 2(ξ_sb/l_be )³.............._^Nsbz ,₄ = ξ_sb [-(ξ_sb/l_be )²+(ξ_sb/l_be )]

_^Nsbθx,1 =1 - ξ_sb/l_{be ^Nsbθx,2} = ξ_sb/l_be ,

where ξ_sb denotes the longitudinal distance between the left node of the ith bridge element and the discrete spring.

Figure 3:
Sleeper attached to ith bridge element of by ballast: (a) frontal view, (b) left side view, (c) top view.

Similarly, one end point of the lateral spring for a ballast connecting a sleeper and the ith bridge element has an independent DOF, i.e., the lateral displacement, y _s , of the sleeper, while the other end point has a dependent DOF depending on the lateral displacement, y _b , and the rotation, ϑ_xb , about the x-axis of the ith bridge element (Figure 3 (b) and (c)). The stiffness matrix of the lateral discrete spring for a ballast, k^{Y ,e} _bal , can then be derived as follows,

(5)

with

Nsby ,1 = 1-3(ξsb/lbe )2 + 2(ξsb/lbe )3 Nsby ,2 = ξsb [1-2(ξsb/lbe )+ (ξsb/lbe )2]

Nsby ,3 = 3(ξsb/lbe )2 - 2(ξsb/lbe )3 Nsby ,4 = ξsb [(ξsb/lbe )2-(ξsb/lbe )].

The vertical and lateral damping matrices, c^{Z, e} _bal and c^{Y, e} _bas , of the discrete damper for a ballast can be obtained simply by replacing "k_sbz " and "k_sby " in the corresponding stiffness matrices, k^{Z, e} _bal and k^{Y ,e} _bal , using "c_sbz " and "c_sby ", respectively.

2.4 Formulation of Stiffness and Damping Matrices of Bearing

For both the vertical and lateral discrete springs and damper that represent a bearing, one end point connects a bridge element, while the other end point connects a node of the pier element, as shown in Figures 1 and 4. If we take the model of a vertical discrete spring and bearing connecting the ith bridge element and the ith node of pier element as an example (Figure 4 (a) and (b)), the upper end point has a dependent DOF depending on the vertical displacement, zb, and rotation, ϑxb, about the x-axis of the ith bridge element. In addition, the lower end point also has a dependent DOF relating to the vertical displacement, zp,i, and rotation, ϑxp,i, about the x-axis of the ith node of the pier element. In this case, the stiffness matrix of the vertical discrete spring for a bearing, ,k^{Z, e} _bea , can be expressed as

(6)

with

Nbbz,1 =1-3(ξbb/lbe )2 + 2(ξbb/lbe )3 Nbbz,2 = ξbb [-1+2(ξbb/lbe )- (ξbb/lbe )2

Nbbz,3 =3(ξbb/lbe )2 - 2(ξbb/lbe )3........ Nbbz,4 = ξbb [-(ξbb/lbe )2+(ξbb/lbe )

_^Nbbθx,1 =1- ξ_bb/l_{be ^Nb bθx,2} = ξ_bb/l_be

where ξ_bb denotes the longitudinal distance between the left node of the ith bridge element and the discrete spring.

Figure 4:
Bridge element attached to the ith node of pier element by bearing: (a) frontal view, (b) left side view.

Similarly, one end point of the lateral spring for a bearing connecting the ith bridge element and the ith node of pier element has a dependent DOF in relation to the lateral displacement, yb, and rotation, ϑ xb, about the x-axis of the ith bridge element. In addition, the other end point has na independent DOF, i.e., the lateral displacement, yp,i, of the corresponding node of the pier element (Figure 4 (b)). The stiffness matrix of the lateral discrete spring for a bearing, k^{Y ,e} _bea , can then be expressed as

(7)

with

Nbby ,1 = 1-3(ξbb/lbe )2 + 2(ξbb/lbe )3 Nbby ,2 = ξbb [1-2(ξbb/lbe )+ (ξbb/lbe )2]

Nbby ,3 = 3(ξbb/lbe )2 - 2(ξbb/lbe )3 Nbby ,4 = ξbb [(ξbb/lbe )2-(ξbb/lbe )]

The vertical and lateral damping matrices, c^{Z, e} _bea and c^{Y, e} _bea , respectively, of the discrete damper for a bearing can then be obtained simply by replacing "k_bbz " and "k_bby " in the corresponding stiffness matrices, k^{Z, e} _bea and k^{Y, e} _bea , with "C_bbz " and " ", respectively.

3 3D EQUATIONS OF MOTION FOR A TTBI SYSTEM WITH PROPOSED ELEMENT

Figure 5 shows a train consisting of a series of four-wheelset vehicles moving with a constant speed, v_t , on a ballasted track structure that rests on a multi-span continuous beam bridge.

Figure 5:
3D model for TTBI system: (a) frontal view, (b) jth vehicle moving on rail-bridge coupling elements of unequal lengths, (c) left side view, (d) top view (without bridge).

The train consists of N_v identical vehicles numbered 1, 2, ...N_v , from right to left. Each vehicle in the train is modeled as a mass-spring-damper system consisting of one carbody, two bogies, four wheelsets, and two-stage suspensions. The carbody is modeled as a rigid body with mass, m_c , and three moments of inertia, I_cx, I_cy , and I_cz . Similarly, each bogie is considered as a rigid body with mass, m_t , and three moments of inertia, I_tx, I_ty , and I_tz , and each wheelset is considered as a rigid body with mass, m_w , and two moments of inertia, I_wx and I_wz . The secondary suspension between the carbody and each bogie is characterized by a three-dimensional system of springs with stiffnesses k_sx, k_sy , and k_sz and dampers with damping coefficients c_sx, c_sy , and c_sz . Likewise, the springs and shock absorbers in the primary suspension for each wheelset are characterized by k_px, k_py , and k_pz and c_px, c_py , and c_pz , respectively. By neglecting longitudinal displacements, the motions of the carbody of the jth vehicle with respect to its center of gravity may be described by y_cj, z_cj , ϑ_cj, φ_cj , and ψ_cj . Similarly, the motions of both the front and rear bogies of the jth vehicle may be described by y_t1j, z_t1j , ϑ_t1j, φ_t1j , and ψ_t1j and y_t2j, z_t2j , ϑ_t2j, φ_t2j , and ψ_t2j , respectively. In addition, the motion from right to left of the hth (h = 1-4) wheelset of the jth vehicle may be described by y_whj, z_whj , ϑ_whj , and ψ_whj , respectively. In this paper however, it is assumed that no jumps occur between the vehicle's wheels and the rails; that is, the vertical and rolling displacements of each wheelset are constrained by the corresponding displacements of the rails. Consequently, each vehicle has 23 independent DOFs.

