Numerical Analysis Method for Analyzing Seismic Vehicle Behavior Up to and After Derailment

Abstract

The authors are conducting research with the aim of establishing a numerical analysis method capable of evaluating vehicle behavior before and after derailment of a train set during an earthquake. In this paper, as a basic study, an analysis method is proposed that can represent seismic vehicle behavior before and after the derailment of a single vehicle in a stationary state. Then, in order to consider the coupling of multiple vehicles, the proposed method is also extended to include dynamic models for connecting members between vehicles such as couplers and inter-car yaw dampers. Furthermore, the influence of the interaction between vehicles on the derailment limit is investigated through trial calculations.

1. Introduction

In recent years, there have been derailments of Shinkansen and other trains due to frequent large earthquakes as shown in the reference [1]. The derailment of high-speed trains may lead to large scale damage. Thus, countermeasures for tracks and vehicles [2, 3] are being developed to prevent derailed vehicles from running too far off the tracks or on to adjoining tracks with oncoming trains. However, taking an experimental approach to study this phenomenon is difficult. There are studies which analyze vehicle behavior during an earthquake, including post-derailment [4] using numerical analysis methods. However, such studies only express the phenomenon by replacing the shapes of the rails and wheels with simple shapes such as rectangles. As such, currently, there is no established method to assess the behavior of vehicle in the moments leading up to and after derailment.

The objective of this study is therefore to establish such an analysis method capable of evaluating the behavior of vehicles before and after derailment during an earthquake. As a basic study to achieve this objective, we propose an analysis method that can represent vehicle behavior of a single stationary vehicle before and after derailment. In addition, we extend the proposed model by considering the dynamic models for connecting members between vehicles such as couplers and inter-car yaw dampers [5, 6], making it possible to represent the behavior of a multi-car train. Moreover, we will examine the influence of the interaction between vehicles on derailment behavior by trial calculation.

2. Proposed analysis method of train behavior during an earthquake

In this section, we explain the proposed analysis method that can represent the behavior of a stationary single car train during an earthquake in the moments leading up to and after derailment. We then extend this model by considering the dynamic model for the connecting members between vehicles, making it possible to represent the behavior of a multiple-car train. The present method is developed using the nonlinear structural analysis software Ansys LS-DYNA (R13.1.1) [7].

2.1 Methods for analyzing the behavior of single car train in moments leading up to and after derailment during an earthquake

2.1.1 Overview of the analysis method

Because the phenomenon in question during an earthquake can last for a long period time, from tens to a few hundred seconds, modelling it requires ensuring a certain level of accuracy while reducing the computational burden as much as possible. On the other hand, when evaluating the vehicle behavior that is of interest in this study, it is critical to properly account for the contact phenomenon between complex shapes of parts on the vehicle side and the track side, as typified by wheel / rail contacts.

Thus, in this study, we base the modeling of the vehicle on the multibody dynamic theory, which provides a low computational load and has been verified as appropriate for the evaluation of the behavior of railway vehicles during an earthquake [8]. In addition, we developed a method to represent the contact phenomenon between a vehicle and track parts (e.g. the rail and the track slab) before and after derailment using rigid finite elements. We show the constructed model in Fig. 1. As a basic study, in the present study we constructed a model that includes the vehicle, track (rail, track slab, CA mortar, concrete roadbed) and the superstructure, assuming a situation where a single car train is stopped on a slab track on an elevated bridge.

Fig. 1　Analysis model based on the proposed method

2.1.2 Modeling the vehicle

Figure 2 shows the dynamic model of the vehicle. In the model, we considered a carbody of a vehicle, bogie frames, and wheelsets as rigid bodies and connected these rigid bodies with springs and dampers. In an actual vehicle, stoppers are installed between vehicle components to suppress the occurrence of significant relative displacement. We therefore included this in our model. The right and left wheels, which are part of the wheelset, are modeled separately using rigid finite elements (average element length of approximately 10mm), which replicate the wheel shape in detail to represent the contact. We do so to represent the contact with the rail and the fall onto the tack slab after derailment (see Fig. 1). It should be noted that since we assume that the vehicle is in a stationary state, we constrained the wheelset rotational degree of freedom in the circumferential direction of the wheel (Y axis rotation: the coordinate system is shown in Fig. 1 and Fig. 2).

