Vertical stiffness limit of 400 km/h high-speed railway simple-supported bridge considering excitation randomness

Abstract

Vertical stiffness of the bridge is of paramount importance in guaranteeing optimal driving performance for high-speed railway (HSR) trains. Initially, the Auto Regression and Moving Average model with eXogenous input (ARMAX) model is employed as the surrogate model for the train-bridge (TB) coupling system. Then, the framework for analyzing vertical bridge stiffness is proposed employing the surrogate model, and the relationship between the train response affected by excitation randomness and the vertical stiffness indicators of the bridge is established. Finally, the influence of various factors such as train speed, train type, car body mass, and bridge span on stiffness limit of 400 km/h HSR simple-supported bridge (SSB) is examined. The results indicate that the surrogate model offers a notable advantage in computational efficiency, while maintaining a satisfactory accuracy in predicting the train response. With higher train speeds, lighter car body masses, and longer spans, the demand for stricter bridge stiffness limits becomes more pronounced. Based on the driving performance of the train types analyzed, the recommended stiffness limit is proposed for the prestressed concrete simply supported beam (SSB) bridges used in HSR, with a span length of less than 40 m and operating at a speed of 400 km/h.

Keywords

vertical stiffness excitation randomness surrogate model high-speed railway bridge train-bridge coupling

Introduction

In recent years, with the construction of the Chengdu-Chongqing high-speed railway (HSR) and the Moscow-Kazan HSR, the development of 400 km/h HSR has become a priority. As the train speed increases, the coupling vibration response between the train and the railway substructure becomes more prominent. To ensure the optimal performance of the train, the smoothness requirements of the track need to be further enhanced (Yang et al., 2021; Zhai et al., 2023). In the railway network of China, bridges account for an average proportion of over 50% and serve as crucial substructures supporting the tracks. In the design of railway bridge structures, controlling the vertical stiffness of the bridge is typically necessary to ensure the smoothness of the track, the safety of train operations, and the comfort of passengers.

The bridge design codes of various countries, including Korea, Japan, China, and Eurocode, have established specific provisions for the vertical stiffness limits of bridges. However, the current design codes are applicable only for maximum train speeds of up to 360 km/h. Further research is needed to establish the vertical stiffness limits for bridges operating at higher train speeds. Researchers commonly employ the train-bridge (TB) coupling vibration analysis method to investigate the relationship between bridge vertical stiffness and the driving performance of high-speed wheel-rail trains (Li et al., 2020; Zhai and Wang, 2012), light rail trains (Wang et al., 2016), medium-low-speed maglev trains (Li et al., 2020; Wang et al., 2020), and some simplified train models (Jeon et al., 2016). Track irregularities constitute one of the primary excitation sources for the TB system, and their inherent randomness leads to stochastic dynamic responses for both the train and the bridge. The research (Xiang et al., 2022) employed an amplification factor to account for the influence of excitation randomness on the dynamic response of trains. Nonetheless, using the amplification factor to analyze the influence of excitation randomness is a rough approach, which can affect the accuracy of determining the bridge stiffness limits based on the dynamic response of the train.

Conducting the random vibration analysis of TB coupling system is essential to ensure the accuracy of bridge stiffness evaluation. The direct Monte Carlo simulation (MCS) approach, employing the motion equations of the TB coupling system (Rocha et al., 2014; Salcher et al., 2019), enables the assessment of the train’s dynamic response considering track random irregularities. This method often suffers from poor calculation efficiency. Alternatively, several random vibration analysis methods have been developed, including the pseudo-excitation method (Wu and Zhang, 2022), the probability density evolution method (Mao et al., 2021), the explicit time-domain method (Yu et al., 2021). These methods exhibit high efficiency in obtaining probabilistic information about the dynamic response of the TB coupling system, such as mean values, standard deviations, or the probability density function. Due to the inherent nature of the random vibration response of the TB coupling system, it does not strictly conform to a specific distribution. As a result, accurately estimating the extreme value of the response at the tail of the probability distribution can be challenging (Xiang et al., 2020).

The surrogate model utilizes a mathematical model as a substitute for the TB motion equations, resulting in a significant improvement in solution efficiency. Deep learning based surrogate models were utilized to investigate the impact of random track irregularities on the dynamic response of both the train and the bridge (Li et al., 2021, 2023). The time-varying autoregressive with exogenous variable (TVARX) model was utilized to estimate the TB dynamic interaction effect of a train passing over a bridge (Matsuoka et al., 2020). The neural network (Han et al., 2019) and the time series model (Han et al., 2023) were also used to build surrogate models to predict the car body acceleration time history. The surrogate model can be utilized to predict the dynamic response at a low exceedance probability due to its excellent predictive accuracy and high computational efficiency.

