Exceeding Probability of Earthquake-Induced Dynamic Displacement of Rail Based on Incremental Dynamic Analysis

Abstract

When an earthquake occurs, it can strongly shake high-speed railway bridges. Consequently, the dynamic displacement of the rail on the bridge may exceed the allowable standard. However, few studies have evaluated the probability of rail displacement exceeding the allowable standard, compared to the rich variety of research on the vulnerability of other components of the high-speed railway track-bridge system or other structures. In this paper, incremental dynamic analysis (IDA) is applied to calculate the exceeding probability of rail displacement under different earthquake excitations. A finite element model (FEM) of a high-speed railway track-bridge system is established, which consists of a finite length CRTS II ballastless track laid on a five-span simply supported girder bridge. Records from five stations in the PEER NGA−West2 strong ground motion dataset are selected as seismic excitation. Based on the simulation, the characteristics of the vertical displacement of the rail under different seismic excitations are investigated, and the probability of the vertical displacement of the rail exceeding the allowable standard is calculated using IDA. The results show that: (1) the vertical displacement of the rail above the abutment is significantly smaller than that above other parts of the bridge; (2) the vertical irregularity of the rail caused by earthquakes has a wavelength close to the length of a simply supported girder; (3) under some excitations, two bumps are observed in the Fourier displacement spectrum in the frequency range of 1.3–2.5 Hz and 10–12 Hz, respectively, which may indicate the resonance of the model to the excitation; and (4) the vertical displacement amplitude probability of the rail exceeding 2 mm is 44%, 89%, and 99% when PGA = 0.01 g, 0.20 g, and 0.40 g, respectively. The exceeding probability of the rail above the mid-span is larger than that above other parts of the bridge. Within the mid-span, the exceeding probability of the rail is the largest above the center of the bridge.

1. Introduction

High-speed railway (HSR) has become the future development direction for the railway system in China due to the high passenger carrying capacity and operating speed of the high-speed train [1,2]. At present, the HSR mainly operates on bridges instead of on roads [3]. For many high-speed railways, however, the construction has to pass high-intensity areas. An earthquake may cause strong shaking of a high-speed railway bridge, leading to the dynamic displacement of the rail on the bridge exceeding the allowable standard. Generally, the rail irregularity caused by the earthquake can deteriorate safety, passenger comfort [4], and interaction with other infrastructures [5]. Therefore, the dynamic displacement of the rail exceeding the allowable standard may also threaten the running safety of the high-speed train.

The rail is the component of the HSR in direct contact with the vehicle. Therefore, a lower dynamic displacement of the rail is key to the high-speed train running safely on the bridge under seismic excitation. The rail of the China HSR has very strict smoothness requirements. The limit for vertical displacement of the rail is ±2 mm on the China HSR, with a design speed of 200–350 km/h [6]. However, the existing studies mainly investigate the dynamic displacement regularities of the rail on the bridge caused by the influence of the track-bridge system and earthquakes. For example, under seismic excitation, earthquakes cause vibration displacement of the bridge [7], which greatly influences the vertical displacement of the rail [3]. The height of the pier will also influence the vertical displacement of the rail [8]. A high proportion of the CRTS II slab ballastless track structure (CRTS II SBTS), an important component of the high-speed railway of China, operates on bridges [9]; providing a high degree of rail smoothness and ensuring high comfort, stability, and safety of the high-speed train. The loss of CRTS II SBTS performance will deteriorate the performance of the rail under seismic excitation. For example, the cement asphalt mortar (CA mortar), an interlayer structure of the CRTS II SBTS, affects the vertical vibration of the rail due to an increase in the elastic modulus of the CA mortar or the loss of the adhesion performance of the CA mortar [10,11]. The fastener, an important component of the CRTS II SBTS, restrains the rail to the track plate. The displacement amplitude of the rail will increase obviously under earthquake excitation due to the increased longitudinal resistance of fasteners, which would lead to the exceeding failure of the rail [12,13]. Not only does the track-bridge system affect the dynamic displacement of the rail, but the earthquake is also a critical factor affecting the vertical displacement of the rail in the seismic analysis. Many researchers have investigated the dynamic displacement of the rail as influenced by pulse-type ground motions [12,13], the input component of the ground motions [14,15,16], the critical input direction of the ground motions [16], the non-uniform excitation of the ground motions [17], and the spatial variability of ground motions [18]. Moreover, earthquakes are a single random process that cannot be reproduced, with considerable variability and uncertainty [19], and the typical dynamic analysis cannot connect the intensity of ground motion and the degree of structural damage to estimate the dynamic capacity of the rail against the earthquake load. The vulnerability analysis may be a suitable method by giving a fragility curve to evaluate the probability that structures exceed the allowable standard.

