Effects of near-fault pulse-type ground motion on train–track–bridge coupled system with nonlinear supports

Researches have revealed that the excitations of near-fault pulse-type (NFPT) earthquakes lead to significant challenge of high-speed railway. This paper mainly focuses on the dynamic responses of train–track–bridge coupled system (TTBCS) under NFPT ground motions. An innovative effective TTBCS model (TTBCM) considering nonlinearities of supports is proposed herein, and with that model, numerical simulation and data analysis are conducted to reveal the potential effect on TTBCS arisen by NFPT ground motion from both qualitative and quantitative perspectives. Conclusions can be summarized below: The comparison between linear model and nonlinear model reveals the necessity of nonlinearity when conducting TTBCS numerical simulation under NFPT ground motions. Moreover, the analysis of pulse parameters of NFPT ground motions reveals that the relationship between dynamic responses of train group and amplitude of velocity pulse is approximately to the linear positive correlation relationship, while the dynamic responses of train group and period of velocity pulse exhibits negative correlation as inverse proportional function. Based on parameters analysis, a multiple-parameter regression between spectral intensity and pulse parameters is conducted, with the combination of running safety assessment (RSA) criteria including Nadal, wheel unloading, and wheel lift, suggestion from RSA perspective under NFPT ground motions are proposed as well.


Introduction
On January 8, 2022, an earthquake with 6.9 magnitude struck Menyuan in Qinghai province, China, the severe damage of HSR bridges and equipment directly led to temporarily suspension of operation of Lanzhou-Tibet high-speed railway (HSR). With rapid development and construction of HSR all over the world, the HSR damage or casualties caused by earthquakes are seem to be inevitable. Hence, the safety of HSR under earthquakes, especially for the HSR lines in the adjacent of typical seismic belt, should be regarded as an issue with significance in infrastructure development.
In recent years, the engineers and scholars have been diligently making contribution to seismic safety of HSR, and basically, the past researches can be roughly divided into two categories, which are the earthquake resistance of HSR bridges and trains running safety assessment (RSA) under seismic excitation. The seismic dynamic investigations of different bridge types are abundant and mature [1][2][3][4], which provides theory basis for the design concept of 'replacing roads with bridges,' and in these variety of bridge types, the simply supported bridges composite the highest proportion in the construction of HSR bridges. The researches of HSR simply supported bridge (HSRSSB) under earthquakes can be dating back to the 1990s. Anderson discussed the potential reasons that led to partial collapse of the Shinkansen HSRSSB under Kobe earthquake, it was found that the brittle manners of these collapsed bridges were mainly determined by the deficiencies of confinement and shear reinforcement [5]. Nowadays, the relevant researches for different components of HSRSSB have been gradually in-depth, which covers the seismic performance of superstructure, piers, interlayer components, as well as supports [6][7][8]. These comprehensive researches as well as the technology innovation of seismic isolation [9][10][11] make the resistance of HSRSSB reach to a new level.
From another perspective, researches aiming to ensure safety of trains running under earthquake are pushing forward simultaneously [12][13][14][15][16][17]. Yang and Wu [18] preliminarily proposed a safety boundary of operation speed of train under four representative seismic ground motions with different peak ground accelerations (PGAs). And in this research, the derailment criterion was based on wheel-set lateral to vertical force ratio (YQ ratio). According to a continuous girder bridge with seven spans, Xia et al. [19] built a train-bridge coupled model. In the numerical simulation, the ground motions were imposed on model as non-uniform excitations. And it was concluded that neglect of the traveling wave effect may lead to unsafe simulation results, while the train speed should be taken into account when conducted RSA due to its significant influence on dynamic response of the train vehicles. Based on the concept of energy balance, Luo proposed spectral intensity (SI) for RSA index. The SI was the integration of spectral velocity with periods ranging from 0.1 to 2.5 s. Moreover, to validate the feasibility of SI, Luo compared the accuracy and stability of RSA with, respectively, utilization of SI and peak velocity. [20,21]. Full-scale experiment conducted by Nishimura et al. [22] revealed that under seismic excitations, higher operation speeds leaded to higher risk of running train due to faster start of wheel slide. Montenegro et al. [23] discussed the jeopardy of running trains under moderate earthquakes with high probability of occurrence, and proposed running safety charts with the combination of return periods of earthquake and operation speeds. Jin et al. [24] took the contribution of vertical ground motion component into account when conducted vehicle-rail-bridge coupled system simulation, and found that the negative impact of vertical ground motion components on the train running safety on bridge is significant and non-negligible. Zeng and Dimitrakopoulos [25] imposed twenty sets of historic records of earthquake into the vehicle-bridge interaction system based on a curved two-way railway bridge prototype. The simulation results revealed that even frequent earthquakes with moderate intensity can make running trains be at risk. Moreover, the position of the running vehicles on bridge significantly affected the dynamic response of the vehicle under seismic excitation as well. However, different with seismic resistance of HSRSSB investigation, only few researches of RSA [14,16] aforementioned considered nonlinearity of bridge components in their corresponding train-bridge models. Whether and how the nonlinearity of bridge components affects the train running are still need further investigation.
Among the researches of seismic safety of HSR, there is a special type of earthquake that are notorious due to its extreme destructiveness to structures, which is near-fault ground motions [26][27][28]. In June 2021, the HSR successfully entered the Tibet region in China, as the HSR lines there are close to the Himalayan seismic belt and Qinghai-Tibet plateau seismic belt, the HSR safety and RSA under near-fault ground motions become a new issue with significant research value [4,13,14,29]. In order to improve the theoretical system of HSR safety under near-fault ground motions and to offer a glimpse into features of running trains under near-fault ground motions from RSA perspective, this paper proposes an innovative train-bridge coupled system with nonlinear bridge components taken into account. Based on the model and numerical simulation, the significance of nonlinear bridge components in the train-bridge coupled system under NFPT ground motion, as well as the potential effects on dynamic responses of train running, will be investigated and discussed. Moreover, suggestion from RSA perspective will be attempted to propose when NFPT ground motions are imposed as excitations.
2 Numerical model for TTBCS

