1 Introduction

China is situated in the transitional zone between the Circum-Pacific seismic belt and the Eurasian seismic belt, characterized by active seismic activity and a wide distribution range with high seismic zone density. With the continuous expansion of high-speed railway lines nationwide, it is inevitable that these lines will traverse regions prone to frequent earthquakes [1]. The vibration and damage caused by earthquakes to high-speed railway track structures will significantly impact train operations [2, 40,41,42]. By utilizing the D’Alembert’s principle and the multibody dynamics method, the dynamic equations of the vehicle system in matrix form can be established as follows:

$${\varvec{M}}_{{\text{v}}}{\ddot{\varvec{X}}}_{{\text{v}}}+{\varvec{C}}_{{\text{v}}}{\dot{\varvec{X}}}_{{\text{v}}}+{\varvec{K}}_{{\text{v}}}{\varvec{X}}_{{\text{v}}}={\varvec{F}}_{{\text{vg}}}+{\varvec{F}}_{w},$$
(1)

where \({\varvec{M}}_{{\text{v}}}\) represents the mass matrix of the vehicle subsystem; \({\varvec{C}}_{{\text{v}}}\) and \({\varvec{K}}_{{\text{v}}}\) represent the dynamic dam** and stiffness matrices of the vehicle subsystem, respectively; \({\ddot{\varvec{X}}}_{{\text{v}}}\), \({\dot{\varvec{X}}}_{{\text{v}}}\), and \({\varvec{X}}_{{\text{v}}}\) denote the acceleration, velocity, and displacement vectors of the vehicle subsystem, respectively; \({\varvec{F}}_{{\text{vg}}}\) and \({\varvec{F}}_{{\text{w}}}\) represent the self-weight load of the vehicle system and the wheel–rail force load applied to the vehicle subsystem, respectively.

Fig. 2
figure 2

Spatial multibody train model: a side view; b front view; c top view; d sign convention of vehicle

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In the subsequent operational condition calculations, a single-car vehicle is considered, with the vehicle component parameters as indicated by Table 1. The nonlinearity of primary suspension and secondary suspension is also given in the value in brackets is the critical displacement with nonlinearity.

Table 1 Main parameters for the trains
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2.2 Track model

In this study, the CRTS II type ballastless track structure is used, and the dynamic equations of the track structure are established based on the finite element direct stiffness method. The main components and their dimensions of the track structure are shown in Fig. 3. Using the ANSYS finite element platform, the steel rail is modeled using beam elements, the track slab and baseplate are modeled using shell elements, the fasteners are modeled using discrete spring-damper elements, and the CA mortar layer and sliding layer are simulated using uniformly distributed spring-damper elements. Based on the stiffness matrix, mass matrix, and dam** matrix of the track structure, the dynamic equations of the track structure are directly derived as follows:

$${\varvec{M}}_{{\text{r}}}{\ddot{\varvec{X}}}_{{\text{r}}}+{\varvec{C}}_{{\text{r}}}{\dot{\varvec{X}}}_{{\text{r}}}+{\varvec{K}}_{{\text{r}}}{\varvec{X}}_{{\text{r}}}={\varvec{F}}_{{\text{r}}}+{\varvec{F}}_{{\text{e}}},$$
(2)

where \({\varvec{M}}_{{\text{r}}}\), \({\varvec{C}}_{{\text{r}}}\), and \({\varvec{K}}_{{\text{r}}}\) represent the overall mass, dam**, and stiffness matrices of the track subsystem, respectively; \({\ddot{\varvec{X}}}_{{\text{r}}}\), \({\dot{\varvec{X}}}_{{\text{r}}}\), and \({\varvec{X}}_{{\text{r}}}\) represent the overall acceleration, velocity, and displacement vectors of the track subsystem, respectively; \({\varvec{F}}_{{\text{r}}}\) represents the wheel–rail force load vector applied to the track structure, and \({\varvec{F}}_{{\text{e}}}\) represents the seismic load vector obtained by simulating the seismic excitation using the lumped mass method [\(\Delta Q\) and the static load per wheel \({Q}_{0}\) is given by

$$\frac{\Delta Q}{{Q}_{0}}=\frac{{Q}_{0}{-Q}_{{\text{dyn}}}}{{Q}_{0}},$$
(3)

where \({Q}_{{\text{dyn}}}\) is the dynamic vertical load of the wheel. According to the “Specifications for Whole Train Testing of High-Speed EMUs” [50], considering the limit of the dynamic wheel unloading rate, a threshold value of 0.8 is set as V1 to determine the occurrence of vertical wheel–rail separation trend, and a threshold value of 1.0 is set as V2 to determine the occurrence of actual wheel–rail separation behavior.