However, few studies have evaluated the probability of the earthquake-induced displacement of the rail exceeding the allowable standard, compared to the wealth of research on the vulnerability analysis of other components of the high-speed railway track-bridge system [15,20,21,22] and other structures [23,24,25,26,27]. In the field of high-speed railway track-bridge systems, researchers mainly investigate the vulnerability of the bridge [20,22], the component of the ballastless track structure [21], and the uplift of wheels from the rail [15]. In contrast, the study of the rail exceeding the allowable standard is almost blank. At present, the incremental dynamic analysis (IDA) proposed by Bertero [28] is widely used to evaluate the vulnerability of structures, which scale the earthquake records step by step and incrementally. The important advantage of this method is the better investigation of the structural behavior during severe earthquakes, as well as showing the changes in structural response with increasing earthquake intensity [29,30]. Although the computational cost of the IDA method is very high, IDA provides a better estimation of the dynamic capacity of the structure against the earthquake load [29]. The seismic vulnerability curves calculated based on the IDA describe the relationship between the intensity of ground motion and the degree of structural damage, which is a powerful tool for assessing structural seismic risk. Therefore, the main purpose of this study is the use of the IDA method to evaluate the exceeding probability of the earthquake-induced dynamic displacement of rail, also called the probability of exceeding certain limits of earthquake-induced dynamic displacement of rail. This study may be the first attempt to evaluate the exceeding probability of the earthquake-induced dynamic displacement of the rail and may be an extension that applies the IDA to the vulnerability analysis of the high-speed railway.

In this paper, the IDA is applied to calculate the exceeding probability of rail displacement under different earthquake excitations. A finite element model (FEM) of a high-speed railway track-bridge system is established, which consists of a finite length CRTS II SBTS laid on a five-span simply supported girder bridge. Ground motion records from five stations in PEER NGA−West2 strong ground motion dataset are selected as seismic excitation. Based on the simulation, the characteristics of the vertical displacement of the rail under different seismic excitations are investigated, and the probability of the vertical displacement of the rail exceeding the allowable standard is calculated using IDA.

2. Methodology

In the typical IDA of structures [31,32], the ground motion records are scaled up until the structure is destroyed, which is evaluated by given damage rules. However, in this study, evaluating the exceeding probability of earthquake-induced dynamic displacement of the rail is the primary objective. High-speed railways have higher requirements for smoothness than ordinary railways; therefore, taking the whole destruction state as the end sign of IDA is unsuitable. In order to investigate the exceeding probability of earthquake-induced displacement of the rail of the high-speed railway, peak ground acceleration (PGA) is selected as the intensity measure of the ground motion records according to the Code for Seismic Design of Railway Engineering [33]. Furthermore, scaling the PGA of ground motion records within the range of PGA ≤ 0.40 g is proposed according to the Code for Seismic Design of Buildings [34]. The steps of the IDA-based exceeding probability of earthquake-induced dynamic displacement of the rail include:

• Select the ground motion records and the intensity measure of the ground motion records;
• Scale the PGA of the selected records to different excitation levels within the range of PGA ≤ 0.40 g;
• Establish a FEM model for the exceeding probability of the earthquake-induced dynamic displacement of the rail;
• Perform a non-linear dynamic response analysis;
• Obtain the vertical displacement of the rail;
• Perform a linear regression analysis and construct response probability functions of earthquake-induced dynamic displacement of the rail;
• Define the limit states;
• Construct the exceeding probability curves of earthquake-induced dynamic displacement of the rail.

