Study on train safety control of high-speed railway bridge under the action of near-fault earthquake

Abstract

In order to study the effect of the velocity pulse on the dynamic response of the train-bridge system of the high-speed railway simple supported beam bridge, the velocity pulse is simulated by the trigonometric function method and superimposed with the far-field earthquake without pulse to synthesize the pulse with different pulse types, pulse periods and pulse peaks. A 10$\times$32m typical high-speed railway simple supported beam bridge is considered an case illustrating study. Then, the dynamic response of train-track-bridge coupling system is calculated by train-track-bridge seismic analysis software TTBSAS. Afterwards, the influence of pulse near-field earthquakes parameters and vertical components on dynamic response of train-bridge system and the safety of the train on the bridge are discussed in detail.The new derailment evaluation index is adopted to evaluate driving safety under earthquakes. The train safety control of simply supported beam bridge under the action of near-field earthquake is studied. The results show that the impact of pulse ground motion on the dynamic response of the train-track-bridge coupling system is significantly higher than that of no-pulse ground motion, especially the impact on the bridge and rail subsystem is more significant than that of train subsystem. Under the excitation of ground motion intensity of 0.05g$\sim$0.15g, the safe speed threshold of pulse near-field ground motion is smaller than that of far-field ground motion. When the ground motion intensity is 0.20g$\sim$0.30g, the safe speed threshold of pulse near-field ground motion and far field ground motion is 200km/h. So, the pulse near-field earthquake poses a greater threat to the safety of the train on the bridge than the far-field earthquake. Therefore, the influence of pulse near-field earthquakes should be considered in the seismic design. The research results of this paper can provide theoretitcal support for the design of a high-speed railway bridge in the near-field area.

1 Introduction

Earthquakes in China are densely distributed with high frequency and high intensity. With the continuous encryption of the high-speed railway network, it is inevitable that the high-speed railway bridge crosses the fault or areas within 20km of the fault, and the earthquakes that occur in this area are called “near-field earthquakes”. The Code for Seismic Design of Railway Engineering (2009 edition) points out that when a bridge must cross a fault, it is appropriate to use a simple-supported beam bridge with a small span and low pier. Therefore, simply supported beam bridges have been widely used in high-speed railways in the near-field area. This type of bridge has a high probability of being subjected to near-field earthquakes in their designed service life.

Domestic and foreign scholars are paying more and more attention to near-field earthquakes due to their particularity. For the first time, Some scholars pointed out that near-field earthquakes contain energy pulses (Housner et al. 1965; Hudson and Housner 1958). Since then, some scholars proposed that the near-field vibration contains the characteristics of large values and long periods and pointed out that the peak values of acceleration, velocity and displacement are the main influencing parameters (Bertero et al. 1978; Hall et al. 1995; Malhotra 1999; Somerville 2002). The research on near-field impulse earthquakes at home and abroad is mostly limited by few historical records and scattered recording stations. Artificial simulation of near-field impulse earthquakes is an important way to break through this bottleneck. Some simplified methods have been presented for the simulation of a near-field pulse earthquake. Among them, Alavi and Krawinkler (2004) first, indicated to represent the near-field motion with simplified pulse based on the similarity of the structure dynamic response under the pulse near-field earthquake, and then pointed out the concept of the equivalent pulse. Makris and Chang (2000) used trigonometric functions to simulate three typical velocity pulse waveforms and verified that the pulse waveform had a high degree of agreement with the actual near-field seismic records. Subsequently, Wang et al. (2012) superimposed trigonometric function pulses based on the far-field vibration to simulate the near-field vibration with different parameters. Tian et al. (2007) expressed the equivalent velocity pulse as a simplified function and superimposed the low-frequency and high-frequency components of the seismic wave to synthesize the near-field vibration. Fan et al. (2008) adopted time-frequency analysis tools to simulate near-field impulse ground motions with local characteristics of the site and near-field pulse earthquakes. From the above researches, the current near-field vibration simulation methods can be classified into two categories: one is to simulate near-fault ground motion by a mathematical model or numerical model; The other is to synthesize near-fault ground motion by superposition of the low-frequency components simulated by impulse function and the high-frequency components recorded. The former is highly theoretical and has a amount of calculation, which is not convenient for application; the latter can simulate near-fault ground motions more accurately, but at present, this method cannot effectively react to the influence of site characteristics. In this paper, the second type of method is adopted to simulate the pulse near-field vibration.

