Train Derailment Process Analysis on Heavy Haul Railway Bridge under Ship Impact

Abstract

In order to research train derailment law under ship impact, the spatial vibration calculation model of a freight train-track-bridge (FTTB) system is used to establish the vibration model of the FTTB system under ship impact. Meanwhile, the calculation method of a train derailment process under ship impact is proposed based on the random analysis method of train derailment energy. Further, the train derailment process on a bridge under ship impact is calculated, and the variation law of the FTTB system spatial vibration response under different impact loads and speeds is analyzed. The results show that the ship impact load has a great influence on wheel lift value. When the impact load is greater than 15 MN, the wheel derails more easily. With the increase of impact load, the derailment coefficient, wheel load reduction rate, and lateral relative displacement of bogie and rail, the lateral displacement of the bridge increases significantly, but the limits of them make it difficult to determine whether the wheel has derailed. The lateral relative displacement of the bogie and rail considering the safety factor is calculated at the moment of derailment, which is taken as the early warning threshold of train derailment. The above conclusions can provide a reference for controlling train safety under ship impact.

1. Introduction

With the rapid construction of heavy haul railways, it has become inevitable for railway lines to cross rivers, lakes, and seas, resulting in frequent ship impacts [1,2,3,4]. Ship impact not only causes damage to bridge structures, but also affects traffic safety and can lead to train derailment accidents in serious cases. In 1983, the passenger ship Drew Ivanov of Arneshan collided with the Volga Railway Bridge, where four carriages were derailed and fell into water, killing 240 people. In 1993, the CSX railway bridge was seriously struck by a tugboat fleet, resulting in a huge displacement of the bridge structure and derailment of the train, while 47 people died [5]. Wuhan Yangtze River Bridge has also seen a ship collision accident which interrupted the railway line. In fact, as the wheel derailment under ship impact destroys the symmetry of the wheel-rail system, it is necessary to carry out further research concerning running safety.

On the problem of running safety caused by ship impact, Wilson et al. [6] established the static model of a bridge anti-collision plate by finite element software ANSYS and studied the mechanical properties of the anti-collision plate under ship impact. Consolazio et al. [7,8,9,10] used the finite element method to simulate a ship collision course with different barge types and velocity. Yuan et al. [11] used LS-DYNA to simulate the process of a single ship and fleet impacting a bridge pier, revealing the mechanism of ship impacting a bridge pier and pointing out the limitations of AASHTO specification [12]. Sha et al. [13,14] made a cylindrical reinforced concrete bridge pier model, assessing equivalent ship collision processes by drop hammer impact, and addressing the link between impact force and velocity. Meanwhile, they simulated the impact process by LS-DYNA software considering the elastic-plastic deformation of piers. Fan et al. [15,16] used a multi-degree of freedom analysis model to simulate the ship-bridge collision process. Gholipour et al. [17,18] carried out research concerning the collision process considering the interaction of pile and soil and proposed that the greater the flexibility of the lower structure, the greater the deformation of the pier during the collision.

In addition, Xuan et al. [19] evaluated the risk of train derailment on a bridge and pointed out that further study of train derailment under ship impact load is necessary. Chen et al. [20] used LS-DYNA software to simulate the interaction between ship and pier, and the time history curve of ship collision force under different impact conditions was obtained. Wang et al. [21] simulated and analyzed the collision process between cargo ships and piers considering the HJC concrete model with strain rate. Fang Hai et al. [5] simulated the process of a ship impacting the bridge at different angles, calculated the lateral vibration responses of the beam and pier, and took the peak lateral acceleration of pier 1 m/s² as the standard to judge the derailment of trains. Xia et al. [4,22] used the dynamic model to calculate and analyze the responses of a (32 + 48 + 32) m double-line prestressed concrete continuous girder bridge under impact and proposed the safety threshold curve of vehicle speed–impact strength.

However, the existing research mainly focuses on the ship impact force and the dynamic response of a vehicle-bridge system. There are few studies on train derailment caused by ship collision. In fact, the most ideal measure to control the derailment accident caused by ship collision is to avoid the occurrence of ship collision. Yet, it is difficult to completely eliminate this incident due to complex environments or mismanagement among other factors. In the operation of freight trains, derailment accidents caused by ships hitting bridges are sometimes difficult to detect in time due to the large number of vehicles and secondary accidents of vehicles rushing out of the bridge and falling. In order to minimize the derailment accidents caused by ship collision with bridges, it is necessary to ensure that the train wheels alarm or stop in time before derailment.

Therefore, according to the spatial vibration calculation model of the FTTB system and random analysis method of train derailment energy [23], a model of the FTTB system and the calculation method of the train derailment process under ship impact is established. Further, the calculation of FTTB system responses in the process of train derailment is realized, and the variation law of system response is analyzed. The threshold value with wheel derailment alarm information is obtained from the angle of the wheel-rail contact state and its relative position, which provides a reference for the development of a train wheel derailment warning device under ship impact.

2. Materials and Methods

2.1. Model of FTTB System under Ship Impact

2.1.1. Model of FTTB System

The FTTB system is a dynamic coupling system composed of freight trains, tracks, and bridges. At time t, a freight train consisting of one locomotive and M wagons is running on the track with the calculation length of L.