Basic theory and dynamic equation of train-
track-bridge coupled model (TTBCM) In this paper, the three-dimensional TTBCM based on high-speed railway simply support bridge (HSRSSB) is constituted with train model and track-slab-bridge structure (TSBS), where one train is simulated with a mass-spring-damping system with 31°of freedom (DOF), which includes the DOF of lateral, vertical, vertical, rolling, pitch, and yawing of car-body and of two bogies, as well as lateral, vertical, and yawing of four wheel-sets. The connections between car-body and bogies, as well as bogies and wheel-sets are the secondary suspension and primary suspension, which are modeled as springs in model. Given springs connected and 31 DOF of train, the dynamic equation of the train can be expressed in accordance with the principle of energy variation.
In Eq. (1), the subscript T denotes those parameters or constants are describing dynamic characteristics of the train. The M and C are mass and damping matrix, and stiffness, while € X, _ X, X, and F are vectors of acceleration, velocity, displacement, and force, respectively. The F R X T ; _ X T À Á is the resilience force of train, and if the train model is assumed to be linear, this matrix is equal to K T X T , where K T is the stiffness matrix of train.
Same to the dynamic equation of the train, the dynamic equation of the TSBS shown as Eq. (2) can be formulated on the basis of the finite-element theory and principle of energy variation, where subscript B denotes the matrixes and vectors are derived from TSBS. The C B is solved by Rayleigh damping with damping ratio equaling to 0.3. The Rayleigh damping is widely adopted in simulation of train-track-bridge system and HSR bridge system, even for those models in which nonlinear stiffness is taken into account, Rayleigh damping is still the damping choice [14,[30][31][32]. Noteworthily, when nonlinear connection is considered and is simulated as nonlinear spring in the TSBS, the model of resilience model and the detailed form of F R X B ; _ X B À Á are based on the choice of hysteresis rules (e.g., Bouc-Wen model [33] and Ozcebe model [34]). The coupling between train and TSBS are based on wheel-rail contact with referring to knife-edge contact constraints, which can be found in past research conducted by Zeng [35] and Liu [36]. For wheel-rail contact forces, mainly three kinds of forces are considered herein, the lateral contact force is calculated with stiffness is 1:617 Â 10 7 N/m [37], the normal force calculation is on the basis of Hertz contact theory [38], and the creep force are assumed as Kalker model [39]. In this way, Eqs. (1) and (2) are coupled and are as the dynamic equation group of the TTBCM.