Because high-speed trains have extremely high requirements on the smoothness and stability of the lines on the bridge, the large displacement caused by near-field seismic velocity pulse may be very unfavorable to the high-speed railway bridge. The seismic problem of railway bridges caused by near-field seismic velocity pulses has been widely concerned. Ellsworth and Champion et al. pointed out that large ground motion pulses would cause large displacements and that balance strength and ductility should be fully considered in structural design and analysis. They also showed that changes in the pulse period and ductility had a significant impact on the predicted damage capability (Ellsworth et al. 2004; Champion and Liel 2012). Zhai et al. (2016) examined the effect of the near-fault earthquakes on the response of nonstructural components. The results indicate that the amplification factors of nonstructural components of primary structural period can significantly increase under near-fault pulse ground motions. Ju S H, Mu D et al. established a finite element model based on contact mode and separation mode under the premise of considering the track irregularity and carried out a numerical simulation of the train-bridge interaction under seismic load and studied the influence of track mass and the dynamic response of the train-bridge system. The research findings suggest that the gap of the simple supported beam under seismic load will produce derailment coefficient (Ju et al. 2012; Mu et al. 2016). Rail irregularity can increase the dynamic response of the train-bridge system. As the train speed increases, the impact of strong seismic load on the train-bridge system decreases gradually. Lei Hujun and many other researchers had systematically studied the safety of high-speed trains under earthquakes and provided a reasonable safety threshold, which provided a valuable reference for the current engineering application (Lei 2015; Zhang 2009; Chen et al. 2014; Chen and Jiang 2013; Xia et al. 2017). Liang (2017) concluded that the train operation safety indicators gradually increase with the increase of the train speed, and the appropriate track spectrum excitation should be selected in the analysis of the train earthquake alarm threshold and operation safety. Liu et al. (2019) established the model of high-speed railway continuous girder bridge and inputted three near-field earthquakes and three far-field earthquakes for simulation calculation, respectively, and studied the isolation effect of friction pendulum support. The results show that the pulse effect of near-field earthquakes would aggravate the collision between adjacent beams of the continuous girder bridge. Chen et al. (2020) took the Chi-Chi seismic records in Taiwan as input to analyze the seismic vulnerability of high-speed rail continuous girder bridges, The results show that near-field earthquakes have destructive force. Chen et al. (2014; 2021) inputted a group of pulsed near-field earthquakes and a group of far-field earthquakes, respectively, and compared and analyzed the elastoplastic seismic response of the high-speed railway simple supported beam bridge. The results show that the nonlinear seismic response of the bridge will be intensified when the pulse period approaches the elastoplastic period of the structure. Subsequently, this paper also researched the elastoplastic seismic response of light rail transit bridges under the combined action of near-field vertical and horizontal earthquakes. The results show that the vertical component has a impact on the seismic response of light rail vehicles in low-frequency operation safety. In addition, Jiang et al. (2020) systematically explained the research status and prospect of the damage characteristics and damage mechanism of the high-speed railway track-bridge system under the action of earthquakes. Numerous researches have shown that earthquakes not only threaten the safety of bridges but also induce significant traffic safety problems (Zhai and Xia 2011; Jin et al. 2016; Li et al. 2020). However, up to now, there have been few research on the effect of near-field seismic velocity pulse on the dynamic response of high-speed railway simple supported beam bridges and trains on the bridges, which is very dangerous to the safety of high-speed railway bridges and running safety of the train in the near-field area.

Based on previous researches, this paper first adopts the method of bottom wave superimposed velocity pulse to synthesize near-field pulse earthquakes with different parameters. It takes a typical high-speed railway simple supported beam bridge with a span of 32m as an example and inputs the near-field vibrations of different pulse types, pulse periods and pulse amplitudes into self-made train-track-bridge- seismic analysis program TTBSAS to carry out the simulation calculation. After that, it discusses the influence of pulse parameters and vertical effects on the dynamic response of the high-speed railway train-track-bridge system and obtains the safe speed threshold when the train passing the bridge. The research results can provide theoretical support for the design of simple supported beam bridge of high-speed railway in the near-field areas.

2 Near-field pulse ground motion simulation

Due to the small number of actual near-field earthquake records, the response spectrum of each record is different, so that its response to the structure will also be different, which will affect the reliability of the research results. Therefore, the synthetic near-field ground motion has been recognized by more scholars and has opened up a new way to study the near-field earthquakes. In this paper, We adopted the superposition of velocity pulse and far-field earthquakes as the bottom wave to synthesize pulsed near-field earthquakes (Wang et al. 2012). The selection of the pulse model, the synthesis and verification process of seismic wave are described in detail below.

• (1) The typical pulse model proposed by Makris et al. was adopted (Makris and Chang 2000; Tian et al. 2007), and according to the regression Eqs.1 and 2 proposed by Somerville (1998) and Mavroeidis and Papageorgiou (2003), the pulse period ${T_{\text {P}}}$ and pulse amplitude ${V_{\text {P}}}$ could be determined by the magnitude and epicentral distance R. Then, it simulated three kinds of velocity pulses, which are the single-half wave, double-half wave and three-half wave.

• (2) The acceleration time history is downloaded from the PEER Ground Motion Database (Pacific Earthquakes Engineering Research Center) and integrated. The low-frequency components <1Hz are filtered to get the high-frequency velocity time history as the bottom wave.

  $$\begin{aligned} \ln ({V_{\text {p}}}) = - 2.31 + 1.15M_{\text {w}} - 0.5\ln (R) \end{aligned}$$

  (1)
  $$\begin{aligned} Lg({T_{\text {p}}}) = - 2.9 + 0.5{M_{\rm{w}}} \end{aligned}$$

  (2)

• (3) Use MATLAB to superimpose the peak time of the first wave of the pulse and the peak of the bottom wave velocity to obtain the velocity time history that containing the pulse. Then the acceleration time history and displacement time history of the synthesized wave are obtained by differentiating and integrating them respectively.

• (4) The least square method is used to correct the baseline of the acceleration time history of the composite wave.

• (5) The response spectrum, Fourier spectrum and power spectrum are compared with the seismic wave of Taiwan Chi-Chi as the historical record wave. The feasibility of synthetic near- field earthquakes is verified. Comparison of Fourier spectrum, and power spectrum is shown in Fig. 1.

Fig. 1

Fourier spectrum and power spectrum

It can be seen from Fig. 1 that the Fourier spectrum amplitudes of the measured wave and synthetic wave of the near-field vibration are both between 0 and 0.5 Hz, and the peak value is concentrated in the low-frequency range. This can also verify that the near-field earthquakes mainly affect the long-period structure. The power spectrum energy of the measured wave and synthetic wave of near-field earthquakes is most significant between 0 and 0.5 Hz, concentrated in the low-frequency range.

In addition, the PGV/PGA of all near-field earthquakes simulated in this paper is greater than 0.2, which meets the basic judgment criteria of near-field earthquakes. Therefore, it can be verified that the synthetic method and the pulse near-field earthquakes generation program compiled in this paper are effective.