A numerical framework is proposed aiming to investigate the vertical stiffness limit of the 400 km/h HSR simple-supported bridge (SSB) based on the surrogate model. Firstly, the ARMAX model is utilized to construct the surrogate model to replace the TB coupling system for MCS calculation. Then, the limit indicators of bridge stiffness and the exceedance probability of acceleration exceeding the limit are determined, then the analysis framework of vertical bridge stiffness considering excitation randomness is proposed. The last section investigates the effects of train speed, train type, and car body mass on the stiffness limit and provides the recommended value for the vertical stiffness limit specific to the 400 km/h HSR SSB.

Surrogate model of TB coupling system

TB coupling system

The dynamic model of the 400 km/h HSR train can be simplified in Figure 1(a), composed of eight carriages. Each train consists of one car body, two bogies, and four wheelsets, which are connected by two suspension systems with springs and dampers. Both car body and bogie have two degrees of freedom (DOF): floating and pitching, and there is no independent DOF on the wheelsets. Therefore, the vertical model of the train has six independent DOFs, denoted as (Z_c, φ_c, Z_t1, φ_t1, Z_t2, φ_t2) ^T, where the subscripts c, t1, and t2 stand for the car body, front bogie, and rear bogie, respectively. The k_p, c_p, k_s, and c_s are the linear spring stiffness and linear damping factor of the primary and secondary suspension systems, respectively. L_c is half the distance between two bogies, and L_t is half distance between two wheelsets. Based on the D'Alembert’s principle, the motion equation of the train system is expressed as:

M_{v} {\ddot{U}}_{v} + C_{v} {\dot{U}}_{v} + K_{v} U_{v} = F_{v}

(1)

where

M_{v}

C_{v}

, and

K_{v}

are the mass, damping, and stiffness matrices of the train;

{\ddot{U}}_{v}

{\dot{U}}_{v}

U_{v}

are the acceleration, velocity, and displacement vectors of the train, respectively; and

F_{v}

is the vertical excitation of the train system.

Figure 1.

Vertical TB dynamic model: (a) Train model traveling on bridge, (b) Girder cross section of 32 m SSB, (c) Displacement of each wheelset.

According to the information provided in the general reference map of the 350 km/h HSR SSB, the cross-section of the bridge is shown in Figure 1(b). The span of the SSB is set to 32 m, with a pier height of 50 m, considering that the pier height has a minor impact on the vertical dynamic response of both the beam and the train. The beam is constructed using C50 concrete with an elastic modulus of 3.55 × 10¹⁰ N/m², a damping ratio of 0.02, a cross-sectional area of 8.839 m², and a vertical bending moment of inertia of 10.897 m⁴. The bridge model was built in commercial finite element software ANSYS, and the first-order vertical natural frequency of the bridge is 5.135 Hz. The motion equation of the bridge system can be written as:

M_{b} {\ddot{U}}_{b} + C_{b} {\dot{U}}_{b} + K_{b} U_{b} = F_{b}

(2)

where

M_{b}

C_{b}

, and

K_{b}

are the mass, damping, and stiffness matrices of the bridge;

M_{b}

and

K_{b}

can be derived from the finite element software, and

C_{b}

is calculated by the Rayleigh damping method;

{\ddot{U}}_{v}

{\dot{U}}_{v}

U_{v}

are the acceleration, velocity, and displacement vectors of the bridge, respectively; and

F_{b}

is the force vector on the bridge system.

Considering wheel-rail adhesion, the wheel-rail interaction relationship between the train subsystem and the bridge subsystem is shown in Figure 1(c). The displacement at each wheelset consists of two parts: the vertical deformation of the bridge and vertical track irregularity. Following the coordinate transformation in Figure 1(b), it is possible to determine the vertical displacement of the track center and to calculate the displacement at each wheelset using the equation (3). F_v is expressed as the equation (4).

Z_{w i} = Z_{r i} + Z_{irr i} = Z_{b i} + e_{x} θ_{b i} + Z_{irr i}

(3)

F_{v} = [\begin{array}{c} 0 \\ 0 \\ k_{p} (Z_{r 1} + Z_{r 2}) + c_{p} ({\dot{Z}}_{r 1} + {\dot{Z}}_{r 2}) \\ k_{p} l_{t} (Z_{r 2} - Z_{r 1}) + c_{p} l_{t} ({\dot{Z}}_{r 2} - {\dot{Z}}_{r 1}) \\ k_{p} (Z_{r 3} + Z_{r 4}) + c_{p} ({\dot{Z}}_{r 3} + {\dot{Z}}_{r 4}) \\ k_{p} l_{t} (Z_{r 4} - Z_{r 3}) + c_{p} l_{t} ({\dot{Z}}_{r 4} - {\dot{Z}}_{r 3}) \end{array}] + [\begin{array}{c} 0 \\ 0 \\ k_{p} (Z_{irr 1} + Z_{irr 2}) + c_{p} ({\dot{Z}}_{irr 1} + {\dot{Z}}_{irr 2}) \\ k_{p} l_{t} (Z_{irr 2} - Z_{irr 1}) + c_{p} l_{t} ({\dot{Z}}_{irr 2} - {\dot{Z}}_{irr 1}) \\ k_{p} (Z_{irr 3} + Z_{irr 4}) + c_{p} ({\dot{Z}}_{irr 3} + {\dot{Z}}_{irr 4}) \\ k_{p} l_{t} (Z_{irr 4} - Z_{irr 3}) + c_{p} l_{t} ({\dot{Z}}_{irr 4} - {\dot{Z}}_{irr 3}) \end{array}]

