Vulnerability Analysis of Catenary-Bridge-Track System

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Introduction
With the swift progression of high-speed rail in China, the utilization of continuous welded rail structures on bridges is becoming increasingly prevalent [1]. Generally, China's railway trains apply an electric traction system. According to the 2021 railway statistical bulletin, the national railway electrifcation proportion in China has now reached 75.4%. Te electric traction system is a pivotal component of the electrifed railway, which comprises of catenary, pantograph, traction substation, and other technical apparatus, with the catenary being the frailest link as it is right open and has no backup due to external environmental factors [2].
China's geographical location is quite unique, making it particularly vulnerable to frequent earthquake disasters. About half of the existing high-speed railway systems are located within the North China earthquake zone, an area renowned for its high earthquake intensity and frequency. Furthermore, these railway systems have been steadily extended to more seismically active zones such as the Qinghai-Tibet Plateau and the southeast coastal seismic zone. Consequently, earthquake actively endangers the safe operation of trains, making practitioners attach the utmost importance to researching the seismic performance of continuous welded rail bridges. As such, performancebased seismic designs have increasingly gained traction, with seismic vulnerability analysis being pivotal to the design process, looking to ofer immense value to seismic designs and seismic performance evaluations.
Scholars have largely focused their research on the vulnerability of highway bridges, with only limited attention being focused on that of high-speed railway bridges. For instance, Tavares et al. [3] employed the Monte Carlo method to quantify the seismic vulnerability of highway bridges; Yang et al. [4] evaluated the seismic performance of diferent bridge types according to their respective vulnerability curves; Siqueira et al. [5] studied the infuence of seismic isolation devices on piers and foundations by plotting their vulnerability curves; and Li-feng et al. [6] proposed a seismic vulnerability analysis approach based on classical reliability theory, which was then applied to examine the seismic performance of medium multispan concrete continuous girder bridges. Yang et al. [7] examined the efect of pier height on the vulnerability of a simplysupported girder bridge based on the 32 m simply-supported girder bridge along the Sichuan-Tibet railway line. Song et al. [8] utilized the Copula function to formulate the vulnerability curve of the continuous girder bridge bearings and piers of high-speed railways, and proposed a vulnerability analysis method for bridge systems considering the correlation of seismic demands of components. In addition, Dong et al. [9] studied the vulnerability of bridge piers and supports under three-dimensional earthquake action, taking high-pier and long-span railway bridges as the research object.
Regarding the research on the dynamic performance of catenary, there has been some progress. For instance, Kim et al. [10] investigated the impact of catenary spacing on its seismic response by constructing a fnite element model of the catenary system. Gregori et al. [11] studied the dynamic performance of high-speed railway overlapping catenary systems, considering double cantilevers and tensioning devices, with numerical simulations. Liu et al. [12] examined the infuence of train speed on catenary fatigue and safety reliability based on dynamic stress test data and twodimensional fnite element simulation data of the Re250−0 simple chain suspension catenary. Zhang and Xie [13] investigated the infuence of diferent connection forms and contact line tension on the seismic response of the structure through a four-column three-span fnite element model of catenary.
At present, there is a relative lack of studies on the vulnerability of continuous welded rail on high-speed railway bridges, especially those that consider the catenary, track type, and bridge structure when assessing their seismic vulnerability.
To explore the infuence of line longitudinal resistance on the vulnerability of the catenary-bridge-track system, this paper takes a 7−32 m double-line simply-supported box girder bridge and establishes a fnite element simulation model, which includes catenary pillar (including wrist arm structure), track structure (ballasted track, ballastless track, and small resistance fastener), and bridge structure. It reveals the seismic vulnerability of each component of the system and discusses the infuence of the longitudinal resistance of the line on the vulnerability of the fasteners and catenary. Te response patterns of the components under seismic motion are also analyzed.