3 Earthquake-train-bridge system coupled vibration model

This paper adopts the TTBSAS program for simulation calculation. The TTBSAS program (Lei 2015) is developed based on the BDAP program (Li 2000) and is widely used in the analysis of vehicle-bridge coupled vibration under earthquake action. The earthquake-train-track-bridge dynamic model is the basic model of the TTBSAS program. It is a strongly nonlinear time-varying system composed of 6 parts: vehicle model, track model, bridge model, wheel-rail relationship model, bridge-rail relationship model and boundary conditions of seismic force. The seismic force directly acts on the bridge piers, and the support points of the left and right subgrade through the ground support points, as shown in Fig. 2.

Fig. 2

Train-track-bridge dynamic model under earthquake

3.1 Vehicle model

The vehicle model in this paper is a four-axle vehicle with a two-stage suspension. Each vehicle has of 35 dofs, including one car body, two bogies, four wheel sets and a total of seven rigid bodies. Five dofs are considered for each rigid body: transverse motion, sink and float, roll, nod, and shake head. The rigid bodies are connected by spring-damping elements. The motion equation of the vehicle subsystem can be deduced by D ’Alembert principle:

$$\begin{aligned} {M_{\text {v}}} \ddot{u}_{\text {v}} + {C_{\rm{v}}}{\dot u_{\rm{v}}} + {K_{\text {v}}}{u_{\text {v}}} = {P_{{\text {tv}}}} \end{aligned}$$

(3)

Where ${M_{\text {v}}}$, ${C_{\text {v}}}$ and ${K_{\text {v}}}$ are, respectively, the mass, damping and stiffness matrices of the vehicle subsystem; $\ddot{u}_{\text {v}}$, ${\dot u_{\rm{v}}}$ and ${u_{\text {v}}}$ are the acceleration, velocity and displacement column vectors of the vehicle subsystem respectively; ${P_{{\text {tv}}}}$ is the load column vector of the track acting on the vehicle subsystem.

3.2 Track model

The track model adopts the slab ballastless track, which is composed of steel rails, fasteners, prefabricated track slabs, cement asphalt mortar adjustment layer, concrete support layer or reinforced concrete base and other parts. The vibration of the track reflects on the vibration of the steel rail and the track slab (Zhai and Xia 2011).The effect of the concrete base is considered in the bridge dynamics model through the form of vibrational mass. Consider the vertical vibration of the track plate according to the thin rectangular plate of equal thickness on the elastic foundation. The lateral vibration is regarded as the rigid body motion. The motion equation of the track subsystem is:

$$\begin{aligned} {M_{\text {t}}} \ddot{u}_{\text {t}} + {C_{\rm{t}}}{\dot u_{\rm{t}}} + {K_{\text {t}}}{u_{\text {t}}} = {P_{{\text {vt}}}} + {P_{{\text {bt}}}} + {P_{{\text {gt}}}} \end{aligned}$$

(4)

Where ${M_{\text {t}}}$, ${C_{\text {t}}}$ and ${K_{\text {t}}}$ are, respectively, the mass, damping and stiffness matrices of the track subsystem; $\ddot{u}_{\text {t}}$, ${\dot u_{\rm{t}}}$ and ${u_{\text {t}}}$ are the acceleration, velocity, and displacement column vectors of the track subsystem respectively; ${P_{{\text {vt}}}}$ and ${P_{{\text {bt}}}}$ are the load sequence vector of the vehicle and bridge subsystem acting on the track; ${P_{{\text {gt}}}}$ is the seismic force acting on the track on the left and right-side roadbed.

3.3 Bridge model

The bridge model uses spatial beam elements to simulate the main beams and piers. The top of the pier and the main beam is connected by the principal and subordinate degree of freedom, and the bottom of the pier is rigidly consolidated. The motion equation of the bridge subsystem is:

$$\begin{aligned} {M_{\text {b}}} \ddot{u}_{\text {b}} +{C_{\rm{b}}}{\dot u_{\rm{b}}} +{K_{\text {b}}}{u_{\text {b}}}={P_{{\text {tb}}}}+{P_{{\text {gb}}}} \end{aligned}$$

(5)

Where ${M_{\text {b}}}$, ${C_{\text {b}}}$ and ${K_{\text {b}}}$ are, respectively, the mass, damping and stiffness matrices of the bridge subsystem; $\ddot{u}_{\text {b}}$, ${\dot u_{\rm{b}}}$ and ${u_{\text {b}}}$ are the acceleration, velocity and displacement column vectors of the bridge subsystem respectively; ${P_{{\text {tb}}}}$ is the load acting on the bridge by the track subsystem Column vector; ${P_{{\text {gb}}}}$ is the seismic force of the foundation acting on the bridge subsystem.

3.4 The wheel-rail relationship and the bridge-rail relationship model

The wheel-rail relationship and the bridge-rail relationship connection are composed of the vehicle subsystem, the track subsystem, the bridge subsystem and the track subsystem, respectively, including the wheel-rail contact geometric relationship and the wheel-rail force relyearationship. The calculation assumes rigid contact between the wheel and the rail, and disengagement is allowed. The normal contact force and tangential force are solved by Hertz nonlinear elastic contact theory (Johnson 1987) and Kalker linear creep theory (Kalker 1991) respectively. The nonlinear correction is based on Shen Zhiyun-Hedrick-Elkins theory. ${P_{{\text {tv}}}}$ and ${P_{{\text {vt}}}}$ are the interaction forces between vehicle and track determined by the above theory, respectively. The displacement of the support point of the track slab is obtained by interpolation of the displacement of the bridge nodes, and the lateral force and vertical force acting on it are obtained from the relative displacement of the track slab, and then obtain the interaction forces ${P_{{\text {bt}}}}$ and ${P_{{\text {tb}}}}$ between the track and the bridge.