(4)

where

Z_{w i}

is the vertical displacement of each wheelset (i = 1, 2, 3, 4);

Z_{r i}

is the vertical track displacement, consisting of two components: one is the vertical deformation

Z_{b i}

generated by the train’s action on the bridge, and the other is the vertical deformation arising from the torsional deformation

θ_{b i}

of the bridge due to the eccentricity

e_{x}

;

Z_{irr i}

is the track irregularity.

Considering the potential decrease in track smoothness with the extended service years of the railway lines, a less smooth German track spectrum was employed to simulate track irregularities. The AR method (Han et al., 2023) was used to generate track irregularities for the German low interference spectrum. The time history curve of track irregularity and the comparison between simulated spectrum and target spectrum is shown in Figure 2.

Figure 2.

Vertical track irregularity: (a) Time-history curve, (b) Comparison between simulated spectrum and target spectrum.

With a space step of 0.2 m, the Newmark-β method is used to iterate the train motion equation (1) and the bridge motion equation (2) separately, and the dynamic response of the train and the bridge can be obtained.

Surrogate model based on ARMAX

The ARMAX model, which is a surrogate model of dynamic system, is used replace the model of TB coupling system to predict train responses. The main advantage of ARMAX model is that (Lakshmi and Rama Mohan Rao, 2016), it inherently mitigates for signals with noise from various sources, providing unbiased parameter estimates. The structure of the ARMAX model is shown in Figure 3 (Fassois, 2001), where B(q)/A(q) and C(q)/A(q) reflect the vibration and noise characteristics of a dynamic system. The ARMAX model can be written into the equation (5).

A (q) y (t) = B (q) u (t - n_{k}) + C (q) e (t)

(5)

where A(q)y(t) is autoregressive (AR) item, C(q)e(t) is moving-average (MA), and B(q)u(t-n_k) is exogeneous (X). y(t) is the output sequence of the system at time t, corresponding to the response of the TB coupling system; u(t-n_k) is the input sequence of the system at time t, where the n_k is the initial order of input affects the output; e(t) is the noise sequence, which is supposed to be a normally independently distributed process. A(q), B(q), and C(q) are the polynomials in the delay operator q and are formulated as follows:

A (q) = 1 + a_{1} q^{- 1} + . . . + a_{n_{a}} q^{- n_{a}}

(6)

B (q) = b_{1} + b_{2} q^{- 1} + . . . + b_{n_{b}} q^{- n_{b} + 1}

(7)

C (q) = 1 + c_{1} q^{- 1} + . . . + c_{n_{c}} q^{- n_{c}}

(8)

where [a₁, a₂, …,

a_{n_{a}}

], [b₁, b₂, …,

b_{n_{b}}

], and [c₁, c₂, …,

c_{n_{c}}

] denote the matrix parameters of AR, X, and MA; n_a, n_b, and n_c are the order of AR, X, and MA. q is the backshift operator, as given by:

q^{- 1 \cdot n} y (t) = y (t - n) (n = 1, 2, \dots, m)

(9)

where m can be n_a, n_b+1, and n_c. Combined with equation (5) to equation (9), the difference equation of the ARMAX model is obtained as shown below (Mei et al., 2019):

y (t) + \sum_{m = 1}^{n_{a}} a_{m} y (t - m) = \sum_{m = 1}^{n_{b}} b_{m} u (t - n_{k} - m + 1) + \sum_{m = 1}^{n_{c}} c_{m} e (t - m) + e (t)

(10)

Figure 3.

ARMAX model structure.

Choose appropriate input and output sequences for training based on the characteristics of the TB coupling system, and then use the trained ARMAX model to predict the response sequence when only the input sequence is provided.

Prediction of train responses based on surrogate model

Research has shown that the influence of random track irregularities on the train response is more pronounced compared to the bridge response (Mao et al., 2019). Therefore, in the surrogate model, the output sequence y(t) primarily relies on the dynamic response time history of the train. In the subsequent analysis, the surrogate model will be employed to investigate the stochastic behavior of the train dynamic response under different stiffness conditions. The process of using the surrogate model to predict acceleration is illustrated in Figure 4.

Figure 4.

Flow chart of the surrogate model.