3.1.3 Wheel lift

For the wheel lift behavior caused by wheelset tilting, the wheel lift is used as an evaluation index. However, the previous definition of wheel lift did not differentiate between wheel climbing and wheel hop**. To accurately reflect the actual running state, it is necessary to distinguish between these two behaviors. Since the climbing behavior has already been described by the lateral displacement of the wheel–rail contact point, this study uses the wheel hop as a measure of the wheel lift caused by wheelset tilting (as shown in Fig. 8), following Ref. [51]. The expression for wheel hop is defined as follows when there is no wheel–rail contact:

$${Z}_{{\text{up}}}={Z}_{0}-{Z}_{{\text{wr}}},$$
(4)

where \({Z}_{0}\) represents the distance from the lowest point of the wheel flange to the center of the rail top surface at the initial moment, and \({Z}_{{\text{wr}}}\) represents the distance from the lowest point of the wheel flange to the center of the rail top surface at any given moment.

Fig. 8
figure 8

Definition of wheel lift

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When \({Z}_{{\text{wr}}}\) reaches 0, it reaches a critical state, known as the dangerous limit of wheel hop, denoted as V3. Based on the selected LMA wheel model and the dimensions of the CN60 rail profile used in this study, the limit is determined to be 28 mm. Additionally, to prevent the risk of vehicle overturning due to excessive tilting, an overturning limit for wheel hop, denoted as V4, is defined based on the vehicle's geometric parameters and set to 300 mm.

3.2 Discriminant domain of wheel–rail contact state

To comprehensively characterize the lateral and vertical wheel–rail contact states under seismic actions, including both before and after wheel–rail separation, the limits of wheel–rail contact safety index obtained according to Sects. 3.1.1–3.1.3 are presented in Table 3.

Table 3 Definitions of wheel–rail contact safety index limit
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Based on the wheel–rail contact safety index limits defined in Table 4, a two-dimensional coordinate system is established with the lateral displacement of the wheel–rail contact point on the x-axis and the wheel unloading rate and wheel lift on the y-axis. Different domains are delineated based on the limits of each wheel–rail contact safety index, representing distinct wheel–rail contact states. This results in the creation of a wheel–rail contact state discriminant domain, as illustrated in Fig. 9. The lower part of the x-axis indicates the contact state between the wheel and rail, so the wheel unloading rate is used for evaluation. The upper part of the x-axis indicates the vertical separation between the wheel and rail, so the wheel lift value is used for description.

Table 4 Wheel–rail contact state and characteristics under sine excitation
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Fig. 9
figure 9

Discriminant domain of wheel–rail contact state under earthquake

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3.3 Evolution characteristics of wheel–rail contact state and derailment form

To analyze the vehicle derailment forms and their wheel–rail contact state characteristics under seismic conditions, the seismic vehicle–track coupled dynamic analysis model established in Sect. 2 is adopted as the derailment analysis model. Due to the complex nature of actual seismic excitations, it is difficult to establish a direct relationship between the vehicle derailment behavior and the input excitation. However, seismic motion can be considered as the composition of sine waves of different frequencies. Therefore, sine waves with different frequencies are chosen as excitations. Considering the main frequency range of seismic events and the natural frequency range of the vehicle, a frequency range of 0.5–3.0 Hz is considered, and a 5-cycle sine wave is constructed as the excitation [52]. The vehicle speed is set uniformly at 300 km/h.

First, we focus on the variation of the wheel–rail contact state of the vehicle under sine excitation at a frequency of 0.5 Hz with increasing excitation amplitude. Here, the attention to the wheel–rail contact state is not the instantaneous contact state of a specific wheel at a particular moment during vehicle operation. Instead, it reflects the overall wheel–rail contact states (including wheel flange contact, wheel climbing, and wheel lift) that have adverse effects on the safe operation of the vehicle throughout the entire motion process. Figure 10 presents the time-history curve of the wheel–rail contact state indexes under a sine excitation with a frequency of 0.5 Hz and an amplitude of 0.15g. According to Fig. 10a, it can be observed that the wheel unloading rate varies periodically under the sine excitation, reaching a maximum value of 0.81, indicating a tendency of wheel–rail separation, primarily occurring in the wheel tread contact region. Based on Fig. 10b, it can be observed that the wheel–rail contact point continuously changes, with a maximum value of 37.5 mm, indicating that the extreme contact position occurs at the wheel flange root, without wheel flange contact occurring. The corresponding indexes of the wheel–rail contact state are as follows: H < H1, V1 < V < V2.