Closed-loop model
The closed-loop model utilized herein is based on the past research conducted by Jiang [12] and Zhou [31], which is aimed to enhance calculation efficiency and reduce total DOF of model by boundary condition adjustment of the middle span structures. As shown in Fig. 1a, with a pseudo-element being introduced and coupled in corresponding position, the mass and stiffness matrices of longitudinal components including rail and CRST II track slab are rewritten. In this way, the entire TSBS can be regarded as a circle where train group can continually and cyclically operate, like shown in Fig. 1b. The connections between girder and pier are shown in Fig. 1c, compared with conventional model, the pier (drawn in dotted line) at the end of right side is omitted, and the supports at end of the girder are connected with the pier at the left side.
The span number of closed-loop TTBCM is preliminarily determined by following the principle of avoiding superposition of vibrations arisen by train group repeatedly operating through. As the transverse dynamic response caused by train operation is significantly smaller than that in vertical direction, the vertical displacement of girder is set as criterion for span number determination. In the case of eight trains in train group with 350 km/h operation speed, the span number of closed-loop TTBCM is set as seventeen, and the time-history of vertical girder displacement is shown in Fig. 2.
Herein, as ensuring the running safety of trains under transverse ground motions is essential requirement of HSR structures [21], the validation of feasibility of closed-loop model under transverse seismic excitation is demonstrated below. The span number of conventional TTBCM is fifty. The displacement time-histories of car-bodies under transverse seismic excitation of two kinds of models are approximately in Fig. 3, and the slightly deviation is due to different boundary condition given by subgrade of conventional model. When the train group are running into the middle span, the displacement timehistories of both models are in consistent gradually (between the fifth second and the sixth second). Thus, conclusion can be made that the closed-loop model is feasible for train-track-bridge coupled system modeling under transverse seismic excitations.

TTBCM under NFPT earthquakes considering supports nonlinearity
In this section, the dynamic responses of TTBCM under near-fault ground motions with pulse effect are investigated and discussed. The pulse portion of near-  fault ground motions can be regarded as velocity waves, the potential impacts on HSR running safety arisen by them are key point of this section. To obtain significant conclusion and make suggestion for HSRSSB design and construction procedure under NFPT ground motions from running safety perspective, two pulse parameters, which are A (amplitude of velocity pulse) and T p (period of velocity pulse), are set as research variables.

Synthesis method of NFPT ground motions
Due to scarcity and stochasticity of recorded NFPT ground motions, for pulse parameters A and T p , it is difficult to separately investigate their independently effects on HSR running safety. Hence, an artificial near-fault records (ANRs) synthesis method based on research conducted by Mavroeidis [40] is introduced. This ANRs synthesis method can be regarded as a simple and effective analytical model for numerical simulation of near-fault ground motions with impulsive character. As the impulsive character of NFPT ground motions is represented as pulses in the timehistory velocity of the seismic records, the purpose of ANRs synthesis method is to accurately simulate and represent the time-history velocity of original records (ORs) of NFPT ground motions with simple mathematical expression.
