1 Introduction

In recent years, there has been a rapid development in high-speed rail construction worldwide. Such a development of railway infrastructure is often accompanied by the severe derailment accidents. Over-speed, impractical railway safety measures, poor weather conditions, and mechanical problems are common causes of train derailment accidents in many countries [1]. These accidents may cause significant casualties and property losses (Figure 1). Derailed high-speed trains, in particular, can collide with surrounding structures, leading to severe secondary disasters. The Amagasaki rail crash was a typical accident. In 2005, having failed to slow down on a curved track properly, five carriages of a high-speed train suddenly derailed on the JR West Fukuchiyama Line in Amagasaki, just in front of the Amagasaki Station on the way to D¨­shisha-mae. The train rushed into a residential building after colliding with a car. This accident killed 107 people and injured 562 others, according to Wikipedia [2].

In addition to the Amagasaki rail crash, there have been many collision accidents of derailed trains with buildings or bridges, as listed in Table 1. It has become evident that the secondary disasters caused by the collision of a derailed train with buildings or bridges near the tracks are more severe than the train derailment itself. The corresponding collision risk, collision pattern and collision load are closely related to the running attitude of the train after derailments.

Problems associated with train derailment have received increased attention since Nadal [9] firstly proposed the criterion for derailments. Brabie [1] systematically reviewed the relevant studies on the derailment of high-speed trains. In recent years, real accident analyses and early detections of train derailment have been intensively studied [10-12]. However, research to date was primarily focused on how to prevent and monitor train derailments. In contrast, studies of the running attitude of trains after derailment are very limited.

Table 1 Derailed trains collide with buildings leading to secondary disasters

Therefore, an in-depth study of the running attitude of a train after derailment is essential to improve the design, operation, and maintenance of railway systems. The work presents a numerical simulation of the running attitude of a train after derailment based on nonlinear finite element analysis. Using the simulation results, the running attitude of the train after derailment was analyzed with the goal of preventing the secondary disasters caused by derailed trains.

2 Finite element model

2.1 Track model

A key issue when simulating the running attitude of a train after derailment is how to simulate the contact between the derailed train wheels, carriages, tracks and the ground (including the concrete sleeper). Hence, this study used the commercial finite element software package MSC.MARC, which is widely used in contact computation to establish a finite element model [13]. Xu et al. [13-14] have analyzed the collision between over-height truck and bridge superstructure using MSC.Marc. The method of establishing the train model in this work is similar to that Xu et al.¡¯s work [13-14]. In constructing the models, solid elements are used for all main contact bodies mentioned above to ensure the accuracy and reliability of contact simulation. The contact module of MSC.Marc is utilized to definite the contact between different components. Specifically, the contact between the train body, the wheelset, the track and the ground were defined separately. The contact table of MSC.Marc defines the coefficients of friction between the train body, the wheelset, the track and the ground. For the normal contact condition, the contact distance tolerance is set to be zero to avoid any penetrability. For the tangential contact condition (i.e., the friction condition), MSC.Marc provides an arctangent model instead of the Coulomb friction model to improve the convergence efficiency [15]. The expression of which is shown in Equation (1).

where t is the distance of tangential relative sliding. The value of RVCNST can be seen as the value of the relative velocity below which sticking occurs. The value of RVCNST is important in determining how closely the mathematical model represents the step function.

Specifically, the tracks are simulated as a steel material based on an ideal elastic-plastic material according to the von Mises criterion. The concrete sleeper and the ground under the tracks are simulated as concrete material, which is also based on an elastic-plastic material model [16]. The finite element models of the tracks, the concrete sleeper and the ground are shown in Figures 2 and 3, respectively. The shapes of tracks are determined according to their actual sizes. Ballastless tracks are used in this simulation. The tracks and ground are connected through sharing nodes in the finite element model. Han & Koo [17] and Bae et al. [18] showed that the simplification of the tracks has little effect on calculation results.

2.2 Finite element model of the train

The finite element model of the train has a significant influence on the computational results. To obtain representative results, the parameters of the train model are determined according to a typical Chinese-manufactured train that has eight carriages. The running speed of the train during the derailment accident is set to 80 km/h, which is the highest speed limit in China for a train arriving at the station at the turnout. Hence, the simulation represents the most unfavorable derailed train speed. Solid elements are used to simulate the wheels of the train according to their actual size. The rims of the wheels are simulated as running inside the tracks to more accurately simulate actual conditions on the tracks. The train wheel size is given according to the size of the Chinese-manufactured train wheels. The contacts between wheels and tracks are shown in Figure 4. In MSC.Marc, the initial travel speed of the train can be defined through the Initial Conditions module. Han & Koo [17] showed that the coupler could be simulated with a link unit. Consequently, this work follows Han & Koo [17] method to simulate the coupler as a simple link, and the coupler will not break after the derailment.