3. Selection and Scale of Ground Motion Records

Earthquakes are selected with the M_w ≥ 6, the standard magnitude in the China Earthquake Network of great earthquakes. All the selected ground motions records are near-fault ground motions with the R_JB < 25 km. The ground motion records are also selected with PGA > 0.05 g, the peak ground motion acceleration corresponding to the six-degree area in the Code for Seismic Design of Buildings [34]. At this stage, the great near-fault ground motions are selected to ensure the intensity of the earthquake. Moreover, according to Chandler’s classification [35], the ground motion records are divided into three sets based on the ratio of PGA and peak ground velocity (PGV). Records classified in the low PGA/PGV range with PGA/PGV < 0.8 g/(m/s), whereas records classified in the high PGA/PGV range with PGA/PGV > 1.2 g/(m/s). Records classified as the intermediate range with PGA/PGV between 0.8 and 1.2 g/(m/s). Each category can create resonance responses for specific structures [35]. Therefore, in this paper, earthquakes are selected from all three categories to have a chance to create more critical responses to the rail laid on the bridge.

Ground motion records selected from five stations in the PEER NGA−West2 strong ground motion dataset are shown in

Table 1. Ground motion records selected from the PEER NGA−West2 strong ground motion dataset.

Earthquakes	Stations	Labels	MW	RJB (km)	Component	PGA (g)	PGV (m/s)	PGA/PGV g s/(m)
Imperial Valley−02	El Centro Array #9	ELC	6.95	6.09	H1	0.28	0.31	0.90
					H2	0.21	0.31	0.68
					UP	0.18	0.09	2.00
Coalinga−01	Pleasant Valley P.P. −yard	PVY	6.36	7.69	H1	0.60	0.60	1.00
					H2	0.53	0.39	1.36
					UP	0.37	0.16	2.31
Chalfant Valley−02	Bishop−LADWP South St	LAD	6.19	14.38	H1	0.25	0.20	1.25
					H2	0.18	0.20	0.90
					UP	0.14	0.07	2.00
Superstition Hills−02	Westmorland Fire Sta	WSM	6.54	13.03	H1	0.17	0.23	0.74
					H2	0.21	0.32	0.66
					UP	0.23	0.09	2.56
Kobe Japan	Kakogawa	KAK	6.90	22.5	H1	0.24	0.21	1.14
					H2	0.32	0.27	1.19
					UP	0.17	0.11	1.55

and Figure 1. In this paper, earthquakes are represented by labels in Table 1. Each ground motion record includes two horizontal components and one vertical component. An input group consists of one horizontal component and one vertical component of ground motion, which are inputted into the model. For example, the first input group consists of ELC−H1 and ELC−UP, and the second consists of ELC−H2 and ELC−UP. Response spectra of the ground motions are also depicted to show the differences between those selected ground motions, as shown in Figure 2. According to the Code for Seismic Design of Railway Engineering [33], the intensity measure of PGA was selected to scale the acceleration amplitude of ground motion records. According to the Code for Seismic Design of Buildings [34], the PGA of earthquakes is then scaled to the excitation levels of PGA = 0.01 g, 0.025 g, 0.05 g, 0.10 g, 0.15 g, 0.20 g, 0.25 g, 0.30 g, 0.35 g, and 0.40 g, respectively. Then ten input groups make a total of 100 seismic excitations groups, inputted to the bottom of the piers. The horizontal component of ground motion is applied in the X direction, and the vertical component of ground motion is applied in the Y direction. The X direction is along the bridge and the Y direction is vertical to the bridge, as illustrated in Figure 3.