By using the energy principle, such as the principle of the stationary value of total potential energy of a dynamic system (Zeng, 2000; Lou and Zeng, 2005), it is possible to derive the 3D equations of motion written in a sub-matrix for a TTBI system that is shown in Figure 5, as

(8)

where the subscripts "t", "r", "s", "b", and "p" denote the train, rail, sleeper, bridge, and pier, respectively; M, C, and K denote the mass, damping, and stiffness sub-matrices, respectively; and X and F denote the displacement and force sub-vectors, respectively. The formation of equation (8) from terms in equations (2), (3), (4), (5), (6), and (7) is further explained below.

In order to build up equation (8), the stiffness matrices, k^{LZ, e}_fas , in equation (2) and, k^{LY, e}_fas , in equation (3) can be partitioned into four parts as follows,

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

and

(18)

Elements in matrices k^{LZ, e 1} _fas and k^{LY, e 1} _fas should be placed in the stiffness sub-matrix K_Lrr (see equation (27)); elements in matrices k^{LZ, e 2} _fas and k^{LY, e 2} _fas should be placed in the stiffness sub-matrix K_Lrs ; elements in matrices k^{LZ, e 3} _fas and k^{LY, e 3} _fas should be placed in the stiffness sub-matrix KsLr; and elements in matrices k^{LZ, e 4} _fas and k^{LY, e 4} _fas should be placed in the stiffness sub-matrix Kss. Furthermore, in a similar manner to k^{LZ, e}_fas and k^{LY, e}_fas , the damping matrices, c^{LZ, e}_fas and c^{LY, e}_fas , can be partitioned into four parts and placed in the damping sub-matrices, C_Lrr , C_Lrs , C_sLr , and C_ss , respectively.

The stiffness matrix, k^{RZ, e}_fas , can be partitioned into four parts and used as follows in building equation (8). Elements in the first six rows and the first six columns should be placed in the stiffness sub-matrix K_Rrr ; elements in the first six rows and the last two columns should be placed in the stiffness sub-matrix K_Rrs ; elements in the first six columns and the last two rows should be placed in the stiffness sub-matrix K_sRr ; and the remaining elements should be placed in the stiffness sub-matrix K_ss .

The stiffness matrix, k^{RY, e}_fas , can be partitioned into four parts and used as follows in building up equation (8). Elements in the first six rows and the first six columns should be placed in the stiffness sub-matrix K_Rrr ; elements in the first six rows and the last column should be placed in the stiffness sub-matrix K_Rrs ; elements in the first six columns and the last row should be placed in the stiffness sub-matrix K_sRr ; and the remaining element should be placed in the stiffness sub-matrix K_ss . In a similar manner as k^{RZ, e}_fas and k^{RY, e}_fas , the damping matrices, c^{RZ, e}_fas and c^{RY, e}_fas , can be partitioned into four parts and placed in the damping sub-matrices, C_Rrr , C_Rrs , C_sRr , and C_ss , respectively.

Furthermore, the stiffness matrix, k^{Z ,e} _bal , in equation (4) can be partitioned into four parts and used as follows in building up equation (8). Elements in the first two rows and the first two columns should be placed in the in the stiffness sub-matrix K_ss ; elements in the first two rows and the last six columns should be placed in the stiffness sub-matrix K_sb ; elements in the first two columns and the last six rows should be placed in the stiffness sub-matrix K_bs ; and the remaining elements should be placed in the stiffness sub-matrix K_bb .

The stiffness matrix, k^{Y ,e} _bal , in equation (5) can be partitioned into four parts as follows and used as follows in building up equation (8). Elements in the first row and the first column should be placed in the in the stiffness sub-matrix K_ss ; elements in the first row and the last six columns should be placed in the stiffness sub-matrix K_sb ; elements in the first column and the last six rows should be placed in the stiffness sub-matrix K_bs ; and the remaining elements should be placed in the stiffness sub-matrix K_bb . In a similar manner as k^{Z, e} _bal and k^{Y ,e} _bal , the damping matrices, c^{Z, e} _bal and c^{Y, e} _bal , can be partitioned into four parts and placed in the damping sub-matrices, C_ss , C_sb , C_bs , and C_bb , respectively.

The stiffness matrix, k^{Z ,e} _bea , in equation (6) can be partitioned into four parts and used as follows in building up equation (8). Elements in the first six rows and the first six columns should be placed in the stiffness sub-matrix K_bb ; elements in the first six rows and the last two columns should be placed in the stiffness sub-matrix K_bp ; elements in the first six columns and the last two rows should be placed in the stiffness sub-matrix K_pb ; and the remaining elements should be placed in the stiffness sub-matrix K_pp .

The stiffness matrix, k^{Y ,e} _bea , in equation (7) can be partitioned into four parts and used as follows in building up equation (8). Elements in the first six rows and the first six columns should be placed in the stiffness sub-matrix K_bb ; elements in the first six rows and the last column should be placed in the stiffness sub-matrix K_bp ; elements in the first six columns and the last row should be placed in the stiffness sub-matrix K_pb ; and the remaining elements should be placed in the stiffness sub-matrix K_pp . In a similar manner as K^{Z ,e} _bea and k^{Y ,e} _bea , the damping matrices, c^{Z ,e} _bea and c^{Y ,e} _bea , can be partitioned into four parts and placed in the damping sub-matrices, C_bb , C_bp , C_pb , and C_pp , respectively.

The displacement sub-vectors, the mass, damping, and stiffness sub-matrices, as well as the force sub-vectors of the train, rail, sleeper, bridge, and pier are explained briefly in the following sections, and a detailed explanation is found in Lou (2005, 2007) and Lou and Zeng (2005).

3.1 Displacement Vectors

The displacement sub-vector of the total train, X_t , of order T_dof ×1(T_dof =23×N_v ) can be written as

(19)

where the superscript "T" denotes the transpose of the matrix, and x_vj (j = 1, 2, ..., N_v ) are the displacement vectors of the jth vehicle, which can be expressed as

X_vj = [y_cj z_cj θ_xcj θ_ycj θ_zcj y_t1j z_t1j θ_xt1j θ_yt1j θ_zt1j y_t2j z_t2j θ_xt2j θ_yt2j θ_zt2j y_w1j θ_zw1j y_w2j θ_zw2j y_w3j θ_zw3j y_w4j θ_zw4j ]

The displacement sub-vector of the rail, X_r , of order , can be written as, and XRr of the right rail of order , which is composed of the displacement vectors XLr of the left rail of order

(20)

where denotes the total number of DOFs of each rail.

The displacement sub-vector of the sleeper,X_s, of order can be written as

(21)

where X_si (i = 1, 2, ..., N_s ) of order 3 denotes the displacement vector of the ith sleeper, N_s denotes the total number of sleepers, and . The displacement vector, Xsi, can be expressed as denotes the total number of DOFs of all sleepers with

Xsi =[ysi zsi θxsi].