Solid elements are also used to model the car body to achieve an accurate simulation of the train-wheel-track-ground contact. The train models are simplified according to the simulation requirements. The train bogies are simplified using truss elements. The train models are shown in Figures 5 and 6. The lengths of carriages along the x-axis (Figure 5) are 15500 mm. The wheel diameter is 700mm.

2.3 Boundary conditions

The bottom of the ground was fully fixed. The tracks were connected to the ground through sharing nodes in the finite element model. The entire model was subjected to gravity loads, which were assigned by defining the gravitational acceleration in a vertically downward direction.

2.4 Mesh generation and time step

The x-direction is the running (i.e., longitudinal) direction of the train; the y-direction is transverse to the running direction of the train; and the z-direction is the vertical direction. The sizes of mesh are as follows: the lengths of the elements along the y and z-axes (Figure 6) of the train are approximately 200 mm. The lengths of the elements along the x-axis (Figure 6) of the train are 200 mm to 500 mm. The size of the elements in the wheels is approximately 60 mm. The lengths of the elements along the x-, y-, and z- axis are 2000 mm, 73mm and 175 mm, respectively (Figure 3). This study adopts an implicit algorithm to solve the dynamic equations, and the time step is set to 0.250 ms.

2.5 The train derailment conditions

There are many potential causes of a train derailment because the actual behavior of the wheel-rail system is so complicated. Hence, this study introduces an artificial obstacle on the track, in front of the running train, to induce the train derailment. With this condition, the running attitude of the train after derailment can be simulated. The condition causing the train derailment is shown in Figure 7.

3 Influence of the friction coefficient on the running of derailed train

At low speed and under dry conditions, the wheel-rail friction coefficient is close to 0.6 [19]. When contaminants, such as water, iron oxide, and leaves, are unintentionally presented on the rail, the friction coefficient will be sharply reduced, to values ranging from 0.05 to 0.1 or even smaller. The relevant information is shown in Table 2. However, when the train contacts the ground after the derailment, the wheel-rail friction coefficient will increase sharply. Therefore, when this work studied the influence of the friction coefficient on the running of the derailed train, the friction coefficients 1.0 and 2.0 were considered to cover a larger parameter range. After the train derails, the wheelsets are in contact with the concrete ground. Rabbat & Russell¡¯s study [20] found that the coefficient of friction between concrete and steel can reach 0.7. Many studies have analyzed the rolling of wheels using different wheel-rail friction coefficients and investigated the influence of different friction coefficients on the motion of wheels [21-23]. Numerous studies have shown that the friction coefficient has a direct influence on wheel-rail contact, which further affects the traction and braking of the train [24]. Therefore, it is necessary to discuss the running attitude of the train after derailment using different friction coefficients. After the train derails, the wheelset will be in contact with the ground, and the friction coefficient between the wheel and the ground is required during the contact process.

Table 2 Friction coefficient with different conditions [19]

3.1 The influence of different friction coefficients on the running speed of the derailed train

According to the range of wheel-rail friction coefficients and the influence of the ballast on the ground after the derailment, five representative friction coefficients were chosen for analysis in this study; these are 0.1, 0.2, 0.5, 1.0 and 2.0.

The variations of running speed after derailment with different wheel-rail friction coefficients are illustrated in Figure 8. The friction coefficient greatly affects the running speed of the train after derailment. With a higher friction coefficient, the running speed of the train after the derailment will decrease, and the elapsed time before the train stops will also be reduced. However, running speeds after derailment are roughly similar when the friction coefficients are 1.0 and 2.0. This shows that when the friction coefficient is greater than 1.0, it has little influence on the running speed and the running time of the train after derailment.

Figure 9 shows the speeds of the first carriage and the train tail after derailment when the wheel-rail friction coefficient is 2.0. The curves are similar to each other. This illustrates a significant pulling-pushing interaction between the front and rear carriages, which causes the whole train to run with nearly the same speed.

3.2 The influence of the friction coefficient on the running displacement of the derailed train

From the above analysis, the running speed of a train after derailment will change corresponding to different wheel-rail friction coefficients. Similarly, the running displacement of the train after derailment will also change with different wheel-rail friction coefficients. The running displacement changes after derailment are shown in Figure 10.

Figure 10 shows that the friction coefficient has a significant influence on the running displacement of the derailed train. With an increase in the friction coefficient, the displacement of the derailed train will decrease gradually, which is consistent with actual experience. When the friction coefficient is greater than 1.0, the derailed train stops after 45 m. When the friction coefficient is smaller than 1.0, the influence of the friction coefficient is more significant. When the friction coefficient is 0.5, the derailed train stops after 60 m. When the friction coefficient is 0.1, the derailed train does not stop even after running for 120 m.