4. FEM Model of the High-Speed Railway Track-Bridge System

The 2D FEM model of a high-speed railway track-bridge system is established using ABAQUS, consisting of a five-span simply supported girder bridge with CRTS II SBTS. The length of each span is 32.4 m, and the total length of the model is 282.1 m. The pier is 8-m-high and made of C35 concrete. The girder bridge has a single box section made of C50 concrete. The friction plate is 52-m-long and made of C40 concrete. The rail is made of 60 kg/m steel, connected to the track plate through fasteners with a spacing of 0.65 m. The CRTS II SBTS includes the track plate, CA mortar layer, and base plate. The track plate is a 0.2-m-thick precast slab made of C60 concrete. The base plate is a 0.19-m-thick slab made of C40 concrete. A 0.03-m-thick CA mortar layer is filled between the track plate and the base plate. The sliding layer is between the base plate and the bridge, and between the base plate and the friction plate. The sliding of the base plate on the bridge is limited by the shear slot. The configuration of the FEM model of the track-bridge system is illustrated in Figure 3. In the model, the rail, track plate, base plate, box girder, and pier are all established by a beam element. The fastener, CA mortar layer, sliding layer, shear slots, fixed bearing, sliding bearing, and the interaction between pier and foundation are simulated by spring. The vertical stiffness of the shear slot, the fixed bearing, and the sliding bearing is rigid [36]. Sections, densities, elasticity modulus, damping, and force-displacement curves (F−D curves) for various elements are shown in

Table 2. Sections, densities, elasticity modulus, and damping coefficients for various components.

Component	Sections Width × Height (m)	Densities (kg/m3)	Elasticity Modulus (Pa)	Damping Coefficients
				α	β
Rail	—	7850	2.06 × 1011	7	1.00 × 10−6
Track plate	2.55 × 0.10	2500	3.65 × 1010	—	—
Base plate	2.95 × 0.19	2500	3.25 × 1010	—	—
Box girder	10.21 × 3.03	3300	3.55 × 1010	0.26	0.91 × 10−2
Friction plate	9.00 × 0.40	2500	3.25 × 1010	—	—
Ground	—	2100	1.50 × 108	—	—
Pier	5.66 × 1.94	2500	3.30 × 1010	—	—

and

Table 3. F−D curves in the X direction and damping of spring elements.

F−D Curve	Spring Elements	F (kN)	D (mm)	Damping (N·s/m)
	Fastener	15	2.00	4.770 × 104
	CA mortar layer	42	0.50	1.000 × 105
	Sliding layer of bridge	6	0.50	—
	Shear slots	1465	0.12	—
	Fixed bearing	1000	2.00	3.475 × 106
	Sliding bearing	100	2.00	3.475 × 106
	Sliding layer of friction plate	14	2.00	—
	Foundation	—	—	1.080 × 105

and Figure 4 [36,37,38,39].

The first 10 natural vibration frequencies of the FEM model of the CRTS II slab ballastless track-bridge system are shown in

Table 4. The first ten natural vibration frequencies of the FEM model.

Orders	1	2	3	4	5
Frequency (Hz)	1.498	1.888	1.992	2.431	3.0568
Orders	6	7	8	9	10
Frequency (Hz)	3.663	3.939	4.029	4.307	4.540

.

The vertical displacement of the rail is mainly compared in this paper at positions 0–5, as shown in Figure 5, where the red dot represents the compared positions of the vertical displacement of the rail. Position 0 is above the abutment; Positions 1, 3, and 5 are above the mid-span; and Positions 2 and 4 are above the pier.

5. Vertical Displacement of the Rail

5.1. Vertical Displacement of the Rail in Time History Curves

Time history curves of the vertical displacement of the rail above the abutment, mid-span, and the pier are compared after inputting 100 seismic excitation groups into the bottom of each pier. Limited by the article space, three cases of the time history curves of vertical displacement of the rail are shown, including the ELC−H1 with PGA = 0.01 g and 0.40 g, and the KAK−H2 with PGA = 0.40 g, as shown in Figure 6. The light green solid line represents the vertical displacement time history curve of the rail at position 0, corresponding to the position above the abutment.