The freight train is divided into M + 1 units, and each unit includes 26 degrees of freedom. Each unit moves at a constant speed, and the car body and bogie have front and rear, left and right symmetry. A model of each unit is shown in Figure 1. In Figure 1, the linear springs and viscous dampers are used to simulate the suspension between bogie and wheelset and the suspension between car body and bogie. Among them, the coefficients of spring and damping between bogie and wheelset are K_1x, K_1y, K_1z, C_1x, C_1y, and C_1z, respectively. Meanwhile, the coefficients of spring and damping between car body and bogie are K_2x, K_2y, K_2z, C_2x, C_2y, and C_2z, respectively.

According to the above assumptions, the displacement mode of unit is shown in Equation (1).

$$\begin{array}{ll}{\left\{\delta \right\}}_{\mathrm{V}}=& [X_{\mathrm{c}},Y_{\mathrm{c}},Z_{\mathrm{c}},{\theta }_{\mathrm{c}},{\varphi }_{\mathrm{c}},{\psi }_{\mathrm{c}},X_{\mathrm{t}1},Y_{\mathrm{t}1},Z_{\mathrm{t}1},{\theta }_{\mathrm{t}1},{\varphi }_{\mathrm{t}1},{\psi }_{\mathrm{t}1},X_{\mathrm{t}2},Y_{\mathrm{t}2},Z_{\mathrm{t}2},{\theta }_{\mathrm{t}2},{\varphi }_{\mathrm{t}2},{\psi }_{\mathrm{t}2}\\ & {Y_{\mathrm{w}1},Y_{\mathrm{w}2},Y_{\mathrm{w}3},Y_{\mathrm{w}4},Z_{\mathrm{w}1},Z_{\mathrm{w}2},Z_{\mathrm{w}3},Z_{\mathrm{w}4}]}^T\end{array}$$

(1)

In Equation (1), the subscripts $X_{\mathrm{c}},Y_{\mathrm{c}},Z_{\mathrm{c}},{\theta }_{\mathrm{c}},{\varphi }_{\mathrm{c}},{\psi }_{\mathrm{c}}$ are respectively the longitudinal, lateral, vertical, rolling, pitching, and yawing vibrations of car body. The subscripts $X_{\mathrm{t}1},Y_{\mathrm{t}1},Z_{\mathrm{t}1},{\theta }_{\mathrm{t}1},{\varphi }_{\mathrm{t}1},{\psi }_{\mathrm{t}1}$ are respectively the longitudinal, lateral, vertical, rolling, pitching, and yawing vibrations of the front bogie. The subscripts $X_{\mathrm{t}2},Y_{\mathrm{t}2},Z_{\mathrm{t}2},{\theta }_{\mathrm{t}2},{\varphi }_{\mathrm{t}2},{\psi }_{\mathrm{t}2}$ are respectively the longitudinal, lateral, vertical, rolling, pitching, and yawing vibrations of the rear bogie. The subscripts $Y_{\mathrm{w}1},Y_{\mathrm{w}2},Y_{\mathrm{w}3},Y_{\mathrm{w}4}$ are respectively the lateral vibrations of each wheelset. The subscripts $Z_{\mathrm{w}1},Z_{\mathrm{w}2},Z_{\mathrm{w}3},Z_{\mathrm{w}4}$ are respectively the vertical vibrations of the wheelset.

According to the Equation (1), the vibration potential energy (Π_vi) of each unit can be derived [23]. The vibration potential energy Π_V of train can be obtained by adding the Π_vi of each unit which is shown in Equation (2). The derivation process is shown in Reference [23].

$${\Pi }_{\mathrm{V}}={\displaystyle \sum _{i=1}^{M+1}{\Pi }_{\mathrm{Vi}}}$$

(2)

In addition, the typical simple-supported double-T girder bridge of a heavy haul railway is taken as an example. The beam element is used to simulate the rail, sleeper, and bridge beam. Meanwhile, the linear spring and viscous damper are used to simulate the fastener, ballast, and bearing. Among them, the coefficients of spring are K₁, K₂, K₄, K₅, K₆, and K₇, respectively, and the coefficients of damping are C₁, C₂, C₄, C₅, C₆, and C₇, respectively. The model of the track-bridge system is shown in Figure 2.

Furthermore, the beam of the bridge is divided into n elements with the adjacent diaphragm spacing. Each element included 50 degrees of freedom, as shown in Equation (3). The numbers 1 and 2 represent the left and right end nodes of the element, respectively.

$$\left\{\delta \right\}={\left\{\begin{array}{c}{\left\{\delta \right\}}_1\\ {\left\{\delta \right\}}_2\end{array}\right\}}_{50\times 1}$$