Referring to research of Mavroeidis [40], totally ten typical ORs of NFPT ground motions, including Imperial Valley, Northridge, Tabas, San Fernando, etc., are gathered from Pacific Earthquake Engineering Research Center (PEER) [41]. These ORs with basic information are listed in Table 1.
v g ðtÞ ¼ v g;BGR ðtÞ þ v g;PTR ðtÞ ð 3Þ ; with ; c [ 1: According to Eq. (3), the velocity time-history of ORs can be decomposed into two parts, which are high-frequency background records (BGRs) and lowfrequency pulse-type records (PTRs). To extract the velocity pulse, Butterworth fourth-order low-pass filter with cut-off frequency calculated through Eq. (4) is utilized. a here is an empirical coefficient determined by trial and error to ensure the period of extracted velocity pulse equal to pulse periods given in Table 1. Once pulse extraction is finished and PTRs are obtained, the artificial PTRs (APTRs) can be obtained by synthesis procedures in accordance with Eq. (5). The fitting degree of synthesis is controlled by five synthetic parameters, which are A p , the amplitude of artificial time-history velocity; f P , the prevailing frequency; t 0 , time of the envelope's peak; c, the parameter to define the oscillatory character; and v, the phase of the amplitude-modulated harmonic. According to fitting criterion given by Mavroeidis [40], five parameters should be adjusted to make sure the pseudo-velocity response spectra of PTRs and APTRs have same peak pseudospectral velocity, and the peak value exhibits at the same natural period, while the pulse velocity time-history of PTRs and APTRs should be in consistency, as well. At last, the APTRs can be combined with BGRs to get ANRs. After parameters optimizing, the fitting results of ANRs, including velocity time-history comparison and pseudo-velocity response spectra comparison between PTRs and APTRs, as well as acceleration response spectra comparison between ORs and ANRs, are shown in Fig. 4. The value of a and the five synthetic parameters aforementioned are listed in Table 2.
The A and T p , which are the two pulse parameters needed investigate and discuss, are listed in Table 2 as well. To realize the single-variable control, the synthetic parameters are adjusted to make the value of A and T p variate separately. Basically, according to Eq. (5) and the research of Mavroeidis [40], the amplitude of velocity pulse A is affected by synthetic parameters A p only, while the f P is directly control the pulse T p . The shape of velocity pulse oscillations is affected by the synthetic parameters c and v being constant. This indicates for instance, the ANRs with only parameters A as single variable can be obtained by only adjusting the values of synthetic parameters A p after the values of other synthetic parameters have been determined. Take E1 for instance, the ANR that is in consistency with E1 OR has 67.37 cm/s for A and 4.13 s for T p , and in this paper, this ANR is recorded as E1 -Base . Meanwhile, the synthetic parameters are adjusted to get ten different ANRs, their A value changing from 33.69 to 101.06 cm/s (ranging from 50% Â 67:37 to 150% Â 67:37 cm=s with 10% interval), and their T p value are the same, 4.13 s. All these ANRs with only variation of A are recorded as E1 -A-No. (i.e., E1 -A-1 is the ANRs based on E1 with 33.69 cm/s amplitude of velocity pulse, E1 -A-10 is the ANRs based on E1 with 101.06 cm/s A). The synthesis procedures of T p variation ANRs are the same as aforementioned (ranging from 50% Â 4:13 to 150% Â 4:13 s with 10% interval), and the ANRs are recorded as E1 -T-No. . Herein, totally 210 ANRs with pulse effect are obtained and are imposed on TTBCM as excitations for dynamic simulation.