3.5 Boundary conditions of seismic forces

From the mechanism of earthquake load and the way of force transmission, it can be known that as long as the part in contact with the ground must directly bear the action of the earthquake load. In the train-track-bridge coupling system, the parts in direct contact with the ground include the support points of the bridge piers (bridge abutments) and the support points of the subgrade at the transition section of the road and bridge. Therefore, the boundary conditions of seismic force include the boundary of bridge seismic force and subgrade seismic force. In subsequent calculations, the relative motion method is used to input the acceleration and velocity time history of seismic waves according to uniform excitation, and it is assumed that the time when the earthquake occurs is the same as the time when the train enters the bridge. After the various forces acting on the vehicle, track, and bridge subsystems are determined, the dynamic responses of each subsystem can be obtained by numerical integration from Eqs. 3 to 5 (Lei 2015).

4 Case study

4.1 Computational condition

Taking a high-speed railway simple supported beam bridge with a span of 10 holes of 32m as an example, the design speed is 350km/h. The main girder is the single box and single chamber, the length of the box beam is 32.6m, the height is 3.5m, the width of the top slab is 12m, and the width of the bottom slab is 5m. The piers are cylindrical piers with a height of 10m and a diameter of 4m. The bridge analysis model is established by Midas Civil 2020 and imported into the TTBSAS program to obtain the simulated bridge model. The single-span simple supported beam bridge model is shown in Fig. 3. The track adopts a slab ballastless track, and the vehicle adopts an 8-section high-speed train (Li et al. 2016).

Fig. 3

Model of single-span simple supported beam bridge

Table 1 Detailed information of far-field seismic waves

According to the site type of the bridge, four far-field seismic waves are taken as the bottom waves, shown in Fig. 4. The near-field seismic waves with different parameters are synthesized by these four bottom waves. The detailed information of these far-field seismic waves as shown in Table 1. Among them, the synthesized wave contains three pulse types, and each pulse type has seven pulse periods to be considered. In addition, each pulse period is divided into 11 pulse amplitudes, and four bottom waves synthesize 924 pulse near-field ground motions.

Fig. 4

The acceleration time history of far-field seismic wave

4.2 Influence analysis of pulse type

In order to research the variation law of the response of the train-track-bridge coupling system under the action of the near-field earthquake with different pulse types, the velocity pulse types are divided into single-half wave type, double-half wave type, and three-half wave type, and the no pulse is assumed to be the far-field earthquake wave. KC-MNB, NT-MNB, BB-MNB and EL-MNB are near-field seismic waves synthesis of four far-field seismic waves-Kern Country, Northridge, Big Bear and EL Centro as bottom waves, respectively. In order to facilitate the comparison and analysis, the pulse period is ${T_{\text {P}}}$=3.0s and the peak value of the pulse velocity is ${V_{\text {P}}}$=1.0m/s in the calculation. The acceleration response spectrum and Fourier spectrum change with the pulse type as shown in Fig. 5. When other parameters are fixed, the pulse type mainly affects the pulse peak of the long period of the acceleration response spectrum and the peak value of the Fourier spectrum in the low-frequency range.

Fig. 5

Comparison of acceleration response spectra and Fourier spectra of near-field earthquakes of different pulse types

Figure 6 shows the variation of train safety indices with different pulse types under the action of near-field seismic ground motions. (index 1: derailment coefficient; index 2: wheel load reduction; index 3: wheelset lateral force). The design speed of the train crossing the bridge is 300km/h, and the simulated wave synthesized by 4 bottom waves is used as the earthquake input for comparative analysis of the dynamic response. In the figure, N, S, D, and T represent no-pulse, single-half wave pulses, double-half wave pulses, and three-half wave pulses respectively.

Fig. 6

The train safety index of different parameters vary with pulse type: (a) Derailment coefficient; (b) Wheel load reduction; (c) Wheelset lateral force

Referring to these figures, the influence of near-field seismic ground motions on train safety indices can be summarized as follows: (1) The train safety indices considering velocity pulse are larger than that without velocity pulse. The maximum increase of index 1-3 are 16.50%, 0.65%, 23.37%, respectively. (2) The train safety indices have a trend of increasing first and decreasing then, but the change is not significant. (3) Among the three train safety indices, the pulse type has the greatest impact on index 3, which is 23.37% higher than that without pulse. It can be seen that the pulse types have a more significant effect on bridge and rail mid-span displacement than train safety indices.

Figure 7 shows the dynamic response of each subsystem under the action of different pulse types of earthquakes.

Fig. 7

The amplitude of different parameters vary with pulse type: (a) bridge mid-span displacement; (b) bridge mid-span acceleration; (c) rail displacement; (d) acceleration of the car body’s center of mass

Comparing Fig. 7, as the pulse type changes from single-half wave type to three-half wave type, the amplitude of dynamic response of each subsystem generally increases at first and then decreases. The dynamic response of each subsystem under various pulse types of earthquakes increases relative to that under the action of no pulse earthquake. Under the action of 4 kinds of simulated waves, KC-MNB, NT-MNB, BB-MNB and EL-MNB, the bridge’s mid-span displacement response increased by 56.85%, 27.88%, 110.86%, and 24.79%, respectively. The acceleration response of the bridge span increases by 40.08%, 28.25%, 35.75%, and 5.67%, respectively. The displacement response of the mid-span track increases by 49.76%, 30.10%, 35.75%, and 15.05%, respectively. The acceleration response of the vehicle mass center increases by 12.37%, 22.22%, 10.96%, and 1.55%, respectively.