Fig. 2　Dynamic model of the vehicle

2.1.3 Modeling the track and the superstructure

As for the track and the superstructure, we modeled each of them individually just below each wheelset using rigid finite elements to represent their shapes, as shown in Fig. 1. We used rigid bonding between the track slabs, CA mortars, roadbed concrete, and the superstructure. We used linear springs and dampers that are equivalent to fastening devices, to connect the rails to the track slabs. We replicated the shape of the rails in detail to account for the contact with the wheels. Input of seismic movement was set as forced displacement to the superstructure.

2.1.4 Modeling the wheel and track contact

We performed three-dimensional contact calculations between the finite elements that replicate each shape in detail to model the contact between the wheel and the rail and also the contact between the wheel and the track slab after derailment. We represented modeling of the contact force in the normal direction of the contact surface using the penalty method. Coulomb friction (with friction coefficient set to 0.3) was used to model the contact force in the tangential direction of the contact surface. It should be noted that, in this study, we do not take into consideration the influence of the creep force generated by the rotation of the wheels, since we assume that the vehicle is in a stationary state.

2.2 Method for the analysis of a multiple-car train during an earthquake

2.2.1 Overview of the analysis method

In this section, we improve the single-car train analysis method established in Section 2.1 to a method capable of computing the behavior of a multiple-car train during an earthquake by inserting a dynamic model that expresses the connecting members between vehicles, which allows the connection of multiple vehicles. Figure 3 shows an example of the connecting members between vehicles in a Shinkansen train [6]. Couplers and inter-car yaw dampers are considered as the connecting members in this study, as shown in the figure. The details of this dynamic model are explained below.

Fig. 3　Example of connecting members between vehicles in a Shinkansen

2.2.2 The dynamic model of the coupler

Figure 4 shows the conceptual diagram for the dynamic model of the coupler. We constructed the dynamic model of the coupler using a hinge connection between the non-linear spring, which represents the characteristics of the buffer fixed onto each carbody, and the rigid bar, which connects the buffers on each carbody. We assume that the buffer springs always follow the movements of the carbody, expanding and contracting only in the direction of the carbody axis. In addition, we use a multilinear model where the spring characteristics are defined by the force and the stroke. Figure 5 shows the characteristics of the buffer springs used in this study. Based on the characteristics of the shock absorber springs used in Shinkansen [9], we set ours as a bilinear type, which has higher rigidity on the extension side.

Fig. 4　Dynamic model of the coupler

Fig. 5　Characteristics of the buffer spring in the model

2.2.3 The dynamic model of the inter-car yaw damper

Figure 6 shows the conceptual diagram for the dynamic model of the inter-car yaw damper. We constructed the inter-car yaw damper by arranging in parallel a damper representing damping characteristics and a spring representing the stopper characteristics and connecting both ends by hinges to an arbitrary position on each vehicle.

Fig. 6　Dynamic model of inter-car yaw damper

Figure 7 shows the damping characteristics and the stopper characteristics of the inter-car yaw damper used in this study. We set the damping characteristics referencing the catalog [10]. However, for the inter-car yaw damper we characterized, no damping force is generated when the piston stroke δ exceeds the threshold (in this study the threshold is ±40 mm), as shown in Fig. 7(a). We do so because there is a system that releases the hydraulic pressure so that no damping force is generated when the yaw damper piston stroke becomes large when turning a curve. It should be noted that we assume that the damping force returns when the piston stroke value is back within the threshold after the threshold has been exceeded. For the characteristic of the stopper spring, we assume that the stopper will operate when the piston stroke exceeds 300 mm, as shown in Fig. 7(b).

Fig. 7　Characteristics of the inter-car yaw damper in the model

3. Evaluation of the influence of the interaction between vehicles on the derailment limit

In this section, we consider the influence of the interaction between vehicles on the derailment limit that occurs due to the presence of the connecting members. We do this by conducting an excitation test of a three-car train set using the analysis method proposed in Section 2.

3.1 Analysis method

Figure 8 shows the analysis model that we used for the examination. The multiple-car train set analysis model consists of three coupled vehicles (hereinafter, multiple-car model), which is constructed by lining up three vehicles of the single-car model in a stationary state and inserting the dynamic model of the connecting members (the couplers and the inter-car yaw dampers) between each vehicle, as described in the previous section. We used the dynamic model and the characteristics of the connecting members as described in Section 2.2.