Significance of nonlinearity of supports under NFPT ground motions
Herein, Chinese HSR EMU CRH2 train group constituted with eight trains is adopted in the threedimensional TTBCM, and the operation speed of train is 350 km/h. Properties of CRH2 train group are listed in Table 3. The TSBS is based on the prototype of HSR simply supported bridge with CRTS II track plate, which is nowadays widely adopted for HSR design in China, and the cross section of girder and pier can be found in research conducted by Zhou [31]. The TSBS is modeled as four-layer beam with 32.5 m per span, as shown in Fig. 5, and based on the finite-element theory, the element length is set as the distance interval between fasteners, 0.65 m. The interlayer component including fasteners, CA mortar layer, sliding layer, and shear alveolar are simulated as springs. The shear keys, based on the relative research [8], which are aimed to limit the transverse displacement of girder, are set as spring with 3-cm gap between pier and      . Supports connection are adopted between pier and girder, according to the support-arrangement shown in Fig. 6, the supports modeling is simplified. In transverse direction, each side of beam is connected with girder by one fixed support and one movable support. Herein, the fixed support damage under seismic excitation is taken into account, which means, the fixed supports are simulated as nonlinear springs, the displacement-force curve of fixed supported is shown in Fig. 7, while the displacement-force curve of suspensions is shown in Fig. 8, which are set as piecewise linear stiffness. As aforementioned, due to high computational memory cost and low simulation efficiency caused by nonlinear spring modeling, some researches considering only linear springs or omitting some components of TSBS to realize the simplification of TTBCM, hence, hereinafter one comparison   It is obvious that from girder displacement at midspan shown in Fig. 9a, the adoption of nonlinear fixed supports in TTBCM has significantly effects on simulation result of track-bridge structure. Due to stiffness degradation of fixed supports (springs into nonlinear stage), the maximum value of girder exceeds that when linear springs is utilized. The positive maximum of nonlinear condition reach to 19.9 mm, while linear condition is 14.0 mm only, the negative maximums of these two models also have a difference of 4.3 mm (deviation is reach to 37.4%), the relatively small dynamic response of linear condition compared to nonlinear condition can lead to high safety thresholds in HSR design and result in safety hazards of structures. Moreover, due to the pulse effect of ANR E2 -T-1 , the displacement time-history after the fifth second of nonlinear condition exhibits unidirectional oscillation, which means the oscillation axis is not the zero axis, while with the ground motion receding, the girder displacement does not converge to zero value, but generate a residual displacement of 6 mm. In contrast, these phenomena are not represented in the linear condition TTBCM. The dynamic properties of girder aforementioned transfer to the rail through interlayer components, and then to the train group through wheel-rail interaction, which is shown in Fig. 9b. Similar to the girder displacement comparison, deviation of first car-body between linear and nonlinear condition is exhibited, as well. The positive maximum of nonlinear condition is 8 mm larger than that of linear condition, while negative maximum has a difference of 9.9 mm compared with linear condition. The unidirectional deviation of car-body displacement time-history is caused by velocity pulse and fixed supports damage, simultaneously, residual displacement of girder is reflected in Fig. 9b from the twentieth second to the end. Figure 10 illustrates three kinds of assessment criteria of train running safety, which are Nadal, wheel unloading, and wheel lift, respectively. The maximum Nadal value of nonlinear condition is 1.15, while of linear condition is 1.09 only, and the timehistories from the fourth second are not in consistent.
The maximum values of wheel unloading and wheel lift are close; however, it can be obviously found that although the ground motion has receded after the twentieth second, the wheel lift comparison shown in Fig. 10c still demonstrates the deviation between nonlinear condition and linear condition; this is  increasing, the wheel unloading maxima of nonlinear condition increase rapidly than that of linear condition, and when the change rates of A exceeds 10%, the wheel unloading of nonlinear TTBCM exceeds that value of linear one simultaneously. As from Nadal perspective, the result is distinct, with higher Nadal values of nonlinear condition, TTBCM considering nonlinear supports reveals that the train group is at a significant risk of derailment during NFPT seismic excitations, while TTBCM with linear supports may underestimate the probability of derailment with relatively smaller Nadal values compared with nonlinear condition.
With combination of time-histories and maxima comparison, it can be concluded that under transverse excitation of NFPT ground motions, considering nonlinear processing of supports is necessary. From TSBS perspective, TTBCM with nonlinear springs is more consistent with dynamic characteristics of HSR bridge in reality and can better analyze and assess the pulse effect on TSBS (larger maximum value of displacement due to stiffness degradation and residual displacement due to supports damage). From train RSA perspective, adoption of linear springs only leads to problems such as distorted wheel unloading curves, relatively small calculated values of Nadal and relatively small wheel lifts due to neglected seismic rail irregularities compared with nonlinear model. These may result in higher safety thresholds of HSR design and create unnecessary safety hazards. Thus, herein the following simulation will base on the TTBCMs with nonlinear processing of fixed supports.

