Dynamic Analysis of Train–Bridge Coupling System for a Long-Span Railway Suspension Bridge Subjected to Strike–Slip Fault

Abstract

Long-span railway bridges crossing active faults are more vulnerable owing to the joint combination of pulse ground motions and surface dislocation. To study the dynamic effects resulting from the coupling of long-span railway suspension bridges crossing strike–slip fault and trains, a nonlinear model in which wheel–rail contact was established based on Hertz’s nonlinear theory and Kalker creep theory. To generate the ground motions across strike–slip fault, an artificial synthetic method, which considers both the fling-step effect with a single pulse and the directivity effect with multiple pulses, is employed. The effects of fault-crossing angles (FCAs) and permanent ground rupture displacements (PGRDs) are systematically investigated based on wheel–rail dynamic (derailment coefficient, lateral wheel–rail force, and wheel–load reduction rate). Conclusions are drawn and can be applied in the practical seismic design and train running safety assessment of long-span railway suspension bridges crossing strike–slip fault.

Full text

Citation: Chen, S.; Kang, W.; Yang, J.;
Dai, S.; Zheng, S.; Jia, H. Dynamic
Analysis of Train–Bridge Coupling
System for a Long-Span Railway
Suspension Bridge Subjected to
Strike–Slip Fault. Appl. Sci. 2023,13,
10422. https://doi.org/10.3390/
app131810422
Academic Editor: Diogo Ribeiro
Received: 14 August 2023
Revised: 13 September 2023
Accepted: 14 September 2023
Published: 18 September 2023
Copyright: © 2023 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
applied
sciences
Article
Dynamic Analysis of Train–Bridge Coupling System for a
Long-Span Railway Suspension Bridge Subjected to
Strike–Slip Fault
Sijie Chen 1, Wei Kang 1,2, Jian Yang 3, Shengyong Dai 4, Shixiong Zheng 1and Hongyu Jia 1,*
1School of Civil Engineering, Southwest Jiaotong University, Chengdu 610031, China;
zhengsx@swjtu.edu.cn (S.Z.)
2China Railway First Survey and Design Institute Group Co., Ltd., Xi’an 710043, China
3Guizhou Transportation Planning Survey & Design Academe Co., Ltd., Guiyang 550081, China
4China Railway Eryuan Engineering Group Co., Ltd., Chengdu 610031, China
*Correspondence: hongyu1016@swjtu.edu.cn
Abstract:
Long-span railway bridges crossing active faults are more vulnerable owing to the joint
combination of pulse ground motions and surface dislocation. To study the dynamic effects resulting
from the coupling of long-span railway suspension bridges crossing strike–slip fault and trains, a
nonlinear model in which wheel–rail contact was established based on Hertz’s nonlinear theory and
Kalker creep theory. To generate the ground motions across strike–slip fault, an artificial synthetic
method, which considers both the fling-step effect with a single pulse and the directivity effect
with multiple pulses, is employed. The effects of fault-crossing angles (FCAs) and permanent
ground rupture displacements (PGRDs) are systematically investigated based on wheel–rail dynamic
(derailment coefficient, lateral wheel–rail force, and wheel–load reduction rate). Conclusions are
drawn and can be applied in the practical seismic design and train running safety assessment of
long-span railway suspension bridges crossing strike–slip fault.
Keywords:
train–bridge interaction; fault-crossing ground motion; train running safety; strike–slip
fault; long-span railway suspension bridge
1. Introduction
Given the intricate temporal and spatial distribution of earthquakes across fault lines,
the limited warning time, and the significant risk of earthquake-induced damage, there is
an urgent need to address the challenges related to the operational performance of bridges
during sudden seismic events and the prompt restoration of bridges post-earthquakes. Con-
sequently, there is a growing demand for enhanced seismic resilience and operational safety
of bridges. It is, therefore, essential to undertake comprehensive research on the theoretical
framework for seismic resilience of bridges in regions prone to intense seismic activity
across fault lines, as well as conduct thorough assessments of their operational safety.
While earthquakes have inflicted considerable damage on bridges crossing faults and
have the potential to result in derailments, the existing research primarily concentrates
on bridge responses. However, there is a paucity of studies investigating the impact of
train–bridge coupling vibrations. Nonetheless, researchers worldwide have made no-
table advancements in this area, yielding a range of valuable scientific research findings.
Yang et al. [1]
developed near-fault pulse seismic ductility spectra based on machine learn-
ing to investigate the influence of pulse parameters on structures in near-fault zones.
Chen et al. [2] utilized a parameter sensitivity approach to investigate the influence of the
velocity pulse period and velocity pulse count on high-pier bridges. The study involved
conducting experiments on a prototype bridge using a shaking table, followed by calibrat-
ing and validating the corresponding numerical models based on recorded experimental
Appl. Sci. 2023,13, 10422. https://doi.org/10.3390/app131810422 https://www.mdpi.com/journal/applsci

Appl. Sci. 2023,13, 10422 2 of 16
data. Ucak et al. [
3
] delved into the consequences of a fault rupture zone intersecting a
seismically isolated bridge. The research demonstrated that the response of the seismically
isolated bridge was notably affected by the position of the fault crossing along the bridge’s
span and the orientation of the fault relative to the bridge’s longitudinal direction. These
findings emphasize the criticality of incorporating fault characteristics into the design
and analysis of seismically isolated bridges, particularly in areas prone to intense seismic
activity. Zhang et al. [
4
] and Gu et al. [
5
] investigated the impact of FCAs and PGRDs on
the seismic responses of bridges. Their investigations revealed that the seismic responses of
fault-crossing bridges surpassed those of bridges located near faults, particularly in terms
of girder and bearing displacements. Moreover, the results indicated that, among the vari-
ous FCAs analyzed, a 90
◦
angle resulted in smaller pylon displacement and longitudinal
bearing displacement while leading to larger transverse bearing displacement. Lin et al. [
6
],
Yang et al. [
7
], and Zeng et al. [
8
] examined both ordinary and seismically isolated bridges
crossing strike–slip faults. Their study systematically explored various factors, including
FCA and fault-crossing location. Their consistent findings indicated that the most favorable
outcomes were generally observed when the FCA was set at 90
◦
and the fault-crossing
location was positioned at the middle span of the bridge.
It is worth noting that the majority of existing studies have primarily focused on
highway bridges crossing strike–slip fault, with relatively fewer investigations conducted
on railway bridges. However, considering the substantial influence of train–bridge inter-
action on the dynamic characteristics of high-speed railway systems, further research in
this domain becomes imperative. As previously mentioned, it is essential to account for
the influence of trains when studying bridges crossing slip faults. Given the pressing need
to ensure the safety of high-speed railway bridges and traffic, research in this area holds
great significance.
Gong et al. [
9
] employed a real-time co-simulation approach to address the train–track–
bridge dynamic interaction. Zhang et al. [
10
] considered the excitation effects of track
irregularity and earthquake action on the train–bridge dynamic interaction system. The
dynamic response of each subsystem of the vehicle and bridge was obtained, and the
vibration mechanism was analyzed. Jiang et al. [
11
] developed a finite element model
to study train–bridge interaction in a high-speed railway context. They focused on a
simply supported beam bridge with CRTS II plate and CRH2C high-speed train as the
subjects of their research. Jin et al. [
12
] and Liu et al. [
13
] took into account the randomness
of earthquakes and track irregularity and evaluated the dynamic index of trains during
and after earthquakes from a probabilistic perspective. Hu et al. [
14
] considered seismic
residual deformations and studied the influence of additional lateral irregularities of
the rail caused by the lateral defection of the girder and train speed on train safety and
stability. Zeng et al. [
15
], Zhu et al. [
16
], Zhang et al. [
17
], Xia et al. [
18
], Gao et al. [
19
],
and Wang et al. [
20
] studied the effects of the seismic wave incidence angle, seismic
wave propagation velocity, seismic intensity, and train speed on the train–bridge dynamic
interaction. Ma et al. [
21
] and Gong [
22
] investigated the influence of spatial variation on
the dynamic performance of a train passing through a bridge and pointed out that the
wave passage effect and incoherence effect should not be disregarded in this context, and
the tri-directional spatially correlated ground motions may underestimate the random
response and running safety. Qiao et al. [
23
,
24
] examined the impact of topographic
effects on the seismic response of train–bridge systems and found that topography and
the incident angle of seismic waves have a significant influence on the system’s seismic
response.
Chen et al. [25,26]
, Zhou et al. [
27
], and Yu et al. [
28
] investigated the dynamic
response of train–track–bridge coupling systems under the influence of pulse parameters
from near-fault pulse-type earthquakes. The analysis revealed that the dynamic response of
the train group is approximately linear and positively correlated with the amplitude of the
velocity pulse, and the dynamic response of the train group is negatively correlated with
the velocity pulse period. Jiang [
29
] assessed the safe running speed limit under different