Fig. 10
figure 10

The time-history curve of the wheel–rail contact state indexes under sine excitation with a frequency of 0.5 Hz and an amplitude of 0.15g: a wheel unloading rate; b lateral displacement of wheel–rail contact point

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Figure 11 presents the time-history curve of the wheel–rail contact state indexes under sine excitation with a frequency of 0.5 Hz and an amplitude of 0.20g. According to Fig. 11a, it can be observed that there is wheel–rail separation, with the maximum wheel lift reaching 15 mm, and the lowest point of the wheel flange not exceeding the center of the rail head. Figure 11b shows that during the wheel lift process, the wheel–rail contact disappears. Based on the position of the wheel–rail contact point, it can be determined that the wheel lift and drop occur in the wheel tread contact area. Additionally, it can be observed that when one wheel is lifted, the maximum value of the non-lifting wheel–rail contact point reaches 38.1 mm and remains stable for a period, indicating that flange contact occurs at this time and remains in a flange contact state throughout the wheel lift period. The corresponding indexes of the wheel–rail contact state are as follows: H1 < H < H2; V2 < V < V3.

Fig. 11
figure 11

The time-history curve of the wheel–rail contact state indexes under sine excitation with a frequency of 0.5 Hz and an amplitude of 0.2g: a wheel lift; b lateral displacement of wheel–rail contact point

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Figure 12 presents the time-history curve of the wheel–rail contact state indexes under sine excitation with a frequency of 0.5 Hz and an amplitude of 0.25g. As shown in Fig. 12a, the maximum wheel lift reaches 84 mm, and the lowest point of the wheel flange significantly exceeds the center of the rail head. Based on Fig. 12b, it can be determined that both the wheel lift and drop occur in the tread area. This indicates that when the wheel experiences a significant lift and the lowest point of the wheel flange surpasses the center of the rail head, there is a notable risk of derailment. According to Fig. 12b, the maximum displacement of the wheel–rail contact point is 38.5 mm, indicating an increased trend of lateral relative motion between the wheel and rail during wheel lift, and the non-lifting side wheel gradually exhibits a climbing tendency, although it has not reached the point of wheel climb initiation. The corresponding indexes of the wheel–rail contact state are as follows: H1 < H < H2; V3 < V < V4.

Fig. 12
figure 12

The time-history curve of the wheel–rail contact state indexes under sine excitation with a frequency of 0.5 Hz and an amplitude of 0.25g: a wheel lift; b lateral displacement of wheel–rail contact point

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Figure 13 presents the time-history curve of the wheel–rail contact state indexes under sine excitation with a frequency of 0.5 Hz and an amplitude of 0.30g. As shown in Fig. 13a, the alternating wheel lift behavior on both sides intensifies significantly. From Fig. 13b, it can be observed that when the wheel lifts, there is not only contact between the wheel and rail in the tread area but also, during the lift phase, the lateral relative displacement between the wheel and rail causes the wheel to drop into the flange contact area. Eventually, in the fourth instance of wheel lift on the right side, the wheel directly derails as it drops to the outer side of the rail. Therefore, it can be seen that the derailment in this case is primarily caused by wheel lift, known as jum** derailment. The corresponding indexes of the wheel–rail contact state are as follows: H2 < H < H3; V3 < V < V4.

Fig. 13
figure 13

The time-history curve of the wheel–rail contact state indexes under sine excitation with a frequency of 0.5 Hz and an amplitude of 0.30g: a wheel lift; b lateral displacement of wheel–rail contact point

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Next, we focus on the vehicle's running condition under sine wave excitation with a frequency of 0.8 Hz. Before the excitation amplitude reaches 0.30g, the vehicle running state and the characteristics of wheel–rail contact and their variation process are similar to those at 0.5 Hz and are therefore omitted. When the excitation amplitude reaches 0.30g, Figure 14 presents the time-history curve of the vehicle running state under this excitation. According to Fig. 14a and b, there is a noticeable wheel lift behavior on both sides of the wheelset, accompanied by significant lateral relative motion between the wheel and rail. The wheels eventually fall to the wheel flange contact position, inducing wheel climbing behavior, but this phenomenon does not lead to derailment. The corresponding indexes of the wheel–rail contact state are as follows: H2 < H < H3; V3 < V < V4.