Fig. 8　The three-car analysis model used in the present examination (Example of when excitation is applied only to Vehicle-1)

For the excitation condition, three different cases are considered: when the same excitation is applied to all three vehicles, when the excitation is applied only to Vehicle-1 (lead-vehicle), when the excitation is applied only to Vehicle-2 (middle-vihicle). An extreme excitation condition was deliberately set to evaluate the influence of the interaction between vehicles, although these are not conditions that can occur in reality. The excitation was performed by inputting five periods of sinusoidal oscillations in the lateral direction (Y-axis direction: see Fig. 1 and Fig. 8 for the coordinate system) with a constant frequency and amplitude. The sinusoidal wave frequency was between 0.5 Hz and 3.0 Hz and increased by 0.1 Hz and the amplitude was increased by 5 mm to determine whether derailment occurred. The relative lateral displacement of ±70 mm between wheel and rail [11] was used as the criterion for derailment during an earthquake. This value is at present the threshold to determine the criterion for derailment during an earthquake. Here, the computation time for the three-car train model was about 45 minutes per case when using parallel computing at 0.5 Hz (15 seconds duration) on a dual-core desktop computer.

In addition to the above, we also performed a sinusoidal oscillation analysis with only one vehicle to evaluate the difference in vehicle behavior with the multiple-car model. Furthermore, we performed an oscillation analysis under the same conditions (five sinusoidal oscillations between 0.3 Hz and 3.0 Hz) as the Vehicle Dynamics Simulator (VDS) [8] to verify the validity of our proposed method. The validity of VDS has been verified by comparison with the results of a full-scale shaking table test [12]. We compared the results with those of the proposed method. Here, VDS simulates the vehicle movement up until just before derailment and the computation ends when one of the wheels reaches the derailment criteria explained above.

3.2 Results of the examination

3.2.1 Verification of validity of the proposed method

Figure 9 shows the comparison between the VDS derailment limit diagram of the single-car model and our proposed method. Here, the derailment limit diagram shows the vibration amplitude just before reaching derailment for each excitation frequency. In the figure, we also show the lines for each acceleration amplitude (5 m/s², 10 m/s² and 15 m/s²) to serve as a guide to know which combination of frequency and amplitude corresponds with which level of acceleration input. From the figure we can confirm that the limit value is slightly different between the VDS and our proposed method at 0.3 Hz and 0.4 Hz, respectively, but the values are well matched for the other excitation frequencies. Therefore, this fact leads us to say that a derailment limit diagram that is equivalent to that of VDS can be computed using our proposed method.

Fig. 9　Derailment limit diagram for five sinusoidal oscillations (single car model)

In additionally, Fig. 9 shows not only the results when the derailment criterion is set at a relative lateral displacement of ±70 mm between wheel and rail, but also the results when the wheels fall off the rail. From the figure, it can be seen that the overall results of both cases are the same. This means that derailment occurs in most cases when the relative lateral displacement between wheel and rail reaches ±70 mm. This indicates the validity of the conventional derailment criterion.

3.2.2 Evaluation of the influence of the interaction between vehicles

Figure 10 shows the time-history waveform of vehicle response for Vehicle-1 in the three-car train model (excitation applied only to the Vehicle-1, the excitation frequency: 0.5 Hz, the excitation amplitude: 520 mm) in a way that compares with the single-car model. Focusing on the relative vertical displacement between wheel and rail as shown in Fig. 10 (a), a significant increase in response over time can be observed only in the single-car model while almost no relative vertical displacement occurs in the multiple-car model. It can be seen that, in the single-car model, a displacement of around -180 mm occurs around 12 seconds. This is due to the derailment of the wheelset, it can be seen that our method can represent the sequence of vehicle behavior before and after derailment. In addition, it is possible to confirm that displacement rapidly increases around 12 seconds before derailment for the lateral displacement shown in Fig. 10 (b).

Fig. 10　Comparison of vehicle behavior in the single-car model and multiple-car model (when excitation is applied only to Vehicle-1, excitation frequency: 0.5 Hz, excitation amplitude: 520 mm)

Next, we focus on the vertical displacement of the center of gravity of the carbody in Fig. 10 (c) as well as the lateral displacement of the center of gravity of the carbody and roll angle in Fig. 10 (d). From the figures, it is possible to verify a phenomenon in which the amplitude in each response increases gradually over time in the single-car model. In contrast, in the multiple-car model, the vertical and lateral responses are generally in agreement up to around 3 seconds when the first sinusoidal wave ends, then after that the amplitude in each response remains constant overall over time. In addition, it is also possible to see the difference in oscillation periods between the multiple-car model and the single-car model, where the oscillation period in the multiple-car model is shorter than that of the single-car model.