Step 1: Get the input and output sequences. The primary sources of excitation for equivalent loads mainly originate from the deformations caused by the train’s interaction with the bridge and the track irregularity. When the vertical track irregularity is set to 0, the dynamic response of TB coupling system is conducted via equation (1) and equation (2), in which the vertical displacement and velocity of each wheelset are obtained. This indicates that $Z_{r i}$ in equation (4) has been acquired. Select any track irregularity (expressed as Sample 1), and superimpose it with the displacement and velocity of each wheelset as the equivalent track irregularity. Then calculate the equivalent load by equation (4) and use it as the input sequence. The train response (expressed as Target 1) corresponding to Sample 1 is calculated by equation (1) and equation (2), and it is taken as the output sequence.

Step 2: Train ARMAX model. First, conduct a stationarity test on the input and output sequences. If the results indicate non-stationarity, apply differencing to make the sequences stationary. Training ARMAX with stationary time series data leads to better training and prediction accuracy. The input and output sequences are tested for stationarity, and it is necessary to deal with the difference when the sequence is non-stationary. Adjust the order of the model (n_a, n_b, n_c, and n_k) until there is little difference between the predicted train response (Predicted 1) obtained by the ARMAX model and the Target 1.

Step 3: Predicted the dynamic response of the train. Select any vertical track irregularity (expressed as Sample 2) for the TB coupling vibration calculation, to obtain the equivalent load again and use it as the input sequence. When only the input sequence is provided, the vertical dynamic response of the train is predicted (expressed as Predicted 2) using the trained ARMAX model. The train dynamic response (expressed as Target 2) is calculated for sample 2 using the TB motion equations. If a significant error is observed between Predicted 2 and Target 2, it is advisable to revisit and repeat step 2 of the process.

Step 4: MCS calculation. When track irregularities are generated, the surrogate model can be employed for MCS calculation to evaluate the car body acceleration at each stiffness of the bridge. In other words, under diverse bridge stiffness scenarios, the utilization of surrogate model becomes viable for addressing the impact of excitation randomness on train response.

Analysis method of stiffness limit considering excitation randomness

At present, there exists no established vertical stiffness limits for 400 km/h HSR bridges. References (Jeon et al., 2016; Li et al., 2020; Wang et al., 2016, 2020; Xiang et al., 2022; Zhai and Wang, 2012) employed the analysis method of TB coupling vibration to investigate the correlation between the train response and the bridge stiffness. However, this approach remains deterministic in nature. To further account for the influence of excitation randomness, the surrogate model is employed under diverse vertical stiffness conditions instead of the TB coupling system to obtain the train response. This method is aimed at evaluating bridge stiffness based on the limit of the train response within a defined exceedance probability.

Exceedance probability of train response

The presence of excitation randomness introduces significant variability in the train response, consequently affecting the accuracy of assessing vertical stiffness limits for bridges. To accommodate the impact of excitation randomness, it is crucial to determine the exceedance probability for the train response limit (denoted as P_f). Following the calculation method for the exceedance probability of the ten-minute average wind speed (Ding and Chen, 2013), the P_f is determined based on the occurrence of the train response exceeding the limit in one pair of trains out of the total number of train pairs during the statistical period. Preventive rail grinding is one of the essential tasks to ensure the smoothness of the railway track, and its cycle can be used as the statistical period for the train response (Xiang et al., 2020). The total number of train pairs passing through during the preventive grinding cycle can be calculated as shown in equation (11).

P = P_{D} \times T = P_{D} \times \frac{M}{m \times P_{D}} = \frac{M}{m}

(11)

Where P_D is daily train operation pairs; T is the preventive grinding cycle; M is the total weight of trains allowed to pass during the preventive grinding cycle; m is the total weight of each train pair; P_f can be expressed as the reciprocal of the total number of train pairs.

P_{f} = \frac{1}{P} = \frac{m}{M}

(12)

The value of M can be set as 30 Mt. HSR passenger trains are commonly equipped with four axles, and the individual axle loads typically range from 14t to 17t. These trains are composed of either eight or 16 train sets. The value of m can be obtained by multiplying the number of axles, axle load, and the number of train sets. From the equation (12), it can be observed that as m decreases, P_f becomes smaller, indicating a lower exceeding probability of the train response limit. This leads to a more conservative evaluation of the bridge stiffness limit. The result of P is 66,964, which can be approximated as 70,000 pairs when rounded up. P_f is established to be 1/70,000.

Limit indicators of bridge stiffness

According to the Code for Design on Railway Bridges and Culverts (DRBC Code 2017), one of the objectives of controlling bridge stiffness is to ensure good operational performance. Stiffness design indicators include deflection, beam end rotation angle, and vertical fundamental frequency. The first two indicators represent the deformation limits of the bridge under the static live load of trains (ZK load), while the vertical fundamental frequency directly reflects the stiffness of the bridge. The DRBC Code specifies the limits of deflection, beam end rotation angle, and vertical frequency, seen in Table 1.