In summary, as the pulse type changes from single-half wave type to three-half wave type, the influence of pulse type on each subsystem of the train-track-bridge coupling system mainly increases at first and then decreases or gradually increases, because the single-half wave type is caused by directivity effect and permanent ground displacement effect. However, the double-half wave type and the three-half wave type pulse are both caused by the directional effect, which have a greater influence on the structure. The pulse type mainly affects the displacement response of the bridge and track. However, it has little effect on the dynamic response of train subsystem and acceleration response of the bridge. Therefore, the deformation coordination between the track and the bridge is verified.

4.3 Influence analysis of pulse period

In order to explore the influence of the pulse period on the dynamic response of the train-track-bridge coupling system, the pulse period is set as 0.0, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, and 5.0s respectively, and it is assumed that 0.0s means no pulse. For the convenience of comparison, the peak value of velocity pulse is kept at 1.0m/s during calculation, and the pulse type is assumed to be double-half wave type. The acceleration response spectrum and Fourier spectrum change with the pulse period are shown in Fig. 8.

Fig. 8

Comparison of acceleration response spectra and Fourier spectra of near-field earthquakes with different pulse periods

When other parameters are fixed, the pulse period mainly affects the pulse peak and the long period of the acceleration response spectrum and the frequency corresponding to the Fourier spectrum peak in the low-frequency range.

Shown in Fig. 9 are the variation of train safety indices with pulse period under the action of near-field seismic ground motions. The design speed of the train crossing the bridge is 300km/h, and the dynamic response is compared and analyzed with the simulated wave synthesized by four bottom waves as the seismic input.

Fig. 9

The train safety index of different parameters vary with pulse period: (a) Derailment coefficient; (b) Wheel load reduction; (c) Wheelset lateral force

Referring to Fig. 9, the influence of far-field seismic ground motions on train safety indices can be summarized as follows: (1) Generally, the larger the pulse period, the smaller the train safety indices, but the variation is not significant. (2) Comparing to no pulse, the train safety indices have the largest amplification when the pulse period ${T_{\text {P}}}$=2.0s, which increase by 29.46%, 15.18%, 36.12%. It can be seen that train safety indices are not sensitive to the change of near-field seismic velocity pulse for high-speed railway simply supported beam. For the condition of this paper, there has the largest train safety indices when the pulse period ${T_{\text {P}}}$=2.0s.

Figure 10 shows the dynamic response of each subsystem under the action of earthquakes with different pulse periods.

Fig. 10

The amplitude of different parameters vary with pulse period: (a) bridge mid-span displacement; (b) bridge mid-span acceleration; (c) rail displacement; (d) acceleration of the car body’s center of mass

Comparing Fig. 10, it can be shown that as the pulse period gradually increases from ${T_{\text {P}}}$=2.0s to ${T_{\text {P}}}$=5.0s, the dynamic response amplitude of each subsystem generally shows a gradually decreasing trend. The dynamic response amplitude of the system is increased compared with that under the action of the earthquake when ${T_{\text {P}}}$=0.0s and the increase is the most significant when ${T_{\text {P}}}$=2.0s. Under the action of the four simulations waves of KC-MNB, NT-MNB, BB-MNB, and EL-MNB, the bridge’s mid-span displacement response increased by 127.31%, 78.55%, 154.44%, and 59.95%, and the bridge’s mid-span acceleration response increased by 66.06%, 42.39%, 92.31%, and 26.35% respectively; when the pulse period ${T_{\text {P}}}$=2.0s, under the action of KC-MNB, the mid-span rail displacement amplitude increases most significantly, which is 92.09%; on the same pulse period, the acceleration response of the car body’s center of mass under the action of three analog waves of KC-MNB, NT-MNB and BB-MNB increase most significant, the maximum increase is 16.49%, 35.80%, and 34.25%, respectively. Under the action of EL-MNB, it also shows a gradual decrease with the increase of the pulse period, the maximum increase is 4.74%.

In summary, as the pulse period gradually increases from ${T_{\text {P}}}$=2.0s to ${T_{\text {P}}}$ =5.0s, the influence of the pulse period on each subsystem of the train-track-bridge coupling system mainly shows a decreasing trend, indicating that the smaller the pulse period is, the greater the response to each substructure will be. The pulse period mainly affects the displacement and acceleration response of the bridge structure, with the maximum increase of 127.31% and 92.31% relative to ${T_{\text {P}}}$=0s, respectively. However, it has little influence on the dynamic response of train subsystem, mainly because the support part of the bridge structure is the direct action part of the earthquake load. Then the vibration of the rail subsystem is transmitted upward through the vibration and deformation of the bridge, the vibration of the vehicle subsystem is finally transmitted upward through the dynamic wheel-rail relationship. On the other hand, because the bridge has low-pass filtering characteristics, it will reduce the high-frequency components of the seismic wave. Although the dynamic response of the vehicle subsystem will increase due to the seismic action, the dynamic response of the bridge increases much more than that of the structure because of its low-pass filtering characteristics.

4.4 Influence analysis of pulse amplitude

In order to explore the influence of pulse amplitude on the dynamic response of the train-track-bridge coupling system, the pulse period remains at 3.0s, and the pulse amplitude is set as 0.0, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4 and 1.5m/s in turn. It is assumed that the pulse amplitude is 0.0, indicating no pulse. The acceleration response spectrum and the Fourier spectrum change with the pulse peak value are shown in Fig. 11. When other parameters are fixed, the pulse peak mainly affects the pulse peak of the long period of the acceleration response spectrum and the peak of the Fourier spectrum in the low-frequency range.

Fig. 11

Comparison of acceleration response spectra and Fourier spectra of near-field earthquakes with different pulse peak

Shown in Fig. 12 are the variation of train safety indices with pulse peak under the action of near-field seismic ground motions. The design speed of the train crossing the bridge is 300km/h, and the dynamic response is compared and analyzed with the simulated wave synthesized by four bottom waves as the seismic input.

