1. Introduction
On 1 October 1964, with the opening of the Tokaido Shinkansen in Japan, the world’s first high-speed railway was officially put into operation. There has subsequently been a wave of building high-speed railways around the world, and high-speed railways have developed rapidly [
1,
2,
3]. Due to the limitations of computational methods, the initial research on vehicle bridge vibration problems mainly focused on independent analysis of vehicle and bridge models, and the models were too simplistic to best reflect the real vehicle bridge dynamic response [
4]. The emergence of electronic computers and the development of finite element technology have further promoted the research on vehicle–bridge coupling vibration [
5]. Matsuura Akio [
6] used the energy method to derive the motion equation of the dynamic interaction between vehicles and bridges in high-speed railways. Chu K.H. et al. [
7] first studied the vehicle–bridge coupling vibration system, considering that the vehicle body is a 3-degree of freedom rigid body, and established a vehicle–bridge coupling vibration calculation model. Dhar C.L. [
8] established a vehicle model consisting of a vehicle body, a bogie, and a wheelset. The train–bridge coupled system was studied using a spring connection between the vehicle body and the bogie, and between the bogie and the wheelset, with the wheelset always in contact with the steel rail. Subsequently, train analysis models constructed with multiple rigid bodies were gradually recognized and adopted by scholars. This paper is also based on this to construct a 38-degree-of-freedom train model.
Since the 1960s, there have been multiple high-speed train derailments worldwide, which caused serious casualties and economic losses. One of the reasons for derailment is earthquakes [
9,
10,
11]. Therefore, it is necessary to consider seismic effects in the study of the dynamic response of train–track–bridge coupled systems. Currently, many scholars have conducted extensive research on the dynamic response of vehicle–bridge coupling systems under seismic excitation. Lei et al. [
12] used the relative motion method to solve seismic influences and analyzed the operational safety of high-speed trains passing over rigid-frame bridges with high piers under seismic influences. Xiang et al. [
13,
14] studied the dynamic response of trains and bridges under emergency braking during earthquakes, as well as the safety of train operation. Zeng [
15], Han [
16], Guo [
17], Variyavwala J.P. [
18] and others used cable-stayed bridges as examples to study the coupling dynamic interaction between trains and cable-stayed bridges under earthquake action, and obtained valuable conclusions.
In order to meet the requirements of smooth and stable operation of high-speed trains, land saving, and not affecting ground transportation, the proportion of high-speed railway bridges is constantly increasing. Many studies have shown that the characteristics of bridges significantly affect the dynamic response and safety of train–track–bridge coupled systems. It is necessary to study the effect of bridge characteristics on the train–track–bridge coupled system. Bridge characteristics include bridge structural stiffness, bridge deformation, abutment settlement, pier height, and so on. Zhai [
19,
20] analyzed the impact of bridge structural stiffness on the dynamic response of the coupling system. Their study shows that when the beam stiffness or lateral stiffness of bridge piers is insufficient, the main dynamic indicators of trains and bridges significantly increase with the decrease in stiffness. Fan et al. [
21] found that the damping coefficient of bridges has great influence on the vertical acceleration and mid-span acceleration of vertical and horizontal beams. Chen et al. [
22,
23,
24] theoretically derived an analytical expression for the mapping relationship between pier settlement and rail deformation in a dual block ballastless track–bridge system, and proposed a method for determining the safety threshold for high-speed railway pier settlement. Zhang et al. [
25] studied the impact of differential settlement of high-speed railway bridge piers on various railway performance-related criteria. The wheel load reduction rate increases with differential settlement of bridge piers, and the vertical acceleration increases with differential settlement of bridge piers and train speed, respectively. Guo et al. [
26] determined the mapping relationship between bridge deformation and train operation safety. This provides a convenient method for engineers to evaluate and maintain high-speed railway bridges. Feng et al. [
27] studied the impact of uneven settlement of side piers on the stability and safety of train operation. Their study shows that under the condition of uneven settlement of side piers, the stability of train operation will be significantly affected.
At present, most bridge piers are of equal height, but in mountainous areas, due to geographic factors, an equal height of bridge piers cannot be guaranteed when constructing bridges, and there is an effect of pier height on the dynamic response of the system. Therefore, in this paper, three kinds of pier height conditions are set up, and the seismic effect is considered to investigate the influence law of unequal-height piers on the dynamic response of high-speed train–track–bridge coupled systems.
4. Dynamic Response of the System
This paper conducted a study on the influence of three pier height working conditions on the coupled vibration response and running safety of the vehicle bridge system at five speeds of 150, 200, 250, 300, and 350 km/h. The constant height pier working conditions were selected to explore the influence of vehicle speed and seismic action on the dynamic response of the high-speed train–track–bridge system.
