Transverse-Direction Post-Seismic Running Safety of Longitudinally Connected Ballastless Track–Continuous Girder Bridge Systems Considering Earthquake Damage State

Abstract

The transverse-direction post-seismic running safety of a longitudinally connected ballastless track-continuous girder bridge (LCBTCGB) system considering earthquake damage state (EDS) was studied. In this study, a simulation model of an LCBTCGB was established, and the post-earthquake damage law of the LCBTCGB was analyzed by selecting the ground motion that had the greatest influence from within the existing studies. The EDS of key interlayer components and the residual deformation law of each layer structure of the LCBTCGB system were defined. Subsequently, the residual deformations and EDS from the simulation model were imported into a coupled dynamic model of the train, track, and bridge. Evaluation of running safety evaluation after an earthquake was carried out with and without considering EDS, and a running safety guidance diagram for after an earthquake is provided. The results revealed that under conditions of rare earthquakes, without considering EDS, the running safety judgment after the earthquake were underestimated, and the risk increased by 13.6%. Following the designed earthquake, the running safety risk after the earthquake increased by 18.7% if EDS was not considered. The risk of the running safety index exceeding the limit did not increase linearly with earthquake intensity with and without considering EDS. When the EDS was considered, derailment coefficients and wheel axle lateral forces exceeded the safety limit value at an earthquake intensity of 0.2 g, whereas these limit values were only exceeded at an earthquake intensity of 0.3 g when EDS is ignored. When the earthquake intensity reached 0.5 g, the influence on the derailment coefficient was greater but the difference in the wheel axle lateral forces was not significant with or without considering EDS. It is suggested that EDS should be considered when post-seismic running safety of LCBTCGBs are analyzed; otherwise, it will lead to misjudgment of running safety after an earthquake.

1. Introduction

The sudden onset and destructive nature of seismic activity pose a significant threat to the running safety of high-speed railways. Such events not only endanger the safety of trains and passengers in operation but may also lead to prolonged running disruptions, severely impacting the normal functioning of railway systems. As critical lifeline engineering, high-speed railways play an essential role in post-seismic emergency response and disaster relief. Following a seismic event, key components of the LCBTCGB inevitably sustain various levels of seismic damage, including stiffness degradation and residual deformations. Assessing whether safe operations can be maintained under these damage conditions, determining how to resume operation, and identifying which types of trains are suitable for operation are crucial considerations for effective disaster response and recovery efforts.

In recent years, several scholars have investigated the running rules of LCBTCGB under seismic-induced damage, yielding valuable insights. Liu et al. [1] conducted seismic analyses on track–bridge systems affected by near-field seismic activity, finding that track irregularities induced by seismic events significantly impact derailment coefficients and lateral accelerations, with these effects intensifying as train speed increased. However, the influence on the vertical dynamic performance of high-speed trains was found to be minor. Hu et al. [2] defined levels of the damage to track by refining the values of the train operation indicator and examined the impact of track lateral displacement amplitude and train speed on the safety and stability of trains, proposing that lateral track displacement be used as an indicator for assessing structural track damage. Guo et al. [3] used a 1:10 scale train–track–bridge model and a four-array shaking table and conducted vibration tests to study the effect of near-fault vertical seismic action on train derailment when crossing high-speed rail bridges. Their results indicated that vertical seismic excitations reduced the minimum vertical force between the wheel and the rail, increasing the possibility of jump derailment. Jiang et al. [4] accounted for track irregularities and seismic randomness and developed a sample database for the dynamic effects of trains on track–bridge systems under seismic conditions. They analyzed the probabilistic characteristics of these dynamic effects and proposed standard values for the influence coefficient of track irregularities based on the probabilistic rate of assurance. Yu et al. [5] conducted running tests under seismic effects to explore the influence of train speed and the vertical component of ground motion on structural responses. Their findings showed that the vertical component of ground motion could cause vertical separation between the girder body and its bearings. Lai et al. [6] studied the effects of different types and amplitudes of lateral bridge deformations on train performance, concluding that the mutual lateral displacement of double-girder bridges had the most significant impact on train operation. Jiang et al. [7] developed a coupled dynamic model for a train–CRTS III track–bridge system. When the track fasteners’ lateral stiffness was set at 60 kN/mm and the isolation layer friction coefficients were 0.9 and 0.8, the speed limits for safe running under frequent and designed seismic events were increased to 250 km/h and 100 km/h, respectively. Mao et al. [8] conducted stochastic seismic dynamic analyses of the interaction between high-speed trains and a cable-stayed bridge in a region of seismic activity, discussing the impact of train speed and seismic intensity on train safety and reliability. Liu et al. [9] investigated the influence of void length in two different CA mortars on the dynamic performance of high-speed trains, finding that the void length threshold in both the mid-span and girder-end gaps of the CA mortar significantly decreased as train speed increased. Wu et al. [10] performed scaled indoor experiments on high-speed trains operating on bridges and explored the dynamic characteristics of a high-speed train–track–bridge system under conditions of near-fault pulse-type ground motion. The results indicated that lateral wheel–rail forces and derailment coefficients were the first to exceed safety limits. Liu et al. [11] proposed recommended deformation thresholds for girders for CRTS III track–bridge systems based on running safety and ride comfort criteria.