Appl. Sci. 2023,13, 10422 3 of 16
seismic levels based on the wheel–rail dynamic indicators of CRH3 high-speed train and
CRTS III ballastless track deformation indicators.
In this study, a train–bridge coupling model was developed to capture the intricate
nonlinear behavior between the wheel and rail using ANSYS/LS-DYNA [
30
]. To investigate
the train–bridge dynamic behavior under the influence of earthquakes, we have adopted
the CRH2 high-speed train and a high-speed railway double-track steel truss suspension
bridge (HSRDTSTSB) across strike–slip fault ground motions (ASSFGMs). To generate the
ASSFGMs, an artificial synthetic method, which considers both the fling-step effect with a
single pulse and the directivity effect with multiple pulses, was employed. The effects of
FCAs and PGRDs were systematically investigated. This paper specifically concentrates on
the nonlinear seismic response of the train–bridge system under the combined influence
of both train load and seismic load rather than considering the train load in isolation.
As a result, the impact of variations in train speed is not discussed in this study due to
space constraints.
2. Train–Bridge System under Earthquake
The relative dislocation of strike–slip fault can result in varying or even opposite
ground motion characteristics on either side of the fault. The internal forces of a structure
are influenced not only by the ground vibration but also by the relative dislocation of
fault planes. The equations of motion can incorporate the impact of surface dislocations
on the seismic response of structures. This allows for an accurate representation of the
internal forces and residual deformation of the piers of crossing-fault bridges subjected
to ground motions. In this study, the multi-support excitation displacement input model
is used as a more appropriate ground motion input mode. The dynamic equations of the
train–track–bridge system subjected to earthquakes can be denoted as [31]
Mv
..
uv+Cv
.
uv+Kvuv=Rv(1)
Mt
..
ut+Ct
.
ut+Ktut=Rt(2)
Mb
..
ub+Cb
.
ub+Kbub=Rb+Rbg (3)
where M, C, and K represent the mass, damping, and stiffness matrices, respectively;
..
u
,
.
u
, and
u
represent the generalized acceleration vector, generalized velocity vector, and
generalized displacement vector, respectively; R represents the generalized load vector
with the subscripts v, t, b, and bg representing the train, track, bridge, and earthquake.
The vehicle Equation (1) and the track Equation (2) are coupled through the
wheel–rail
relationship, while the track Equation (2) and the bridge Equation (3) are coupled through
the bridge–rail relationship, thus forming a comprehensive and complex system of dynamic
equations. The train–track–bridge system contains strong nonlinear links such as wheel–rail
contact geometric nonlinearity and wheel–rail force nonlinearity; it is part of a complex
nonlinear dynamic system. The LS-DYNA explicit integral method [
30
] is used to simulate
the system dynamics behavior of the bridge under seismic action, which is an effective way
to analyze and evaluate the safety of trains crossing the bridge.
3. Numerical Analysis
3.1. Configurations of HSRDTSTSB
To study the train–bridge dynamic behavior of the HSRDTSTSB subjected to strike–slip
fault, a long-span HSRDTSTSB with a main span length of 1060 (m) is adopted herein. The
overall bridge has a total length of 1280 (m) with a layout of 130 (m) + 1060 (m) + 90 (m),
whose elevation layout view is shown in Figure 1a.

Appl. Sci. 2023,13, 10422 4 of 16
Appl. Sci. 2023, 13, x FOR PEER REVIEW 4 of 15
The main girder of the bridge has a total width of 30 (m) and a height of 12 (m) and
adopts a stiffening girder with a three-span continuous system. The stiffening girder is
designed as a fully floating system in the longitudinal direction, with transverse wind-
resistant bearings installed at the bridge tower to enhance stability. The cross-section of
the bridge features a double-track railway ballasted deck arrangement using a CHN60 rail
type, with a spacing of 5 (m) between the middle lines of the tracks. The cross-section
layout of the stiffening girder and ballasted railway are shown in Figure 1c,d, respectively.
The interval distances of the main cable and slings are 41 (m) and 30 (m) in the transversal
direction, respectively. The main tower is an H-type concrete structure with a height of
262.8 (m) and a longitudinal width of 41.0 (m) at both sizes shown in Figure 1b. The main
tower foundation adopts 40 bored cast-in-place piles with a diameter of 3.0 (m). The di-
mension of the cushion cap is 33.0 (m) × 29.0 (m). Two gravity-type anchorages are located
on the bank. The expanded foundation placed on the bedrock is adopted. In addition, the
ASSFGMs are displayed in Figure 1.
Figure 1. General schemata and details of ballasted railway suspension bridge: (a) Distribution of
geological conditions and schematic view of Bridge, (b) Elevation of main tower, (c) Layout of stiff-
ening girder, (d) Layout of ballasted railway, (e) Key sections of main tower (GM : Ground motion
in the up direction; GM : Ground motion in the fault-normal direction).
3.2. Bridge and Track Model
This study employs Timoshenko beam elements to model the steel truss main girder,
main tower, and sleeper, which are characterized by slender to moderately stubby beam
structures. The rail is treated as an infinitely long Euler–Bernoulli beam with discrete elas-
tic point supports, and its vertical, lateral, and rotational degrees of freedom are taken into
account. The ballast is distributed in mass blocks according to the actual distance between
the sleepers, taking into account their vertical and lateral vibrations (as shown in Figure
Figure 1.
General schemata and details of ballasted railway suspension bridge: (
a
) Distribution
of geological conditions and schematic view of Bridge, (
b
) Elevation of main tower, (
c
) Layout of
stiffening girder, (
d
) Layout of ballasted railway, (
e
) Key sections of main tower (GM
UP
: Ground
motion in the up direction; GMFN: Ground motion in the fault-normal direction).
The main girder of the bridge has a total width of 30 (m) and a height of 12 (m) and
adopts a stiffening girder with a three-span continuous system. The stiffening girder is
designed as a fully floating system in the longitudinal direction, with transverse wind-
resistant bearings installed at the bridge tower to enhance stability. The cross-section of
the bridge features a double-track railway ballasted deck arrangement using a CHN60
rail type, with a spacing of 5 (m) between the middle lines of the tracks. The cross-section
layout of the stiffening girder and ballasted railway are shown in Figure 1c,d, respectively.
The interval distances of the main cable and slings are 41 (m) and 30 (m) in the transversal
direction, respectively. The main tower is an H-type concrete structure with a height of
262.8 (m) and a longitudinal width of 41.0 (m) at both sizes shown in Figure 1b. The
main tower foundation adopts 40 bored cast-in-place piles with a diameter of 3.0 (m).
The dimension of the cushion cap is 33.0 (m)
×
29.0 (m). Two gravity-type anchorages
are located on the bank. The expanded foundation placed on the bedrock is adopted. In
addition, the ASSFGMs are displayed in Figure 1.
3.2. Bridge and Track Model
This study employs Timoshenko beam elements to model the steel truss main girder,
main tower, and sleeper, which are characterized by slender to moderately stubby beam
structures. The rail is treated as an infinitely long Euler–Bernoulli beam with discrete
elastic point supports, and its vertical, lateral, and rotational degrees of freedom are
taken into account. The ballast is distributed in mass blocks according to the actual
distance between the sleepers, taking into account their vertical and lateral vibrations (as