Analysis of effects on train operation arisen from pulse parameters
In this section, to directly analyze the effects of pulse parameters A and T p on the train running over HSRSSB, some typical simulation cases are chosen. For these cases, time-history dynamic responses of important TTBCM components (e.g., girder, carbody, and wheel-set) are demonstrated and discussed.

Time-histories comparison and investigation
For parameter A, cases of ANRs E6 -A-1 and E6 -A-10 being imposed on TTBCM with T b = 0.904 s are gathered and compared. As aforementioned, these two ANRs are based on the same OR E6, with the same T p values but different amplitudes of velocity pulse, A = 12.46 cm/s for E6 -A-1 while A = 37.37 cm/s for E6 -A-10 . Figure 12a is the acceleration time-histories of E6 -A-1 and E6 -A-10 , while Fig. 12b is showing the detailed comparison of these two ANRs. According to the ANRs synthesis method, the variation of velocity time-histories and acceleration time-histories occur only in time period from t 0 À 1 2 c Á T p to t 0 þ 1 2 c Á T p as shown in Fig. 12b. Therefore, it is obvious that the dynamic response difference simulated by TTBCM in this time period is directly derived from the variation of pulse parameter A, which is the focus of discussion hereinafter.
As being derivative of velocity, acceleration timehistories are affected by variation of amplitude of velocity pulse, in other word, ANR E6 -A-10 has It can be concluded that higher A value leads to larger displacement response of TSBS. During the time period from 4.76 to 6.52 s, totally three groups of data are compared, including two positive value groups (crest) and one negative value group (trough). For crest near the fifth second, the displacement under E6 -A-1 is 13.33 mm, while that under E6 -A-10 is 16.49 mm, for crest near the sixth second, the displacement under E6 -A-1 is only 53.9% of that under E6 -A-10 , and for the wave trough, the absolute displacement value of E6 -A-0 is 16.39 greater than E6 -A-1 . Moreover, similar to the situation of girder, higher A value results in larger deformation of fixed supports as well. From Fig. 14, it is known that the deformation variation range under E6 -A-10 is from -14.54 to 12.78 mm while that under E6 -A-1 is -5.92 to 9.87 mm only.
For acceleration time-histories of the first car-body shown in Fig. 15a, the positive value (crest) variation will be discussed. In general, the occurrence tendency of crests and troughs of acceleration of car-body is basically consistent with that of acceleration of ANRs, the first crest shown in Fig. 15a occurs nearly the 5.64 s, when the acceleration of E6 -A-10 reaches to peak value. However, the crest value under E6 -A-10 is 15.28 m/s 2 , while under E6 -A-1 is only 9.88 m/s 2 . While from the 6 to the 6.52 s, there is another crest with value of 13.51 m/s 2 under E6 -A-10 , but this situation is not exhibited into the car-body acceleration time-history under E6 -A-1 . At the point of time corresponding to the occurrence of the crests in acceleration of first car-body, the troughs of displacement time-histories of the first car-body occur, as shown in Fig. 15b. The absolute displacement values of 88.65 mm and 90.59 mm under E6 -A-10 is significantly greater than the value of 51.43 mm under E6 -A-1 .
Thus, from the comparison demonstrated from Figs. 13, 14 and 15, it is found that with the increase in A, the dynamic response of both TSBS and running train is increase simultaneously. To get further verification, the comparisons of wheel-rail contact force and related running safety criteria are shown below.  [42], higher lateral contact force leads to higher Nadal value, as results, the crest Nadal value under E6 -A-10 is 1.79 and 2.03, while that under E6 -A-10 is 0.84, which can be found in Fig. 17a.  For Fig. 17b Fig. 18, it is found that the acceleration of E6 -T-1 , whose T p value is relatively smaller than the other, has larger maximum acceleration value (also called peak ground acceleration, PGA). This is opposite to the variation tendency given by research of parameter A.
As for the dynamic response of TSBS shown in Figs. 19 and 20, the positive maxima of girder displacement at the mid-span are 17.95 mm under E6 -T-1 and 13.05 mm under E6 -T-10 , respectively, while the absolute negative maxima of girder displacement are 26.00 mm and 8.20 mm. With the increase in swing amplitude of the girders, the deformation range of the fixed supports increase simultaneously, which can be got from Fig. 20. The support deformation under E6 -T-1 is from -21.4 to 17.24 mm, while that under E6 -T-10 is only -3.97 to 9.09 mm. The ANR whose T p value is relatively smaller than the other leads to larger dynamic response of train group as well. The maximum values of acceleration and displacement of the first car-body under E6 -T-1 are significantly greater than that under E6 -T-10 , which can be easily observed from Fig. 21. For instance, the positive maximum of displacement of car-body under E6 -T-10 is 28.31 mm, which is only 55% of the value under E6 -T-1 .
It is found similar with the situation of dynamic response aforementioned when simulation result from wheel-rail interaction perspective is being investigated. When the displacement of the first carbody reaches the negative maximum and produces a positive movement tendency, the positive maximum of lateral contact force of the left side wheel-set appears. As shown in Fig. 22a Thus, conclusion can be got here that being opposite to parameter A, dynamic response of both TSBS and operation trains are larger when T p value is smaller, and with other pulse parameters being same, being imposed under seismic excitation with smaller T p will lead to higher risk of derailment due to higher running safety criteria.