(3)

$$\begin{array}{l}{\left\{\delta \right\}}_1=[U_{1\mathrm{R}}^{\mathrm{T}},V_{1\mathrm{R}}^{\mathrm{T}},W_{1\mathrm{R}}^{\mathrm{T}},{\theta }_{\mathrm{X}1\mathrm{R}}^{\mathrm{T}},{\theta }_{\mathrm{Y}1\mathrm{R}}^{\mathrm{T}},{\theta }_{\mathrm{Z}1\mathrm{R}}^{\mathrm{T}},{\gamma }_{1\mathrm{R}}^{\mathrm{T}},\\ \begin{array}{ccccc}& & & & \end{array}{U}_{1\mathrm{L}}^{\mathrm{T}},V_{1\mathrm{L}}^{\mathrm{T}},W_{1\mathrm{L}}^{\mathrm{T}},{\theta }_{\mathrm{X}1\mathrm{L}}^{\mathrm{T}},{\theta }_{\mathrm{Y}1\mathrm{L}}^{\mathrm{T}},{\theta }_{\mathrm{Z}1\mathrm{L}}^{\mathrm{T}},{\gamma }_{1\mathrm{L}}^{\mathrm{T}},\\ \begin{array}{ccccc}& & & & \end{array}{V}_1^{\mathrm{S}},W_{1\mathrm{R}}^{\mathrm{S}},W_{1\mathrm{L}}^{\mathrm{S}},\\ \begin{array}{ccccc}& & & & \end{array}{V}_{1\mathrm{U}}^{\mathrm{B}},{\theta }_{\mathrm{Z}1\mathrm{U}}^{\mathrm{B}},V_{1\mathrm{D}}^{\mathrm{B}},{\theta }_{\mathrm{Z}1\mathrm{D}}^{\mathrm{B}},W_{1\mathrm{R}}^{\mathrm{B}},{\theta }_{\mathrm{Y}1\mathrm{R}}^{\mathrm{B}},W_{1\mathrm{L}}^{\mathrm{B}},{\theta }_{\mathrm{Y}1\mathrm{L}}^{\mathrm{B}}]\end{array}$$

(4)

$$\begin{array}{l}{\left\{\delta \right\}}_2=[U_{2\mathrm{R}}^{\mathrm{T}},V_{2\mathrm{R}}^{\mathrm{T}},W_{2\mathrm{R}}^{\mathrm{T}},{\theta }_{\mathrm{X}2\mathrm{R}}^{\mathrm{T}},{\theta }_{\mathrm{Y}2\mathrm{R}}^{\mathrm{T}},{\theta }_{\mathrm{Z}2\mathrm{R}}^{\mathrm{T}},{\gamma }_{2\mathrm{R}}^{\mathrm{T}},\\ \begin{array}{ccccc}& & & & \end{array}{U}_{2\mathrm{L}}^{\mathrm{T}},V_{2\mathrm{L}}^{\mathrm{T}},W_{2\mathrm{L}}^{\mathrm{T}},{\theta }_{\mathrm{X}2\mathrm{L}}^{\mathrm{T}},{\theta }_{\mathrm{Y}2\mathrm{L}}^{\mathrm{T}},{\theta }_{\mathrm{Z}2\mathrm{L}}^{\mathrm{T}},{\gamma }_{2\mathrm{L}}^{\mathrm{T}},\\ \begin{array}{ccccc}& & & & \end{array}{V}_2^{\mathrm{S}},W_{2\mathrm{R}}^{\mathrm{S}},W_{2\mathrm{L}}^{\mathrm{S}},\\ \begin{array}{ccccc}& & & & \end{array}{V}_{2\mathrm{U}}^{\mathrm{B}},{\theta }_{\mathrm{Z}2\mathrm{U}}^{\mathrm{B}},V_{2\mathrm{D}}^{\mathrm{B}},{\theta }_{\mathrm{Z}2\mathrm{D}}^{\mathrm{B}},W_{2\mathrm{R}}^{\mathrm{B}},{\theta }_{\mathrm{Y}2\mathrm{R}}^{\mathrm{B}},W_{2\mathrm{L}}^{\mathrm{B}},{\theta }_{\mathrm{Y}2\mathrm{L}}^{\mathrm{B}}]\end{array}$$

(5)

In Equations (4) and (5), the U, V, W, θ, and γ are the longitudinal, transverse, and vertical linear displacement, angular displacement, and the change rate of the longitudinal torsional angle along the element, respectively; the subscripts R and L represent the right and left side of element respectively; the superscripts T, S and B are respectively the displacements of rail, sleeper and bridge girder. In addition, the subscripts U and D represent the upper and lower flanges of the bridge girder respectively, the subscripts X, Y and Z are the element direction which is longitudinal, transverse and vertical direction.

At the same time, the pier is divided into P elements according to the cross-sectional characteristics of the pier. The node displacement of the mode is shown in Equation (6).

$${\{\delta \}}_{\mathrm{D}}={\left\{\begin{array}{l}{\{\delta \}}_{\mathrm{D}1}\\ {\{\delta \}}_{\mathrm{D}2}\end{array}\right\}}_{8\times 1}$$

(6)

$${\{\delta \}}_{\mathrm{D}1}={[V_1^{\mathrm{D}},W_1^{\mathrm{D}},{\theta }_{\mathrm{X}1}^{\mathrm{D}},{\theta }_{\mathrm{Z}1}^{\mathrm{D}}]}^{\mathrm{T}}$$

(7)

$${\{\delta \}}_{\mathrm{D}2}={[V_2^{\mathrm{D}},W_2^{\mathrm{D}},{\theta }_{\mathrm{X}2}^{\mathrm{D}},{\theta }_{\mathrm{Z}2}^{\mathrm{D}}]}^{\mathrm{T}}$$

(8)

In Equations (7) and (8), V, W, and θ are the transverse, vertical, and angular displacement of the element, respectively; the superscript D is the displacement of the pier.