Appl. Sci. 2023,13, 10422 5 of 16
shown in Figure 2a,b). A link element is used to model the main cable and sling, with
the assumption that the stress is uniformly distributed across the entire element. Shell
elements are appropriate for simulating orthogonal plates, and the mesh size of the plates
should correspond to the spacing of the sleepers. The transverse pounding generated
by horizontal seismic motion significantly influences the dynamic response of bridge
structures. Consequently, there are lateral anti-wind connections between the steel truss
girders and the towers [
32
,
33
]. Figure 3shows the detailed FE model of the HSRDTSTSB.
The present study adopts the well-established properties for the connection between track
structures [
34
]. In addition, 300 m of ballastless tracks are constructed on each side of the
ballasted tracks as boundary conditions and the parameters related to Jiang [
29
]. Table 1
tabulates the material properties adopted in this study.
Appl. Sci. 2023, 13, x FOR PEER REVIEW 5 of 15
2a,b). A link element is used to model the main cable and sling, with the assumption that
the stress is uniformly distributed across the entire element. Shell elements are appropri-
ate for simulating orthogonal plates, and the mesh size of the plates should correspond to
the spacing of the sleepers. The transverse pounding generated by horizontal seismic mo-
tion significantly influences the dynamic response of bridge structures. Consequently,
there are lateral anti-wind connections between the steel truss girders and the towers
[32,33]. Figure 3 shows the detailed FE model of the HSRDTSTSB. The present study
adopts the well-established properties for the connection between track structures [34]. In
addition, 300 m of ballastless tracks are constructed on each side of the ballasted tracks as
boundary conditions and the parameters related to Jiang [29]. Table 1 tabulates the mate-
rial properties adopted in this study.
Figure 2. Ballasted track structure: (a) Transverse section of ballasted track, (b) Longitudinal section
of ballasted track, (c) Transition zone.
Figure 3. Finite element model of the bridge.
Table 1. Material properties.
LS-DYNA Model Component Parameter Value
* MAT_ELASTIC (MAT_001)
Main girder/Rail
Young’s modulus 210 GPa
Poisson’s ratio 0.3
Mass density 7850 kg/m
Main tower Young’s modulus 34.5 GPa
Poisson’s ratio 0.2
Figure 2.
Ballasted track structure: (
a
) Transverse section of ballasted track, (
b
) Longitudinal section
of ballasted track, (c) Transition zone.
Table 1. Material properties.
LS-DYNA Model Component Parameter Value
* MAT_ELASTIC (MAT_001)
Main girder/Rail
Young’s modulus 210 GPa
Poisson’s ratio 0.3
Mass density 7850 kg/m3
Main tower
Young’s modulus 34.5 GPa
Poisson’s ratio 0.2
Mass density 2420 kg/m3
Cushion cap
Young’s modulus 33.5 GPa
Poisson’s ratio 0.2
Mass density 2410 kg/m3
Foundation
Young’s modulus 31.5 GPa
Poisson’s ratio 0.2
Mass density 2390 kg/m3
* MAT_CABLE_DISCRETE_BEAM
(MAT_071)
Main cable Young’s modulus 200 GPa
Mass density 8500 kg/m3
Sling Young’s modulus 200 GPa
Mass density 8400 kg/m3
* MAT_LINEAR_ELASTIC_DISCRETE
_BEAM (MAT_066) Track
Transversal stiffness of the fastener Kph 3×107N/m
Transversal damper of the fastener Cph 6×104N·S/m
Vertical stiffness of the fastener Kpv 7.8 ×107N/m
Vertical damper of the fastener Cpv 5×104N·S/m
Transversal stiffness of the sleeper-ballast Kbh 3×107N/m
Transversal damper of the sleeper-ballast Cbh 6×104N·S/m
Vertical stiffness of the sleeper-ballast Kbv 1.2 ×107N/m
Vertical damper of the sleeper-ballast Cbv 7.5 ×104N·S/m
Stiffness of adjacent ballast Kw7.8 ×107N/m
Damper of adjacent ballast Cw8×104N·S/m
Vertical stiffness of the ballast-bridge Kfv 1.7 ×107N/m
Vertical damper of the ballast-bridge Cfv 3.1 ×104N·S/m

Appl. Sci. 2023,13, 10422 6 of 16
Appl. Sci. 2023, 13, x FOR PEER REVIEW 5 of 15
2a,b). A link element is used to model the main cable and sling, with the assumption that
the stress is uniformly distributed across the entire element. Shell elements are appropri-
ate for simulating orthogonal plates, and the mesh size of the plates should correspond to
the spacing of the sleepers. The transverse pounding generated by horizontal seismic mo-
tion significantly influences the dynamic response of bridge structures. Consequently,
there are lateral anti-wind connections between the steel truss girders and the towers
[32,33]. Figure 3 shows the detailed FE model of the HSRDTSTSB. The present study
adopts the well-established properties for the connection between track structures [34]. In
addition, 300 m of ballastless tracks are constructed on each side of the ballasted tracks as
boundary conditions and the parameters related to Jiang [29]. Table 1 tabulates the mate-
rial properties adopted in this study.
Figure 2. Ballasted track structure: (a) Transverse section of ballasted track, (b) Longitudinal section
of ballasted track, (c) Transition zone.
Figure 3. Finite element model of the bridge.
Table 1. Material properties.
LS-DYNA Model Component Parameter Value
* MAT_ELASTIC (MAT_001)
Main girder/Rail
Young’s modulus 210 GPa
Poisson’s ratio 0.3
Mass density 7850 kg/m
Main tower Young’s modulus 34.5 GPa
Poisson’s ratio 0.2
Figure 3. Finite element model of the bridge.
3.3. Train and Wheel–Rail Contact Model
The CRH2 high-speed train, comprising 8 carriages, is simulated using a mass-spring
damping system with 31 degrees of freedom (DOFs) per carriage, including the lateral,
floating, rolling, yawing, and nodding DOFs of the car-body and 2 bogies, as well as the
lateral, vertical, and yawing DOFs of the 4 wheelsets. The car body is linked to the bogie
through a secondary suspension system, while the bogie is connected to the wheelset
through a primary suspension system. Indeed, the suspension system comprises both
springs and dampers. In particular, the car body, bogies, and wheelsets can be considered
rigid bodies and ignore the coupling effect between trains (as shown in Figure 4). Details
of train parameters can be found in the work by Zhou [
27
]. The actual running speed of the
train is 250 km/h.
Appl. Sci. 2023, 13, x FOR PEER REVIEW 6 of 15
Mass density 2420 kg/m
Cushion cap
Young’s modulus 33.5 GPa
Poisson’s ratio 0.2
Mass density 2410 kg/m
Foundation
Young’s modulus 31.5 GPa
Poisson’s ratio 0.2
Mass density 2390 kg/m
* MAT_CABLE_DIS-
CRETE_BEAM (MAT_071)
Main cable Young’s modulus 200 GPa
Mass density 8500 kg/m
Sling Young’s modulus 200 GPa
Mass density 8400 kg/m
* MAT_LINEAR_ELASTIC_DIS-
CRETE
_BEAM (MAT_066)
Track
Transversal stiffness of the fastener
K

 3×10
N/m
Transversal damper of the fastener C

 6×10
N∙S/m
Vertical stiffness of the fastener
K

 7.8 × 10N/m
Vertical damper of the fastener C

 5×10
N∙S/m
Transversal stiffness of the sleeper-ballast
K

3×10
N/m
Transversal damper of the sleeper-ballast C 6×10
N∙S/m
Vertical stiffness of the sleeper-ballast
K
 1.2 × 10N/m
Vertical damper of the sleeper-ballast C 7.5 × 10N∙S/m
Stiffness of adjacent ballast
K
 7.8 × 10N/m
Damper of adjacent ballast C 8×10
N∙S/m
Vertical stiffness of the ballast-bridge
K
 1.7 × 10N/m
Vertical damper of the ballast-bridge C 3.1 × 10N∙S/m
3.3. Train and Wheel–Rail Contact Model
The CRH2 high-speed train, comprising 8 carriages, is simulated using a mass-spring
damping system with 31 degrees of freedom (DOFs) per carriage, including the lateral,
floating, rolling, yawing, and nodding DOFs of the car-body and 2 bogies, as well as the
lateral, vertical, and yawing DOFs of the 4 wheelsets. The car body is linked to the bogie
through a secondary suspension system, while the bogie is connected to the wheelset
through a primary suspension system. Indeed, the suspension system comprises both
springs and dampers. In particular, the car body, bogies, and wheelsets can be considered
rigid bodies and ignore the coupling effect between trains (as shown in Figure 4). Details
of train parameters can be found in the work by Zhou [27]. The actual running speed of
the train is 250 km/h.
Figure 4. Train and finite element model.