Fig. 12

The train safety index of different parameters vary with pulse peak: (a) Derailment coefficient; (b) Wheel load reduction; c Wheelset lateral force

Referring to Fig. 12, the influence of near-field seismic ground motions on train safety indices can be summarized as follows: (1) Generally, the train safety indices increase with the increase of the pulse peak, but the growth is less than the bridge mid-span displacement. (2) Under the action of four simulated waves, the maximum increased amplitude of train safety indices changes with pulse peak in 20.16%, 1.32%, 28.31%, respectively. It can be seen that the pulse peak has an effect on train safety indices, but the effect is less than bridge, rail subsystem.

Figure 13 shows the dynamic response of each subsystem under the action of earthquakes with different pulse amplitudes.

Fig. 13

The amplitude of different parameters vary with pulse peak: (a) bridge mid-span displacement; (b) bridge mid-span acceleration; (c) rail displacement; (d) acceleration of the car body’s center of mass

Comparing Fig. 13, we can see that as the pulse peak value gradually increases from ${V_{\text {P}}}$=0.5m/s to ${V_{\text {P}}}$=1.5m/s, the dynamic response amplitude of each subsystem shows a gradually increasing trend. The dynamic response amplitude of each subsystem increases relative to that under the action of an earthquake when ${V_{\text {P}}}$=0m/s. The increase is the most significant when the pulse peak value is ${V_{\text {P}}}$=1.5m/s. Under the action of 4 kinds of simulated waves of KC-MNB, NT-MNB, BB-MNB and EL-MNB, the displacement response of the bridge mid-span increased by 92.79%, 55.35%, 202.58% and 49.09% respectively, the acceleration response of the bridge mid-span increased by 67.51%, 40.91%, 57.97% and 5.67% respectively; the mid-span displacement response increased by 73.02%, 56.97%, 1.81% and 50.94% respectively; the acceleration response of the car body’s center of mass increased by 15.46%, 32.10%, 28.77% and 1.05%, respectively.

When other parameters are fixed, the pulse period mainly affects the pulse peak and the long period of the acceleration response spectrum and the frequency corresponding to the Fourier spectrum peak in the low-frequency range.

In summary, as the pulse peak value gradually increases from ${V_{\text {P}}}$=0.5m/s to 1.5m/s, the influence of pulse peak on each subsystem of the train-track-bridge coupling system mainly presents an increasing trend, indicating that the larger the pulse amplitude is, the greater the influence on the response of each substructure will be. Relative to ${V_{\text {P}}}$=0m/s,the maximum increase is 202.58%, 67.51% and 73.02% respectively. However, it has little influence on the dynamic response of train subsystem.

4.5 Influence of vertical effect on the dynamic response of train-track-bridge coupling system

The ratio of the vertical and horizontal acceleration peak of the near-field seismic wave is generally greater than 2/3, or even more than 1, which is called the vertical effect of the near-field vibration. Based on the research in the previous section, the influence of the vertical effect of near-field earthquakes on the dynamic response of the train-track-bridge coupling system is explored. The peak ratios of vertical and horizontal accelerations are respectively set to 0.65, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4 and 1.5.

Shown in Fig. 14 are the variation of train safety indices with acceleration peak ratio under the action of near-field seismic ground motions.

Fig. 14

The train safety index of different parameters vary with acceleration peak ratio: (a) Derailment coefficient; (b) Wheel load reduction; c Wheelset lateral force

Referring to Fig. 14, comparing to 0.65, under the action of four simulated waves, the maximum increased amplitude of train safety indices is 0.92%, 1.60%, 0.49%, respectively. Therefore, the change of train safety indices with the increasing of acceleration peak ratio is not significant.

Comparing Fig. 15, it can be seen that as the peak acceleration ratio gradually increases from 0.65 to 1.5, it has a significant influence on the vertical displacement amplitude and vertical acceleration in the mid-span of the bridge, and the amplitude presents a trend of gradual increase. Compared with the specified value of 0.65, the increase is the most significant when the peak acceleration ratio is 1.5. Under the action of the four simulated waves of KC-MNB, NT-MNB, BB-MNB and EL-MNB, the bridge mid-span displacement response increases by 120.72%, 134.92%, 117.28%, and 131.49% respectively. The maximum increase in the velocity response of the bridge span is 125.90%, 112.74%, 133.52% and 121.72%, respectively. As the acceleration peak ratio gradually increases from 0.65 to 1.5, the mid-span rail displacement tends to be stable. Therefore, it can be concluded that the acceleration peak ratio has little effect on the rail subsystem. As the peak acceleration ratio gradually increases from 0.65 to 1.5, the lateral acceleration of the vehicle mass center is stable. In contrast, the vertical acceleration amplitude shows a trend of gradual increase. Under the action of four kinds of simulated waves of KC-MNB, NT-MNB, BB-MNB and El-MNB, the maximum amplitude of the vertical acceleration of the vehicle mass center increases by 23.60%, 34.52%, 17.11% and 31.25%, respectively.

Fig. 15

The amplitude of different parameters vary with acceleration peak ratio: (a) bridge mid-span displacement; (b) bridge mid-span acceleration; (c) rail displacement; (d) acceleration of the car body’s center of mass

In summary, under the calculation conditions in this paper, the dynamic response of the train-track-bridge coupling system is mainly reflected in the transverse aspect. Besides, the near-field vibration acceleration peak ratio has little influence on the lateral response of the dynamic system of the train-track-bridge coupling system, so that the vertical effect can be ignored in the calculation. The vertical ground motion is only used for the dynamic analysis by taking 65% of the basic acceleration of the horizontal earthquake according to the specified value of the code.