According to the Equations (3) and (6), the vibration potential energy Π_TBj and Π_Pk can be derived, respectively. The vibration potential energy of track-bridge system Π_TBP is obtained by superimposing each element, which is shown as Equation (9). The derivation process is shown in References [23,24].

$${\Pi }_{\mathrm{TBP}}={\displaystyle \sum _{j=1}^N{\Pi }_{\mathrm{TBj}}}+{\displaystyle \sum _{k=1}^P{\Pi }_{\mathrm{Pk}}}$$

(9)

The wheel-rail connection condition is the link between the train and track-bridge system. In order to realize the whole process of train derailment, the relative motion state between wheel and rail should be considered in Figure 3. When the wheel is in a normal state (Figure 3a) and gradually closes to the rail head (Figure 3b) until it climbs up the top of the rail (Figure 3c), the wheel derailment occurs (as shown in Figure 3).

According to the above motion state, the convergence condition of relative displacement between wheel and rail [18] is established as shown in Equations (10) and (11).

$$\mathrm{\Delta }{Y}_{\mathrm{W}\mathrm{R}}=Y_{\mathrm{W}\mathrm{i}}-Y_{\mathrm{R}}-Y_{\mathrm{I}\mathrm{O}\mathrm{R}}$$

(10)

$$\mathrm{\Delta }{Z}_{\mathrm{W}\mathrm{R}}=Z_{\mathrm{W}\mathrm{i}}-Z_{\mathrm{R}}-Z_{\mathrm{I}\mathrm{O}\mathrm{R}}$$

(11)

In Equations (10) and (11), $Y_{\mathrm{Wi}}$, $Y_{\mathrm{R}}$, and $\Delta Y_{\mathrm{WR}}$ represent the lateral displacement of wheel, rail, and wheel-rail, respectively, where i denotes the number of wheelsets, i = 1~4; $Z_{\mathrm{Wi}}$, $Z_{\mathrm{R}}$, and $\Delta Z_{\mathrm{WR}}$ represent the vertical displacement of wheel, rail, and wheel-rail respectively, where i denotes the number of wheelsets, i = 1~4; $Y_{\mathrm{IOR}}$ and $Z_{\mathrm{IOR}}$ are the lateral and vertical geometric irregularity of rail respectively. Meanwhile, the clearance between wheel flange and gauge line is considered in the FTTB system model.

Thus, the total potential energy of FTTB system can be derived as shown in Equation (12).

$${\Pi }_{\mathrm{FTTB}}={\Pi }_{\mathrm{V}}+{\Pi }_{\mathrm{TB-P}}$$

(12)

According to the principle of total potential energy with stationary value in elastic system dynamics [25], Equation (12) is changed and set to 0, as shown in Equation (13).

$$\delta {\Pi }_{\mathrm{FTTB}}=0$$

(13)

Using the “set in right position” rule [26] that forms the stiffness, damping and mass matrix of FTTB system which are represented by the K, C, and M, respectively. Meanwhile, the bogie frame hunting wave and track vertical geometric irregularity to the FTTB system are used to form the load array which is represented by the P. So, the FTTB system matrix equation at time t is shown in Equation (14).

$$K\delta +C\dot{\delta }+M\ddot{\delta }=P$$

(14)

In Equation (14), $\delta$, $\dot{\delta }$, and $\ddot{\delta }$ are represent the displacement, velocity, and acceleration arrays. The results of Equation (14) is solved by the Wilson-θ step-by-step integration method, and the FORTRAN program of Equation (14) is established for calculation.

2.1.2. Simulation of Ship Impact

Ship impact has the characteristics of short time, large energy, and strong destruction. The ship impact load proposed by AASHTO code of the United States [12] and Code for Design of Railway Bridges of China [27] does not reflect the interaction between ship and pier, and it is difficult use for dynamic analysis. In addition, the European Union and British norms [28] propose that the ship impact on the pier is mainly characterized by nonlinear fluctuations. Chen et al. [20,21] simulated the process of ship collision and obtained a ship collision load with nonlinear wave characteristics. So, in order to calculate the dynamic response of a FTTB system under ship impact, the impact load curve from the Figure 4.4 of reference [20] is used as the external excitation input to the FTTB system.