Figure 11 shows the displacements of the first carriage and the train tail after derailment when the wheel-rail friction coefficient is 0.5. The train¡¯s front and rear carriages have a similar displacement after the derailment. This illustrates a significant pulling-pushing interaction between different carriages.

3.3 The influence of the friction coefficient on the lateral displacement of the derailed train

The friction coefficient has a significant influence on the running speed and displacement of the derailed train. Therefore, it is necessary to consider the impact of the friction coefficient on the lateral oscillation of the derailed train. Table 3 illustrates the maximum lateral oscillation displacements that occur with different friction coefficients. There is a significant lateral oscillation of the derailed train. When the friction coefficient is 0.2, the maximum lateral oscillation displacement is 6.685 m. The maximum lateral oscillation displacements with other friction coefficients approach approximately 3 m, which makes it quite possible that the train will collide with objects and buildings near the track, subsequently inducing secondary disasters.

Table 3 The maximum lateral displacement with different friction coefficients

Using the same wheel-rail friction coefficient, the lateral displacements of the first carriage and the train tail after the derailment are shown in Figure 12. There are significant lateral oscillations of both the front and rear carriages. According to EN1991-1-7 [25], for permanent buildings across or near railways in operation, if the distance between the nearest rail center line and the permanent building is not greater than 5 m, it is necessary to consider the possibility of a train running into these buildings after derailment and to take appropriate measures to protect the buildings. EN1991-1-7 [25] does not consider that a derailed train may impact buildings at a distance of greater than 5 m from the center line. However, Table 3 shows that when the friction coefficient is 0.2, the maximum lateral oscillation displacement is 6.685 m, which is beyond the range specified in EN1991-1-7 [25].

3.4 The influence of the friction coefficient on the lateral oscillation speed of the derailed train

The lateral oscillation speed of the derailed train is a critical factor that determines the value of the force when the train collides with station buildings. The lateral oscillation speeds of a derailed train with different friction coefficients and lateral displacements are presented in Table 4, which can provide a useful reference for determining the lateral collision speed of a derailed train.

Table 4 Lateral speed with different friction coefficients and lateral displacements

The distance between the station building columns in large-scale Chinese railway stations and the nearest track is fixed. The distance from the center of the through line to the surface of the station building columns is 2.44 m to 2.6 m. The distance from the arrival and departure line to the surface of the station building columns is 2.15 m to 2.25 m. Considering the width of the train, it is conservative to anticipate a collision between the derailed train and the station building columns within a range of 0.5 m from both sides of the tracks.

4 The running attitude after the train derailment

The final running attitudes of the train after the derailment, with different friction coefficients, are presented in Figure 13. The derailed train moves with a "snakelike" attitude. The lateral running attitudes and amplitudes are random to some extent. As shown in Table 3, when the friction coefficient is 0.2, the maximum lateral oscillation displacement is 6.685 m. When the friction coefficient is 2.0 or 0.1, the maximum lateral oscillation displacement is approximately 3 m. When the friction coefficient is 1.0 or 0.5, the maximum lateral oscillation displacement is approximately 2.5 m.

5 Conclusions

The nonlinear finite element method was used to analyze the running attitude of a train after the derailment. The influences of different friction coefficients were discussed, and conclusions can be derived as follows:

(1) The wheel-rail friction coefficient has a significant influence on the speed and displacement of the derailed train. The greater the friction coefficient, the faster the speed of the train will reduce. However, the friction coefficient has little influence on the reduction of speed and the maximal displacement of the derailed train when it is greater than 1.0.

(2) The speeds and displacements of the first carriage and the train tail after derailment change similarly. This illustrates a significant pulling-pushing interaction between the front and rear carriages, which makes the speed and longitudinal displacement of the whole train almost identical.

(3) There is a significant lateral oscillation of the train after the derailment, which eventually causes the complete derailment of some carriages. The train may collide with objects within a distance of 6.685 m from both sides of the tracks. Hence, actions need to be taken to protect objects within this range.

The running attitude of a train after derailment analyzed in this work will assist in providing a useful reference for the prevention of secondary disasters caused by the collision of derailed trains with station buildings.

Acknowledgements

The research in the paper is supported by the National Natural Science Foundation of China (NSFC) (71741024), and Southeast University, Key laboratory of concrete and prestressed concrete structure of Ministry of Education (CPCSME2016-11).

Disclosure statement

No potential conflict of interest was reported by the authors.

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