4.6 Research on the safety of high-speed trains crossing the simply supported beam bridge under near-field earthquake

This section takes a 10-span 32m prestressed high-speed railway simple supported box beam bridge as an example to research the influence of the change of near-field earthquake intensity on the evaluation index of the new derailment (Lei 2015). The track irregularity adopts the German low-disturbance track. The track type is slab ballastless track model. The vehicle is German ICE3 high-speed train. It adopts the uniform excitation mode to input the seismic wave according to the horizontal and vertical 1:0.65. This section explores the driving safety of high-speed trains crossing bridges at different speeds and ground motion intensity under the action of near-field earthquakes, and divides the speeds into nine grades: 200, 225, 250, 275, 300, 325, 350, 375, 400km/h and sets the intensity as 0.05, 0.1, 0.15, 0.2, 0.25, 0.3g.

4.6.1 Maximum response analysis

In order to explore the effect of the near-field earthquakes on the safety of the train on the bridge, this section mainly analyzes the dynamic response of the vehicle subsystem under different ground motion intensity and different speeds of the high-speed train crossing the bridge. The maximum value of the derailment coefficient (index 1), overrun duration of derailment coefficient (index 2) and equivalent overrun duration of derailment coefficient (index 3), the maximum value of wheel load reduction (index 4), overrun duration of wheel load reduction (index 5) and equivalent overrun duration of wheel load reduction (index 6), maximum lateral displacement of the contact point (index 7), and maximum wheel vertical rise (index 8) under different ground motion intensity and speed change are shown in Figs. 16, 17, 18 and 19 respectively.

Fig. 16

Maximum value of derailment coefficient and its overrun duration and equivalent overrrun duration

Fig. 17

Maximum value of wheel load reduction and its overrun duration and equivalent overrun duration

Fig. 18

Maximum value of lateral displacement of the contact point

Fig. 19

Maximum value of wheel vertical rise

It can be shown from Fig. 16: (1) The indices 1-3 increase as the seismic intensity and train speed increase. When the seismic intensity keeps a fixed value, the indices 1-3 show an overall growth trend with the increase of train speed. Therefore, the seismic intensity and train speed are two important factors that affect the train running safety. (2) However, with the increase of train speed under different seismic intensity, the influence on indices 1-3 is different. It has the greatest impact when the seismic intensity is 0.3g. (3) When the seismic intensity is 0.05g, the index1 exceeds the traditional limit of 0.8 when the train speed reaches 375 km/h under far-field earthquakes and 350 km/h under pulse near-field earthquakes. When the train speed is 350 km/h, the far-field earthquake intensity reaches 0.20g, and the pulse near-field seismic intensity reaches 0.05g, the index 1 exceeds the traditional derailment limit of 0.8. (4) The index 3 increases gradually with the increase of seismic intensity and train speed. When the seismic intensity is 0.05g, the index 3 exceeds the equivalent overrun duration limit of 15ms when train speed reaches 375km/h under far-field earthquakes and 350km/h under pulse near-field earthquakes. When the train speed is 350km/h, the far-field seismic intensity reaches 0.15g, and the pulse near-field seismic intensity reaches 0.05g, the index3 exceeds equivalent overrun duration limit of 15ms.

It can be shown from Fig. 17: (1) The indices 4-6 increase with the increase of seismic intensity and train speed. When the seismic intensity remains at a fixed value, the indices 4-6 show an overall increasing trend with the increase of train speed. Under the same conditions, the indices 4-6 of the pulse near-field earthquakes have a larger increase compared with that of the far-field earthquakes. (2) When the earthquake intensity is 0.05g and the train speed under the far-field earthquakes and the pulse near-field earthquakes both reaches 375 km/h, the index 4 exceeds the traditional limit of 0.6. (3) The index 3 increases gradually with the increase of seismic intensity and train speed. When the seismic intensity is 0.05g and train speed reaches 400km/h, the index 4 exceeds equivalent overrun duration limit of 15ms under far-field and pulse near-field earthquakes, the train has derailed. When the seismic intensity is 0.3g, the index 4 has already exceeded equivalent overrun duration of 15ms when train speed is 300km/h under far-field and pulse near-field earthquakes, the train has derailed.

It can be shown from Fig. 18: (1) The index 7 shows an overall increasing trend with the increase of seismic intensity and train speed. Under the same conditions, the pulse near-field earthquakes have a larger increase compared with the far-field earthquakes. (2) The amplitude of index 7 changes slighter than index 1 and index 4. (3) When the earthquake intensity is 0.05g, and the train speed under the far-field earthquake and the pulse near-field earthquake reaches 350 km/h, the index 7 exceeds the traditional limit of 38mm, and the train has derailed. When the pulse near-field earthquake intensity is 0.15g, the safety speed threshold is less than 200 km/h; when the far-field earthquake intensity is 0.20g, the safety speed threshold is less than 200 km/h. It can be seen that the index 7 reaches the threshold at a lower seismic intensity under near-field earthquakes.

It can be shown from Fig. 19: (1) The index 8 increases with the increase of seismic intensity and train speed; When the earthquake intensity maintains a fixed value, as the train speed increases, the index 8 generally shows an increasing trend, and the pulse near-field earthquakes have a larger overall increase compared to the far-field earthquakes. (2) In particular, when the earthquake intensity is 0.05g, and the train speed reaches 350km/h, the index 8 rise increases sharply from 5.591 to 23.742, an increase of 324.65%, under the action of pulse near-field earthquakes. At this time, the wheel-rail relationship is destroyed, and the train derails. (3) When the pulse near-field seismic intensity is 0.20g, the safety speed threshold is less than 200 km/h; when the far-field earthquakes intensity is 0.25g, the safety speed threshold is less than 200 km/h.

In summary, under seismic excitation, the safe driving speed and earthquake intensity of pulse near-field earthquakes are smaller than those of far-field earthquakes. The pulse near-field earthquakes have a greater impact on the safety of train on the bridge than far-field earthquakes.