Then, the FTTB system matrix equation under ship impact is established, as shown in Equation (15).

$$\left[\begin{array}{cc}{M}_{\mathrm{ww}}& M_{\mathrm{wc}}\\ M_{\mathrm{cw}}& M_{\mathrm{cc}}\end{array}\right]\left\{\begin{array}{c}{\ddot{\delta }}_{\mathrm{w}}\\ {\ddot{\delta }}_{\mathrm{c}}\end{array}\right\}+\left[\begin{array}{cc}{C}_{\mathrm{ww}}& C_{\mathrm{wc}}\\ C_{\mathrm{cw}}& C_{\mathrm{cc}}\end{array}\right]\left\{\begin{array}{c}{\dot{\delta }}_{\mathrm{w}}\\ {\dot{\delta }}_{\mathrm{c}}\end{array}\right\}+\left[\begin{array}{cc}{K}_{\mathrm{ww}}& K_{\mathrm{wc}}\\ K_{\mathrm{cw}}& K_{\mathrm{nn}}\end{array}\right]\left\{\begin{array}{c}{\delta }_{\mathrm{w}}\\ {\delta }_{\mathrm{c}}\end{array}\right\}=\left\{\begin{array}{c}{P}_{\mathrm{w}}\\ P_{\mathrm{c}}\end{array}\right\}$$

(15)

In Equation (15), the subscripts w and c represent the state of non-impact and impact. The implication of M, C, K is consistent with Equation (14). ${\ddot{\delta }}_{\mathrm{w}}$, ${\dot{\delta }}_{\mathrm{w}}$, and ${\delta }_{\mathrm{w}}$ represent the acceleration, velocity, and displacement vectors of freedom degrees at non-impact. ${\ddot{\delta }}_{\mathrm{c}}$, ${\dot{\delta }}_{\mathrm{c}}$, and ${\delta }_{\mathrm{c}}$ are the acceleration, velocity, and displacement vectors of freedom degrees at impact, while $P_{\mathrm{w}}$ and $P_{\mathrm{c}}$ are the load array of non-impact and impact, respectively.

2.2. Calculation Method of Train Derailment Process on Bridge under Ship Impact

The method of train derailment energy random analysis [24] shows that the wheel derailment is the process of the wheel flange climbing up the midpoint of the rail top from the normal driving state, and the process of wheel climbing caused by wheel lateral movement. The lateral movement of wheel is a kind of the lateral vibration response of FTTB system. Lateral vibration of the FTTB system is caused by many factors. Reference [24] proposed taking the measured or artificial bogie frame hunting wave as the excitation source of FTTB system lateral vibration, and taking it as input energy. Through the measurement and statistics of a large number of bogie frame hunting waves, the standard deviation σ_p of bogie frame hunting waves with 99% probability level at different vehicle speed V is obtained, and the σ_p-V curve is established. According to the σ_p at a certain speed, a bogie frame hunting wave is simulated by the Monte-Carlo method. Then, the bogie frame hunting wave is input into the FTTB system and the lateral vibration response of the FTTB system is calculated. However, σ_p is the input energy of normal train operation, which struggles to reflect the state of abnormal train operation (i.e., train derailment). However, according to the principle of function conversion, the greater input energy, the greater vibration response. Similarly, σ_pn must be greater than σ_p when the train derails. The σ_pn requires carrying out a large number of derailment tests to obtain. The damage caused by derailment tests discourages attempts, and this test is difficult to realize easily. Therefore, σ_pn cannot be obtained through measurement and statistics. With the help of theoretical simulation calculation, starting from the calculation process of the bar stability critical load, the trial algorithm is used to try to calculate the σ_pn of the train derailment under the ship impact, which realizes the calculation of the whole train derailment process. The calculation method is as follows.

Step 1: Taking the ship impact load as an external excitation on pier, the model of the FTTB system under ship impact is established.

Step 2: According to the curve of σ_p-V, the standard deviation σ_p of a certain speed is found. Moreover, a bogie frame hunting wave is simulated by the Monte-Carlo method which is adopted as the lateral vibration source of the FTTB system, and track irregularity is taken as the vertical vibration source.

Step 3: The spatial vibration response under ship impact is calculated. Each time step is calculated to determine whether the value of wheel lift meets the wheel derailment geometric criterion, 25 mm. If it is reached, the calculation is stopped and the whole process of train derailment is completed. The vibration response obtained reflects the whole process of train derailment. If not, the response of the next time step is calculated until the end of the calculation.

Step 4: If the train is not derailed at the end of the calculation, the ship impact load or speed is increased. Step 1–3 are repeated until the whole process of train derailment is completed.

So far, the calculation method of the train derailment process on bridge under ship impact has been expounded. According to the above method, the train derailment process is calculated, and the wheel-rail contact state and relative position at the moment of wheel derailment are analyzed. The corresponding vibration response reflecting wheel derailment information can be obtained. At the same time, it is difficult to install the early warning device on the wheel considering the space at the bottom of the vehicle and the fact that the wheel is the running part of the train. Thus, the vibration response of the bogie closest to the wheel is used to describe the wheel-rail contact state and its relative position. So, ΔY_tr/1.25 is calculated, where 1.25 [29] is the safety factor when Japanese scholars use the derailment coefficient to calculate train safety. The threshold with the derailment warning function is obtained through calculation, which can provide basic data for a wheel derailment warning device installed on bogies.

3. Results

3.1. Calculation of Train Derailment Process under Ship Impact

In view of the derailment accident and calculation results, empty wagons are more likely to derail [24]. the formation of a train is set to be one locomotive and 16 empty wagons, and the speed is 60 km/h. For a heavy haul railway, the 32 m concrete simple supported beam bridge is taken as the object. The bridge type is the reference bridge 2019, the pier height is 10 m, and the pier cross section is round with a diameter of 2.4 m. In addition, the track structure on the bridge is a ballasted track which consists of 60 kg/m rail, type II concrete sleeper, and crushed stone ballast. The time history curve of impact load was adopted from Reference [20], which was normalized according to the impact peak P_{c, max} as 0, 5, 10, 15, and 20 MN, respectively, and then input into the FTTB system. In order to simulate the most unfavorable driving condition on the bridge under the ship impact, the train starts driving on the bridge for 10 s when the bridge is full of vehicles. The impact load is applied on pier 4 in the X direction perpendicular to the bridge (In Figure 4), and the impact position is 5 m away from the bottom of the pier.