Simulation Model of the Catenary-Bridge-Track System
2.1. Simulation of System Structure. In this paper, a simulation model for the catenary-bridge-track system is primarily established, which takes into account various structures such as the catenary, steel rail, fastener, beam, support, bridge pier, and bridge abutment, among others.
Considering that the contact line is prone to signifcant elastic deformation and failure during strong earthquakes, for the sake of simplifcation, this study focused on modeling the catenary pillar and wrist arm system as research objects [14]. To achieve this, all components of the catenary pillar and wrist arm system were simulated using frame units. Te section form of the catenary pillar adopts H-shaped steel column of GH240 model, and its section size is shown in Figure 1(a). It is made of Q235 with a column height of 7.5 m. Meanwhile, the wrist arm system was comprised of a wrist arm support device and a positioner, with the support device selected as the seamless steel tube wrist arm support device (WG type) and the positioner as the limit positioning device. Insulators were used to connect the wrist arm support device and the catenary column. Te relative positions of each component are depicted in Figure 1(b), while the specifc parameters can be found in Table 1. To establish the catenary, the study started from the 1/4 span of the leftmost span of the simply-supported beam, with the catenary being established 50 meters apart from left to right and numbered as 1#, 2#, . . . 5#.
Te steel rail utilized in this study was the CHN60 rail, with Q345 material being selected and simulated using the frame unit. Meanwhile, the fastener was simulated using the nonlinear connection unit, taking into account the resistance in the three-dimensional direction. Te relationship between the resistance and displacement was determined using formulas (1)-(3) [15]: where r L 1 is the longitudinal resistance of the fastener, kN/m per line, and x is the longitudinal relative displacement between the rail and the rail bearing platform, mm.
where r T is the lateral resistance of the fastener, kN/m per line, and y is the lateral relative displacement between the rail and the rail bearing platform, mm.
where r V is the vertical resistance of the fastener, kN/m per line, and z is the vertical relative displacement between the rail and the rail bearing platform, mm. To mitigate boundary conditions, a 200 m track was established at each end of the bridge span structure [16], with the fastener connection unit being linked to the subgrade.
Te beam was a 7−32 m double-line prestressed equalsection simply-supported box girder with C55 concrete and a cross-sectional area of 9.2826 m 2 . Te concrete density is 2250 kg/m 3 , the elastic modulus is 36 GPa, and the Poisson ratio is 0.2, whose cross-sectional is shown in Figure 2. It was simulated using frame units, whose secondary constant loads were considered in the modeling. Te fxed support was modeled using the linear connection element Linear, while the sliding support was based on the ideal elastic-plastic spring model. Te cut of point of elastoplastic deformation was set at 3 mm, with a friction coefcient of 0.3 [17].
Te bridge pier had a hollow thin-walled structure with circular ends, standing at a height of 20 m with a wall thickness of 0.5 m, and made of C40 concrete. Te crosssectional dimensions are shown in Figure 3. Te Mander confned model was employed for simulation, taking into account the constraints imposed by stirrups. Te crosssectional moment-curvature curve can be observed in Figure 4. Te interaction between the pile group foundation and the soil was simplifed as a stifness matrix with 6 degrees of freedom and was simulated using the linear connection element Linear. At both ends of the bridge, the abutments were round-ended solid pier structures with a height of 4 meters, and both the piers and abutments were simulated using frame units.
Te key to establishing a simulation model for the catenary-bridge-track system lies in simulating the contact mode between the bridge, rail, and catenary. Te connection between the bridge and rail was simulated using a nonlinear connection element, while the catenary was connected to the simply-supported beam through a rigid arm. Te simulation model that was established is illustrated in Figure 5.       Shock and Vibration where r L 2 is the longitudinal resistance of ballasted track bed, kN/m per line; r L 3 is the longitudinal resistance of small resistance fasteners, kN/m per line; and x is the longitudinal displacement of the rail relative to the fastener, mm.