Appl. Sci. 2023,13, 10422 7 of 16
Locomotive vehicles experience vibrations due to track irregularities and earthquakes,
which are then transmitted to the track and bridge structures through the contact point
between the wheels and rails. This creates a dynamic interaction process within the
train–bridge system. In order to consider both the vertical and lateral wheel–rail contact
and the nonlinear characteristics of wheel–rail contact state, the Hertz’s nonlinear the-
ory [
35
] and the Kalker creep theory [
36
] are used to obtain the vertical contact stiffness of
1.325 ×109N/m [37] and the lateral contact stiffness of 1.671 ×107N/m [38].
3.4. Rail Irregularity
Track geometry irregularities serve as the primary source of excitation that alters the
dynamic interaction between the wheels and rails, leading to coupled vibrations within the
train–bridge system. This phenomenon represents a crucial factor that directly influences
the safety and comfort of train operations. The presence of track irregularities directly affects
the dynamic behavior of the wheel–rail system by impacting the geometry of the wheel–rail
contact. German Low Disturb Track Spectra are chosen as the track irregularity [39]. Four
groups of irregularities are considered for both the left and right rails, which include left
vertical, right vertical, left lateral, and right lateral irregularities (as shown in Figure 5).
Appl. Sci. 2023, 13, x FOR PEER REVIEW 7 of 15
Figure 4. Train and finite element model.
Locomotive vehicles experience vibrations due to track irregularities and earth-
quakes, which are then transmied to the track and bridge structures through the contact
point between the wheels and rails. This creates a dynamic interaction process within the
train–bridge system. In order to consider both the vertical and lateral wheel–rail contact
and the nonlinear characteristics of wheel–rail contact state, the Her’s nonlinear theory
[35] and the Kalker creep theory [36] are used to obtain the vertical contact stiffness of
1.325 × 10N/m [37] and the lateral contact stiffness of 1.671 × 10N/m [38].
3.4. Rail Irregularity
Track geometry irregularities serve as the primary source of excitation that alters the
dynamic interaction between the wheels and rails, leading to coupled vibrations within
the train–bridge system. This phenomenon represents a crucial factor that directly influ-
ences the safety and comfort of train operations. The presence of track irregularities di-
rectly affects the dynamic behavior of the wheel–rail system by impacting the geometry
of the wheel–rail contact. German Low Disturb Track Spectra are chosen as the track ir-
regularity [39]. Four groups of irregularities are considered for both the left and right rails,
which include left vertical, right vertical, left lateral, and right lateral irregularities (as
shown in Figure 5).
Figure 5. Lateral and vertical track irregularity samples: (a) Lateral track irregularity of the left rail,
(b) Lateral track irregularity of the right rail, (c) Vertical track irregularity of the left rail, (d) Vertical
track irregularity of the right rail.
3.5. Vibration Characteristics
The bridge’s modal analysis is performed using the block-Lanczos method [40]. The
first ten natural frequencies and their corresponding mode shapes are presented in Figure
6. It can be observed that most of these modes exhibit dominant transverse vibration of
the main cable due to its smaller stiffness in the transverse direction. The first-order mode
shows the symmetrical transverse vibration of the main girder, while the second-order
mode shows the symmetric transverse vibration of the main cable. The third-order mode
exhibits the antisymmetric transverse vibration of the main girder. Finally, the fifth- to
tenth-order modes primarily involve cable vibration.
Figure 5.
Lateral and vertical track irregularity samples: (
a
) Lateral track irregularity of the left rail,
(
b
) Lateral track irregularity of the right rail, (
c
) Vertical track irregularity of the left rail, (
d
) Vertical
track irregularity of the right rail.
3.5. Vibration Characteristics
The bridge’s modal analysis is performed using the block-Lanczos method [
40
]. The
first ten natural frequencies and their corresponding mode shapes are presented in Figure 6.
It can be observed that most of these modes exhibit dominant transverse vibration of the
main cable due to its smaller stiffness in the transverse direction. The first-order mode
shows the symmetrical transverse vibration of the main girder, while the second-order
mode shows the symmetric transverse vibration of the main cable. The third-order mode
exhibits the antisymmetric transverse vibration of the main girder. Finally, the fifth- to
tenth-order modes primarily involve cable vibration.

Appl. Sci. 2023,13, 10422 8 of 16
Appl. Sci. 2023, 13, x FOR PEER REVIEW 8 of 15
Figure 6. Modal frequencies and mode shapes.
3.6. Generation of Ground Motions on Both Sides of a Strike–Slip Fault
ASSFGMs are pulse-type seismic motions caused by fling step and rupture directiv-
ity effects. For strike–slip fault, the fling step effect parallel to the fault direction and the
rupture directivity effect perpendicular to the fault direction occur almost simultaneously
and should be considered at the same time. This paper uses the pulse model proposed by
Abrahamson [41] to simulate the low-frequency part and uses an artificial simulation
method based on the phase difference spectrum to synthesize the high-frequency part. At
the same time, the frequency domain method [42] is used to iteratively adjust the synthe-
sized seismic amplitude. The ground motion across faults can be obtained by superim-
posing low-frequency components to high-frequency components, and the synthesized
fling step effect pulse and directivity effect pulse are located in the parallel fault direction
and perpendicular fault direction, respectively. A record from the Landers earthquakes in
1992, with a magnitude of 7.28 and the closest distance of 1.1 km from the surface rupture
of the fault, was selected as the original near-fault ground motion for synthesizing cross-
fault ground motions. Table 2 shows the records of the Landers earthquakes. Figure 7 il-
lustrates the time histories of the synthetic ASSFGMs with different pulse characteristics.
Table 2. Actual near-fault ground-motion records.
Location Data
𝑴𝒘 Station Rupture
Distance (km) Component PGA (𝐜𝐦/𝐬𝟐) PGV (𝐜𝐦/𝐬) PGD (𝐜𝐦)
Landers, CA, USA 28 June 1992 7.2 LUC 1.1 FN 206° 702.5 64.8 64.5
FP 296° 672.8 141.4 258.1
Figure 6. Modal frequencies and mode shapes.
3.6. Generation of Ground Motions on Both Sides of a Strike–Slip Fault
ASSFGMs are pulse-type seismic motions caused by fling step and rupture directivity
effects. For strike–slip fault, the fling step effect parallel to the fault direction and the
rupture directivity effect perpendicular to the fault direction occur almost simultaneously
and should be considered at the same time. This paper uses the pulse model proposed
by Abrahamson [
41
] to simulate the low-frequency part and uses an artificial simulation
method based on the phase difference spectrum to synthesize the high-frequency part.
At the same time, the frequency domain method [
42
] is used to iteratively adjust the
synthesized seismic amplitude. The ground motion across faults can be obtained by super-
imposing low-frequency components to high-frequency components, and the synthesized
fling step effect pulse and directivity effect pulse are located in the parallel fault direction
and perpendicular fault direction, respectively. A record from the Landers earthquakes in
1992, with a magnitude of 7.28 and the closest distance of 1.1 km from the surface rupture of
the fault, was selected as the original near-fault ground motion for synthesizing cross-fault
ground motions. Table 2shows the records of the Landers earthquakes. Figure 7illustrates
the time histories of the synthetic ASSFGMs with different pulse characteristics.
Table 2. Actual near-fault ground-motion records.
Location Data MwStation Rupture
Distance (km) Component PGA (cm/s2)PGV (cm/s) PGD (cm)
Landers, CA, USA 28 June 1992 7.2 LUC 1.1
FN 206◦702.5 64.8 64.5
FP 296◦672.8 141.4 258.1