The process of train derailment under ship impact is calculated and the vibration responses of the FTTB system is obtained. In order to analyse the response of the FTTB system, some representative indicators are listed. This included ΔZ, Q/P, ΔP/P, B_h, D_h, and ΔY_tr, which represent the wheel lift value, derailment coefficient, wheel reduction rate, lateral displacement of beam and pier, and lateral displacement between bogie and rail. The time history curves of the above indicators are shown in Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10. The “0 line” in the diagram represents the centerline of model.

Figure 5 reflects the left wheel lift value time history curve of the 1st axletree of the 7th wagon under different ship impact load. In Figure 5, the wheel does not suspend before the ship impact. When the train runs to the 10^th s, the ship begins to hit the pier, and ΔZ increases significantly. The maximum value of ΔZ increases with P_{c, max}. The maximum values of ΔZ are 8.7, 11.5, 22.6, and 25.0 mm, respectively, when P_{c, max} is 5, 10, 15, and 20 MN. Among them, when P_{c, max} is 15 MN, the maximum value of ΔZ is close to the wheel derailment geometric criterion limit, 25 mm [24]. When P_{c, max} is 20 MN, the maximum value of ΔZ is equal to 25 mm, which reaches the limit value of the wheel derailment geometric criterion [24]. At this time, the wheel is derailed. Then, it can be seen that the train wheel is more prone to derailment when P_{c, max} is greater than or equal to 15 MN. It is necessary to control train derailment in time before the peak value of impact load is reached.

In addition, the maximum value of ΔZ increases with P_{c, max}. The maximum values of ΔZ are 8.7, 11.5, 22.6, and 25.0 mm, respectively, when P_{c, max} is 5, 10, 15, and 20 MN. Among them, when P_{c, max} is 15 MN, the maximum value of ΔZ is close to the wheel derailment geometric criterion limit, 25 mm [24]. When P_{c, max} is 20 MN, the maximum value of ΔZ is equal to 25 mm, which reaches the limit value of the wheel derailment geometric criterion [24]. At this time, the wheel is derailed. Then, it can be seen that the train wheel is more prone to derailment when P_{c, max} is greater than or equal to 15 MN.

Figure 6 reflects the derailment coefficient time history curve of the 1^st axletree of the 7^th wagon under different ship impact load. In Figure 6, the Q/P changes gently before ship impact, and the maximum value of Q/P is 0.32. When the train runs to 10^th s, the ship begins to hit the pier, Q/P increases with the increase of P_{c, max}. Among them, the maximum values of Q/P are 0.45, 0.77, 1.03, and 1.15, respectively, when P_{c, max} are 5, 10, 15, and 20 MN. When P_{c, max} is 15MN, the maximum value of Q/P exceeds the specification limit 1.0 [30], but the train does not derail. When P_{c, max} is 20 MN, the maximum value of Q/P is equal to 1.15 which also exceeds the specification limit 1.0 [30], but the train derails. Then, the derailment coefficient increases, induced by ship impact, affecting running safety. However, the derailment coefficient limit cannot determine the train wheel derailment.

Figure 7 reflects the wheel reduction rate time history curve of the 1st axletree of the 7^th wagon under different ship impact load. In Figure 7, the ΔP/P changes gently before ship impact, and the maximum value of ΔP/P is 0.15. When the train runs to 10^th s, the ship begins to hit the pier, and ΔP/P increases with the increase of P_{c, max}. Among them, the maximum value of ΔP/P were 0.27, 0.55, 0.75, and 0.81, respectively, when P_{c, max} were 5, 10, 15, and 20 MN. When P_{c, max} is equal to 15 MN, the maximum value of ΔP/P exceeds the specification limit 0.65 [30], but the train does not derail. When P_{c, max} is equal to 20 MN, the maximum value of ΔP/P is is equal to 0.85 which also exceeds the specification limit 1.0 [30], but the train derails. From above analysis, the derailment coefficient increases significantly, induced by ship impact, but the wheel reduction rate limit cannot determine the train wheel derailment. In addition, Xiao et al. [31] also proposed that the derailment coefficient and wheel load reduction rate limit are conservative when assessing the influence of a complex environment on the wheel derailment of high-speed trains.

Figure 8 and Figure 9 reflect the transverse displacement time history curve of the midspan of 3^rd span and top of 4^th pier under different ship impact load. In Figure 8, the B_h changes gently before ship impact, and the maximum value of B_h is 1.1 mm. When the train runs to 10^th s, the ship begins to hit the pier, B_h increases with the increase of P_{c, max}. Among them, the maximum values of B_h are 12.4, 24.2, 35.7, and 47.3 mm, respectively, when P_{c, max} are 5, 10, 15, and 20 MN. Besides, in Figure 9, the D_h changes gently before ship impact, and the maximum value of D_h is 0.2 mm. When the ship begins to hit the pier, D_h increases significantly with the increase of P_{c, max}. Among them, the maximum values of D_h are 19.5, 38.7, 58.1, and 77.5 mm respectively when P_{c, max} are 5, 10, 15, and 20 MN.