Figure 11 shows the time-history waveform of the buffer spring on Vehicle-1 side in the coupler between Vehicle-1 and Vehicle-2 in the multiple-car model (excitation applied only to Vehicle-1, excitation frequency: 0.5 Hz, excitation amplitude: 520 mm). The time-history waveforms are shown by separating into longitudinal, lateral and vertical directions (X-axis, Y-axis, and Z-axis in Fig. 8, respectively). Focusing on the longitudinal direction, it can be seen that the force amplitude on the positive side (extended side) is larger than on the negative side (compressed side). This corresponds to the excitation condition under which a pull force is generated between vehicles; that is, the relative discrepancy between vehicles increases on both Vehicle-1 and Vehicle-2 by oscillating only Vehicle-1. It should be noted that forces on the lateral and vertical directions are extremely small in contrast to the force on the longitudinal direction. This is because the forces other than those in the longitudinal direction of the carbody are absorbed by the hinge (see Fig. 4), which is the connection point between the coupler and the buffer spring, instead of the buffer spring.

Fig. 11　Time-history waveform of the buffer spring

Figure 12 shows the time-history of the damping force of the inter-car yaw damper between Vehicle-1 and Vehicle-2 and the piston stroke in the multiple-car model (excitation applied only to Vehicle-1, excitation frequency: 0.5 Hz, excitation amplitude: 520 mm). In the figure, we also show the ±40 mm threshold line of the piston stroke, at which point there is no damping force due to oil pressure release. The figure shows that the piston stroke exceeds the +40 mm line multiple times and that the damping force becomes zero at these instances. It can also be seen that the damping force occurs again when the piston stroke enters the ±40 mm range afterwards.

Fig. 12　Time-history waveform of the inter-car yaw damper

To evaluate the influence of the interaction between vehicles on the derailment limit, Fig. 13 shows a comparison of derailment limit diagram for oscillating frequency between 0.5 Hz and 2.0 Hz for the following cases: excitation in the single-car model, excitation in-phase excitation of all cars in the multiple-car model, excitation of only Vehicle-1 (lead-vehicle) in the multiple-car model, and excitation of only Vehicle-2 (middle-vehicle) in the multiple-car model. It should be noted that derailment occurred on the vehicles that were excited under all excitation condition. We also show a diagram that converts Fig. 13 into the ratio of a single-car derailment limit amplitude in Fig. 14 so that it is easier to capture the influence of the interaction between vehicles. This diagram means that, when the amplitude ratio is larger than 1.0, it is harder to derail compared to the single-car case and that, when the amplitude ratio is smaller than 1.0, it is easier to derail compared to the single-car case. From both figures, it is possible to see that the results of the excitation of only one car is mostly consistent with those of the in-phase excitation of the multiple-car train. This result is consistent with those in existing study [6]. It can also be seen that, when oscillation is added only to the first car (lead-vehicle) in the multiple-car model the limit value is mostly the same compared to the single-car model while the value tends to decrease slightly when it is 1.2 Hz or below. In contrast, in the case of excitation of only Vehicle-2 (middle-vehicle) in a multiple-car train, the limit value is reduced at some excitation frequencies but, overall, the limit value tends to increase compared to the case of only one car. In the figure, we show the range in which the ratio of the derailment limit between the single-car model and the multiple-car model is ±10%. It can be confirmed that it is generally within this range. In the present study we deliberately set extreme excitation conditions to make it easier to understand the impact of interaction between vehicles. In the case of a real earthquake, the degree of influence is expected to be even smaller.

Fig. 13　The influence of the interaction between vehicles on the derailment limit

Fig. 14　Ratio of derailment limit amplitude in contrast to the single-car model

The examination results shown above are only an example. It should be noted that it may be possible that the influence of the interaction between vehicles on derailment limits may differ greatly depending on the characteristics of the vehicle and the characteristics of the connecting members between vehicles.

4. Conclusions

This study aimed to establish an analysis method that can evaluate the behavior of vehicles during an earthquake in the moments leading up to and after derailment. As a basic study towards this objective, this paper proposes two methods: one which can analyze vehicle behavior before and after derailment in a single stationary state, the other which can express the behavior of a multiple-car train. In addition, we examined the influence of the interaction between vehicles on the derailment limit during an earthquake using the proposed method. We summarize our study as follows:

(1)We proposed an analysis method that combines a multi-body for representing vehicle motion and rigid finite elements for representing the contact between vehicle members and track members. This method can efficiently calculate the behavior of a single-car train in a stationary condition during an earthquake before and after derailment.

(2)We proposed a new dynamic model that represents the coupler and the inter-car yaw damper, which are the connecting members between vehicles, and we incorporated this model into the single car method explained above for the sake of representing the behavior of a train set during an earthquake.