Lai [6] and Hu [2] analyzed train running safety at various speeds and under different lateral deformations, using the lateral deformation assumption, revealing that track lateral deformation significantly affects running safety. However, post-seismic safety assessments may overlook residual deformation and damage to tracks after seismic events [12]. For instance, the 2022 M6.9 Menyuan seismic event caused severe damage to a high-speed railway bridge, with maximum lateral displacements, out-of-plane rotations, and torsional angles of 212.6 cm, 3.1 degrees, and 19.9 degrees, respectively, resulting in significant damage to bearings and lateral blocks [13]. Seismic damage analysis indicated that residual displacement of bearings was the primary cause of track misalignment, and fastener damage was concentrated around the 2~4 fasteners near the girder ends during design and rare seismic activity, according to Tang et al. [14]. The analysis also highlighted that neglecting damage to the bridge–track system could lead to misjudgment of post-seismic train safety. Therefore, it is necessary to consider the EDS (key interlayer components’ stiffness defects within the track–bridge system) when performing analysis of post-earthquake running safety. Key interlayer components of the track–bridge system include fixed bearings, sliding bearings, sliding layers, lateral blocks, etc.

In this study, the post-earthquake damage law of LCBTCGB was analyzed. The EDS of key interlayer components was defined and the residual deformation law of each layer structure of LCBTCGB was analyzed. The residual deformations and EDS from the simulation model were imported into a coupled dynamic train–track–bridge model. An evaluation of running safety after the earthquake was carried out with and without EDS, and a running safety guidance diagram for after an earthquake is provided. The results can provide a reference and suggestions for train running safety after an earthquake.

2. Post-Seismic Damage Analysis of LCBTCGB

2.1. Simulation Model and Ground Motion Input for LCBTCGB

Based on the standard engineering diagrams for high-speed rail in regions with seismic fortification intensity of 8 degrees [15], a simulation model was developed with a longitudinally connected ballastless track–continuous girder bridge system, as shown in Figure 1. The main bridge was a (32 + 48 + 32) m double-track, single-box, single-cell variable-height continuous box girder bridge, while the approach bridges on both sides consisted of 32 m simply supported concrete box girders with single-box single-cell cross sections. The subgrade segment was 20 m long. In Figure 1, 1# to 4# represent piers, 0# and 5# represent abutments, 1F, 2F, and 3F represent fixed bearings, and 1S to 13S represent sliding bearings. The pile–soil interaction was modeled using a six-spring model at the pier base, with spring stiffness calculated using the “m” method. The spring simulated the elastic response of the soil, representing the reverse force of the soil on the pile by a certain stiffness, to more realistically reflect the support provided by the soil on the bottom of the pile.

Table 1. Parameters of key components of the LCBTCGB.