According to the specification [32], the limit values of lateral displacement of the mid-span beam and the lateral displacement of the pier top are 3.56 and 0.44 mm, respectively. The maximum values of B_h and D_h meet the limit requirements before ship impact. When P_{c, max} is 5, 10, and 15 MN, the maximum value of B_h and D_h exceeds the specification limit [32]. However, there is no wheel derailment. So, it can be seen that the transverse displacement of the beam span and the pier top increase significantly after the ship impacts the pier, but it does not affect the running safety when it exceeds the limit value. The mapping relationship between existing limits and train wheel derailment remains to be further studied.

Figure 10 reflects the lateral displacement between front bogie and rail time history of the 7^th wagon under different ship impact load. In Figure 10, the ΔY_tr changes gently before ship impact, and the maximum value of ΔY_tr is 8.7 mm. When the train runs to 10th s, the ship begins to hit the pier, ΔY_tr increases with increase of P_{c, max} and the wheel has a large displacement towards the line side. Among them, the maximum values of ΔY_tr are 20.2, 37.7, 56.5, and 75.1 mm, respectively, when P_{c, max} are 5, 10, 15, and 20 MN. When P_{c, max} is 20 MN, the maximum value of ΔY_tr is 75.1 mm. This value is the vibration response of train derailment and reflects the relative position of wheel and rail. In order to predict wheel derailment in advance, ΔY_tr/1.25 is calculated as 60.1 mm, which can be used as a warning threshold of train wheel derailment.

3.2. Influence of Impact Load on the Train Derailment Process

In order to analyze the train derailment process at different ship loads and speeds, the V is from 60 to 80 km/h, and the train formation, track and bridge structure, impact load, and action time are consistent with Section 3.1. The vibration responses reflecting the whole process of train derailment are obtained. The ΔZ, Q/P, ΔP/P, B_h, D_h, ΔY_tr, and ΔY_tr/1.25 are listed in

Table 1. Calculation results.

V (km/h)	Pc,max (MN)	ΔZ (mm)	Derailment Wagon and Wheel	Q/P	ΔP/P	Bh (mm)	Dh (mm)	ΔYtr (mm)	ΔYtr/1.25 (mm)
60	5	8.7	No	0.45	0.27	12.4	19.5	20.2	---
	10	11.5	No	0.77	0.55	24.2	38.7	37.7	---
	15	22.6	No	1.03	0.75	35.7	58.1	56.5	---
	20	25.0	7th wagon, 1st axletree, left wheel	1.15	0.81	47.3	77.5	75.1	60.1
65	5	8.9	No	0.67	0.31	13.8	20.6	22.5	---
	10	12.1	No	0.86	0.59	25.4	40.5	39.4	---
	15	23.1	No	1.13	0.78	36.1	60.5	57.9	---
	20	25.0	8th wagon, 1st axletree, left wheel	1.32	0.84	48.5	79.6	76.3	61.0
70	5	9.6	No	1.28	0.36	14.1	21.5	24.9	---
	10	15.5	No	1.46	0.68	26.1	41.6	43.2	---
	15	23.6	No	1.52	0.82	37.4	61.6	58.1	---
	20	25.0	9th wagon, 1st axletree, left wheel	1.68	0.86	49.7	80.4	78.5	62.8
75	5	10.2	No	1.49	0.52	15.6	22.2	25.6	---
	10	17.9	No	1.62	0.71	28.2	43.1	44.9	---
	15	23.9	No	1.74	0.84	38.9	63.1	59.3	---
	20	25.0	10th wagon, 1st axletree, left wheel	1.82	0.87	51.6	81.5	80.5	64.4
80	5	11.7	No	2.29	0.55	16.2	23.4	35.4	---
	10	19.8	No	2.41	0.73	29.7	45.1	48.1	---
	15	24.7	No	2.52	0.88	40.8	65.9	60.6	---
	20	25.0	11th wagon, 1st axletree, left wheel	2.63	0.92	52.8	83.6	81.7	65.4

, and their changes with V and P_{c, max} are shown in Figure 11, Figure 12, Figure 13, Figure 14, Figure 15 and Figure 16. The red dotted line in the figure represents the specification limits of different indicators.

Figure 11 reflects the ΔZ changes with V and P_{c, max}. In Figure 11, when the impact load is constant, with the increase of V, ΔZ increases gradually, but the increase is small. For example, the ΔZ at other speeds increased by 2.2%, 4.4%, 5.8%, and 9.3% compared with 60 km/h when P_{c, max} is 15 MN. Meanwhile, when V is constant and P_{c, max} is less than 15 MN, ΔZ increases with P_{c, max}, but the relative limit requirements still have a large margin. When P_{c, max} is great than or equal to 15 MN, ΔZ increases obviously which approaches or reaches 25 mm. Among them, when the V is from 60 to 80 km/h and P_{c, max} is 20 MN, train derailment occurs. Then, it can be seen that the impact load is the main factor causing the derailment of train wheels, and the speed has little effect.