4.1. Validation of the System
In order to verify the reliability and accuracy of the train–track–bridge coupled vibration system model established in this paper, an example of Ref. [
38] is used for the reliability analysis, the parameters of the bridge and the train are set to be the same as those in Ref. [
38], and the train is set to run at a speed of 240 km/h. Comparing the vertical displacement of the bridge spans, as can be seen in
Figure 6, the vertical time-dependent response of the bridge spans obtained via the two models are very close and almost overlap, which means that the train–track–bridge coupled vibration system model is reliable.
4.2. Train Running Safety Indicators
4.2.1. Derailment Coefficient
French scientist Nadal first began to study the wheel climbing phenomenon, and in 1896, according to the climbing wheel climbing tendency of static equilibrium conditions, deduced the minimum derailment coefficient
Q/P [
39]. The wheel–rail contact diagram as shown in
Figure 7.
where
Q denotes the wheel rail lateral force, P denotes the wheel weight,
denotes the angle between tangent line and horizontal line at the contact point of wheel and rail, that is, rim contact angle; and
denotes the coefficient of friction between wheel rim and rail side.
The derailment factor varies from country to country. According to Chinese standard TB10761-2013 [
40], the derailment factor should meet the following requirements:
4.2.2. Rate of Wheel Load Reduction
In addition to the derailment coefficient, people also use the rate of wheel load reduction to determine the train running safety. When the wheels are substantially loaded down, that is, the vertical force on the wheels and rail is very small, the corresponding lateral force on the wheels and rail is often very small, and due to the impact of measurement error, it is difficult to calculate the derailment coefficient, especially when the wheel and rail are detached, which provides the derailment coefficient to assess the train running safety, addressing this serious problem.
Therefore, the increase in the rate of wheel load reduction can be used to comprehensively and effectively assess the derailment stability of the train’s running. The rate of wheel load reduction is defined as the ratio of the wheel weight reduction
on the reduced side to the average static wheel weight
of the wheelset. The rate of wheel load reduction standard stipulated by China is as follows [
41]:
4.3. The Influence of Train Speed on System Dynamic Response
We compare the dynamic response of the system under no seismic excitation with that under seismic excitation, and analyze the influence of an earthquake on the dynamic response of the train–track–bridge system. The train passed over the bridge at speeds of 150, 200, 250, 300, and 350 km/h, respectively, and we recorded the peak dynamic response of the system, as shown in the following figures.
From
Figure 8 and
Figure 9, it can be seen that seismic excitation has a significant impact on the dynamic response of bridges, with a particularly prominent impact on the acceleration response. The lateral dynamic response of the bridge increases more than the vertical dynamic response. This is because when there is no earthquake, the lateral dynamic response of the train mainly comes from the lateral track’s irregularity, and the vertical dynamic response mainly comes from the train’s gravity. The seismic excitation is input according to the vertical seismic wave strength, which is 0.65 times the horizontal seismic wave strength. Therefore, the seismic excitation has a more intense and sensitive impact on the lateral dynamic response of the bridge than the vertical dynamic response, and under earthquake excitation, some responses exceed the safety index limit, seriously reducing the smoothness and safety of the train. Overall, the dynamic response of the bridge increases with the increase in vehicle speed.
From
Figure 10, it can be seen that the train response generally increases with the increase in operating speed, and the lateral response is more affected by seismic excitation than the vertical response, for the same reason as the above bridge response influence law. From
Figure 11, it can be seen that under seismic excitation, the wheel rail load reduction rate and derailment coefficient of trains generally increase with the increase in vehicle speed. When the operating speed is 350 km/h, the two safety indicators of trains approach the limit values [
42]. When the seismic acceleration is greater, the vehicle speed is faster, and the bridge structure is more flexible, the risk of train derailment will greatly increase, and the safety and comfort of trains will be greatly reduced.
4.4. The Influence of Pier Height on System Dynamic Response
This paper only analyzes the influence of unequal pier heights on the dynamic response and running safety of the high-speed train–track–bridge system under seismic excitation when the train passes over the bridge at a speed of 300 km/h. The first- to fourth-order natural frequencies of the bridge as shown in
Table 6.
The working conditions of the bridge piers are shown in
Table 4 in
Section 3.2. From
Figure 12, it can be seen that compared to equal-height piers, the peak lateral dynamic response of bridges with unequal-height piers (gradually increasing) decreases, while the peak vertical dynamic response of bridges increases. The peak lateral dynamic response of bridges with unequal-height piers (Sharply increasing) increases sharply, while the peak vertical dynamic response of bridges decreases. From
Figure 13, it can be seen that compared to equal-height piers, the peak lateral dynamic response of trains with unequal-height piers (gradually increasing) decreases, which is beneficial for stabilizing the vehicle body. The peak lateral dynamic response of trains with unequal-height piers (sharply increasing) increases sharply, seriously reducing passenger comfort. Moreover, the vertical dynamic response of trains under both unequal-height pier conditions increases, and the safety indicators of equal-height piers are significantly better than those under the two unequal-height pier conditions. From the data in the above figures, it can be concluded that considering the comfort and safety of the train, the optimal choice is to design equal-height bridge piers, followed by gradually increasing pier heights, and avoiding steep increases in pier heights.