Table 1.

Stiffness design indicators.

Train speed/(km/h)	Deflection/(m)	Beam-end rotation angle/(rad)	Vertical frequency/(Hz)
250	L/1400	1.5‰	23.58L^−0.592
300	L/1500
350	L/1600

L is the span of the bridge. The deflection limits specified in Table 1 are applicable to spans not exceeding 40 m. When the suspension length of the beam end is less than 0.55 m, the rotation angle limit of one of the two adjacent spans is set at 1.5‰ rad. The vertical fundamental frequency limit of the SSB is 23.58L^−0.592 Hz with the L being 20 m to 128 m. The three stiffness indicators are applicable for train speeds not exceeding 350 km/h. To ensure consistency in standards, the analysis of stiffness limits for 400 km/h HSR SSB will still follow the indicators.

Analytical frame for stiffness limit based on surrogate model

The evaluation of the train response includes both ride comfort and running safety (Arvidsson et al., 2019). The literature (Xiang et al., 2022) indicates that, during the decrease in vertical bridge stiffness, the vertical car body acceleration of the CRH400 train first exceeds the limit. This demonstrates that car body acceleration is the most sensitive indicator for train dynamic response to stiffness variations. Meanwhile, the deflection limit of the Shinkansen Railway bridge shows that riding comfort is stricter than the limit determined by running safety (Sogabe et al., 2005). Therefore, the vertical stiffness limit of the bridge is evaluated using the car body acceleration indicator.

When the stiffness of the bridge changes, the vertical car body acceleration varies accordingly. This variation is associated with changes in the vertical fundamental frequency of the bridge, as well as the deflection and the beam end rotation angle occurring under the influence of the ZK load. Thus, it is possible to establish a relationship between the acceleration and three stiffness indicators. The vertical bridge stiffness limit can be analyzed using the framework of Figure 5(a), which can be summarized as the subsequent four steps.

Figure 5.

Analysis frame diagram of stiffness limit: (a) Flow chart, (b) Exceedance probability curves of acceleration for various stiffness, (c) Relationship between stiffness reduction facto and acceleration, (d) Deformation of the bridge due to the ZK load.

Step 1: Reduce the vertical stiffness of the bridge. By decreasing the vertical bending moment of inertia of the bridge, the stiffness of the structure is reduced. Equation (13) defines the stiffness reduction factor as the ratio between the initial stiffness and the reduced stiffness.

F_{s} = \frac{{Iyy}_{0}}{Iyy}

(13)

Step 2: Compute the vertical acceleration at the P_f for each stiffness. For each stiffness reduction coefficient, the process involves first obtaining the input and output sequences. Then, the surrogate model is trained using these sequences. The trained surrogate model is subsequently used to predict the car body acceleration for the current stiffness. Once the predicted samples of vehicle dynamic responses meet the requirements, the exceedance probability curve of car body acceleration can be generated. The corresponding vertical acceleration at the P_f (known as $a_{v, P_{f}}$ ) can be extracted, seen in Figure 5(b). If the $a_{v, P_{f}}$ is found to be less than 1.3 m/s², the process is repeated by going back to step 1. Conversely, if the $a_{v, P_{f}}$ exceeds 1.3 m/s², the analysis progresses to the next step.

Step 3: Calculate the limit of the stiffness reduction factor (F_s,limit). The acceleration at the P_f is correspondingly matched with the stiffness reduction factor. When the acceleration is equal to 1.3 m/s², the stiffness reduction factor of the bridge is derived by linear interpolation in Figure 5(c), which is the limit of the stiffness reduction factor.

Step 4: Obtain the vertical fundamental frequency, deflection, and beam end rotation angle of the bridge at the F_s,limit. By adjusting the stiffness of the bridge to the F_s,limit, it is possible to ascertain the vertical fundamental frequency of the bridge. Subsequently, by subjecting the bridge to the ZK load after reducing its stiffness, as depicted in Figure 5(d), the limits for the deflection-to-span ratio and rotation angle can be determined.

Validation of stiffness analysis method

Efficient and accurate calculation of car body acceleration is a crucial step in the analysis framework for determining the bridge stiffness limit. To verify the adaptability of the surrogate model to the analysis of bridge stiffness, it is necessary to evaluate the efficiency and accuracy of obtaining the acceleration at a small exceedance probability.

The train model, bridge model and track irregularities in section 2 are used in study case, and the train speed is 400 km/h. The order of the ARMAX model is set as n_a = 10, n_b = 10, n_c = 10, and n_k = 0, and 10⁴ track irregularities were analyzed by using MATLAB software running on the computer with Core (TM) I 7-8700 CPU @3.20 GHz. Table 2 compares the calculation efficiency between TB motion equations and the surrogate model. The single time-consuming is equal to the ratio of the total time-consuming and sample numbers in Table 2. TB motion equations require approximately 133 s to calculate the time history of the acceleration, while the surrogate model achieves the same prediction in just 0.09 s. This indicates that the surrogate model is three orders of magnitude more efficient in terms of computational time.