Component	Vertical Stiffness (Mpa)	Transverse Stiffness (Mpa)	Yield Point (mm)
Fixed bearing	1 × 104	33.2	2
Lateral block	1.375 × 103	238.68	2
Sliding layer	1.5 × 103	12	0.5
Friction plate	1 × 106	4.26 × 106	0.5

lists the parameters of the key components of the track, which are analyzed in the following. For detailed modeling and validation processes, refer to references [16,17,18].

From the 100 ground motions selected from the literature [19], 5 ground motions with large seismic responses were chosen for analysis. These selected ground motions are listed in

Table 2. Five selected ground motions.

Symbol	Seismic Area Name	Station Name	RSN	PGA (g)
R1	“Irpinia_Italy-01”	“Auletta”	RSN284_ITALY_A-AUL000.AT2	0.05513
R2	“Superstition Hills-02”	“Parachute Test Site”	RSN723_SUPER.B_B-PTS225.AT2	0.43182
R3	“Imperial Valley-06”	“El Centro Array #3”	RSN178_IMPVALL.H_H-E03140.AT2	0.26739
R4	“Coyote Lake”	“Gilroy Array #4”	RSN149 _COYOTELK_G04270.AT2	0.23291
R5	“Loma Prieta”	“Gilroy Array #3”	RSN767_LOMAP G03000.AT2	0.55912

. The peak ground acceleration (PGA) was adjusted according to different earthquake intensity levels, as follows: frequent earthquakes of 8 degrees were represented by 0.1 g, designed earthquake of 8 degrees were 0.3 g, and a rare earthquake of 8 degrees was 0.57 g. The seismic inputs were applied in the transverse direction. In Fugure 1, the transverse bridge direction is the y direction. This study analyzed the damage to key components of the LCBTCGB under transverse seismic excitation, providing the basis for the subsequent analysis of the coupled dynamics of the train–LCBTCGB model considering EDS.

2.2. Post-Seismic Damage State Analysis of Key Interlayer Components in LCBTCGB

Under the transverse earthquake, the damage to the key track components of the LCBTCGB mainly occurred in the fixed bearing, lateral block, sliding layer, and friction plate. Extracted from the literature review, the limit values of transverse damage for key interlayer components in the LCBTCGB are summarized in

Table 3. Damage indexes of key interlayer components in LCBTCGB (Unit: mm)

Components	Indexes	Limit Value of Transverse Damage
		Slight	Moderate	Severe	Complete
Fixed bearings	Displacement	2	4	6	8
Sliding layer	Displacement	0.5	1.0	1.5	2
Friction plate	Displacement	0.5	1.0	1.5	2
Fastener	Displacement	2	3	4	5
Lateral block	Displacement	2	3	4	5

[19,20].

(1)

Fixed Bearings

For the fixed bearings, LV-2 represents a slight damage limit value of 2, and LV-8 represents a complete damage limit value of 8. As shown in Figure 2, under frequent earthquakes of 8 degrees, the maximum residual lateral displacement of the fixed bearings was 0.27 mm, and all fixed bearing deformations remained within the linear stage. Under designed earthquakes of 8 degrees, most fixed bearings exhibited slight damage, with some showing signs of complete damage. The maximum residual lateral displacement reached 39 mm. Under rare earthquakes of 8 degrees, the majority of fixed bearings were completely damaged, with the maximum residual lateral displacement reaching 44 mm. The residual displacement values for 2F were found to be larger than those for 1F and 3F in Figure 2. This was due to the stronger support at the ends of the bridge, producing a smaller bridge response, and the relatively flexible center of the bridge, which produced larger displacement. The residual displacements of 2F and 3F of R2 as shown in Figure 2b were larger than those of R2 as shown in Figure 2c. From Figure 1, it can be seen that bearing 1F was located in the bridge abutment, and the residual of 1F showed a larger increase when the PGA increased from 0.3g to 0.57g, indicating that the transition section between the bridge and the roadbed has underwent significant damage; part of the load originally carried by the transition section was transferred to the rest of the bridge, and the redistribution of the load reduced the displacements of 2F and 3F. Compared with R1 in Figure 3b, the incremental values of residual displacements of 2F and 3F for R1 in Figure 3c are very high, which is because the frequency and amplitude of the bridge sway under high lateral seismic acceleration can lead to a large transient load, causing large deformations of the 2F and 3F bearings.