Variation tendency and sensitivity analysis
In Sect. 4.2, the conclusion about dynamic responses of TTBCM under seismic excitations with pulse parameters is based on the comparison between ANRs having maxima and minima of A and T p , respectively, but have the same BGR. This kind of comparison only exhibits simple numerical relationships between pulse parameters and dynamic responses, but the variation tendency of dynamic responses under different pulse parameters values is still vague only through the cases of E6 -A and E6 -T . Numerous simulation data are needed to get the target conclusion about variation tendency; thus, for all the four TTBCMs with different first natural periods (T a , T b , T c , and T d ), 210 ANRs as aforementioned are imposed on these models, The assessment criterion average variation (AV) of dynamic responses of train group is calculated by Eqs. (6) and (7). Five typical kinds of dynamic responses are chosen for variation tendency analysis of train group under ANRs with different pulse parameter values, which are maximum value of acceleration, velocity, and displacement of car-body, as well as lateral contact force and vertical contact force, these maximum values are collected after comparing all corresponding values of train group (e.g., for acceleration of car-body, the maximum is chosen from all eight car-bodies in the train group) and is denoted as R 1n ; R 2n ; R 3n ; R 4n ; R 5n ð Þ , where n denote the number of ORs. The superscripts A or T p denote the response values are collected under variation of A or T p , the response values B 1n ; B 2n ; B 3n ; B 4n ; B 5n ð Þ are calculated by TTBCM under ANR E n-Base , and E is the total ORs in this paper, which is ten. The results are shown in Fig. 24. With the change of A values increasing from ? 50 to ? 50%, the AV A of four kinds of first natural periods increases simultaneously, and the variation relationship between change percentage of A and AV A is approximately to positive linear relationship. However, it is found that when change percentage are negative values, the slopes for all kinds of first natural periods are relatively smaller than the slopes of change percentage are positive values, the potential reason is that the decreasing pulse amplitude indicates the pulse effect of ANRs are being weaken, the dynamic responses of train group are beginning to be dominated by BGRs, which makes the effects on the AV A less obvious, even though the change percentage of A remains to be the same.
As for T p , the values are increasing with interval of 10% of the base value (T p of E n-Base ), but the AV T p are decreasing gradually. Different with Fig. 24a, the decreasing tendency of T p is nonlinear. Take the results of first natural period T b , when T p value is only 50% of base value, the AV T p is 111.89%, and decreases to 58.21% being similar an inverse proportional function when change percentage of T p reaches to 50%. Similar to situation revealed by parameter A, when the T p increasing, the decrease amplitudes of AV T p of four first natural periods are smaller simultaneously, the reason is that the BGRs begin to being the main part of ANRs which make attribution to dynamic response, just as aforementioned.
Thus, it is concluded that generally speaking, the dynamic responses of train group under NFPT ground motions have positive linear relationship with the pulse parameter A, but inverse proportional function relationship with the pulse parameter T p . Not only the variation tendency relationship, but the sensitivity of dynamic responses of train group to these two pulse parameters are investigated as well.
Referring to the local sensitivity analysis conducted by Ma [43] e R , root-mean-square rate error is defined and calculated by Eq. (8) and Eq. (9). For all curves illustrated in Fig. 25, the maximum of e R , e R , are gathered and set as the criterion for sensitivity comparison. The sensitivity analysis diagraph is shown in Fig. 25, and the values of e R are listed in Table 5. It is found that the dynamic responses of train groups running over TSBS with the first natural period of T b have the higher sensitivity to pulse parameter T p due to its maximum e R value, while pulse parameter A with first natural period of T d has the minimum e R value and is the parameter whose sensitivity level is the lowest. Overall, the e R values of T p under all four kinds of first natural periods are larger than e R values of A. Conclusion can be got that the dynamic response of train groups are relatively more sensitive to pulse parameter T p than pulse parameter A. But the potential relationship between sensitivity and the natural periods of TSBS is obscure, more simulation comparison and further research are needed to be conducted.

RSA and suggestion
In this section, the analyses between intensity measures (IMs) and running safety criterion are conducted based on the numerical values of Nadal, wheel unloading, and wheel lift. Moreover, suggestions of HSR construction with the consideration of NFPT ground motions will be proposed as well. Hereinafter, the simulation results of all 210 ANRs imposed on TTBCS with the first natural period T a = 0.455 s are illustrated as example.
Referring to probabilistic seismic demand model (PSDM) conducted by Jin [13], linear regression based on Eq. (10) are conducted between the three kinds of RSA criteria and IMs including PGA, PGV, PGD, PSA, PSV, PSD and spectrum intensity (SI).
where C denotes the criterion, a and b are regression parameters, and IMs are spectrum intensity candidates. However, it can be found from Fig. 26, due to high dispersion of data point, the R 2 values, which describes the degree of fitting between IMs and criteria, are not high for all three diagraphs in Fig. 26. Moreover, the separately utilization of IMs does not reflect the potential relationship between pulse parameters (A and T p ) and RSA criteria. Thus, a multiple-parameter regression is conducted here to obtain expression of RSA criteria with the combination of IMs and pulse parameters, and the main  procedures of the multiple-parameter regression are described below.