Fig. 14
figure 14

The time-history curve of the wheel–rail contact state indexes under sine excitation with a frequency of 0.8 Hz and an amplitude of 0.30g: a wheel lift; b lateral displacement of wheel–rail contact point

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Figure 15 presents the time-history curve of the vehicle running status under a sine excitation with a frequency of 0.8 Hz and an amplitude of 0.35g. According to Fig. 15a and b, compared to the scenario with an amplitude of 0.30g, as the severity of the first wheel lift behavior on the right side intensifies, the maximum wheel lift increases from 40 to 58 mm. The wheel similarly falls to the wheel flange contact area, but this time the descent position is closer to the top surface of the rail. It induces wheel climbing behavior and ultimately leads to derailment under the lateral relative motion between the wheel and rail, manifested as the jum** and climbing coupled derailment. The corresponding indexes of the wheel–rail contact state are as follows: H > H3; V3 < V < V4.

Fig. 15
figure 15

The time-history curve of the wheel–rail contact state indexes under sine excitation with a frequency of 0.8 Hz and an amplitude of 0.35g: a wheel lift; b lateral displacement of wheel–rail contact point

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Finally, focus on the vehicle running condition under sine wave excitation with a frequency of 1.0 Hz. Figure 16 presents the time-history curve of the vehicle running status under a sine excitation with a frequency of 1.0 Hz and an amplitude of 0.25g. According to Fig. 16a, it is observed that the maximum wheel unloading rate is 0.87, indicating a risk of vertical separation between the wheel and rail. Figure 16b shows that the wheel–rail contact point undergoes continuous changes with a maximum value of 38.1 mm, indicating wheel flange contact, but without the occurrence of a climbing trend. The corresponding indexes of the wheel–rail contact state are as follows: H1 < H < H2; V1 < V < V2.

Fig. 16
figure 16

The time-history curve of the wheel–rail contact state indexes under sine excitation with a frequency of 1.0 Hz and an amplitude of 0.25g: a wheel unloading rate; b lateral displacement of wheel–rail contact point

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Figure 17 presents the time-history curve of the vehicle running status under a sine excitation with a frequency of 1.0 Hz and an amplitude of 0.35g. In Fig. 17a, it is evident that the wheelset on the left side experiences wheel–rail separation, with a maximum wheel lift of 6 mm, indicating the occurrence of wheel lift behavior but not significantly pronounced. Figure 17b shows that the lateral displacement of the wheel–rail contact point reaches a maximum of 39.5 mm, reaching the initiation point of wheel climbing, signifying the onset of wheel climbing behavior. The corresponding indexes of the wheel–rail contact state are as follows: H2 < H < H3; V2 < V < V3.

Fig. 17
figure 17

The time-history curve of the wheel–rail contact state indexes under sine excitation with a frequency of 1.0 Hz and an amplitude of 0.35g: a wheel lift; b lateral displacement of wheel–rail contact point

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Figure 18 presents the time-history curve of the vehicle running status under a sine excitation with a frequency of 1.0 Hz and an amplitude of 0.40g. In Fig. 18a, it is observable that the wheels exhibit pronounced lift behavior, with the maximum wheel lift reaching 20.4 mm; however, the lowest point of the wheel flange has not surpassed the center of the rail head. Figure 18b shows that the left wheel induces wheel climbing behavior as it descends to the wheel flange contact area after lifting, and this does not result in derailment. However, the right wheel directly experiences climbing behavior and leads to derailment upon approaching the rail for the second time, defining this derailment form as climbing derailment. The corresponding indexes of the wheel–rail contact state are as follows: H > H3; V2 < V < V3.

Fig. 18
figure 18

The time-history curve of the wheel–rail contact state indexes under sine excitation with a frequency of 1.0 Hz and an amplitude of 0.40g: a wheel lift; b lateral displacement of wheel–rail contact point

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Similarly, analyzing the wheel–rail contact states at different stages of the vehicle under different frequency sine excitations, all the wheel–rail contact states during the transition from a safe running condition to a derailment condition are summarized and listed in Table 4. The numbered wheel–rail contact states in the table correspond to those shown in Fig. 9 in Sect. 3.2. Considering the limitations of space, all computed results are presented in Table 5.