The displacement sub-vector, Xb, of order for multi-span continuous beams used to model the bridge can be written as

(22)

where X_bi (i = 1, 2, ..., N_b ) denotes the displacement vector of the ith bridge, N_b denotes the total number of bridges, and _bi , of order , can be expressed as, and the number of DOFs, denotes the total number of DOFs of all bridges. The displacement vector, X

where ith bridge. denotes the total number of DOFs of the

The displacement sub-vector of the pier, X_p , of order can be written as

(23)

where X_pi (i = 1, 2, ..., N_p ) denotes the displacement vector of the ith pier, N_p denotes the total number of piers, and _pi , of order denotes the total number of DOFs of all piers. The displacement vector, X, can be expressed as and the number of DOFs,

where ith pier. denotes the total number of DOFs of the

3.2 Sub-Matrices of Train

The sub-matrices of the train are marked with the subscript "tt". The mass sub-matrix, M_tt , of the train of order (23 × N_v ) × (23 × N_v ) can be written as

(24)

where M_vj (j = 1, 2, ..., N_v ) of order 23 × 23 denotes the mass matrix of the jth vehicle, and can be expressed as

M_vj = diag [m_c m_c I_cx I_cy I_cz m_t m_t I_tx I_ty I_tz m_t m_t I_tx I_ty I_tz m_w I_wz m_w I_wz m_w I_wz m_w I_wz ].

The stiffness sub-matrix, K_tt , of the train of order (23 × N_v ) × (23 × N_v ) can be written as

(25)

where K_vj (j = 1, 2, ..., N_v ) of order 23 × 23 denotes the stiffness matrix of the jth vehicle, and can be expressed by the stiffness of the suspension systems of the jth vehicle.

The damping sub-matrix, C_tt , of the train of order (23 × N_v ) × (23 × N_v ) can be obtained by simply replacing k in the corresponding stiffness sub-matrix, K_tt , with c.

3.3 Sub-Matrices of Rail

The matrices of the rail are marked with the subscript "rr". The mass sub-matrix of the rail, M_rr , of order , composed of the mass matrices of the left and right rails, MLrr and MRrr, respectively, both of order , can be written as

(26)

with

where M_rr1 , M_rr2 , and M_rr3 denote the overall mass matrices in the xy plane, in the xz plane, and rotation about the x-axis of the rail itself, respectively; M_rr4 denotes the overall mass matrix induced by the wheel masses of all vehicles; A_r denotes the cross-sectional area of the rail; I_rx denotes the torsional moment of inertia of the rail about the x-axis; n_r denotes the total number of elements of each rail; ξ denotes the local coordinate measured from the left node of a rail element; N_ry,eg , N_rz,eg , and N_rθ,eg of order xy plane, the xz plane, and in rotation about the x-axis for the gth rail element, respectively. In addition, each element in N_ry,eg , N_rz,eg , and N_rθ,eg is zero except those corresponding to the DOFs, respectively, in the xy plane, the xz plane, and in rotation about the x-axis of the two nodes of the gth rail element; N_zjh of order xz plane for the rail element, which is evaluated at the position of the hth wheelset of the jth vehicle; ξ_j1, ξ_j2, ξ_j3 , and ξ_j4 denote, respectively, the distances between the 1st, 2nd, 3rd, and 4th wheelsets of the jth vehicle and the left node of the rail element on which the wheelsets are acting. Furthermore, each element in N_zjh is zero except for those corresponding to the four DOFs in the xz plane of the two nodes of the rail element on which the hth wheelset of the jth vehicle is acting. denotes the mass per unit length of the rail; denotes the time-dependent shape function vector in the are the shape function vectors in the

The stiffness sub-matrix of the rail, K_rr , of order , which is composed of the stiffness matrices of the left and right rails, KLrr and KRrr, of order , can be written as, and the left rail-right rail interaction stiffness matrices, KLrRr and KRrLr, of order

(27)

with

where R_rr1 , R_rr2 , and R_rr3 denote the overall stiffness matrices in the xy plane, the xz plane, and in rotation about the x-axis of the rail itself, respectively; R_rr4 denotes the overall stiffness matrix induced by the vertical displacement of all vehicles; R_rr5 denotes the overall stiffness matrix induced by the train's weight; R_rr6 , R_rr7 , and R_rr8 denote, respectively, the lateral, vertical, and torsional stiffness matrices induced by the stiffness of all fasteners; K_LrRr denotes the left rail-right rail interaction stiffness matrix induced by the train's weight; E_r denotes Young's modulus of the rail; G_r denotes the shear modulus of the rail; I_ry and I_rz denote the flexural moments of inertia about the y- and z-axes of the cross section of the rail, respectively; h_rt1 denotes the vertical distance between the top surface and torsional center of the cross section of the rail; W_axle denotes the axle weight of each vehicle; b ₀ denotes half of the transverse distance between the contact points of the wheel and rail; λ denotes the slope of the wheel tread which is a variable depending on the wheel-rail contact position; n_f denotes the total number of the fastener underneath each rail; ξ_{r, p} denotes the distance between the pth fastener and the left node of the rail element containing the pth fastener; N_yjh and N_θjh of order xy plane and in rotation about the x-axis for the rail element, when evaluated at the position of the hth wheelset of the jth vehicle; N_ry,p , N_rz,p , and N_rθ,p (p = 1, 2, ..., n_f ) denote, respectively, the time-independent shape function vectors in the xy plane, the xz plane, and in rotation about the x-axis for the rail element, when evaluated at the position of the pth fastener. denote, respectively, the time-dependent shape function vectors in the

By omitting the damping of the rail itself, the damping sub-matrix of the rail, C_rr , of order Zhang et al., 2010), the damping of the primary suspension of all vehicles, and the lateral, vertical, and torsional damping of all fasteners. can be derived according to the lateral creep between the rails and the wheels of all vehicles (

3.4 Sub-Matrices of Sleeper

The sub-matrices of the sleeper are marked with the subscript "ss". The mass sub-matrix, M_ss , stiffness sub-matrix, K_ss , and the damping sub-matrix, C_ss , of order of all of the sleepers can be written respectively, as

(28)

(29)

(30)

where m_s and J_sx denote the mass and the moment of inertia about x-axis of each sleeper, respectively.

3.5 Sub-Matrices of Bridge and Pier

The sub-matrices of the bridge and pier, marked with the subscripts "bb" and "pp", respectively, can be obtained in a similar way to derivation of the sub-matrices of the rail. It should be noted that the damping property is assumed to be of a Rayleigh type (Yang et al., 2004) in the derivation of the damping sub-matrices of the bridge and pier.