Figure 12 reflects the Q/P changes with V and P_{c, max}. In Figure 12, when P_{c, max} is 5 and 10 MN and the V is from 70 to 80 km/h, all Q/P exceed the limit [30]. Besides, when P_{c, max} is 15 and 20 MN and the V is from 60 to 80 km/h, the corresponding of Q/P also exceed the limit [30]. However, the train wheel does not derail when the P_{c, max} is less than or equal to 15 MN, and the wheel derails when P_{c, max} is equal to 20 MN. So, this further shows that the limit value of derailment coefficient is difficult to determine whether the wheel is derail. However, it should be noted that with the increase of V and P_{c, max}, the greater the derailment coefficient is, the more unsafe the train is. For example, when V is 80 km/h and P_{c, max} is 20 MN, the Q/P is up to 2.63, which is 2.63 times of the specification limit 1.0.

Figure 13 reflects the ΔP/P changes with V and P_{c, max}. In Figure 13, when P_{c, max} are 0, 5, and 10 MN and the V is from 70 to 80 km/h, all ΔP/P exceed the limit [30]. Besides, when P_c,max is 15 and 20 MN and V is from 60 to 80 km/h, the corresponding ΔP/P exceed the limit [30]. However, the train wheel does not derail when the P_{c, max} is less than or equal to 15 MN, and the wheel derails when P_{c, max} is equal to 20 MN. This shows that the limit value of wheel load reduction rate makes it difficult to determine whether the wheel has derailed. However, it should be noted that the ΔP/P increases significantly when P_{c, max} is great than or equal to 15 MN, the maximum value of ΔP/P is 0.92. At this point, the train wheels are close to full load shedding. The impact load makes it easy to cause a significant reduction of wheel load, which affects safety.

Figure 14 and Figure 15 reflect the B_h and D_h changes with V and P_{c, max}, respectively. In Figure 14, when V is 80 km/h, B_h under other impact loads increased by 92%, 182%, and 257%, respectively, compared with P_{c, max} of 5 MN. When P_{c, max} is 20 MN, the B_h corresponding to other speeds increased by 2.5%, 5.1%, 9.1%, and 11.6% respectively, compared with the V of 60 km/h. Besides, in Figure 15, D_h increases with V and P_{c, max}. When the V is 80 km/h, D_h increases by 83%, 151%, and 226% under other impact loads, respectively, compared with P_{c, max} of 5 MN. When P_{c, max} is 20 MN, the D_h corresponding to other speeds increased by 2.7%, 3.7%, 5.2%, and 7.9% respectively, compared with the V of 60 km/h. Then, it can be seen that speed has little effect on the transverse displacement of beam span and pier top under impact load.

Figure 16 reflects the ΔY_tr changes with V and P_{c, max}. When V is 80 km/h, ΔY_tr are 35.4, 48.1, 60.6, and 81.7 respectively. Compared with P_{c, max} of 5 MN, ΔY_tr increased by 35%, 71%, and 131% respectively under the other P_{c, max}. When P_{c, max} is 20 MN, ΔY_tr increased by 1.5%, 4.5%, 7.2%, and 8.8%, respectively, under the other V, compared with V of 60 km/h. So, it can be seen that the impact load increases significantly, especially when the train wheel derails, when the P_{c, max} is 20 MN, and the corresponding of maximum ΔY_tr is 81.7 mm. Therefore, it is necessary to predict the derailment information of train wheels in advance to control the derailment of train wheels under impact load. Considering safety 1.25, the early warning threshold ΔY_tr/1.25 with derailment information is obtained, and the specific values are shown in Table 1.

4. Conclusions

In this paper, the model of a FTTB system under ship impact is established, and the whole calculation method of train derailment on a heavy haul railway bridge under ship impact is proposed. The variation law of the FTTB system response in the whole process of train derailment is analyzed. The main conclusions are as follows.

(1) Ship impact has a great influence on wheel lift value, and the peak impact load occurs synchronously with the maximum wheel lift. When the impact load peak is greater than or equal to 15 MN, the train wheel is more prone to derailment. It is necessary to take early warning measures in time.

(2) With the increase of ship impact load, derailment coefficient and wheel load reduction rate increase significantly, but its limit cannot determine whether the train wheel has derailed.

(3) The lateral displacement of the mid-span of beam and the top of pier increase significantly under ship impact, respectively. Since the pier is directly impacted, the lateral displacement of the pier top is greater than that of the beam span. When the peak value of impact load is less than 15 MN, although the above indexes exceed the specification limit, the train does not derail. Therefore, it is suggested to further refine the mapping relationship between specification limits and train safety.

(4) The increase of vehicle speed under ship impact has little effect on the response of the FTTB system.

(5) The lateral relative displacement of bogie and rail increases significantly under ship impact, and the lateral relative displacement of bogie and rail is obtained when the wheel is derailed. The lateral relative displacement of bogie and rail is divided by the safety factor as the warning threshold reflecting the train wheel derailment information. In addition, a scaled model test will be carried out in subsequent articles based on this threshold.