Table 5 The wheel–rail contact state and derailment form at each stage under sine wave excitation
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By analyzing the types of vehicle derailment and their corresponding wheel–rail state changes based on Tables 4 and 6, three main forms of derailment can be identified under different frequency sine excitations: jum** derailment, jum** and climbing coupled derailment, and climbing derailment. The characteristics of each derailment form are described as follows:

  1. (1)

    Jum** derailment (JD) The derailment characteristics are characterized by significant wheel lift due to lateral tilting of the wheelset, accompanied by a certain degree of rail climbing tendency. Ultimately, the derailment occurs when the wheel jumps and lands in the dangerous zone. The progression of wheel–rail contact states in this type of derailment is as follows: A → B → C → D → E.

  2. (2)

    Jum** and climbing coupled derailment (JCCD) The derailment characteristics manifest as the wheel jum** while accompanied by lateral movement of the wheelset, causing the wheel to land on the wheel flange. This induces wheel rail climbing behavior, ultimately leading to derailment. The progression of wheel–rail contact states in this type of derailment is as follows: A → B → C → D → E.

  3. (3)

    Climbing derailment (CD) The derailment characteristics are observed when there is no wheel jum** or when the wheel jump occurs within the safe contact area, directly resulting in climbing derailment. The progression of vehicle operating states in this type of derailment is as follows: A → F → G.

Table 6 Derailment risk of different wheel–rail contact states in JD and JCCD
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Thus, the analysis reveals the various forms of derailment that can occur during earthquakes and the corresponding changes in wheel–rail contact states from a safe state to a derailment state.

4 Running safety risk domain under earthquake

4.1 Derailment evolution path of high-speed trains under earthquake

To establish the risk domain of vehicle derailment under seismic conditions, a key issue is to determine the magnitude of derailment risk associated with different wheel–rail contact states. To address this, based on the preconditions for different derailment forms and the derailment evolutionary path (the process of transitioning from a safe state to a derailment state with corresponding changes in wheel–rail contact states), the relative risks of derailment associated with each wheel–rail contact state are analyzed, and different risk levels are assigned to different derailment forms. Based on the analysis results in Sect. 3.3, the evolutionary paths of vehicle derailment under seismic are represented in the established domain for wheel–rail contact states, as shown in Fig. 19. The evolutionary path for JD and JCCD is depicted by the red line in Fig. 19, while the evolutionary path for CD is represented by the blue line in Fig. 19.

Fig. 19
figure 19

Derailment evolution path of high-speed train under earthquake

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4.2 Risk level and risk domain of high-speed train derailment under earthquake

To accurately assess the risk of vehicle derailment, it is necessary to further integrate the prerequisites for derailment occurrence and the variation paths of wheel–rail contact states under different derailment forms. This will help determine the risk levels associated with different wheel–rail contact states.

For derailment forms such as JD and JCCD, the prerequisites for vehicle derailment are as follows: wheel lifting causes the lowest point of the wheel flange to exceed the top center of the rail, accompanied by significant lateral relative motion between the wheel and rail, resulting in the wheel descending into the dangerous contact zone. Therefore, in these cases, derailment behavior is jointly controlled by the vertical and lateral wheel–rail contact states. By considering the variation process of wheel–rail contact states in such derailment forms, the derailment risk of each contact state is analyzed, and the relative magnitudes of derailment risk for different wheel–rail contact states are used to determine the derailment risk levels, as presented in Table 6.

For the CD, the prerequisite condition is the significant climbing behavior of the wheels. Therefore, this type of derailment is primarily controlled by the lateral wheel–rail contact state. By considering the variations in the wheel–rail contact states in this derailment form, the derailment risk of each contact state is analyzed. The relative magnitudes of the derailment risks are used to determine the derailment risk levels, which are presented in Table 7. To maintain consistency in the risk levels between the CD and the JD or JCCD, contact discrimination domains F and G are defined as risk level III and risk level IV, respectively.

Table 7 Derailment risk of different wheel–rail contact states in CD
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Through defining the risk of derailment based on different wheel–rail contact states, the map** relationship between the wheel–rail contact state and the derailment risk, as well as the map** relationship between the wheel–rail contact state and the derailment form, has been established. This ultimately leads to the establishment of the derailment risk domain under seismic conditions (as shown in Fig. 20). Based on this, accurate assessment of train safety under seismic conditions can be achieved.