As shown in Figure 6, the vertical displacement of the rail at position 0, corresponding to the position above the abutment is comparable to that at positions 1–5, corresponding to positions above the mid-span and the pier under seismic excitation, when the vertical displacement of the rail is positive. That may be because the distance between the abutment and the bridge is small; CRTS II SBTS is longitudinally continued. Under seismic excitation, the CRTS II SBTS above the abutment moves together with that above the bridge because the CRTS II SBTS above the abutment is adjacent to and is restrained to that above the bridge. Then the rail moves together with the CRTS II SBTS. Therefore, the difference between the vertical displacement of the rail above the abutment and other parts of the bridge is small when the vertical displacement is positive. However, the vertical displacement of the rail above the abutment is obviously smaller than that above other parts of the bridge under seismic excitation when the vertical displacement of the rail is negative. That may be due to the hard compression of the components above the abutment when the vertical displacement of the rail is negative.

Figure 6 also shows that the displacement variation trend is comparable between vertical displacement time history curves of the rail at the same position when PGA = 0.01–0.40 g. However, the displacement amplitude of the rail is different. For example, in the action of ELC−H1 with PGA = 0.01–0.40 g, the maximum positive vertical displacement of the rail increases from approximately 1.5–60 mm, the maximum negative vertical displacement of the rail above the abutment increases from approximately 0.7–28 mm, and the maximum negative vertical displacement of the rail above the mid-span and the pier increases from approximately 1–41 mm. Variation trends of the vertical displacement time history curves of the rail vary when earthquakes are different, which may be because the PGA, the frequency, the energy, and other parameters of various strong motions are different.

The amplitude and frequency of the vertical displacement time history curves of the rail changed obviously to the action of the PVY−H2 with PGA = 0.20 g compared to that under the action of the PVY−H2 with PGA = 0.15 g, as shown in Figure 7.

In the action of PVY−H2 with PGA = 0.20 g, an obvious increase in the vertical displacement amplitude of the rail can be observed. In addition, a high-frequency vibration in the vertical displacement of the rail can be observed at positions 1–5, corresponding to positions above the mid-span and the pier, which is not obvious at position 0, corresponding to the position above the abutment. The change in vertical displacement amplitude and the vibration frequency can be observed from the action of PVY−H2 with PGA = 0.20 g rather than PVY−H2 with PGA = 0.15 g, which may be because of the frequency of the PVY−H2 with PGA = 0.20 g close to a natural frequency of the model. Therefore, in the action of PVY−H2 with PGA = 0.20 g, the resonance of the model occurs. Moreover, the resonance frequency may be the natural frequency of the lower vibration mode of the model and significantly influences the vertical displacement of the rail.

5.2. Vertical Irregularity along the Bridge of the Rail

The vertical irregularity of the rail along the bridge is shown at the moment of the largest vertical displacement of the rail under seismic excitation at position 5. Limited by the article space, six cases of the vertical irregularity curves of the rail along the bridge are shown, including the ELC−H1 with PGA = 0.01 g and 0.40 g, the LAD−H1 with PGA = 0.01 g and 0.40 g, and the KAK−H2 with PGA = 0.01 g and 0.40 g, respectively. As shown in Figure 7 and Figure 8, the vertical irregularity curves of the rail along the bridge are shown by the positive and negative vertical displacement of the rail, respectively.