Table 2.

Comparison of calculation efficiency.

Program type	Time-consuming categories	Single time-consuming/(s)	Total time-consuming/(s)
TB equations	Calculations of TB equations	132.95	1.33 × 10⁶
Surrogate model	Two calculations of TB equations	265.91	1.12 × 10³
	Train	1.12
	Prediction	0.09

As seen in Table 2, the surrogate model can predict the car body acceleration only after two calculations of TB motion equations and one training. The surrogate model requires a prediction time of 1.12 × 10³ s to estimate the acceleration for 10⁴ time histories, whereas the calculation time of the TB motion equations amounts to 1.33 × 10⁶ s. The total computation time of the surrogate model is also three orders of magnitude lower than that of the motion equations.

When the bridge is of original stiffness, Figure 6 shows acceleration time history curves calculated by the surrogate model (recorded as the prediction) and TB motion equations (recorded as the target), with both curves almost coinciding. Figure 6(b) demonstrates that the error ratio for the acceleration peak value is less than 0.01%, confirming the excellent accuracy of the surrogate model in predicting acceleration.

Figure 6.

Time history curve of car body acceleration: (a) Comparison of prediction and target acceleration time history curve, (b) Error of acceleration time history curve.

The deformation of the girder will increase as the bridge stiffness decreases, so the prediction accuracy of the surrogate model will also be affected. It is necessary to compare the error between the predictive acceleration and the target acceleration at different exceedance probabilities. According to the reference (Bucher, 2009), the number of MCS calculation samples should be at least 10 times the reciprocal of the exceedance probability. As a result, the surrogate model should calculate 10⁶ samples to obtain the car body acceleration with the P_f of 1/70,000. Using the TB motion equations, Table 2 has shown that it takes 1.33 × 10⁶ seconds to perform 10⁴ direct MCS calculations, so 10⁶ direct MCS calculations are too time-consuming. When the bridge stiffness decreases to 1/10 time of the initial stiffness (i.e., the stiffness reduction factor is 10), 10⁶ MCS calculations are performed using the surrogate model. Track irregularities corresponding to car body accelerations at exceedance probabilities of 10⁻⁵ and 10⁻⁶ are located (recorded as samples 3 and 4), and the TB motion equations are then utilized to compute the car body acceleration corresponding to these two samples (recorded as target values 3 and 4). The exceedance probability curves of target and predictive car body accelerations are depicted in Figure 7, and Table 3 shows the acceleration prediction errors at various exceedance probability levels.

Figure 7.

Comparison of exceedance probability.

Table 3.

Error between target and predictive accelerations.

Acceleration	Exceedance probability
Acceleration	10⁻⁴	10⁻⁵	10⁻⁶
Target value (m·s⁻²)	1.330	1.445	1.505
Predictive value (m·s⁻²)	1.351	1.465	1.524
Error	1.6%	1.4%	1.3%

When the stiffness reduction factor is 10 and there are 10⁴ samples, there is a slight deviation at the tail of the target and the predictive exceedance probability curves in Figure 7. When the exceedance probability is 10⁻⁴, 10⁻⁵, and 10⁻⁶, the target value and predictive value of the acceleration are extracted, and Table 3 demonstrates that the error of both values is less than 2%. When there are 10⁶ samples, the red dotted line in Figure 7 representing the P_f intersects with the predictive exceedance probability curve. The acceleration at the intersection is 1.442 m/s² more than the acceleration limit of 1.3 m/s², revealing that the proposed surrogate model for the bridge stiffness is accurate enough to account for the acceleration as stiffness decreases.

Result discussions

Effects of train type and train speed

The train parameters of various train types differ significantly, which influences the dynamic response of trains. The CRH3 train and the CRH400 train are used to investigate the effect of the train type on the bridge stiffness limit. The effect of track irregularity on acceleration is evaluated using a total of 10⁶ samples for each stiffness reduction factor. When conducting train-bridge coupling vibration analysis, it is essential to consider the impact of train resonance or bridge resonance on dynamic responses. Previous studies (Xia et al., 2006) have indicated that the resonant train speed under the influence of regularly arranged bridge spans and their deflections is given by V_vr = 3.6f_vL_b. The vertical frequencies for the CRH400 train and CRH3 train are 0.714 Hz and 0.919 Hz, respectively, with corresponding resonant train speeds of 82 km/h and 106 km/h. Within the speed range of 250 km/h to 400 km/h investigated in the study, the trains do not experience resonance. Therefore, the surrogate model can be employed to assess the impact of excitation randomness on the car body accelerations.