Fig. 20
figure 20

Risk domain of train derailment under earthquake

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The evaluation method based on the derailment risk domain established in this study is compared with the current evaluation methods based on single index in terms of method characteristics, consideration of different derailment forms during seismic events, and assessment of the magnitude of derailment risk during seismic events (as shown in Table 8). Thus, it can be observed that the assessment method based on the derailment risk domain effectively addresses the current issues of incomplete reflection of vehicle derailment forms and wheel–rail contact states under seismic, leading to misjudgments. Simultaneously, it improves the existing methods by addressing the challenge of accurately assessing the safety margin for vehicle operation.

Table 8 Comparison of different running safety assessment methods under earthquake
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4.3 Method validation

In order to validate the effectiveness of the assessment method based on derailment risk domain in the evaluation of vehicle running safety under actual seismic excitations, it is essential to verify whether the derived vehicle derailment forms and the variation paths of wheel–rail contact states based on the derailment risk domain analysis align with the results obtained from actual dynamic simulations. To achieve this, a single CRH2 train running on the CRTS II type ballastless track structure is selected as the case study.

Three typical seismic waves are selected from PEER (https://ngawest2.berkeley.edu/, Pacific Earthquake Engineering Research Center), namely the Kobe wave, El-Centro wave, and Tian** wave, as external excitations. The acceleration time histories of these earthquake waves are shown in Fig. 21. Under different seismic motion intensities, the wheel with the maximum wheel unloading rate, wheel lift, and lateral displacement of the wheel–rail contact point is chosen as the representative in the most unfavorable state.

Fig. 21
figure 21

Acceleration time histories of earthquakes: a Kobe; b El-Centro; c Tian**

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When the Kobe wave is used as the excitation, the variation of train derailment risk with seismic motion intensity is shown in Fig. 22a. When the seismic motion intensity is less than 0.4g, the train is in wheel–rail contact state A, indicating a Safe condition. As the seismic motion intensity reaches 0.50g, it transitions to wheel–rail contact state B, corresponding to risk level I. With further increase in the seismic motion intensity to 0.60g, it enters wheel–rail contact state C, corresponding to risk level II. Within the range of 0.60g–0.70g, the wheel–rail contact state changes from C to D, and the train derailment risk increases from risk level II to risk level III. In this range, the lateral rolling behavior of the wheelset intensifies, leading to a significant increase in wheel lift, and gradually generating a wheel–rail climbing tendency in the lateral direction. When the seismic motion intensity reaches 0.75g, a significant lateral relative motion between the wheel and rail occurs, resulting in wheel–rail contact state E and entering risk level IV. At this point, there is a risk of jump derailment or jump-climb coupled derailment. Finally, when the seismic motion intensity reaches 0.80 g, the wheel–rail contact point exceeds the derailment threshold, resulting in derailment. Comparing the wheel lift before and after the train reaches the derailment state, it is observed that the wheel lift is actually smaller at the moment of derailment. This suggests that the train derailed before the maximum wheel lift was reached. Under this seismic excitation, as the seismic motion intensity increases, the wheel–rail contact state of the train mainly goes through the process of A → B → C → D → E, indicating a derailment form of JD or JCCD. Observing the actual dynamic derailment behavior of the train under this seismic excitation, it is found that the derailment is caused by the wheel of the first right wheelset falling onto the wheel flange contact area before reaching the maximum wheel lift, which induces wheel climbing and eventually leads to derailment, indicating a JCCD form.

Fig. 22
figure 22

The evolution of train derailment risk and wheel–rail contact status with seismic motion intensity: a Kobe; b El-Centro; c Tian**

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When the El-Centro wave is used as the excitation, the variation of train derailment risk with seismic motion intensity is shown in Fig. 22b. When the seismic motion intensity is less than 0.4g, the wheel–rail contact is in state A, indicating a safe condition for the train. As the seismic motion intensity ranges from 0.50g to 0.55g, the wheel–rail contact transitions to state B, corresponding to risk level I. With further increase in the seismic motion intensity within the range of 0.60g–0.75g, the wheel–rail contact state progresses from state C to D, and the derailment risk increases from risk level II to risk level III. When the seismic motion intensity increases from 0.75g to 0.80g, the tendency for wheel climbing increases significantly. The wheel–rail contact state changes from D to E, reaching risk level IV, indicating a risk of JD or JCCD. When the seismic motion intensity reaches 0.85g, the wheel–rail contact point exceeds the derailment threshold, resulting in derailment. Under this seismic excitation, as the seismic motion intensity increases, the wheel–rail contact state of the train mainly goes through the process of A → B → C → D → E, indicating a derailment form of JD or JCCD. Observing the actual dynamic derailment behavior of the train under this seismic excitation, it is found that the derailment is caused by the wheel of the first left wheelset falling onto the wheel flange contact area during the first wheel lift behavior, inducing wheel climbing and eventually leading to derailment, indicating a JCCD form.