3.6 Sub-Matrices of Train-Rail-Sleeper-Bridge-Pier Interaction

The sub-matrices of train-rail interaction, marked with the subscripts "tr" or "rt", are induced by the interaction of the wheel and rail, and consist of the train-left rail and train-right rail interaction matrices, which are marked with the subscripts "tLr" and "tRr", respectively. The stiffness sub-matrices, K_tr and K_rt , of order T_dof × 2N_r , and the damping sub-matrices, C_tr and C_rt , of order T_dof × 2N_r , for train-rail interaction can be written respectively, as

(31)

(32)

where the stiffness matrices, K_tLr and K_tRr , and the damping matrices, C_tLr and C_tRr , of order T_dof × 2N_r can be expressed respectively, as

in which hth wheelset of the jth vehicle and the left and right rails; hth wheelset of the jth vehicle and the left and right rails; and hth wheelset of the jth vehicle and the left and right rails (Zhang et al., 2010; Kalker, 1967). represent, respectively, the damping matrices induced by the vertical interaction between the and and represent, respectively, the damping matrices induced by the lateral creepage between the represent, respectively, the stiffness matrices induced by the vertical interaction between the

The sub-matrices of rail-sleeper interaction, marked with the subscripts "rs" or "sr", are induced by the stiffness and damping of all fasteners between the rail and sleeper. The sub-matrices of sleeper-bridge interaction, marked with the subscripts "sb" or "bs", are induced by the stiffness and damping of all ballasts between the sleeper and bridge. In addition, the sub-matrices of bridge-pier interaction, marked with the subscripts "bp" or "pb", are induced by the stiffness and damping of all bearings between the bridge and pier. To reduce repetitions, the deviations of all sub-matrices for rail-sleeper-bridge-pier interaction are not listed here, but can be calculated according to equations (1) to (32).

3.7 Load Sub-Vectors of Train, Rail, Sleeper, Bridge, and Pier

The load sub-vector of the train, F_t , of order T_dof ×1 can be written as

(33)

The load vector of the jth vehicle, F¹ _vj and F² _vj , of order 23 × 1can be written respectively, as

where r(_^xVjh ), r(_^xCjh ), r(_^xAjh ), and r(_^xGjh ) denote the track elevation, cross level, alignment, and gauge irregularities, respectively, at the hth wheel-rail contact point of the jth vehicle; b ₂ denotes half of the transverse distance between the vertical primary suspension system and L_t denotes half of the bogie axle base; f ²² _Ljh denotes the lateral creep coefficient between the left rail and the hth (h = 1-4) wheelset of the jth vehicle, and f ²² _Rjh denotes the lateral creep coefficient between the right rail and the corresponding wheelset. denotes the first derivative of track irregularity;

The load sub-vector of the rail, F_r , of order 2N_r × 1 can be written as

(34)

where _F ^{L 0} _r , _F ^{L 1} _r , _F ^{L 2} _r , _F ^{L 3} _r , and _F ^{L 4} _r of order N_r × 1 represent the load vectors of each wheelset acting on 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; _F ^{L 5} _r , _F ^{L 6} _r , _F ^{L 7} _r , and _F ^{L 8} _r of order N_r × 1 represent the load vectors of each wheelset acting on the left rail caused by the velocities of the track elevation, cross level, alignment, and gauge irregularities, respectively; and _F ^{L 9} _{r F} ^{L 10} _r of order N_r × 1 represent the load vectors of each wheelset acting on the left rail that are caused by accelerations in the track elevation and cross level irregularities, respectively; and _F ^{R 0} _r to _F ^{R 10} _r similarly represent the load vectors of each wheelset acting on the right rail.

Each of the elements of the load sub-vector of the sleeper, F_s , of order , and the load sub-vector of the pier, Fp, of order , the load sub-vector of the bridge, Fb, of order are zero.

4 NUMERICAL VERIFICATION

To verify the theory presented in this paper, the vertical dynamic responses of a TTBI system, which were obtained using the proposed 3D rail-bridge coupling element (3D element) and the 2D rail-bridge coupling element presented by Lou et al. (2012) (2D element), respectively, are used. A train consisting of five identical vehicles is considered to run over a single-track bridge along the centerline of the bridge, with no consideration made for torsional action. The railway track is assumed to be smooth and continuous throughout and has a total length of 100 m and LRE = 0.625 m. The central part of the railway track is supported on a 3-span continuous bridge with spans of 20 m and LBE = 5.0 m, while the left and right parts of track are supported on embankments, both with lengths of 20 m. The vertical parameters of vehicle, track, and bridge can be found in Lou et al. (2012), the spatial parameters of the identical vehicle can be found in Yang et al. (2004), and the spatial parameters of the identical track and bridge are listed in Table 1. To solve the equation of motion for the TTBI system, the Wilson- ϑ method is used with ϑ = 1.4 and a moving length of the vehicles of 0.1 m along the track for each time step. The analysis is performed by applying the train speeds from 25 m/s to 200 m/s at 25 m/s intervals. The vertical dynamic responses of bridge, sleeper, rail, and vehicle obtained by the 3D element and the 2D element at various train speeds are shown in Table 2, where the term "Carbody acceleration" means the maximum vertical acceleration at the centroid of the last carbody, "Rail displacement" means the maximum vertical displacement of the rail at the middle of the central span, "Sleeper displacement" means the maximum vertical displacement of the sleeper immediately to the right of the middle of the central span, and "Bridge displacement" means the maximum vertical displacement of the bridge at the middle of central span. It can be observed from Table 2 that there are only minimal differences between the solutions obtained using the 3D element and those using the 2D element, where the differences of the displacements of bridge, sleeper, and rail are less than 1.00%, and the differences between the carbody acceleration at various train speeds are not larger than 4.00%. This thus confirms the accuracy of the proposed 3D rail-bridge coupling element in simulating the dynamic responses of a TTBI system.

Table 2:
Comparsion of vertical dynamic responses of TTBI system at various train speeds.