Although the shape of vertical irregularity curves of the rail along the bridge is different, the vertical irregularity of the rail caused by earthquakes has a wavelength close to the length of a simply supported girder. That may be because the vertical displacement of the rail on the bridge is affected by the girder and the vertical displacement of each simply supported girder is comparable under seismic excitations. Although the shape of the vertical irregularity curves of the rail along the bridge is comparable, the vertical displacement amplitude of the rail increases with PGA. For example, the minimum and maximum vertical displacement of the rail along the bridge is 0.02 mm and 0.2 mm when PGA = 0.01 g, respectively. The minimum and maximum vertical displacement of the rail along the bridge is 0.5 mm and 7 mm when PGA = 0.40 g, respectively. The “L” shape of the rail above the connection between the abutment and the bridge can be observed significantly when the vertical displacement of the rail is negative, as shown in Figure 9, which may be due to the component above the abutment being hard to compress. There is an obvious increment of the vertical displacement of the rail at a longitudinal range of 2.6 m from position 0 above the abutment, which is along the bridge. The average vertical increment increases from 4–17 mm when PGA = 0.01–0.40 g.

5.3. Fourier Displacement Spectrum of the Rail

Fast Fourier transform (FFT) is applied to calculate the Fourier displacement spectrum of the rail. Therefore, the Fourier displacement spectrum of the rail compares different earthquakes, PGA, and positions in this paper. Limited by the article space, three cases of the Fourier displacement spectrum curves of the rail are shown in Figure 10, including the WSM−H1 with PGA = 0.01 g and 0.40 g and the ELC−H1 with PGA = 0.40 g.

As shown in Figure 10, an obvious bump may be observed in the Fourier displacement spectrum in the frequency range of 10–12 Hz at positions 1–5, corresponding to positions above the mid-span and the pier; however, the bump may not be observed in the Fourier displacement spectrum at position 0, corresponding to the position above the abutment. Furthermore, the bump amplitude in the Fourier displacement spectrum of the rail at positions 2 and 4, corresponding to positions above the pier, is smaller than that at positions 3 and 5, corresponding to positions above the mid-span. The bump in the Fourier displacement spectrum of the rail may indicate the resonance of the model to the excitation. The bump amplitude in the Fourier displacement spectrum of the rail above the mid-span is larger than that above the pier, which may be due to the larger vertical displacement of the rail above the mid-span easily caused by earthquake excitation.

Figure 7 shows an obvious change in amplitude and frequency of the vertical displacement of the rail with the action of the PVY−H2 with PGA = 0.20 g. Therefore, in this paper, the Fourier displacement spectrum of the rail to the action of PVY−H2 with PGA = 0.20 g is shown in Figure 11, which is compared to that under the action of the PVY−H2 with PGA = 0.15 g.

A bump may be observed in the Fourier displacement spectrum in the frequency range of 10–12 Hz at positions 0, 1, 3, and 5, and a bump can also be observed in the Fourier displacement spectrum in the frequency range of 1.3–2.5 Hz at positions 2–4, which may indicate the resonance of the model to the excitation and indicate that two ground motion frequencies may be approximately the natural frequency of the model. In addition, the change in the amplitude and frequency of the vertical displacement of the rail may indicate that the frequency range of 1.3–2.5 Hz may be approximately the natural frequency of the lower vibration mode of the model, according to Table 4.

The resonance of the model to the excitation may be due to the ground motion frequency being approximately the natural frequency of the model. A total of 100 cases are calculated in this paper. However, the obvious change in amplitude and frequency of the vertical displacement of the rail is only observed from the action of PVY−H2 with PGA = 0.20 g. The result indicates that it is not easy to cause the resonance of the model and cause the vertical displacement of the rail with the obvious change in amplitude and frequency under seismic excitation.

6. Exceeding Probability of the Earthquake-Induced Dynamic Displacement of the Rail

Curves of the vertical displacement amplitude of the rail changing with PGA are shown in Figure 12. The gray area represents the range of the vertical displacement amplitude of the rail at six positions under seismic excitation, and points represent the average of the vertical displacement amplitude of the rail at six positions.