When the Tian** wave is used as the excitation, the variation of train derailment risk with seismic motion intensity is shown in Fig. 22c. When the seismic motion intensity is less than 0.30g, the wheel–rail contact is in state A, indicating a safe condition for the train. As the seismic motion intensity reaches the range of 0.30g–0.35g, the wheel–rail contact transitions to state F, directly placing the train in risk level III. When the seismic motion intensity changes from 0.35g to the range of 0.40g, the wheel–rail contact state reaches state H, and the train enters risk level IV, indicating a risk of CD. Within the range of seismic motion intensity from 0.40g to 0.50g, as the seismic motion intensity increases, both wheel climbing behavior and wheel lift behavior intensify. Finally, when the seismic motion intensity reaches 0.50g, the wheel–rail contact point exceeds the derailment threshold, and the train derails. Under this seismic excitation, as the seismic motion intensity increases, the wheel–rail contact state of the train mainly goes through the process of A → F → G, indicating a derailment form of CD. Observing the actual dynamic derailment behavior of the train under this seismic excitation, it is found that the derailment is caused by significant climbing behavior of the first left wheelset before reaching the maximum wheel lift behavior.

Based on the above analysis, it can be seen that the assessment method based on the derailment risk domain accurately reflects the derailment forms, wheel–rail contact states, and the variation of derailment risk with seismic motion intensity under the excitations of Kobe wave, El-Centro wave, and Tian** wave. It is found that although the actual seismic waves have complex components, the train will experience specific forms of derailment behavior dominated by a certain seismic frequency component, exhibiting similar characteristics and variations in the wheel–rail contact states as under sine wave excitation. The proposed assessment method based on the derailment risk domain accurately captures the train derailment forms, wheel–rail contact states, and the variation of derailment risk with seismic motion intensity under actual seismic motion. This validates the effectiveness of the derailment risk domain-based method in assessing the running safety under seismic.

5 Conclusion

This paper proposes a running safety assessment method based on the derailment risk domain, taking into account the variations in wheel–rail contact states and the characteristics of train derailment behavior under seismic. The method is then applied to analyze the running safety during earthquakes. The main conclusions are as follows:

  1. (1)

    Under seismic conditions, three types of derailment forms occur: jum** derailment (JD), jum** and climbing coupled derailment (JCCD), and climbing derailment (CD). For JD and JCCD, the derailment behavior is controlled by both vertical and lateral wheel–rail contact states. For CD, the derailment behavior is primarily governed by the lateral wheel–rail contact state.

  2. (2)

    At a speed of 300 km/h, JD and JCCD mainly occur under low-frequency excitations (below 1.0 Hz), while CD mainly occurs under high-frequency excitations (above 1.0 Hz). When the excitation frequency exceeds 2.0 Hz, the excitation amplitude required to induce derailment significantly increases.

  3. (3)

    The assessment method based on the derailment risk domain coupled indexes that reflect the lateral and vertical wheel–rail contact states, effectively addressing the limitations of current single-index-based methods in accurately and comprehensively capturing the wheel–rail contact states under seismic conditions.

  4. (4)

    The assessment method based on the derailment risk domain reflects the characteristics of potential JD, JCCD and CD under seismic conditions. It distinguishes between the vehicle’s safe state, transitional state, and hazardous state of the train, revealing the evolutionary path of high-speed train derailment under seismic actions. This method achieves a qualitative assessment of train derailment risk during earthquakes and effectively addresses the challenge of assessing the running safety during seismic conditions, which has been difficult with current evaluation methods.

  5. (5)

    The assessment method proposed in this paper, based on the derailment risk domain, achieves only a qualitative evaluation of vehicle derailment risk under seismic conditions. Subsequent work can further investigate the probability of vehicle derailment from a statistical perspective, introducing quantitative methods for assessing derailment risk.