Appl. Sci. 2023,13, 10422 9 of 16
Appl. Sci. 2023, 13, x FOR PEER REVIEW 9 of 15
Figure 7. Time history of synthetic cross-fault ground motions with different pulse characteristics:
(a–c) Fling-step effect with PGRDs of 0 m, 0.4 m, 0.8 m, 1.2 m, and 1.6 m, respectively; (d–f) Forward
directivity effect. Note: FS and FD represent fling-step effect and forward directivity effect, respec-
tively.
4. Results and Discussion
The effect of FCAs, defined as an intersection angle between the bridge axial and fault
strike–slip directions, is examined in this section. Additionally, the PGRDs are considered
from 15°, 45°, 90°, 135°, and 160° and decomposed along these angles and input along the
two orthogonal axes of the bridge.
4.1. Dynamic Response of Bridge
Figure 8 displays the peak displacements in the middle of the span and shear force
at the boom of the tower with the variation of FCAs and PGRDs. As seen from each row
view, the responses of interest vary with FCAs under different PGRDs. It should be noted
that when FCA = 15°, the movements on both sides of the sinistral strike–slip fault plane
cause the two towers to move backward along the direction of the bridge axis. Yet, when
FCA = 165°, the opposite is found in the present study (as shown in Figure 9). Moreover,
it is also very noteworthy that the peak response of the structure is almost symmetric in
Figure 8, although the FCAs vary from 15° to 165° in a symmetrical way because of the
symmetry of the examined suspension bridge.
In Figure 8a, it can be seen that the transverse displacements of the midspan increase
as the FCA increases from 15° to 90° and decrease as the FCA increases from 90° to 165°.
Yet, the vertical displacement has the opposite distribution in Figure 8b. A similar phe-
nomenon is found in shear force at the boom of the tower in Figure 8c,d. That is, there is
a greater response in the transverse direction at 90°, while vertical displacement and lon-
gitudinal shear force are more sensitive at 15° and 165°.
The displacement response demands of the long-span suspension bridge subjected
to strike–slip fault is comprehensively determined by these deformations of soil layers,
pier deformation, tower, and girder. For instance, the deformations from three soil layers
with a depth of 270 m and tower are responsible for the displacement responses on the
top of the 2# tower on Chengdu Bank. Meanwhile, the maximum transverse displacement
of the midspan and tower top does not occur at FCA = 90° but rather at FCA = 165° because
of the asymmetry of the bridge structure and valley terrain and the characteristics of sin-
istral strike–slip fault.
Figure 7.
Time history of synthetic cross-fault ground motions with different pulse characteristics:
(
a
–
c
) Fling-step effect with PGRDs of 0 m, 0.4 m, 0.8 m, 1.2 m, and 1.6 m, respectively; (
d
–
f
) For-
ward directivity effect. Note: FS and FD represent fling-step effect and forward directivity effect,
respectively.
4. Results and Discussion
The effect of FCAs, defined as an intersection angle between the bridge axial and fault
strike–slip directions, is examined in this section. Additionally, the PGRDs are considered
from 15
◦
, 45
◦
, 90
◦
, 135
◦
, and 160
◦
and decomposed along these angles and input along the
two orthogonal axes of the bridge.
4.1. Dynamic Response of Bridge
Figure 8displays the peak displacements in the middle of the span and shear force at
the bottom of the tower with the variation of FCAs and PGRDs. As seen from each row
view, the responses of interest vary with FCAs under different PGRDs. It should be noted
that when FCA = 15
◦
, the movements on both sides of the sinistral strike–slip fault plane
cause the two towers to move backward along the direction of the bridge axis. Yet, when
FCA = 165
◦
, the opposite is found in the present study (as shown in Figure 9). Moreover,
it is also very noteworthy that the peak response of the structure is almost symmetric in
Figure 8, although the FCAs vary from 15
◦
to 165
◦
in a symmetrical way because of the
symmetry of the examined suspension bridge.
Appl. Sci. 2023, 13, x FOR PEER REVIEW 10 of 15
Figure 8. Peak responses with different FCAs: (a) displacement of midspan in transverse direction;
(b) displacement of midspan in vertical direction; (c) shear force of pier boom in longitudinal di-
rection; (d) shear force of pier boom in transverse direction.
Figure 9. Sketch of tower movement with different FCAs: (a) FCA = 15°, (b) FCA = 90°, (c) FCA =
165°.
As was found during recent major strong earthquakes, the PGRDs induced by sinis-
tral strike–slip fault varied from tens of centimeters to several meters [4,43]. The faulting-
induced dislocation, which affects the responses of fault-crossing bridges through the
quasi-static term in the vibration equation of structures in nature, is a crucial factor in
influencing the seismic behavior of bridges in the immediate vicinity of active seismic
faults. Therefore, it is necessary to study the influence of PGRDs on the responses of long-
span suspension railway bridges crossing faults. In addition, FCA = 15° and FCA = 90° are
two extreme cases for fault-crossing bridges. To save space, these two extreme cases of
FCA = 15° and FCA = 90° are considered herein to study the influence of PGRDs on the
responses of long-span suspension railway bridge crossing faults.
As depicted in Figure 10a,c, the transverse displacement of the midspan barely
changes with the increase in PGRD, which is aributed to the symmetry of the bridge. The
transverse displacement on both sides of the midspan is more sensitive to the increase in
PGRD. Particularly when vertically crossing the fault, the displacement at the edge span
is almost equivalent to the size of the PGRD. As presented in Figure 10b,d, the vertical
displacement of the girder is more sensitive to longitudinal ground motion and almost
barely changes in transverse ground motion. Regarding the vertical displacement of the
girder, it increases throughout the entire girder with the increment in PGRD. This can be
aributed to the fact that longitudinal ground motion has a significant impact on the ver-
tical displacement of the girder, and when the FCA = 15°, the ground motion acts as a
longitudinal load.
Figure 10. Peak displacement in girder: (a) transverse direction at FCA = 15°; (b) vertical direction
at FCA = 15°; (c) transverse direction at FCA = 90°; (d) vertical direction at FCA = 90°.
Figure 8.
Peak responses with different FCAs: (
a
) displacement of midspan in transverse direction;
(
b
) displacement of midspan in vertical direction; (
c
) shear force of pier bottom in longitudinal
direction; (d) shear force of pier bottom in transverse direction.
In Figure 8a, it can be seen that the transverse displacements of the midspan increase as
the FCA increases from 15
◦
to 90
◦
and decrease as the FCA increases from 90
◦
to 165
◦
. Yet,
the vertical displacement has the opposite distribution in Figure 8b. A similar phenomenon
is found in shear force at the bottom of the tower in Figure 8c,d. That is, there is a greater
response in the transverse direction at 90
◦
, while vertical displacement and longitudinal
shear force are more sensitive at 15◦and 165◦.