(2)

Lateral blocks

LV-2 represents the lateral blocks’ slight damage limit value of 2, and LV-5 represents a complete damage limit value of 5. As shown in Figure 3, under frequent and designed earthquakes of 8 degrees, the maximum residual lateral displacements of the lateral blocks were 0.018 mm and 0.85 mm, respectively. These displacements did not exceed the slight damage threshold, indicating that the lateral blocks remained undamaged under these conditions. Under rare earthquakes of 8 degrees, some lateral blocks at both ends of the bridge exceeded the 2 mm slight damage threshold, and a small portion exceeded the 5 mm complete damage threshold. Additionally, it was observed that damage to the lateral blocks under the rare earthquake of 8 degrees primarily occurred at the junction between the bridge ends and the transition section. This was mainly due to the shaking of the bridge caused by seismic forces, especially at the midspan. The inertial forces generated by this shaking were transmitted through the deck and structure to the bridge ends. As a result of the concentration of inertial forces at the ends, the lateral blocks at these locations bore significant impact forces, leading to their complete damage.

(3)

Sliding Layer and Friction Plates

LV-0.5 represents the sliding layer and friction plates’ slight damage limit value of 0.5, and LV-2 represents a complete damage limit value of 2. As shown in Figure 4, under frequent earthquakes of 8 degrees, the maximum residual lateral displacements of the sliding layer and friction plates were 0.024 mm and 0.004 mm, respectively, with both remaining below the slight damage thresholds. Under designed earthquakes of 8 degrees, both the sliding layer and the friction plates experienced slight damage. The maximum residual lateral displacement of the sliding layer was 1.50 mm, still within the 2 mm complete damage threshold, indicating that the sliding layer avoided complete damage. However, the friction plate’s maximum residual lateral displacement was 4.370 mm, exceeding the 2 mm complete damage threshold, meaning that the friction plate experienced complete damage under the designed earthquake of 8 degrees. Under rare earthquakes of 8 degrees, parts of the sliding layer were completely damaged, with a maximum residual lateral displacement of 15.2 mm. The friction plates were all damaged to varying degrees, and trains would need to be limited in speed through this section. Furthermore, it was observed that the damage to the sliding layer and friction plates primarily occurred at the ends of the bridge. This was mainly due to the shaking of the bridge induced by seismic forces, with the inertial forces from this shaking being transmitted through the deck and structure to the bridge ends. Consequently, the sliding layers and friction plates at these locations were subject to high-impact forces, leading to their damage.

2.3. Post-Seismic Residual Deformation Analysis of Each Structural Layer in LCBTCGB

Taking the R3 ground motions as examples, the post-seismic residual deformation of each structural layer in an LCBTCGB was analyzed, as shown in Figure 5.

As shown in Figure 5, the residual displacements of the rails, base plates, and track slabs were similar. In addition, the residual deformations between the rail and the main girder were similar in the frequent and designed earthquakes. However, with rare earthquakes, there were obvious misalignments between the rail and the end of the main girder, with the misalignments at the two ends following the R3 earthquake being 7.22 mm and 9.82 mm, respectively, indicating a potential safety hazard when the train passes between the two ends of the bridge and over the location of the transition section following a rare earthquake.

3. Coupled Dynamic Model of Train–LCBTCGB

3.1. LCBTCGB Model

According to the Lagrange equation, the dynamic equation of an LCBTCGB can be expressed as follows [21]:

$$M_B{\ddot{x}}_B+C_B{\dot{x}}_B+K_B{x}_B=Q_B$$

(1)

where $M_B$, $C_B$, $K_B$, $Q_B$ are the overall mass matrix, damping matrix, stiffness matrix, and total nodal load matrix of the LCBTCGB, and $x_B$, ${\dot{x}}_B$, and ${\ddot{x}}_B$ are the displacement, velocity, and acceleration vectors of the LCBTCGB [22].