It can be observed that the vertical displacement amplitude of the rail increases with the PGA, and the increasing trend is approximately linear. The vertical displacement amplitude of the rail differs from the action of earthquake excitations with the same PGA. For example, the maximum and minimum vertical displacement of the rail are 2.06 mm and 1.31 mm, respectively, from the earthquake excitation with PGA = 0.01 g. Moreover, the maximum and minimum vertical displacement of the rail are 92.91 mm and 52.25 mm, respectively, from the ground motion with PGA = 0.40 g. Finally, an obvious bump can be observed from the excitation of PVY−H2 with PGA = 0.20 g. As shown in Figure 7 and Figure 11, this may be due to a resonance of the model being caused, resulting in a large vertical displacement amplitude of the rail.

In this paper, linear regression is applied to fit the logarithm of PGA to the logarithm of the vertical displacement of the rail [31]. The general linear regression fitted equation between the logarithm of PGA and the logarithm of vertical displacement of the rail is shown in Formula (1).

$$\ln \left(D_{\mathrm{d}}\right)=A+B\ln \left(\mathrm{PGA}\right)$$

(1)

where $D_{\mathrm{d}}$ is the vertical displacement of the rail (mm); PGA is the peak ground motion acceleration (g); and A and B are the coefficients of the fitted equation.

The fitted results of the linear regression are shown in

Table 5. Coefficients and R-square of the vertical displacement fitted equation of the rail.

Positions	Coefficients of Fitted Equation		R-Square
	A	B
0	5.145	1.008	0.9817
1	5.167	1.008	0.9847
2	5.167	1.010	0.9834
3	5.172	1.009	0.9844
4	5.166	1.010	0.9837
5	5.173	1.009	0.9843

, where R-square is the deviation coefficient indicating the correlation degree of the linear regression function. Coefficients A or B at six positions are comparable, and the R-square of the vertical displacement linear regression of the rail are all above 0.98, which indicates an excellent fitted result.

High-speed railways have a high requirement for smoothness. Therefore, the driving safety of the high-speed train may be significantly threatened by the slight displacement of the rail. Consequently, a 2 mm vertical displacement of the rail is selected as the allowable standard of the vertical displacement of the rail [6].

Exceeding probability curves of the vertical displacement can represent the probability that the dynamic vertical displacement of the rail exceeds the allowable standard from the action of earthquake excitations with different PGA. The vertical displacement $D_{\mathrm{d}}$ and bearing capacity $D_{\mathrm{c}}$ of the rail obey the lognormal distribution [21,31]. Thus the exceeding probability can be defined with a standard normal cumulative distribution function [21,31]. The calculation formula is shown in Formula (2).

$$P= \mathsf{\Phi }\left(\frac{\ln \left(\widetilde{D_{\mathrm{d}}}/\widetilde{D_{\mathrm{c}}}\right)}{\sqrt{{\beta }_{\mathrm{c}}^2+{\beta }_{\mathrm{d}}^2}}\right)= \mathsf{\Phi }\left(\frac{\ln \left({\mathrm{e}}^A{(PGA)}^B/\widetilde{D_{\mathrm{c}}}\right)}{\sqrt{{\beta }_{\mathrm{c}}^2+{\beta }_{\mathrm{d}}^2}}\right)$$

(2)

where $\widetilde{D_{\mathrm{d}}}$ is the average response of the structure, which can be calculated by the regression analysis; $\widetilde{D_{\mathrm{c}}}$ is the average structural total bearing capacity, which can be determined by the limit state of the vertical displacement of the rail; e is the natural constant; ${\beta }_d$ is the logarithmic standard deviation of structural reaction; ${\beta }_c$ is the logarithmic standard deviation of structural bearing capacity; and $\mathsf{\Phi }\left(x\right)$ is the standard normal cumulative distribution function. The values of the input parameters are shown in Table 5 and

Table 6. The values of some input parameters in Equation (2).

Positions	$\widetilde{{\mathit{D}}_{\mathit{c}}} \left(\mathbf{mm}\right)$	$\sqrt{{\mathit{\beta }}_{\mathit{c}}^2+{\mathit{\beta }}_{\mathit{d}}^2}$
0	2	1.1913
1	2	1.1895
2	2	1.1920
3	2	1.1904
4	2	1.1915
5	2	1.1904

.