Appl. Sci. 2023,13, 10422 10 of 16
Appl. Sci. 2023, 13, x FOR PEER REVIEW 10 of 15
Figure 8. Peak responses with different FCAs: (a) displacement of midspan in transverse direction;
(b) displacement of midspan in vertical direction; (c) shear force of pier boom in longitudinal di-
rection; (d) shear force of pier boom in transverse direction.
Figure 9. Sketch of tower movement with different FCAs: (a) FCA = 15°, (b) FCA = 90°, (c) FCA =
165°.
As was found during recent major strong earthquakes, the PGRDs induced by sinis-
tral strike–slip fault varied from tens of centimeters to several meters [4,43]. The faulting-
induced dislocation, which affects the responses of fault-crossing bridges through the
quasi-static term in the vibration equation of structures in nature, is a crucial factor in
influencing the seismic behavior of bridges in the immediate vicinity of active seismic
faults. Therefore, it is necessary to study the influence of PGRDs on the responses of long-
span suspension railway bridges crossing faults. In addition, FCA = 15° and FCA = 90° are
two extreme cases for fault-crossing bridges. To save space, these two extreme cases of
FCA = 15° and FCA = 90° are considered herein to study the influence of PGRDs on the
responses of long-span suspension railway bridge crossing faults.
As depicted in Figure 10a,c, the transverse displacement of the midspan barely
changes with the increase in PGRD, which is aributed to the symmetry of the bridge. The
transverse displacement on both sides of the midspan is more sensitive to the increase in
PGRD. Particularly when vertically crossing the fault, the displacement at the edge span
is almost equivalent to the size of the PGRD. As presented in Figure 10b,d, the vertical
displacement of the girder is more sensitive to longitudinal ground motion and almost
barely changes in transverse ground motion. Regarding the vertical displacement of the
girder, it increases throughout the entire girder with the increment in PGRD. This can be
aributed to the fact that longitudinal ground motion has a significant impact on the ver-
tical displacement of the girder, and when the FCA = 15°, the ground motion acts as a
longitudinal load.
Figure 10. Peak displacement in girder: (a) transverse direction at FCA = 15°; (b) vertical direction
at FCA = 15°; (c) transverse direction at FCA = 90°; (d) vertical direction at FCA = 90°.
Figure 9.
Sketch of tower movement with different FCAs: (
a
) FCA = 15
◦
, (
b
) FCA = 90
◦
, (
c
) FCA = 165
◦
.
The displacement response demands of the long-span suspension bridge subjected to
strike–slip fault is comprehensively determined by these deformations of soil layers, pier
deformation, tower, and girder. For instance, the deformations from three soil layers with
a depth of 270 m and tower are responsible for the displacement responses on the top of
the 2# tower on Chengdu Bank. Meanwhile, the maximum transverse displacement of the
midspan and tower top does not occur at FCA = 90
◦
but rather at FCA = 165
◦
because of
the asymmetry of the bridge structure and valley terrain and the characteristics of sinistral
strike–slip fault.
As was found during recent major strong earthquakes, the PGRDs induced by sinistral
strike–slip fault varied from tens of centimeters to several meters [
4
,
43
]. The faulting-
induced dislocation, which affects the responses of fault-crossing bridges through the
quasi-static term in the vibration equation of structures in nature, is a crucial factor in
influencing the seismic behavior of bridges in the immediate vicinity of active seismic
faults. Therefore, it is necessary to study the influence of PGRDs on the responses of long-
span suspension railway bridges crossing faults. In addition, FCA = 15
◦
and
FCA = 90◦
are two extreme cases for fault-crossing bridges. To save space, these two extreme cases of
FCA = 15
◦
and FCA = 90
◦
are considered herein to study the influence of PGRDs on the
responses of long-span suspension railway bridge crossing faults.
As depicted in Figure 10a,c, the transverse displacement of the midspan barely changes
with the increase in PGRD, which is attributed to the symmetry of the bridge. The transverse
displacement on both sides of the midspan is more sensitive to the increase in PGRD.
Particularly when vertically crossing the fault, the displacement at the edge span is almost
equivalent to the size of the PGRD. As presented in Figure 10b,d, the vertical displacement
of the girder is more sensitive to longitudinal ground motion and almost barely changes in
transverse ground motion. Regarding the vertical displacement of the girder, it increases
throughout the entire girder with the increment in PGRD. This can be attributed to the fact
that longitudinal ground motion has a significant impact on the vertical displacement of
the girder, and when the FCA = 15◦, the ground motion acts as a longitudinal load.
Appl. Sci. 2023, 13, x FOR PEER REVIEW 10 of 15
Figure 8. Peak responses with different FCAs: (a) displacement of midspan in transverse direction;
(b) displacement of midspan in vertical direction; (c) shear force of pier boom in longitudinal di-
rection; (d) shear force of pier boom in transverse direction.
Figure 9. Sketch of tower movement with different FCAs: (a) FCA = 15°, (b) FCA = 90°, (c) FCA =
165°.
As was found during recent major strong earthquakes, the PGRDs induced by sinis-
tral strike–slip fault varied from tens of centimeters to several meters [4,43]. The faulting-
induced dislocation, which affects the responses of fault-crossing bridges through the
quasi-static term in the vibration equation of structures in nature, is a crucial factor in
influencing the seismic behavior of bridges in the immediate vicinity of active seismic
faults. Therefore, it is necessary to study the influence of PGRDs on the responses of long-
span suspension railway bridges crossing faults. In addition, FCA = 15° and FCA = 90° are
two extreme cases for fault-crossing bridges. To save space, these two extreme cases of
FCA = 15° and FCA = 90° are considered herein to study the influence of PGRDs on the
responses of long-span suspension railway bridge crossing faults.
As depicted in Figure 10a,c, the transverse displacement of the midspan barely
changes with the increase in PGRD, which is aributed to the symmetry of the bridge. The
transverse displacement on both sides of the midspan is more sensitive to the increase in
PGRD. Particularly when vertically crossing the fault, the displacement at the edge span
is almost equivalent to the size of the PGRD. As presented in Figure 10b,d, the vertical
displacement of the girder is more sensitive to longitudinal ground motion and almost
barely changes in transverse ground motion. Regarding the vertical displacement of the
girder, it increases throughout the entire girder with the increment in PGRD. This can be
aributed to the fact that longitudinal ground motion has a significant impact on the ver-
tical displacement of the girder, and when the FCA = 15°, the ground motion acts as a
longitudinal load.
Figure 10. Peak displacement in girder: (a) transverse direction at FCA = 15°; (b) vertical direction
at FCA = 15°; (c) transverse direction at FCA = 90°; (d) vertical direction at FCA = 90°.
Figure 10.
Peak displacement in girder: (
a
) transverse direction at FCA = 15
◦
; (
b
) vertical direction at
FCA = 15◦; (c) transverse direction at FCA = 90◦; (d) vertical direction at FCA = 90◦.
As shown in Figure 11, when the PGRD increases, the longitudinal and transverse
shear forces remain almost unchanged, indicating that the increase in PGRD has little effect
on the internal forces of the tower. However, for the pier bottom, the increase in PGRD is