3.2. Train Model

The schematic model of the train is illustrated in Figure 6. The train is simplified as a multi-rigid-body system operating on the track, with each train modeled with four axles. The complete train body incorporates 31 DOFs [23]. According to D’Alembert’s principle, the train dynamics equations can be expressed in matrix form as follows [22]:

$$M_c{\ddot{x}}_c+C_c{\dot{x}}_c+K_c{x}_c=Q_c$$

(2)

where $M_c$, $C_c$, $K_c$, $Q_c$ are the mass matrix, damping matrix, stiffness matrix, and external load matrix of the train, and $x_c$, ${\dot{x}}_c$, ${\ddot{x}}_c$ are the displacement, velocity and acceleration of the train.

3.3. Wheel–Rail Contact Relationship

During wheel–rail contact, normal forces and creep forces are generated. The wheel–rail contact is assumed to follow a knife-edge contact constraint, as depicted in Figure 7. The lateral clearance between the wheel and rail is 10 mm, with a lateral contact stiffness of 1.617 × 10⁷ N/m and a contact angle of 1/40 rad. The creep forces are calculated according to Kalker linear creep theory. The normal forces are computed via Hertz’s nonlinear contact theory. The Hertz nonlinear contact theory is as follows:

$$P=G^{-1.5}{\delta }_{wr}^{1.5}$$

(3)

where ${\delta }_{wr}$ is the relative displacement between the wheel and rail, $G$ is the wheel–rail contact stiffness.

Kalker’s linear creep slip theory is as follows [24]:

$$\{\begin{array}{c}{F}_x=-f_{11}{\zeta }_x\\ F_y=-f_{22}{\zeta }_y-f_{23}{\zeta }_{\phi }\\ M_z=f_{23}{\zeta }_y-f_{33}{\zeta }_{\phi }\end{array}$$

(4)

$$\{\begin{array}{l}{f}_{11}=G^{\prime }(ab)C_{11}\\ f_{22}=G^{\prime }(ab)C_{22}\\ f_{23}=G^{\prime }{(ab)}^{3/2}{C}_{23}\\ f_{33}=G^{\prime }{(ab)}^2{C}_{33}\end{array}$$

(5)

where $F_x$ is the longitudinal creep force, $F_y$ is the transverse creep force, $M_z$ is the creep force moment, $f_{11}$, $f_{22}$, $f_{23}$, $f_{33}$ are the creep coefficient values, ${\zeta }_x$ is the longitudinal creep rate, ${\zeta }_y$ is the transverse creep rate, ${\zeta }_{\phi }$ is the spin creep rate, $G^{\prime }$ is the synthetic shear modulus of the wheel–rail material, $a$ is the long semi-axis of the wheel–rail contact ellipse, $b$ is the short semi-axis of the wheel–rail contact ellipse, and $C_{11}$, $C_{22}$, $C_{23}$, $C_{33}$ represent the Kalker coefficient.

3.4. Extraction of EDS of LCBTCGB

According to the post-seismic damage analysis of the LCBTCGB, after the calculations above, the simulation model of the LCBTCGB under earthquake activity was subjected to modal analysis. A full binary file of the overall stiffness, mass, and damping matrices containing seismic damage was obtained. The HBMAT command was utilized to convert the binary file into the Harwell–Boeing format. Subsequently, it was converted into a sparse matrix via programming, and the sparse matrix and node DOF mapping file generated by the HBMAT command were imported into the coupled dynamic train–LCBTCGB model. After rearranging the nodes, the overall matrix was obtained by referring to the node DOF mapping file, the overall mass, stiffness, and damping matrix files of the simulation model. The overall mass, stiffness and damping matrices of the LCBTCGB and the post-earthquake track irregularities were input into the running model. The post-earthquake track irregularities consisted of the residual displacement of the rails and the initial irregularity of the rails.