Seismic Vulnerability Analysis Based on IDA
3.1. Analysis Method and Earthquake Input. Te incremental dynamic analysis (IDA) technique is a seismic evaluation method that accounts for both structural demand and capacity. Tis approach involves scaling ground motion using a proportional coefcient to generate a range of structural responses at diferent seismic intensities. Subsequently, a relationship curve between ground motion intensity parameters and structural damage parameters, known as the IDA curve, is plotted. Tis curve can illustrate the trend of structural response variations with changes in ground motion intensity [18,19]. Based on pertinent standards and prior research experience, 20 appropriate ground motion records were culled from the seismic record database maintained by the Pacifc Earthquake Engineering Research Center [20,21]. Te relevant information of seismic wave is shown in Table 2, and the acceleration response spectra of these records are shown in Figure 6.
Te peak ground acceleration (PGA) was selected as the ground motion intensity parameter, and amplitude adjustment treatment of the seismic wave was performed while preserving its spectral characteristics. Specifcally, the PGA values of 0.05 g, 0.1 g, 0.2 g, 0.4 g, and 0.8 g were utilized, and the structures were then analyzed using the Newmark-β method for nonlinear time course.

Establishment of Vulnerability Curves.
Te probabilistic seismic demand model demonstrates the probabilistic relationship between the seismic demand of a structure and the intensity of ground shaking, thereby laying the groundwork for formulating vulnerability curves.
Te probability of failure of a structure, based on seismic demand and ground shaking intensity, can be indicated in the following formula: where P f is the probability of failure of the structure; D is structural seismic demand or Engineering Demand Parameters, expressed as the seismic response of a structure, i.e. the structural damage parameters in the IDA method; C is the structural capacity, which corresponds to the damage index of the structure in the vulnerability analysis; and IM is the ground motion intensity parameter in the IDA method, which is used in this study as the PGA. As per Cornell et al., it is assumed that the mean seismic demand follows an exponential relationship with IM [22], i.e., where S D is the mean value of the structural seismic demand D and a and b are exponential relationship coefcients. Formula (8) can be obtained by transforming formula (7) into a logarithmic expression.
Formula (8) represents a linear regression that is ftted to the logarithmic IDA curve. Tis formula serves as the probabilistic seismic demand model that is necessary for conducting the susceptibility analysis.
As per the conventional theory of structural reliability, a structure is deemed to have failed if the load efect surpasses the resistance of the structure. In the context of seismic susceptibility analysis using the IDA approach, this can be expressed through the following formula: where S D is the seismic demand of the structure, determined by a probabilistic seismic demand model, and S C is the loadbearing capacity of the structure, determined by damage index. It is shown that both S D and S C can be considered to obey a log-normal distribution, and then the probability of failure of the structure can be expressed in the following formula [23]: where Φ(·) is the standard normal distribution function and β d and β c are the standard deviations of the logarithms of S D and S C , respectively. By substituting formulas (8) into (10), the transcendental probability function for the structure can be obtained, as illustrated in the following formula: Since the independent variable IM was used in this study as PGA, ������ β 2 d + β 2 c was taken to be 0.5 [24]. Based on the calculated beyond probability function, the susceptibility curves were established. Tis study focuses on investigating the vulnerability of the structure by analyzing the fasteners and catenary.