Train speed (m/s)		25	50	75	100	125	150	175	200
Bridge displacement (mm)	3D element	0.165	0.181	0.176	0.196	0.205	0.503	0.517	0.411
	2D element	0.166	0.181	0.176	0.195	0.204	0.501	0.515	0.410
Sleeper displacement (mm)	3D element	0.268	0.289	0.305	0.327	0.342	0.600	0.652	0.559
	2D element	0.270	0.290	0.304	0.325	0.339	0.598	0.649	0.556
Rail displacement (mm)	3D element	0.725	0.730	0.734	0.750	0.774	1.062	1.120	0.997
	2D element	0.729	0.733	0.732	0.747	0.769	1.056	1.116	0.992
Carbody acceleration (m/s2)	3D element	0.008	0.015	0.021	0.025	0.029	0.043	0.041	0.055
	2D element	0.008	0.015	0.021	0.026	0.030	0.044	0.042	0.053

5 ILLUSTRATIVE EXAMPLES

5.1 Parameters of a TTBI System

The proposed 3D rail-bridge coupling element is applied in the following three examples. The first example is shown in relation to an investigation of the influence of the efficiency and accuracy of LRE and LBE on the spatial dynamic responses of a TTBI system. The other two examples are shown in relation to an investigation of the effects of two types of track models and two types of wheel-rail interaction models on the spatial dynamic responses of a TTBI system, respectively. A seven-span continuous beam bridge, with a span length of 40 m + 5 × 60 m + 40 m = 380 m, is considered. The heights of the piers are 20 m, and the length of the pier element (LPE) is 2.5 m. However, to save the length of the paper, the influence of LPE is not considered in this paper. The parameters of the track and bridge already listed in Table 1 are adopted unless otherwise stated. A train consisting of five identical vehicles is considered to move over the bridge from left to right, and the major parameters of each vehicle are listed in Table 3. The PSDs of a German high-speed track spectrum of low irregularity (Zhai and Xia, 2011) are adopted, i.e.,

track elevation irregularity: ,

track alignment irregularity: ,

track cross level irregularity: ,

and track gauge irregularity: ,

where Ω = 2π/λ_r denotes the spatial frequency (rad/m), λ_r denotes the wavelength of the irregularity (m), Ω_c = 0.8246 rad/m, Ω_r = 0.0206 rad/m, Ω_s = 0.438 rad/m, A_V = 4.032 × 10⁻⁷ m·rad, A_A = 2.119 × 10⁻⁷ m·rad, and A_G = 0.532 × 10⁻⁷ m·rad.

Table 3:
Major parameters of vehicle.

Notation	Value	Notation	Value	Notation	Value
mc (kg)	4.4E4	Iwz (kg·m2)	1.1E3	cpy (N·s/m)	0.0
mt (kg)	2.4E3	ksx (N/m)	2.8E5	cpz (N·s/m)	5.0E4
mw (kg)	2.4E3	ksy (N/m)	2.8E5	h1(m)	1.14
Waxle (kg)	1.46E4	ksz (N/m)	3.0E5	h2(m)	−0.14
Icx (kg·m2)	1.0E5	kpx (N/m)	1.5E7	h3(m)	0.24
Icy (kg·m2)	2.7E6	kpy (N/m)	5.0E6	b0(m)	0.75
Icz (kg·m2)	2.7E6	kpz (N/m)	7.0E5	b1(m	0.95
Itx (kg·m2)	1.8E3	csx (N·s/m)	1.0E6	b2(m)	1.0
Ity (kg·m2)	2.2E3	csy (N·s/m)	2.5E4	Lc (m)	8.6875
Itz (kg·m2)	2.2E3	csz (N·s/m)	6.0E4	Lt (m)	1.25
Iwx (kg·m2)	1.1E3	cpx (N·s/m)	0.0	R0(m)	0.4575

The time domain samples of track irregularities with 1 m ≤ λ_r ≤ 120 m are simulated using the method proposed by Zhai and Xia (2011), and an analysis is performed by applying train speeds between 2.78 m/s and 97.2 m/s at 2.78 m/s intervals; that is, from 10 km/h to 350 km/h at 10 km/h intervals.

5.2 Example 1: Influence of the Efficiency and Accuracy of LRE and LBE on the Dynamic Responses of the TTBI System

To illustrate the efficiency and accuracy of the proposed 3D rail-bridge coupling element, the following six cases are studied, as shown in Table 4. The rail-bridge coupling element with LRE = LBE is used in Cases 1-1 to 1-5, while the proposed element is adopted in Case 1-6. The spatial dynamic responses of the TTBI system for Cases 1-1 to 1-6 at various train speeds are plotted in Figures 6 to 19 and the calculation time and differences, D_e1 , for Cases 1-1 to 1-6 at a train speed of 350 km/h are shown in Table 5. Herein, the differences, D_e1 , between the dynamic responses of different calculation cases is defined as D_e1 = (D_yn11-D_yn12 )/D_yn11 × 100%, where D_yn11 and D_yn12 denote the dynamic responses obtained by the proposed element (Case 1-6) and the rail-bridge coupling element of equal length (Cases 1-1 to 1-5), respectively. For convenience hereafter, the "bridge midpoint" means the midpoint of the fourth span for the seven-span bridge; "sleeper", "rail" and "fastener" mean the sleeper, left rail and left fastener immediately on the bridge midpoint, respectively; "carbody" and "bogie" mean the carbody and front bogie of the third vehicle, respectively; and the "derailment factor" and "offload factor" mean the derailment factor and offload factor of the left wheel for the second wheelset of the third vehicle, respectively. The derailment factor is defined as the ratio of the lateral wheel-rail force to the vertical wheel-rail force of the same wheel, while the offload factor is defined as the ratio of the offload in the vertical wheel-rail force to the static vertical wheel-rail force of the same wheel (Xia et al., 2006). Figures 6 and 7 show the maximum lateral and vertical accelerations of the bridge midpoint, respectively; Figures 8 and 9 show the maximum lateral and vertical accelerations of the sleeper, respectively; Figures 10 and 11 show the maximum lateral and vertical accelerations of the left rail, respectively; Figures 12 and 13 show the maximum lateral and vertical accelerations of the carbody, respectively; Figures 14 and 15 show the maximum lateral and vertical accelerations of the bogie, respectively; Figures 16 and 17 show the maximum lateral force and vertical pressure of the left fastener, respectively; and Figures 18 and 19 plot the maximum derailment factor and offload factor, respectively. As is shown, the differences in the dynamic responses between Cases 1-1 to 1-5 appear to decrease as the lengths of the elements are reduced, indicating that the use of a shorter length of element tends to greatly improve the calculation accuracy. However, the corresponding calculation time increases significantly in relation to an increase in the number of DOFs. When the LBE is shorter than 2.5 m (Cases 1-3 to 1-5), the differences in the accelerations of the bridge midpoint (Figures 6 and 7) are much smaller than those of the accelerations of the sleeper (Figures 8 and 9), of the accelerations of the rail (Figures 10 and 11), of the fastener forces (Figures 16 and 17), and of the wheel-rail interactions (Figures 18 and 19), due to the fact that the mass and stiffness of the bridge are much larger than those of sleeper and rail. For instance, the ratios of lateral and vertical flexural rigidity of the bridge to those of the single rail are respectively 4.28 × 10⁶ and 9.99 × 10⁴. Similar phenomenon can also be observed for the lateral and vertical accelerations of the carbody (Figures 12 and 13) and for the lateral acceleration of the bogie (Figure 14), because the vehicle's suspension systems and wheel-rail creepage serve to some extent as an energy dissipating mechanism. It is evident that a sufficiently fine mesh, i.e., LRE = l_sp , should be adopted for the rail if accurate accelerations of sleeper and rail, fastener forces, and wheel-rail interactions are required. By comparing the dynamic responses of Case 1-2 with those of Case 1-6, it can be seen that the influence of LRE on the bridge dynamic responses is also important if the track irregularities are considered, which is different from the case that considers an ideal smooth track (Lou et al., 2010). As shown in Table 4, the major difference between the two calculation cases in modeling rail-bridge interaction is that Case 1-2 uses the rail-bridge coupling element with LBE = LRE = 5.0 m, while Case 1-6 uses the proposed element with LBE = 5.0 and LRE = 0.625 m. Although LBE in Case 1-2 is equal to that in Case 1-6, the differences in the lateral and vertical accelerations of the bridge at a train speed of 350 km/h may reach 8.87% and 13.90%, respectively. It is interesting to note that negligible differences between the bridge, sleeper, rail, and the vehicle dynamic responses can be observed in Cases 1-5 and 1-6. As shown in Table 4, the major differences between the two calculation cases in modeling rail-bridge interaction is that Case 1-5 uses the rail-bridge coupling element with LBE = LRE = 0.625 m, while Case 1-6 uses the proposed element with LBE = 5.0 and LRE = 0.625 m. Although LBE in Case 1-6 is eight times that in Case 1-5, an excellent agreement of the dynamic responses can be obtained because of the high flexural rigidity of bridge. Furthermore, the proposed element helps to save calculation time compared with the rail-bridge coupling element of equal length, due to the reduction of DOFs. For example, the total CPU times for Case 1-5 and Case 1-6 are 1248.2 s and 902.4 s on a 2.8 GHz personal computer, respectively, and the ratio of the latter to the former is 0.723. Therefore, it is concluded that the proposed 3D rail-bridge coupling element with shorter rail elements and longer bridge elements can not only help to save calculation time but can also provide satisfactory results when investigating the spatial dynamic responses of a TTBI system.