3.5. Coupled Dynamic Train–LCBTCGB Model Considering EDS

Based on the wheel–rail contact relationship, and incorporating the dynamic equations of each subsystem, the coupled dynamic equations for the train–LCBTCGB model were formulated as follows:

Not considering EDS:

$$\{\begin{array}{c}{M}_B{\ddot{x}}_B+C_B{\dot{x}}_B+K_B{x}_B=Q_B{=Q}_{\mathrm{b}g}+F_{bl}\\ M_c{\ddot{x}}_c+C_c{\dot{x}}_c+K_c{x}_c=Q_c=Q_{cg}-F_{bl}\end{array}$$

(6)

Considering EDS:

$$\{\begin{array}{c}{M^{\prime }}_B{{\ddot{x}}^{\prime }}_B+{C^{\prime }}_B{{\dot{x}}^{\prime }}_B+{K^{\prime }}_B{x^{\prime }}_B=Q_B{=Q}_{\mathrm{b}g}+F_{bl}\\ M_c{\ddot{x}}_c+C_c{\dot{x}}_c+K_c{x}_c=Q_c=Q_{cg}-F_{bl}\end{array}$$

(7)

where ${M^{\prime }}_B$, ${C^{\prime }}_B$, ${K^{\prime }}_B$ are the overall mass, damping, and stiffness matrices of the LCBTCGB considering EDS, ${{\ddot{x}}^{\prime }}_B$, ${{\dot{x}}^{\prime }}_B$, ${x^{\prime }}_B$ are the acceleration, velocity, and displacement of LCBTCGB considering EDS, $Q_{\mathrm{b}g}$ and $Q_{cg}$ are the gravitational force vectors of the train and LCBTCGB, respectively, and $F_{bl}$ is the wheelset lateral contact force vector [22].

The dynamic response was calculated using a co-simulation approach integrating ANSYS and MATLAB. The simulation process is outlined in Figure 8, which illustrates the workflow of the ANSYS–MATLAB co-simulation. Figure 9 shows a schematic diagram of the coupled dynamic train–LCBTCGB model considering EDS.

3.6. Validation of Coupling Dynamic Model of Train-LCBTCGB Considering EDS

To validate the proposed coupled dynamic train–LCBTCGB model considering EDS, a prestressed continuous girder bridge with three spans (32 + 48 + 32 m) was constructed. The train model used was an ICE3 train. The specific parameters are shown in

Table 4. ICE3 train model parameters.

Parameter	Motor Car	Trail Car	Parameter	Motor Car	Trail Car
$m_c$ (kg)	4.8 × 104	4.4 × 104	$c_{py}$ (N/m·s)	0	0
$I_{cx}$ (kg·m2)	1.15 × 105	1 × 105	$c_{pz}$ (N/m·s)	5 × 104	5 × 104
$I_{cy}$ (kg·m2)	2.7 × 106	2.7 × 106	$k_{sx}$ (kg·m2)	2.4 × 105	2.8 × 105
$I_{cz}$ (kg·m2)	2.7 × 106	2.7 × 106	$k_{sy}$ (kg·m2)	4.8 × 105	5.6 × 105
$m_t$ (kg)	3.2 × 103	2.4 × 103	$k_{sz}$ (N/m·s)	4 × 105	3 × 105
$I_{tx}$ (kg·m2)	3.2 × 103	2.4 × 103	$c_{sx}$ (N/m·s)	1.2 × 105	1.2 × 105
$I_{ty}$ (kg·m2)	7.2 × 103	5.4 × 103	$c_{sy}$ (N/m·s)	3 × 104	2.5 × 104
$I_{tz}$ (kg·m2)	6.8 × 103	5.1 × 103	$c_{sz}$ (N/m·s)	6 × 104	6 × 104
$m_w$ (kg)	2.4 × 103	2.4 × 103	$d_1$ (m)	1.25	1.25
$I_{wx}$ (kg·m2)	1.2 × 103	1.2 × 103	$d_2$ (m)	8.6875	9
$I_{wz}$ (kg·m2)	1.2 × 103	1.2 × 103	$b_1$ (m)	1.025	1.025
$k_{px}$ (N/m)	9 × 106	1.5 × 107	$b_2$ (m)	1.025	1.025
$k_{py}$ (N/m)	3 × 106	5 × 106	$h_1$ (m)	0.36	0.83
$k_{pz}$ (N/m)	1.04 × 106	7 × 105	$h_2$ (m)	0.24	0.15
$c_{px}$ (N/m·s)	0	0	$h_3$ (m)	0.33	0.34

, consisting of eight cars per set (two motorized and six trailer cars), operating at a speed of 275 km/h. Track irregularities were assessed using the Chinese ballastless track spectrum. Since the exact value of the track irregularities described in the literature [25] was not known, some inaccuracies will inevitably have occurred when performing model validation. The mean displacements at the mid-span of the first and second spans of the bridge were compared with the results from the literature [25], as shown in Figure 10.