(3)We examined the influence of the interaction between vehicles on the derailment limit through the analysis of the three-car train modeled by the above method. In the case of in-phase excitation of all vehicles in a multiple-car train, the derailment limit is mostly the same as the single-car case. In the case of excitation of only the middle car in the multiple-car train, the derailment limit is on an increasing trend compared to the single-car case.

The findings explained above are only an example. It is expected that the influence of the interaction between vehicles on derailment limits may differ greatly depending on the characteristics of the vehicle, the characteristics of the connecting members between vehicles, the excitation conditions and other factors. To clarify the extent of these influences, the validity of the analysis method constructed in this study have to be verified. Moreover, it is necessary to conduct various parametric studies under real world conditions. In the future, we plan to study vehicle behavior after derailment since the interaction between vehicles has a large influence on the behavior of the vehicle after derailment.

References

• [1]  Ogura, M., “The Niigata Chuetsu Earthquake - Railway Response and Reconstruction,” Japan Railway & Transport Review, No. 43/44, pp. 46-63, March, 2006.
• [2]  Kumagai, N., “Preparing for Major Earthquakes,” Japan Railway & Transport Review, No. 60, pp. 34-41, October, 2012.
• [3]  Asanuma, K., Sogabe, M., Watanabe, T., Okayama, J. and Wakui, H., “Development of Ballasted Ladder Track Equipped with a Vehicle Guide Device,” QR (Quarterly Report of RTRI), Vol. 50, No. 4, pp. 233-240, November, 2009.
• [4]  Goto, K., Sogabe, M., Tanabe, M., Watanabe, T. and Tokunaga, M., “Simple Method for Analyzing Contact between Wheelset Members and Track Structures Using MBD,” QR (Quarterly Report of RTRI), Vol. 59, No. 1, pp. 55-62, February, 2018.
• [5]  Miyamoto, M., “Coupler of Railway Vehicle,” The Japan Society of Mechanical Engineers, Vol. 103, No. 982, pp. 595-597, September, 2000 (in Japanese).
• [6]  Miyamoto, T., Ishida, H., Sogabe, M., Kawanishi, T. and Murono, Y., “Analysis on Running Safety of Train Quaked by Seismic Motion,” RTRI Report, Vol. 21, No. 12, pp. 41-46, December, 2007 (in Japanese).
• [7]  Ansys, Inc., LS-DYNA R14 Theory Manual, 2024. https://ftp.lstc.com/anonymous/outgoing/web/ls-dyna_manuals/R14/LS-DYNA_Manual_Theory_R14.pdf
• [8]  Miyamoto, T., Ishida, H. and Matsuo, M., “Running Safety of Railway Vehicle as Earthquake Occurs,” QR (Quarterly Report of RTRI), Vol. 38, No. 3, pp. 117-112, September, 1997.
• [9]  Hayase, T. and Chonan, S., “Numerical Analysis on Buckling Train Set,” The Japan Society of Mechanical Engineers, Vol. 66, No. 646, pp. 213-221, June, 2000 (in Japanese).
• [10]  KYB Corp., Rolling Stock Equipment, p. 7,
• [11]  Railway Technical Research Institute, ed., Design Standards for Railway Structures and Commentary (Displacement Limits), Maruzen Publishing Co., Ltd., Tokyo, Japan, pp. 67-69, 2006 (in Japanese).
• [12]  Miyamoto, T., Matsumoto, N., Sogabe, M., Shimomura, T., Nishiyama, Y. and Matsuo, M., “Full-scale Experiment on Dynamics Behavior of Railway Vehicle against Heavy Track Vibration,” Journal of Environment and Engineering, Vol. 2, No. 2, pp. 419-428, March, 2007.

Authors

	Keiichi GOTO, Ph.D. Senior Researcher, Structural Mechanics Laboratory, Railway Dynamics Division Research Areas: Dynamic Interaction between Vehicle and Bridge, Vehicle Running Safety during Earthquake, Numerical Simulation
	Kohei IIDA, Ph.D. Senior Chief Researcher, Head of Vehicle Mechanics Laboratory, Railway Dynamics Division Research Areas: Numerical Simulation of Railway Vehicle, Running Safety during Earthquake, Specification of Vehicle Characteristics
	Munemasa TOKUNAGA, Ph.D. Senior Researcher, Structural Mechanics Laboratory, Railway Dynamics Division Research Areas: Bridge Dynamics, Vehicle Bridge Interaction, Seismic Design, Structural Health Monitoring