Table 4:
Calculation cases.

Calculation case	LBE (m)	LRE (m)	LPE (m)	Number of DOFs
Case 1-1	10.0	10.0		2628
Case 1-2	5.0	5.0		3198
Case 1-3	2.5	2.5	2.5	4338
Case 1-4	1.25	1.25		6618
Case 1-5	0.625	0.625		11178
Case 1-6	5.0	0.625		8518

Figure 6:
Maximum lateral acceleration of bridge midpoint with respect to train speed.

Figure 7:
Maximum vertical acceleration of bridge midpoint with respect to train speed.

Figure 8:
Maximum lateral acceleration of sleeper with respect to train speed.

Figure 9:
Maximum vertical acceleration of sleeper with respect to train speed.

Figure 10:
Maximum lateral acceleration of rail with respect to train speed.

Figure 11:
Maximum vertical acceleration of rail with respect to train speed.

Figure 12:
Maximum lateral acceleration at centroid of carbody with respect to train speed.

Figure 13:
Maximum vertical acceleration at centroid of carbody with respect to train speed.

Figure 14:
Maximum lateral acceleration at centroid of bogie with respect to train speed.

Figure 15:
Maximum vertical acceleration at centroid of bogie with respect to train speed.

Figure 16:
Maximum lateral force of fastener with respect to train speed.

Figure 17:
Maximum vertical pressure of fastener with respect to train speed.

Figure 18:
Maximum derailment factor with respect to train speed.

Figure 19:
Maximum offload factor with respect to train speed.

Table 5:
Calculation time and accuracy for different calculation cases at train speed of 350 km/h.

Calculation case	Case 1-1	Case 1-2	Case 1-3	Case 1-4	Case 1-5	Case 1-6
Calculation time (s)	303.4	358.7	475.9	711.9	1248.2	902.4
De1 of bridge midpoint lateral acceleration (%)	27.45	8.87	1.22	1.03	0.29	0.00
De1 of bridge midpoint vertical acceleration (%)	43.41	13.90	2.56	1.31	0.09	0.00
De1 of sleeper lateral acceleration (%)	52.91	30.15	19.60	5.31	−0.01	0.00
De1 of sleeper vertical acceleration (%)	49.55	26.74	15.96	4.29	−0.01	0.00
De1 of rail lateral acceleration (%)	68.14	47.92	34.20	20.05	−0.01	0.00
De1 of rail vertical acceleration (%)	58.88	44.53	23.26	5.47	−0.01	0.00
De1 of carbody lateral acceleration (%)	6.03	4.17	3.25	1.61	0.00	0.00
De1 of carbody vertical acceleration (%)	11.19	5.53	2.49	0.45	0.00	0.00
De1 of bogie lateral acceleration (%)	6.22	4.22	2.30	1.60	−0.01	0.00
De1 of bogie vertical acceleration (%)	28.00	19.63	9.06	2.05	0.01	0.00
De1 of lateral force of fastener (%)	55.92	27.96	16.67	3.77	−0.26	0.00
De1 of vertical pressure of fastener (%)	51.72	37.90	22.75	3.15	−0.04	0.00
De1 of derailment factor (%)	27.40	25.00	19.09	4.60	0.00	0.00
De1 of offload factor (%)	32.04	26.99	8.92	1.84	0.01	0.00