The calculated results from this study, as shown in Figure 10, were in overall agreement with those presented in the literature [25], thereby confirming the accuracy of the proposed dynamical model. The slight discrepancies observed can be attributed to the influence of the random nature of the initial track irregularities.

4. Post-Seismic Running Safety of LCBTCGB in the Transverse Bridge Direction, Considering and Not Considering EDS

4.1. Impact on Train Running Performance Considering and Not Considering EDS

In agreement with the Code for Design of High-Speed Railways (TB10621-2014) [26], this study adopted two main indicators, i.e., derailment coefficient and wheel axle lateral forces, to evaluate the safety of train operation. According to the specifications, the derailment coefficient limit was 0.8 and the limit value of the wheel axle lateral forces was 64 kN. LV-1.3 represents a derailment coefficient limit value of 0.8, and LV-64 represents a wheel axle lateral force limit value of 64. To illustrate the impact on train running performance considering and not considering EDS, two working conditions were analyzed. Case 1 considered only the post-seismic residual rail deformation without considering EDS (Figure 11a); in Case 2, both the post-seismic rail residual deformation and the EDS were considered (Figure 11b). The initial track irregularity was determined using the Chinese ballastless track spectrum. Figure 12 shows the derailment coefficient and wheel axle lateral forces for the first wheelset of the first carriage under R3 rare earthquakes (0.57 g). Due to misalignment between the track and the main girder at the junctions between the bridge ends and the transition sections, the derailment coefficient and wheel axle lateral forces were significantly elevated. The derailment coefficient and wheel axle lateral forces were noticeably greater in Case 2 compared with Case 1, demonstrating that failing to consider EDS following rare earthquakes (0.57 g) can lead to an underestimation of the train’s post-seismic response. This could result in misjudging the running safety after an earthquake. Moreover, since the misalignment between the track and the main girder predominantly occurred at the junctions between the bridge ends and the transition sections, these areas pose the greatest safety risk when trains pass through.

4.2. Train Running Safety Analysis Considering and Not Considering EDS

An analysis of train running safety was conducted with reference to different speeds and ground motions, considering and not considering EDS. The results are summarized in Figure 13, Figure 14 and Figure 15, which compare the maximum values of derailment coefficient and wheel axle lateral forces for various train speeds and seismic intensities.

As noted in Section 2, the LCBTCGB did not exhibit even slight damage under frequent earthquakes. Similarly, as shown in Figure 13, the train running safety index values, including the derailment coefficient and wheel axle lateral forces, did not exceed their safety limits under frequent earthquakes.

As shown in Figure 14, with a designed earthquake of 0.3 g, the train running safety index exceeded the safety limit values, and the over-limit of the train running safety index considering EDS was stronger than that without EDS. Taking the R2 ground motion as an example, when considering the EDS, the derailment coefficient exceeded the limit value at a train speed of 274 km/h, and the wheel axle lateral forces exceeded the limit value at 265 km/h. In contrast, without considering the EDS, the derailment coefficient and wheel axle lateral forces exceeded the limit values at speeds of 324 km/h and 326 km/h. Analyzed in terms of the derailment coefficient, it was found that the risk increased by 15.4%. Analyzed in terms of the wheel axle lateral forces, it was found that the risk increased by 18.7%. This analysis shows that ignoring EDS can lead to underestimation of trains’ post-seismic running risks, increasing the safety risk by a maximum of 18.7%. Therefore, when assessing trains’ post-seismic performance under designed earthquakes of 0.3 g, it is crucial to account for EDS.