Choosing of IMs
The choice of IMs is based on the results of linear regression aforementioned. As listed in Table 6, acceleration type of IMs and SI has relatively high fitting degree with Nadal and wheel lift than other IMs, and for wheel unloading, PSA, PSV, and PGA are the top three IMs due to R 2 values. To get comprehensive choice, the IMs are ranked according to the values of R 2 , and the sum of rank in Table 6 is the basis for IMs selection with all three RSA criteria. For instance, from the sum of rank results, the PGA is the optimal choice of IM due is smallest sum of rank. Herein, the top three IMs are chosen according to sum of rank for next step, which are PGA, PSA, and SI, respectively.

Calculation of constants
The chosen IMs are assumed as linear relationship with RSA criteria and are substituted into Eq. (11).
where k 1 , k 2 , k 3 , and k 4 are four constants needed to be solved. Noteworthily here, according to variation tendency observed in Fig. 24, the parameter A are assumed as linear relationship with RSA criteria, but for f T p À Á , two forms of candidate functions are  considered, which are exponential function and inverse proportional function. After preliminary calculation, inverse proportional function is chosen and only final results based on inverse proportional function are illustrated here. The results of multipleparameter regression and values of constants are shown in Table 7.

Final expression of multiple-parameter regression
It is obvious from  Table 7, expression with the basis of IM SI is chosen. The diagraphs of final expression are shown in Fig. 27.

Final expression utilization in HSR design
The final expression can be somehow regarded as preliminary calculation formula and be utilized with relatively standard. Take Chinese HSR design code TB 20,002-2017 [44] for instance, the Nadal limit value is 0.8 and the wheel unloading limit value is 0.6, while for wheel lift, according to geometry of wheelset, the limitation of wheel lift is 28 mm [13]. Under the scenario of one HSRSSB with first natural period of 0.455 s, and the operation speed of train group is 350 km/h, the engineers can pulse parameters A and T p , and SI values of NFPT ground motions recorded by the stations in the adjacent of the target construction site. After the parameters are gathered, the C values can be preliminarily calculated and compared with corresponding RSA criteria limit values. Once the calculated C values exceeding the limitation, which indicates train group are at the risk of derailment on this HSRSSB under NFPT ground motions, special consideration such as operation speed decrease or additional seismic isolation design are necessary. Based on the procedures described above, the constants of multiple-parameter regression based on SI of all four kinds of first natural period are listed in Table 8.

Conclusion
HSR, as a convenient and efficient form of transportation, has become a hot topic in both engineering and scientific perspectives while developing rapidly around the world, especially the safety of structures and of running trains under extreme excitations. And in recently decades, the extensive HSR lines constructions make it inevitable that some HSR structures are threatened by NFPT ground motions due to their locations being adjacent to the seismic belt zones. Hence, it is urgent to conduct researches of HSR under NFPT ground motions from RSA perspective. In this paper, an innovative TTBCM taking nonlinearity of supports into account is adopted for investigation of (1) Consideration of nonlinearity of supports in TTBCMs are necessary for dynamic simulation of train-track-bridge coupled system under transverse NFPT ground motions. The adoption of ideal linear supports indicates the damage of supports are omitted, which leads to simulation results including dynamic responses of TSBS and train group deviating from actual situation. Moreover, unidirectional oscillation of TSBS and train group, as well as residual displacement of TSBS, is obtained only when nonlinear supports are adopted in the model, which significantly affect the RSA. (2) The relationship between dynamic responses of train group and pulse parameter A exhibits positive correlation and is approximately to the linear relationship, while for dynamic responses of train group and pulse parameter T p , the relationship exhibits negative correlation and can be regarded as inverse proportional function. Moreover, with the weakening of pulse effects of NFPT ground motions (decrease in A or increase in T p ), the effects on dynamic response of train group arisen by variation of pulse parameters are not significant, it is due to the responses being domination by BGRs of NFPT ground motions.  (3) A multiple-parameter regression is conducted to obtain the expression based on SI and pulse parameters, which is able to calculate three different RSA criteria including Nadal, wheel unloading, as well as wheel lift. In accordance with these, engineers can preliminarily estimate the feasibility and safety of HSR construction from the perspective of RSA perspective.