4.6.2 Discussion on the threshold of safe speed

Through the analysis in the previous section, the thresholds of safe train speed on simple supported beam bridges under different near-site vibration intensities can be obtained according to the overrun duration of equivalent derailment coefficient, overrun duration of equivalent wheel unloading rate, lateral displacement of contact point and wheel vertical rise, which is shown in Fig. 20.

Fig. 20

The threshold of safe driving speed under seismic excitation: (a) Far-field seismic excitation; (b) Near-field seismic excitation

It can be shown from Fig. 20 that the new derailment evaluation index is adopted to evaluate the driving safety, and the overrun duration of equivalent wheel load reduction and the lateral displacement of contact point is the upper and lower limits of the train safe speed threshold respectively (Lei 2015). When the earthquake intensity is 0.05g, the safe speed threshold is 325km/h under the excitation of both near-field and far-field earthquakes. When the intensity of far-field earthquakes is 0.10g and 0.15g, the corresponding safety speed threshold is 300km/h, and the corresponding safety threshold is 200km/h for the range of 0.20g to 0.30g. Under the pulse near-field earthquakes, when the intensity of the earthquake is 0.10g, the corresponding safe speed threshold is 225km/h, and the corresponding safety threshold for 0.20g$\sim$0.30g is 200km/h. Therefore, under the excitation of ground the intensity of 0.05g$\sim$0.15g, the safety speed threshold of pulse near-field earthquakes is smaller than that of far-field earthquakes, which poses a greater threat to the safety of the train on the bridge. Under the calculation condition in this paper, the safety speed threshold of the train crossing the bridge will be overestimated when the train is judged by a single evaluation index.

5 Conclusion

In this paper, based on the vibration theory of train-track-bridge coupling and using TTBSAS program as the calculation tool, through the artificial synthesis of pulsed near-field earthquakes with different parameters, the influence of the vertical component and the changes in near-field earthquake parameters on the dynamic response of the train-track-bridge coupling system are explored. After obtaining the most unfavorable condition, the safety of the train moving on the simple supported beam bridge under near-field and far-field earthquake is studied. Some conclusions are as follows.

• (1) The pulse type mainly affects the pulse peak of the long period of the acceleration response spectrum and the peak value of the Fourier spectrum in the low-frequency range. As the pulse type changes from single-half wave type to three-half wave type, the response of each subsystem of the train-track-bridge coupling system roughly increases first and then decreases; compared to no pulse earthquake, the response amplitudes of bridges, rail, and vehicle subsystems are increased under the action of pulse near-field earthquakes. Among them, bridge mid-span displacement and acceleration of bridge subsystem increase most significantly.

• (2) The pulse period mainly affects the pulse peak and the long period of the acceleration response spectrum and the frequency corresponding to the peak of the Fourier spectrum in the low-frequency range. The shorter the pulse action time, the greater the response to each substructure. As the pulse period changes from ${T_{\text {P}}}$=0.0s to ${T_{\text {P}}}$=5.0s, the response of each subsystem of the train-track-bridge coupling system shows a gradually decreasing trend. Compared with no pulse ground motions, the pulse period mainly affects the displacement and acceleration response of the bridge structure. However, it has little influence on the acceleration of car body’s center of mass.

• (3) The change of the pulse peak does not affect the shape and duration of the pulse, but mainly affects the pulse peak of the long period of the acceleration response spectrum and the peak of the Fourier spectrum in the low-frequency range. As the pulse peak value changes from ${V_{\text {P}}}$=0.5m/s to ${V_{\text {P}}}$=1.5m/s, the response of each subsystem of the train-track-bridge coupling system shows a gradual increasing trend. The displacement of bridge and rail mid-span and the acceleration of the bridge mid-span have a more significant influence than the acceleration of car body’s center of mass on the dynamic response of train-track-bridge coupling system.

• (4) Compared with no-pulse ground motions, the impact of impulsive ground motion on the response of the train-track-bridge coupling system is significantly increased, especially the impact on the structural displacement, acceleration, and track displacement of the bridge is more significant than that of other indicators. According to the calculation conditions in this paper, it can be concluded that when the pulse type is a double half-wave type, ${T_{\text {P}}}$=2.0s and ${V_{\text {P}}}$=1.5m/s are the most unfavorable conditions. As the three parameters of pulse type, pulse period, and pulse peak value change, they have different effects on the response of the coupling system. Therefore, it is recommended to classify and consider the parameter types when analyzing.

• (5) Under the calculation conditions in this paper, as the peak acceleration ratio gradually increases from 0.65 to 1.5, the vertical displacement and acceleration of the bridge and the vertical acceleration of the car body center gradually increase. However, there is little influence on the transverse displacement, transverse acceleration of the bridge, the rail subsystem, and the transverse acceleration of the car body. In addition, because the dynamic response of the train-track-bridge coupling system is mainly reflected in the transverse aspect, the vertical effect cannot be considered in the calculation, and the vertical ground motion effect is only 65% of the horizontal seismic basic acceleration according to the specified value of the code for dynamic analysis.

• (6) Under the excitation of ground motion intensity of 0.05g$\sim$0.15g, the safe speed threshold of pulse type near-field earthquakes is smaller than that of far-field earthquakes, and the maximum difference is 100km/h; when the earthquake intensity is 0.20g$\sim$0.30g, the safe speed threshold of pulse near-field earthquakes and far-field earthquakes are both 200km/h. Therefore, the pulse near-field earthquake poses a greater threat to the safety of the train on the bridge than the far-field earthquakes, and its influence should be considered in the seismic design. This paper only studies the train-track-bridge coupling system of simple supported girder bridges, which is the most widely used in high-speed railway lines in the near-field area, and the impact of near-field seismic velocity pulses on other bridge types needs to be further studied.