Figure 8(a) illustrates the influence of the train speed on the acceleration for different stiffness reduction factors. It reveals that for a given stiffness reduction factor, as the train speed increases, the acceleration also increases. Additionally, higher train speeds correspond to smaller stiffness reduction factors for the car body acceleration limits. Figure 8(b) depicts the acceleration of the CRH3 train with two stiffness reduction factors near the acceleration limit. According to Figure 8(c), the stiffness reduction factor limits of two types of trains decrease as the increase of train speed, which means that the higher the train speed, the higher the vertical stiffness requirements of the bridge. Simultaneously, the difference between the stiffness reduction factors of the CRH3 train and the CRH400 train gradually grows as the train speed rises. Figure 8(d) illustrates that the acceleration exceedance probability curve of the CRH3 train is on the CRH400 train when the train speed is 250 km/h and the stiffness reduction factor is 10 (expressed as 250-10); when the condition is 400-6, the acceleration of the CRH3 train moves up in comparison to that of 250-10, while that of the CRH400 train moves down.

Figure 8.

Car body acceleration at various stiffness reduction factor: (a) CRH400 train, (b) CRH3 train, (c) Stiffness reduction factor limit, (d) Exceedance probability.

Considering the specified maximum design speed of 350 km/h in the DRBC Code, the deflection-span ratio limit for 400 km/h can be estimated as 1/1700 by extrapolating the deflection-span ratio limits obtained for Table 1. Limits of the rotation angle and vertical fundamental frequency are not directly related to the train speed, so these limits can still be applied at the speed of 400 km/h as specified in the DRBC code.

Figure 9(a) depicts that when the train speed increases, the deflection-span ratio limit of the CRH3 train changes more quickly than the CRH400 train. When the train speed is less than 300 km/h, the deflection-span ratio limit of the CRH3 train is more than that of the DRBC Code; when the speed is greater than 300 km/h, the former has a smaller limit, meaning that the deflection-span ratio limit of the CRH3 train is stricter than that of the DRBC Code. The design speed of the CRH400 train is higher than that of the CRH3 train, so the former will operate better in dynamic performance than the latter. The rotation angle limit of the CRH3 train is near to the extrapolated value of the CRBC Code at the train speed of 400 km/h, but it is larger than that of the CRBC Code at other train speeds, and the rotation angle limit of the CRH400 train is larger than that of the CRH3 train, as shown in Figure 9(b). Figure 9(c) depicts that the vertical fundamental frequency limit of the CRH3 train and 400 km/h is lower than that of the standard, and that of the latter is lower.

Figure 9.

Stiffness limits of two trains: (a) Deflection-span ratio, (b) Girder-end rotation angle, (c) Vertical fundamental frequency.

It is known that the design speed of the CRH3 train is 250 km/h, and DRBC Code requires that 1.2 times the design speed be used for dynamic checking. After the checking speed is exceeded, to ensure that the car body acceleration at the P_f does not exceed the limit, the bridge is required to have sufficient vertical stiffness. Therefore, the CRH3 train will be stricter than DRBC Code on the deflection-span ratio limit when the train speed is higher than 300 km/h. Compared to the CRH400 train, the CRH3 train has stricter requirements for the bridge stiffness limit, such as the smaller limits of deflection-span ratio and rotation angle and higher frequency limit. Additionally, the deflection-span ratio limit is stricter than the other two limits for the CRH3 train because it is lower than that of DRBC Code after reaching the speed of 300 km/h, the rotation angle limit is close to that of DRBC Code at the speed of 400 km/h, and the vertical fundamental frequency limit is always lower than that of DRBC Code. At present, the driving performance of HSR trains on the bridge is good, which indicates that DRBC Code has enough reserve for the bridge stiffness limit. If the design stiffness of the bridge is carried out in accordance with the deflection-span ratio limit of the CRH3 train in Figure 9(a) for speeds of 350 km/h or higher, the stiffness requirements will increase and the construction cost will increase. As a result, for the stiffness design of HSR bridges, it is essential to choose the train that matches to the design speed of the bridge.

Effects of car body mass

The axle load of Electric Multiple Units on HSR lines is usually 14 t ∼ 17 t, which can be 14 t, 15 t, 16 t, and 17 t respectively. By altering the car body mass and moment of inertia, the axle load parameters can be changed. The train speed is set at 400 km/h, and the stiffness reduction factors of 32 m SSB are considered as 6, 8, 10, and 12 respectively. The deflection-span ratio limit is stricter than the rotation angle limit and the frequency limit, so the subsequent research primarily uses the deflection-span ratio limit to assess the bridge stiffness limit.