Appl. Sci. 2023,13, 10422 11 of 16
still relatively sensitive. When FCA = 15
◦
, as PGRD increases, the shear force increases by
20%, while when FCA = 90◦, it increases by 88%.
Appl. Sci. 2023, 13, x FOR PEER REVIEW 11 of 15
As shown in Figure 11, when the PGRD increases, the longitudinal and transverse
shear forces remain almost unchanged, indicating that the increase in PGRD has lile ef-
fect on the internal forces of the tower. However, for the pier boom, the increase in PGRD
is still relatively sensitive. When FCA = 15°, as PGRD increases, the shear force increases
by 20%, while when FCA = 90°, it increases by 88%.
Figure 11. Peak shear force in tower: (a) longitudinal direction at FCA = 15°; (b) transverse direction
at FCA = 15°; (c) longitudinal direction at FCA = 90°; (d) transverse direction at FCA = 90°.
4.2. Dynamic Response of Train
Due to the nonlinear behavior of the wheel–rail contact and the randomness of seis-
mic motion, coupled with the superposition of low-frequency pulse components and
high-frequency impulsive components in fault seismic motion, the seismic random anal-
ysis of the train–bridge coupled system is highly complex. To unveil the correlation be-
tween the dynamic indicators of the train and FCAs and PGRDs, the assumption is made
that the earthquake occurs simultaneously while the train is on the bridge. The scenario
before the complete formation of PGRD is defined as case 1, while the scenario after the
complete formation of PGRD is defined as case 2.
The Code for Design of High Speed Railways [44] and the High-Speed Test Train
Power Car Strength Dynamic Performance Specifications [45] of China offer three indica-
tors to assess responses of train wheel–rails: 1) the maximum lateral wheel–rail force Q,
where Q≤10+P
/3 and P
 represents the static wheel load on the load-reduction side;
2) the derailment coefficient Q/P, where Q/P ≤ 0.8 and P denotes the maximum verti-
cal force of the wheel–rail; 3) the wheel-load-reduction rate ∆P/P
, where ∆P/P
≤0.6,
∆P refers to the reduction in wheel load on the load-reduction side, and P
 stands for
the average static wheel load. In this study, these indicators were employed to evaluate
the running safety of the high-speed train.
When the FCA is 90°, the fling-step effect is primarily characterized by lateral excita-
tion. In Figure 12, it is clear that lateral excitation is fatal to train derailment, and all three
indicators show a positive correlation with the PGRD. When the bridge crosses the fault
at a small angle, the fling-step effect is primarily characterized by longitudinal excitation.
However, longitudinal excitation has a minimal impact on train derailment. Nevertheless,
there is no clear tendency between the derailment coefficient and PGRD, which suggests
the intricate nature of the train–bridge interaction.
During fault dislocation, the fault seismic motion is primarily dominated by low-fre-
quency components, which control the internal forces and displacements of the bridge.
Consequently, trains have the potential to pass safely during this phase. Figure 12 illus-
trates that when the bridge crosses the fault at a small angle, all three indicators (the dy-
namic indicators of the train) remain below the specified limit. This suggests that trains
can safely traverse the bridge under such conditions. Similarly, when the bridge is ori-
ented vertically across the fault, it is still safe for trains to pass as long as the PGRD does
not exceed 0.4 m.
Figure 11.
Peak shear force in tower: (
a
) longitudinal direction at FCA = 15
◦
; (
b
) transverse direction
at FCA = 15◦; (c) longitudinal direction at FCA = 90◦; (d) transverse direction at FCA = 90◦.
4.2. Dynamic Response of Train
Due to the nonlinear behavior of the wheel–rail contact and the randomness of seismic
motion, coupled with the superposition of low-frequency pulse components and high-
frequency impulsive components in fault seismic motion, the seismic random analysis of
the train–bridge coupled system is highly complex. To unveil the correlation between the
dynamic indicators of the train and FCAs and PGRDs, the assumption is made that the
earthquake occurs simultaneously while the train is on the bridge. The scenario before the
complete formation of PGRD is defined as case 1, while the scenario after the complete
formation of PGRD is defined as case 2.
The Code for Design of High Speed Railways [
44
] and the High-Speed Test Train Power
Car Strength Dynamic Performance Specifications [
45
] of China offer three indicators to
assess responses of train wheel–rails: (1) the maximum lateral wheel–rail force
Q
, where
Q≤
10
+P0/
3 and
P0
represents the static wheel load on the load-reduction side; (2) the
derailment coefficient
Q/P
, where
Q/P ≤
0.8 and
P
denotes the maximum vertical force of
the wheel–rail; (3) the wheel-load-reduction rate
∆P/Pm
, where
∆P/Pm≤
0.6,
∆P
refers
to the reduction in wheel load on the load-reduction side, and
Pm
stands for the average
static wheel load. In this study, these indicators were employed to evaluate the running
safety of the high-speed train.
When the FCA is 90
◦
, the fling-step effect is primarily characterized by lateral excita-
tion. In Figure 12, it is clear that lateral excitation is fatal to train derailment, and all three
indicators show a positive correlation with the PGRD. When the bridge crosses the fault
at a small angle, the fling-step effect is primarily characterized by longitudinal excitation.
However, longitudinal excitation has a minimal impact on train derailment. Nevertheless,
there is no clear tendency between the derailment coefficient and PGRD, which suggests
the intricate nature of the train–bridge interaction.
Appl. Sci. 2023, 13, x FOR PEER REVIEW 12 of 15
Figure 12. Peak wheel–rail responses with different FCAs and PGRDs in case 1: (a) derailment coef-
ficient; (b) wheel-load-reduction rate; (c) lateral wheel–rail force.
Due to space limitations, only representative scenarios of the bridge crossing the fault
at a small angle (FCA = 15°) and vertically crossing the fault (FCA = 90°) are selected for
the comparison of case 1 and case 2, as shown in Figures 13 and 14. In Figures 13 and 14,
it can be observed that the PGRD generation phase (case 1) accounts for only 40% to 80%
of the influence on the wheel–rail dynamic indicators. After the formation of PGRD (case
2), the train is simultaneously affected by additional track irregularities and the high-fre-
quency components of fault seismic motion. This results in a 31% to 48% increase in the
derailment coefficient, a 45% to 142% increase in the wheel load reduction rate, and a 50%
to 103% increase in the lateral wheel–rail force. It is also noteworthy that case 1 occurs
when the PGRD is about to be fully formed, while case 2 occurs when the train is about to
exit the bridge. This indicates that the train’s dynamic indicators while traveling on large-
span bridges are significantly reduced when the train passes through the bridge before
the complete formation of PGRD. However, the wheel–rail contact is a dynamic accumu-
lation process, and under the combined action of additional track irregularities caused by
PGRD and high-frequency components of fault ground motion, the wheel–rail dynamic
indicators will continue to increase. That is to say, if the train cannot leave the bridge be-
fore the PGRD is fully formed, the risk of derailment will significantly increase.
Figure 13. Time histories of wheel–rail responses at FCA = 15°: (a–e) derailment coefficient; (f–j)
wheel-load-reduction rate; (k–o) lateral wheel–rail force.
Figure 12.
Peak wheel–rail responses with different FCAs and PGRDs in case 1: (
a
) derailment
coefficient; (b) wheel-load-reduction rate; (c) lateral wheel–rail force.

Appl. Sci. 2023,13, 10422 12 of 16
During fault dislocation, the fault seismic motion is primarily dominated by low-
frequency components, which control the internal forces and displacements of the bridge.
Consequently, trains have the potential to pass safely during this phase. Figure 12 illustrates
that when the bridge crosses the fault at a small angle, all three indicators (the dynamic
indicators of the train) remain below the specified limit. This suggests that trains can safely
traverse the bridge under such conditions. Similarly, when the bridge is oriented vertically
across the fault, it is still safe for trains to pass as long as the PGRD does not exceed 0.4 m.
Due to space limitations, only representative scenarios of the bridge crossing the fault
at a small angle (FCA = 15
◦
) and vertically crossing the fault (FCA = 90
◦
) are selected for the
comparison of case 1 and case 2, as shown in Figures 13 and 14. In Figures 13 and 14, it can
be observed that the PGRD generation phase (case 1) accounts for only 40% to 80% of the
influence on the wheel–rail dynamic indicators. After the formation of PGRD (case 2), the
train is simultaneously affected by additional track irregularities and the high-frequency
components of fault seismic motion. This results in a 31% to 48% increase in the derailment
coefficient, a 45% to 142% increase in the wheel load reduction rate, and a 50% to 103%
increase in the lateral wheel–rail force. It is also noteworthy that case 1 occurs when the
PGRD is about to be fully formed, while case 2 occurs when the train is about to exit the
bridge. This indicates that the train’s dynamic indicators while traveling on large-span
bridges are significantly reduced when the train passes through the bridge before the
complete formation of PGRD. However, the wheel–rail contact is a dynamic accumulation
process, and under the combined action of additional track irregularities caused by PGRD
and high-frequency components of fault ground motion, the wheel–rail dynamic indicators
will continue to increase. That is to say, if the train cannot leave the bridge before the PGRD
is fully formed, the risk of derailment will significantly increase.
Appl. Sci. 2023, 13, x FOR PEER REVIEW 12 of 15
Figure 12. Peak wheel–rail responses with different FCAs and PGRDs in case 1: (a) derailment coef-
ficient; (b) wheel-load-reduction rate; (c) lateral wheel–rail force.
Due to space limitations, only representative scenarios of the bridge crossing the fault
at a small angle (FCA = 15°) and vertically crossing the fault (FCA = 90°) are selected for
the comparison of case 1 and case 2, as shown in Figures 13 and 14. In Figures 13 and 14,
it can be observed that the PGRD generation phase (case 1) accounts for only 40% to 80%
of the influence on the wheel–rail dynamic indicators. After the formation of PGRD (case
2), the train is simultaneously affected by additional track irregularities and the high-fre-
quency components of fault seismic motion. This results in a 31% to 48% increase in the
derailment coefficient, a 45% to 142% increase in the wheel load reduction rate, and a 50%
to 103% increase in the lateral wheel–rail force. It is also noteworthy that case 1 occurs
when the PGRD is about to be fully formed, while case 2 occurs when the train is about to
exit the bridge. This indicates that the train’s dynamic indicators while traveling on large-
span bridges are significantly reduced when the train passes through the bridge before
the complete formation of PGRD. However, the wheel–rail contact is a dynamic accumu-
lation process, and under the combined action of additional track irregularities caused by
PGRD and high-frequency components of fault ground motion, the wheel–rail dynamic
indicators will continue to increase. That is to say, if the train cannot leave the bridge be-
fore the PGRD is fully formed, the risk of derailment will significantly increase.
Figure 13. Time histories of wheel–rail responses at FCA = 15°: (a–e) derailment coefficient; (f–j)
wheel-load-reduction rate; (k–o) lateral wheel–rail force.
Figure 13.
Time histories of wheel–rail responses at FCA = 15
◦
: (
a
–
e
) derailment coefficient;
(f–j) wheel-load-reduction rate; (k–o) lateral wheel–rail force.