Vulnerability Analysis of Fasteners.
Te damage state of the fastener was categorized into fve distinct levels, and a quantitative assessment of the deterioration state was achieved by utilizing the deformation-based failure criteria, where the displacement parameter was employed as the damage index [20], as depicted in Table 3.
Fasteners located at the stationary support, middle section, and sliding support of the initial, central (i.e., the fourth), and fnal (i.e., the seventh) spans of the simplysupported beam were designated as representatives for executing vulnerability analysis. Tese fasteners were labeled as follows: 1-fxed end, 1-mid-span, 1-sliding end, 4-fxed end, 4-mid-span, 4-sliding end, 7-fxed end, 7-mid-span, and 7sliding end, correspondingly. Figure 7 presents the vulnerability curves of the fasteners at distinct positions, subjected to varying damage states.
According to Figure 7, within a certain range of seismic intensity, the exceedance probability of fasteners at the same position for a certain damage state increases with the increase of PGA. With the increase of the damage state, the exceedance probability of fasteners at the same position decreases gradually.
Te vulnerability curves of fasteners located at the 1fxed end and 7-sliding end appear to be similar under the same damage state. However, at the same exceedance probability, the PGA of the 7-sliding end is lower, indicating that the vulnerability of fastening elements at both ends of a multispan simply-supported girder bridge for the highspeed railway is signifcantly infuenced by the track structure of the subgrade at both ends. Te vulnerability of fasteners at the 1-sliding end is noticeably higher than that at the 7-fxed end, and the discrepancy increases signifcantly with the augmentation of the damage state, revealing that it is greatly afected by the end fxation of the beam. Te vulnerability of fasteners located at 1-mid-span and 7mid-span is comparable. As the damage state increases, the vulnerability gap gradually increases, indicating that fasteners located at the middle of the span of the simplysupported beam at both ends are afected by the track structure of the subgrade and the fxation of the beam end, and as the damage state increases, the infuence of the fxation of the beam end also increases. Te vulnerability of fasteners on the central span of the beam is relatively low overall, and the middle part of the span is hardly damaged.

Vulnerability
Analysis of Catenary. Tis paper accounted for the limit value of the dynamic lift of the catenary [25], and the displacement of the endpoint of the locator at the fxed contact line position was utilized as the damage index to provide a quantitative depiction of the damage state of the catenary, as presented in Table 4.
Exceedance   Te vulnerability curves of fve catenary groups under various damage states are depicted in Figure 8 for the simulation model of the catenary-bridge-track system established in this study.
Te exceedance probability of the same catenary for a particular damage state tends to increase within a specifc range of seismic intensity as the PGA rises. As the damage state increases, the exceedance probability of the same catenary gradually diminishes.
Te exceedance probability of the catenary varies signifcantly under diverse damage states. When the damage state is relatively small, the vulnerability of each catenary is relatively concentrated. As the damage state increases, the gap in vulnerability between catenaries gradually enlarges. Among all catenaries, the 3# catenary near the middle of the bridge span is the most vulnerable under the same damage state and seismic intensity, whereas the 1# and 5# catenary at the end of the bridge are relatively less vulnerable.

Influence of the Longitudinal Resistance of Line on Fastener Vulnerability
Based on the fastener vulnerability analysis, it can be concluded that for a multispan simply-supported girder bridge, fasteners at both ends are more susceptible to damage, and their vulnerability is somewhat similar. Conversely, fasteners on the mid-span have relatively lower vulnerability. In this study, the fastener at the center of the frst span of the simply-supported beam was chosen as the representative, and various longitudinal resistances of the line were set to investigate their vulnerability changes, as illustrated in Figure 9. For the fastener located at the middle of the 1st span of the simply-supported beam, also known as the middle of the side span, its vulnerability is signifcantly afected by the longitudinal resistance of the line under diferent damage states. Within a certain range of seismic intensity, the exceedance probability of the small resistance fastener is the highest for a certain damage state, followed by ballasted track, and the lowest for the ballastless track. When the exceedance probability is set to 5%, the failure sequence of fasteners is the ballasted track, small resistance fastener, and ballastless track in order when the damage degree is small. With the increase of the damage state, the small resistance fastener will be damaged before the ballasted track.

Influence of the Longitudinal Resistance of Line on Catenary Vulnerability
By analyzing the vulnerability of the catenary, it becomes evident that the vulnerability curves of the catenary on the bridge demonstrate a certain degree of regularity and similarity under diferent damage conditions. Building upon this foundation, the 3# catenary with the highest degree of vulnerability was chosen as the representative, and various longitudinal resistances of the line were established to investigate the alterations of the catenary's vulnerability. Te vulnerability curves are illustrated in Figure 10.
Under diferent damage states, the vulnerability curves of 3# catenary remain relatively similar across diferent longitudinal resistances of the line, suggesting minimal impact of longitudinal resistance on catenary vulnerability. Taking the exceedance probability of 5% as the criterion, catenary failure occurs in the order of ballastless track, ballasted track, and small resistance fastener. It is noteworthy that the catenary of ballastless track proves to be more sensitive to ground motion.