As shown in Figure 15, under the rare earthquake conditions of 0.57 g, the selected ground motions resulted in the train’s running safety index exceeding the limit values. The derailment coefficient considering the EDS was stronger than that without considering the EDS, while the wheel axle lateral forces were similar with and without considering the EDS. Using the R2 ground motion as an example, considering the EDS, the derailment coefficient exceeded the limit values at a speed of 216 km/h, but without considering the EDS, the derailment coefficient exceeded the limit values at 250 km/h. Considering and without considering the EDS, the running speeds corresponding to the wheel axle lateral force overrun were 250 km/h and 253 km/h. Analyzed in terms of the derailment coefficient, it was found that the risk increased by 13.6%. Analyzed in terms of the wheel axle lateral forces, it was found that the risk increased by 1.2%. Therefore, without considering the EDS, the over-limit risk increased by a maximum of 13.6%, slightly lower than the running safety over-limit risk under the designed earthquake conditions of 0.3 g. Therefore, seismic damage in the post-earthquake evaluation of running safety for the LCBTCGB should consider EDS following rare earthquakes. However, with or without considering the EDS, the over-limit risk of the train’s running safety index did not linearly increase with the ground motion intensity. As shown in Figure 15c, when the PGA was 0.57 g, the wheel axle lateral force did not continue to increase when the running speed increased from 300 km/h to 350 km/h, because there is no significant change in the relative sliding velocity between the wheels and rails, there is no significant change in the creep force; thus, the wheel axle transverse force did not increase.

4.3. Post-Seismic Running Safety Threshold Considering and Not Considering EDS

To clarify the relationship between earthquake intensity and train running safety, the R2 ground motion with the largest response in Section 4.2 was selected. The calculation results are shown in Figure 14 and Figure 15.

As can be seen from Figure 16, when considering EDS, the derailment coefficient and wheel axle lateral forces exceeded the limit values when the earthquake intensities reached 0.2 g; the corresponding speeds were 320 km/h and 280 km/h. respectively. When EDS was not considered, the derailment coefficient and wheel axle lateral forces did not exceed the limiting values, which were exceeded only when the seismic intensity reached 0.3 g. When the earthquake intensities reached 0.5 g, the influence on the derailment coefficient overrun was greater, with or without considering EDS. With EDS, when the train speed reached 218 km/h, it exceeded the limit values, while without consideration of EDS, the derailment coefficient overrun corresponded to a speed of 266 km/h. There was little difference between the wheel axle lateral forces over the limit with and without considering EDS, and they all were over the limit at train speeds of 252 km/h and 259 km/h. Therefore, the risk of over-limit increased by a maximum of 18% for 0.5 g earthquakes without considering EDS.

To more directly reflect the relationship between different earthquake intensities and the train running safety speed after the earthquake, the post-earthquake running safety threshold considering and not considering the EDS was established, as shown in Figure 17, to provide a certain theoretical basis for the guidance of LCBTCGB traffic safety after an earthquake.

5. Conclusions

(1)

A coupled dynamic train–LCBTCGB model considering the EDS was established and verified. The calculated results were in good agreement with the literature results, verifying the accuracy of the established coupled dynamic model.

(2)

For rare earthquakes, without considering EDS, the running safety judgment after the earthquake was underestimated, and the risk increased by 13.6%. For designed earthquakes, the running safety risk after an earthquake increased by 18.7% if EDS was not considered. The risk of the train running safety index exceeding the limit did not increase linearly with the earthquake intensities when considering or not considering EDS.

(3)

When considering EDS, the derailment coefficient and wheel axle lateral force exceeded the limit value when the earthquake intensity reached 0.2 g, while when EDS was not considered, the derailment coefficient and wheel axle lateral force exceeded the limit value when the earthquake intensity reached 0.3 g. When the earthquake intensity reached 0.5 g, the influence on the derailment coefficient was greater with or without considering the EDS, but the influence on the wheel axle lateral force was not much different.

(4)

When conducting post-earthquake analysis of running safety for LCBTCGBs, it is suggested to consider the EDS.