5.3 Example 2: Influence of Two Types of Track Models on Dynamic Responses of TTBI System

In this example, two types of track models are considered, with the same train, bridge, and track irregularity as presented in Section 5.1. One is a double-layer track model which has the same sleepers that were considered in Section 5.1, while the other is a single-layer track model in which the sleepers are ignored. The parameters LBE = 5.0 m and LRE = 0.625 m are adopted in both models. The lateral and vertical stiffnesses, k_rby and k_rbz , of the discrete springs between the rail and bridge in the single-layer track model can be obtained by considering k_rsy and k_sby, k_rsz and k_sbz , respectively, in series in the double-layer track model with sleepers ignored, i.e., k_rby = k_rsy ·k_sby /(k_rsy + k_sby ) and k_rbz = k_rsz ·k_sbz /(k_rsz + k_sbz ). Similarly, the lateral and vertical damping coefficients, c_rby and c_rbz , of the discrete dampers between the rail and bridge in the single-layer track model can be obtained as c_rby = c_rsy ·c_sby /(c_rsy + c_sby ) and c_rbz = c_rsz ·c_sbz /(c_rsz + c_sbz ). To investigate the influence of the mass, m_s , of the sleeper on the spatial dynamic responses of the TTBI system, five masses of 170 kg, 255 kg, 340 kg, 425 kg, and 510 kg are applied, which are equal to 0.50, 0.75, 1.00, 1.25, and 1.50 times the normal value, respectively. The other parameters are the same as those in Table 1. It is of note that the mass of the sleeper is added to the dead load of the bridge in the single-layer track model, but that a detailed derivation of the stiffness and damping matrices of the rail-bridge interaction is not given here. However, it can be obtained by following a procedure similar to that given in Sections 2.2 and 2.3. To investigate the influence of the track model on the spatial dynamic responses of the TTBI system, the difference, D_e2 , between the dynamic responses based on the single-layer track model and those based on the double-layer track model can be defined as D_e2 = (D_yn21-D_yn22 )/D_yn21 × 100%, where D_yn21 and D_yn22 denote the dynamic responses obtained by the single-layer track model and by the double-layer track model, respectively. The differences, D_e2 , of the dynamic responses of the TTBI system at various train speeds based on the single-layer track model and the double-layer track model with m_s = 170 kg, m_s = 225 kg, m_s = 340 kg, m_s = 425 kg, and m_s = 510 kg, are plotted in Figures 20 to 27. Figures 20 and 21 show the differences in the maximum lateral and vertical accelerations of the bridge midpoint, respectively; Figures 22 and 23 show the differences in the maximum lateral and vertical accelerations of the rail, respectively; Figures 24 and 25 show the differences in the maximum lateral and vertical accelerations of the carbody, respectively; and Figures 26 and 27 plot the differences in the maximum derailment factor and offload factor, respectively. Differences in the dynamic responses based on the single-layer and double-layer track models can be seen in Figures 20 to 27, and it is evident that the differences in both the maximum lateral and vertical dynamic responses generally increase with an increase in the mass of sleeper and train speed. Although the differences in the maximum lateral and vertical acceleration of the carbody (Figures 24 and 25) are negligibly small (≤ 2%), due to the energy dissipating effect of the vehicle's suspension systems and wheel-rail creepage, the differences in other dynamic responses are quite visible, particularly at higher train speeds. For instance, the differences in lateral acceleration of the bridge (Figure 20), lateral acceleration of the rail (Figure 22), and the derailment factor (Figure 26) are larger than 100%, 10%, and 4%, respectively, in the present calculation cases. Figures 28 to 31 plot the maximum lateral and vertical accelerations of the bridge midpoint and rail with the single-layer track model and double-layer track model with m_s = 170 kg, m_s = 255 kg, m_s = 340 kg, m_s = 425 kg, and m_s = 510 kg, respectively. As can be seen from Figures 28 to 31, the lateral and vertical accelerations of the bridge midpoint tend to increase steadily with an increase in the mass of the sleeper, while the lateral and vertical accelerations of the rail show a trend of slight decrease at higher train speeds. This can be explained by the fact that the sleepers serve as a medium for transmitting the kinetic energy brought by the moving train from the rail to the bridge. The increase in the mass of the sleeper thus increases the train-induced impact effect on the bridge, while reducing the vibration amplitude of the rail. In addition, the two-layer track model allows us to compute not only the bridge responses, rail responses, vehicle responses, and wheel-rail interaction, but also the sleeper responses and the fastener force. However, it is worth noting that the single-layer track model saves calculation time because of the reduction of DOFs. For example, the DOFs of the single-layer and double-layer track models are 6691 and 8518 respectively, while the corresponding calculation times are 720.1 s and 902.4 s on a 2.8 GHz personal computer, respectively. It is thus concluded that the double-layer model, although more time consuming, is shown to be more accurate.

Figure 20:
Differences, D_e2 , in maximum lateral acceleration of the bridge midpoint with respect to train speed.

Figure 21:
Differences, D_e2 , in maximum vertical acceleration of bridge midpoint with respect to train speed.

Figure 22:
Differences, D_e2 , in maximum lateral acceleration of rail with respect to train speed.

Figure 23:
Differences, D_e2 , in maximum vertical acceleration of rail with respect to train speed.

Figure 24:
Differences, D_e2 , in maximum lateral acceleration at centroid of carbody with respect to train speed.

Figure 25:
Differences, D_e2 , in maximum vertical acceleration at centroid of carbody with respect to train speed.

Figure 26:
Differences, D_e2 , in maximum derailment factor with respect to train speed.

Figure 27:
Differences, D_e2 , in maximum offload factor with respect to train speed.

Figure 28:
Maximum lateral acceleration of bridge midpoint with respect to train speed.

Figure 29:
Maximum vertical acceleration of bridge midpoint with respect to train speed.

Figure 30:
Maximum lateral acceleration of rail with respect to train speed.

Figure 31:
Maximum vertical acceleration of rail with respect to train speed.

5.4 Example 3: Influence of Two Types of Wheel-Rail Interaction Models on Dynamic Responses of TTBI System

In this example, two types of wheel-rail interaction models are considered, i.e., the no-jump model and the jump model. The same train, track, bridge, and track irregularity as that presented in Section 5.1 is used, with LBE = 5.0 m and LRE = 0.625 m. The wheels of each vehicle are considered to be in full contact with the rails at all times for the no-jump model (Yang et al., 2004, Lou and Zeng, 2005), while the wheels are free to jump from the rails for the jump model (Zhai and Sun, 1994).

The running safety of trains has been of great concern in railway engineering for a long time, particularly in relation to the development of high-speed railways and the need to upgrade existing railways. Several mechanisms that can result in the derailment of a running train have been identified through analytical and experimental investigations, and a number of indices have been proposed based on these to evaluate the possibility, or risk, of train derailment. One of these indices is the offload factor, PD, (Yang et al., 2004). Large PD values indicate that dynamic vertical wheel-rail force acting on the wheel is substantially reduced. This is detrimental to the lateral stability of the wheelset, and thus a limit needs to be placed on the value of the PD index to prevent the wheelset from derailing. An upper limit of 0.60 on the PD value was used in Chinese specifications for the design of railways (Xia et al., 2006), which implies that jumps between the vehicle's wheels and the rails are not usually permitted in practice. Therefore, the wheels of a vehicle are generally assumed to be in constant contact with the rails (i.e., the no-jump model) when most train-track-bridge interaction problems occur. Based on this assumption, the dynamic contact forces between the wheels and rails are considered as internal forces, and it is thus not necessary to calculate the internal forces when setting up the equations of motion of a TTBI system (Lou and Zeng, 2005). As such, the vehicle response, wheel-rail contact force, track response, and bridge response can be computed with no iterations required. However, in some extreme cases, such as with poor track quality or during an earthquake, the wheels may jump upward and separate from the rails (i.e., the jump model) and the train then has a high risk of derailment. When studying the dynamic responses of a TTBI system using the jump model, two sets of equations of motion can be written, one for the moving train subsystem and the other for the track-bridge subsystem. These equations are coupled with the wheel-rail contact forces existing at the contact points of the two subsystems, and are usually solved using procedures of an iterative nature (Zhai and Sun, 1994). For instance, when first assuming a trial solution for the wheel-rail contact forces, the dynamic responses of the train and track-bridge subsystems can be solved from the two sets of equations of motion, respectively. An improved solution for the wheel-rail contact forces can then be obtained according to the displacements of the wheels and rails at the contact points. By substituting these forces into the equations of motion within a train and track-bridge subsystems, an improved solution for the dynamic responses of the two subsystems can be solved. However, to avoid divergence and improve the convergence rate of iteration, sufficiently small time steps are required in the process of calculation, which may thus result in more computer time.