Structural Response under Ground Motion
Taking into account the seismic fortifcation intensity of 8 degrees and a PGA value of 0.3 g, the probabilities of 1mid-span fastener and 3# catenary reaching diferent damage states under various longitudinal resistances of the line were calculated, as presented in Figure 11. As depicted in Figure 11(a), at a PGA of 0.3 g, the probability of damage for the fastener at the 1-mid-span position is at its highest in the undamaged state. Te ballastless track signifcantly improves the probability of no damage to fasteners, ensuring safety of fasteners. Similarly, the exceedance probability of ballasted track and small resistance fastener is relatively similar under the same damage state, indicating that the efect of these two track structures on fastener damage states is negligible.
As shown in Figure 11(b), 3# catenary is unable to remain undamaged when PGA is 0.3 g. Tere is minimal variation in the exceedance probability of the three track structures under diferent damage states. Te longitudinal resistance of the line has only a minor efect on the probability of diferent damage states occurring in the 3# catenary. In terms of complete damage, the ballastless track has the highest probability of exceeding, followed by ballasted track, and the small resistance fastener track has the lowest probability of exceeding, with probabilities of 85.0%, 84.4%, and 83.1%, respectively.

Conclusions
To examine the impact of longitudinal resistance on the vulnerability of the catenary-bridge-track system, a sophisticated simulation model of a 7−32 m simply-supported beam bridge system was constructed, taking into account various components such as the catenary pillar, wrist arm structure, steel rail, fastener, beam, support, bridge pier, bridge abutment, and pile group foundation. Using the IDA method, this study revealed the damage pattern of fasteners and catenary on high-speed railway bridges under seismic conditions, along with the infuence of longitudinal resistance on fasteners and catenary. Te primary fndings of this study are as follows: (1) Within a certain range of seismic intensity, the probability of fasteners and catenary on the bridge reaching a specifc damage state increases as the ground motion intensity increases. When PGA reaches a certain threshold and the exceedance probability approaches 1, the structure is considered to be damaged. Catenary is found to be more vulnerable to earthquakes compared to fasteners. (2) Te vulnerability of fasteners located near the abutments at both ends of multispan simplysupported beam bridges is greatly afected by the track structure of the subgrade, whereas those near the middle of the side span display similar vulnerability. In contrast, the fasteners located at the midspan exhibit minimal damage. In addition, the vulnerability of fasteners located in the middle of each span is considerably lower than those located at the fxed and sliding ends.
(3) Te exceedance probability of each catenary exhibits a signifcant variation under distinct damage states, and the gap in vulnerability between catenaries widens as the damage state increases. Under the same damage and seismic intensity, the 3# catenary positioned close to the bridge's center displays the highest vulnerability, while the catenaries situated at both ends of the bridge exhibit a relatively lower vulnerability. (4) Te longitudinal resistance of the line exerts a greater infuence on the vulnerability of fasteners than on that of the catenary. When using a 5% exceedance probability as a benchmark for small damage states, the failure order of fasteners is ballasted track, small resistance fastener, and ballastless track. As the damage degree increases, the small resistance fastener will fail before the ballasted track. For the catenary, the damage sequence is the ballastless track, ballasted track, and small resistance fastener. (5) At a peak ground acceleration of 0.3 g, the probability of no damage to the 1-mid-span fastener is highest, while the probability of complete damage to the 3# catenary is maximum. Te ballastless track structure obviously improves the probability of no damage to the 1-mid-span fastener, and the longitudinal resistance of the line has little impact on the 3# catenary.

Data Availability
Te data used to support the fndings of this study are included within the article.

Conflicts of Interest
Te authors declare that they have no conficts of interest.