The probability curves of the vertical displacement amplitude of the rail exceeding 2 mm at positions 0–5 are shown in Figure 13 under seismic excitation. The average probability of the vertical displacement of the rail exceeding 2 mm at six positions is comparable and approximately 44%, 89%, and 99% when the PGA is 0.01 g, 0.05 g, and 0.20 g, respectively. The PGA = 0.05 g is the peak ground motion acceleration corresponding to the areas with a seismic intensity of 6 degrees in the Code for Seismic Design of Buildings [33]. The probability of the earthquake-induced dynamic vertical displacement amplitude of the rail exceeding 2 mm increases with PGA.

The probability of the earthquake-induced vertical displacement with PGA = 0.01 g, 0.05 g, and 0.20 g of the rail exceeding 2 mm is shown in Figure 14, respectively. The exceeding probability of the rail above the mid-span is larger than that above the pier and the abutment, while the exceeding probability of the rail above the abutment is the minimum. Moreover, the exceeding probability of the rail at position 5, which corresponds to the position above the center of the bridge, is the largest among positions above the mid-span. Although there are differences in the exceeding probability of the vertical displacement amplitude of the rail at different positions, the difference is less than 2%.

7. Conclusions

In this paper, the IDA is applied to calculate the exceeding probability of rail displacement under different earthquake excitations. An FEM model of a high-speed railway track-bridge system is established. Based on the simulation, the characteristics of the vertical displacement of the rail under different seismic excitations are investigated, and the probability of the vertical displacement of the rail exceeding the allowable standard is calculated. The main conclusions include:

(1) The vertical displacement of the rail on the bridge is comparable under earthquake excitations; however, the vertical displacement of the rail above the abutment is significantly smaller than that above other parts of the bridge when the vertical displacement of the rail is negative. The variation trend changes with the time of the vertical displacement of the rail on the bridge are comparable under seismic excitations. The increasing trend of the vertical displacement amplitude of the rail is approximately linear. The peak vertical displacement amplitude of the rail is 2.06 mm when PGA = 0.01 g.

(2) The vertical irregularity of the rail caused by earthquakes has a wavelength close to the length of the simply supported girder. An “L” shape irregularity of the rail caused by earthquakes can be observed above the bridge-abutment joint when the vertical displacement of the rail is negative. It may indicate that the mid-span and the bridge-abutment joint are much more hazardous areas that threaten the safety of the high-speed train.

(3) Under some excitation, two bumps are observed in the Fourier displacement spectrum in the frequency range of 1.3–2.5 Hz and 10–12 Hz, respectively, which may indicate the resonance of the model to the excitation. The bump amplitude in the Fourier displacement spectrum in the frequency range of 10–12 Hz above the mid-span is larger than that above the pier. A bump in the frequency range of 1.3–2.5 Hz may be approximately the natural frequency of the lower vibration mode of the model, which may significantly change the amplitude and frequency of the vertical displacement of the rail.

(4) The vertical displacement amplitude probability of the rail exceeding 2 mm is 44%, 89%, and 99% when PGA = 0.01 g, 0.20 g, and 0.40 g, respectively. The exceeding probability of the earthquake-induced dynamic vertical displacement of the rail increases with the PGA. The exceeding probability of the rail above the mid-span is larger than that above other parts of the bridge, while the exceeding probability of the rail above the abutment is the minimum. Moreover, within the mid-span, the exceeding probability of the rail above the center of the bridge is the largest.

This study may be the first attempt to evaluate the earthquake-induced dynamic displacement of the rail based on IDA. However, there may be some shortcomings in this work: (1) The 2D FEM model is just a simply supported girder bridge with the CRTS II slab ballastless track structure, which is not a 3D solid model and does not include other complex bridge types; and (2) Those selected ground motions may not be rich enough. For example, pulse-type near-fault ground motions and far-fault ground motions are out of consideration in this paper.