Appl. Sci. 2023,13, 10422 13 of 16
Appl. Sci. 2023, 13, x FOR PEER REVIEW 13 of 15
Figure 14. Time histories of wheel–rail responses at FCA = 90°: (a–e) derailment coefficient; (f–j)
wheel-load-reduction rate; (k–o) lateral wheel–rail force (note: the time of occurrence is depicted on
the lower horizontal axis for the peak value of case 1, while the time of occurrence is depicted on
the upper horizontal axis for the peak value of case 2).
5. Conclusions
This study provides a beer understanding of the train–bridge interaction of long-
span railway suspension bridges subjected to strike–slip fault. This study presents a de-
tailed discussion of a nonlinear model of wheel–rail contact based on ANSYS/LS-DYNA
software. A refined finite element model of train–bridge coupling system was established
based on the nonlinear finite element software of ANSYS/LS-DYNA. The findings of this
study can provide guidance for the seismic design and train running of long-span railway
suspension bridges crossing faults and help improve the safety and efficiency of the rail-
way transportation system. Therefore, the following conclusions are drawn:
(1) For long-span suspension bridges crossing left-lateral strike–slip faults, it is crucial
to emphasize the substantial impact of the fling-step effect on PGRD, leading to am-
plified bridge responses. This effect is particularly pronounced in the transverse re-
sponse at FCA = 90° and significantly influences the longitudinal response at lower
FCA values. Consequently, PGRD plays a pivotal role in seismic bridge responses,
with a direct correlation between the magnitude of the response and PGRD.
(2) It is imperative to underscore the critical consequences of lateral seismic motion dur-
ing fault dislocation, which profoundly affects the dynamic parameters of trains. In
contrast, the influence of longitudinal seismic motion on the derailment coefficient is
relatively minor. However, as seismic motion persists, longitudinal ground motion
can lead to vertical displacements in the main beam, potentially causing the wheel-
load-reduction rate to surpass permissible limits.
(3) Within the context of strike–slip faults, it is worth noting that PGRD resulting from
fault slips is predominantly governed by the low-frequency components of seismic
waves, accounting for approximately 70% of its impact on train dynamic parameters.
This underscores the necessity for trains to traverse large-span bridges before PGRD
reaches its full formation to significantly mitigate dynamic indicators.
(4) Continuing to emphasize the importance of timing, if trains persist in traversing the
bridge after the complete formation of PGRD, they will encounter a compounded
effect. This effect combines additional track irregularities induced by PGRD with the
Figure 14.
Time histories of wheel–rail responses at FCA = 90
◦
: (
a
–
e
) derailment coefficient;
(
f
–
j
) wheel-load-reduction rate; (
k
–
o
) lateral wheel–rail force (note: the time of occurrence is depicted
on the lower horizontal axis for the peak value of case 1, while the time of occurrence is depicted on
the upper horizontal axis for the peak value of case 2).
5. Conclusions
This study provides a better understanding of the train–bridge interaction of long-
span railway suspension bridges subjected to strike–slip fault. This study presents a
detailed discussion of a nonlinear model of wheel–rail contact based on ANSYS/LS-DYNA
software. A refined finite element model of train–bridge coupling system was established
based on the nonlinear finite element software of ANSYS/LS-DYNA. The findings of this
study can provide guidance for the seismic design and train running of long-span railway
suspension bridges crossing faults and help improve the safety and efficiency of the railway
transportation system. Therefore, the following conclusions are drawn:
(1)
For long-span suspension bridges crossing left-lateral strike–slip faults, it is crucial
to emphasize the substantial impact of the fling-step effect on PGRD, leading to
amplified bridge responses. This effect is particularly pronounced in the transverse
response at FCA = 90
◦
and significantly influences the longitudinal response at lower
FCA values. Consequently, PGRD plays a pivotal role in seismic bridge responses,
with a direct correlation between the magnitude of the response and PGRD.
(2)
It is imperative to underscore the critical consequences of lateral seismic motion
during fault dislocation, which profoundly affects the dynamic parameters of trains.
In contrast, the influence of longitudinal seismic motion on the derailment coefficient
is relatively minor. However, as seismic motion persists, longitudinal ground motion
can lead to vertical displacements in the main beam, potentially causing the wheel-
load-reduction rate to surpass permissible limits.
(3)
Within the context of strike–slip faults, it is worth noting that PGRD resulting from
fault slips is predominantly governed by the low-frequency components of seismic
waves, accounting for approximately 70% of its impact on train dynamic parameters.
This underscores the necessity for trains to traverse large-span bridges before PGRD
reaches its full formation to significantly mitigate dynamic indicators.

Appl. Sci. 2023,13, 10422 14 of 16
(4)
Continuing to emphasize the importance of timing, if trains persist in traversing the
bridge after the complete formation of PGRD, they will encounter a compounded
effect. This effect combines additional track irregularities induced by PGRD with the
high-frequency components of seismic waves, resulting in a progressive escalation of
dynamic parameters that may ultimately breach established safety limits.
It should be noted that the above conclusions are derived from the simulation analysis
in this paper. The wheel–rail contact in the train–bridge system involves nonlinear dynamic
behavior, and it is important to acknowledge that earthquakes are characterized by random
vibrations. Seismic random analysis of the train–bridge coupled system is a highly complex
issue. Therefore, additional numerical calculations and field experiments are necessary to
provide further support for these conclusions.
Author Contributions:
Conceptualization, S.C.; Software, S.C.; Validation, S.C.; Investigation, W.K.,
J.Y. and S.D.; Data curation, S.C.; Writing—original draft, S.C.; Writing—review and editing, H.J.;
Funding acquisition, H.J. and S.Z. All authors have read and agreed to the published version of
the manuscript.
Funding:
The research for this paper was supported partially by the National Science Foundation of
China (No. 52178169 and No. 2021YFB1600300), Science and Technology Plan of Sichuan Science
and Technology Department (No. 2019YJ0243 and No. 2020YJ0081), and Major Systematic Projects
of China Railway Corporation (No. P2018G007). The authors would like to express their sincere
gratitude to all the sponsors for the financial support.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: The data is unavailable due to privacy.
Conflicts of Interest: The authors declare no conflict of interest.
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