According to Figure 10(a), for the same stiffness reduction factor, the car body acceleration increases as the axle load decreases. In Figure 10(b), the limit values of the deflection-span ratio limit corresponding to axle load 14 t, 15 t, 16 t, and 17 t are 1/1338, 1/1081, 1/939, and 1/853, respectively. Whether the initial stiffness or the stiffness reduction factor is 10, the mid-span vertical displacement of the bridge increases with the increase of axle load, but the vibration amplitude exhibits no obvious changes, as depicted in Figure 10(c) and (d). The force of the train system ( $F_{v}$ ) changes little as a result of the vibration amplitude of the bridge being relatively constant under various axle loads. The equation (1) indicates that the force of the train ( $F_{v}$ ) varies little and that the smaller the car body mass ( $M_{v}$ ) is, the larger the car body acceleration there will be. As axle load decreases, the deflection-span ratio limit decreases, while the vertical bridge stiffness becomes more restrictive as the car body mass decreases.

Figure 10.

Influence of car body mass: (a) Limit of stiffness reduction factor, (b) Deflection-span ratio limit, (c) Displacement time history curves of bridge with initial stiffness, (d) Displacement time history curves of bridge with stiffness reduction factor of 10.

Recommended value of stiffness limit

The deflection-span ratio limit of the HSR bridge is specified in the DRBC Code according to the grades of the span and the speed under the action of ZK load. For the analysis of the stiffness limit, spans of 24 m, 32 m, and 40 m can be selected for prestressed concrete SSBs commonly used on HSR lines in China. As illustrated in Figure 11, the deflection-span ratio limit is calculated using 80-span 24 m, 60-span 32 m, and 48-span 40 m SSBs to ensure the same length of track irregularities.

Figure 11.

Recommended value of stiffness limit.

At the same train speed, Figure 11 depicts that the deflection-span ratio limit is smaller the longer bridge span. The vertical deformation of the bridge contributes to a portion of the excitation to the train system. The regular bridge span arrangement will periodically excite the train running on the multi-span SSBs (Xia et al., 2006), and the frequency of excitation is V/3.6L (where V is the train speed). When the train speed is 400 km/h and the bridge stiffness drops to the car body acceleration limit, Figure 12(a) depicts the excitation frequency of the train (F_v(3) as an example), and Figure 12(b) depicts the frequency of the car body acceleration.

Figure 12.

Dominant frequency: (a) Force of the train system; (b) Car body acceleration.

Compared with Figure 12(a) and (b), the dominant frequencies of train excited by SSBs with spans of 24 m, 32 m, and 40 m are consistent with the dominant frequencies of the car body acceleration and equal to the ratio of speed to span, which is 4.63 Hz, 3.47 Hz, and 2.78 Hz, respectively. The findings demonstrate that when the bridge stiffness is reduced until the car body acceleration exceeds the limit, the periodic excitation of the bridge span is the primary factor influencing the car acceleration. The car body acceleration increases with the increasing wavelength, therefore the longer the span, the stricter requirements for the bridge stiffness limit. The Korean design criteria for a railroad reflect this phenomenon (Cho et al., 2016).

Figure 11 reveals that the deflection-span ratio limit is 1/1395, which can be rounded up to 1/1400 when the train is traveling at 400 km/h and the span is 40 m. The deflection-span ratio limit of 1/1400 accounts for the most demanding conditions (the highest speed, smallest car body mass, and the largest span), and it guarantees the low exceedance probability (1/70,000), which has a high safety reserve for the vertical bridge stiffness. Therefore, the deflection limit of L/1400 (L is the span) can be applied to the stiffness limit of the prestressed concrete SSBs with a train speed of 400 km/h and a span of less than 40m. It should be noted that the bridge stiffness limits in different nations are quite variable due to the significant differences in train parameters and the requirements of railway bridge design specifications. The recommended limit of vertical bridge stiffness is only applicable to the train type in this work, so designers need to be clear about the application scope of the stiffness limit before reference.

Conclusions

The surrogate model is used to replace the TB motion equations for MCS calculation, and the analysis framework of the vertical bridge stiffness limit considering excitation randomness is proposed. Based on this, the effects of car body mass, train speed, train type, and other factors on the bridge stiffness limit are examined, and the recommended stiffness limit for the 400 km/h SSB bridge is provided.

(1) The suggested analysis framework has high efficiency and precision for the vertical bridge stiffness limit. The efficiency of MCS calculation by the ARAMX surrogate model is 3 orders of magnitude higher than that by the direct MCS calculation. The prediction error of the surrogate model for the acceleration at the exceedance probability of 10⁻⁶ is less than 2%.

(2) When the car body mass is smaller, the train speed is higher, and the bridge span is larger, the vertical deflection-span ratio limit is smaller, so the requirement for the stiffness limit of the bridge is stricter.

(3) To ensure the running performance of the high-speed train used in the analysis, the deflection-span ratio limit of 1/1400 can be applied for the stiffness limit of the prestressed concrete SSB with the train speed of 400 km/h and a span of less than 40 m.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (51978589, 52322811).

ORCID iDs

Huoyue Xiang

Jin Zhu

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