Seven cases with train speeds of 50 km/h, 100 km/h, 150 km/h, 200 km/h, 250 km/h, 300 km/h, and 350 km/h are considered. The dynamic responses of the TTBI system obtained using the no-jump and jump wheel-rail interaction models at various train speeds are shown in Table 6. As is evident from the table, the solutions obtained using the no-jump model for the seven cases given agree very well with those of the jump model, although the impact response induced by the jump model appears to increase slightly. The differences in bridge acceleration are no larger than 2.00%, due to the relatively larger stiffness and mass of the bridge. However, although the differences in the dynamic responses of the sleeper, rail, and vehicle are slightly larger, all of them are smaller than 5.00%. It should be noted that the total CPU times for the no-jump model and the jump model at a train speed of 350 km/h are 902.4 s and 7183.1 s on a 2.8 GHz personal computer, respectively, and the ratio of the former to the latter is 0.126. Therefore, it is concluded that the no-jump wheel-rail interaction model can be reliably and efficiently used to predict the spatial dynamic responses of a TTBI system.

Table 6:
Calculation results for two wheel-rail interaction models at various train speeds.

Train speed (km/h)		50	100	150	200	250	300	350
Bridge midpoint lateral acceleration (m/s2)	no-jump model	0.005	0.015	0.023	0.052	0.068	0.106	0.152
	jump model	0.005	0.015	0.023	0.053	0.069	0.107	0.153
Bridge midpoint vertical acceleration (m/s2)	no-jump model	0.008	0.036	0.100	0.151	0.253	0.403	0.610
	jump model	0.008	0.036	0.101	0.154	0.257	0.411	0.618
Sleeper lateral acceleration (m/s2)	no-jump model	0.096	0.743	2.378	6.209	10.047	14.180	18.651
	jump model	0.099	0.758	2.432	6.347	10.165	14.459	18.757
Sleeper vertical acceleration (m/s2)	no-jump model	0.194	0.909	3.005	6.865	10.822	20.809	39.986
	jump model	0.200	0.910	3.038	6.878	10.864	21.494	41.097
Rail lateral acceleration (m/s2)	no-jump model	0.557	2.436	5.917	12.511	21.175	36.174	49.852
	jump model	0.564	2.528	5.925	12.730	21.498	37.281	51.437
Rail vertical acceleration (m/s2)	no-jump model	1.151	7.542	18.837	44.083	83.549	148.46	238.90
	jump model	1.162	7.726	19.256	45.507	86.512	154.062	242.197
Carbody lateral acceleration (m/s2)	no-jump model	0.059	0.133	0.172	0.179	0.188	0.237	0.296
	jump model	0.060	0.134	0.173	0.179	0.190	0.241	0.298
Carbody vertical acceleration (m/s2)	no-jump model	0.103	0.150	0.237	0.320	0.368	0.462	0.531
	jump model	0.105	0.151	0.242	0.322	0.373	0.471	0.545
Bogie lateral acceleration (m/s2)	no-jump model	0.663	2.136	4.341	6.170	6.339	7.210	7.946
	jump model	0.678	2.165	4.356	6.193	6.379	7.361	7.996
Bogie vertical acceleration (m/s2)	no-jump model	0.558	1.599	3.434	5.444	8.510	13.558	18.209
	jump model	0.576	1.614	3.561	5.520	8.576	13.694	18.657
Lateral force of fastener (kN)	no-jump model	0.634	1.638	3.059	4.026	4.932	5.242	5.733
	jump model	0.643	1.655	3.135	4.096	5.013	5.386	5.782
Vertical pressure of fastener (kN)	no-jump model	28.543	29.600	31.462	33.908	37.420	41.791	47.013
	jump model	29.407	30.492	31.940	34.678	37.533	41.881	48.01
Derailment factor	no-jump model	0.047	0.092	0.138	0.166	0.227	0.304	0.471
	jump model	0.048	0.092	0.140	0.168	0.227	0.306	0.473
Offload factor	no-jump model	0.038	0.073	0.128	0.222	0.337	0.454	0.591
	jump model	0.039	0.074	0.129	0.228	0.341	0.467	0.603

6 SUMMARY AND OUTLOOK

Based on obvious differences in the flexural rigidity between the rail and bridge, a 3D rail-bridge coupling element of unequal lengths is presented. The spatial dynamic responses of a TTBI system with a seven-span continuous beam bridge are studied using a 3D rail-bridge coupling element of unequal lengths and equal lengths. Furthermore, the effects of two types of track models on the spatial dynamic responses of the TTBI system are investigated, and the following conclusions can be drawn from the numerical results.

(1) The proposed 3D rail-bridge coupling element with shorter rail elements and longer bridge elements not only helps save calculation time, but it also delivers satisfactory results when investigating the spatial dynamic responses of a TTBI system.

(2) In analyzing the spatial dynamic responses of a TTBI system using a 3D rail-bridge coupling element that has the same length as the bridge element, the influence of the length of the rail element is significant, not only on the rail dynamic responses but also on the bridge dynamic responses, when the track irregularities are considered. This differs from the case with an ideal smooth track.

(3) There are differences in the dynamic responses based on the single-layer and double-layer track models, and the differences in both the maximum lateral and vertical dynamic responses generally increase with an increase in the mass of the sleeper and the train speed, particularly with respect to accelerations of the bridge and rail. In addition, the two-layer track model is more accurate.

(4) The no-jump assumption between the vehicle's wheels and the rails can be reliably and efficiently used for most train-track-bridge interaction problems.

(5) Further studies on the efficiency and accuracy of the proposed 3D rail-bridge coupling element are needed to investigate the spatial dynamic responses of a TTBI system during an earthquake, due to the fact that vibrations of the system may be more violent during such an occurrence.

Acknowledgments

The research work described in this paper was supported by the Joint Fund of the National Natural Science Foundation of China (Grant Nos. U1361204, U1334203, and U1434204); the Project of Innovation-driven Plan in Central South University (Grant No. 2015CXS014); the Fundamental Research Funds for the Central Universities of Central South University (No. 2016zzts067).

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Publication Dates

• Publication in this collection
  Dec 2016

History

• Received
  18 Oct 2015
• Reviewed
  01 June 2016
• Accepted
  07 July 2016