A Probabilistic Assessment Model for Train-Bridge Systems: Special Attention on Track Irregularities

In this paper, a probabilistic model devoted to investigating the dynamic behaviors of train-bridge systems subjected to random track irregularities is presented, in which a train-ballasted track-bridge coupled model with nonlinear wheel-rail contacts is introduced, and then a new approach for simulating a random ﬁeld of track irregularities is developed; moreover, the probability density evolution method is used to describe the probability transmission from excitation inputs to response outputs; ﬁnally, extended analysis from three aspects, that is, stochastic analysis, reliability analysis, and correlation analysis, are conducted on the evaluation and application of the proposed model. Besides, compared to the Monte Carlo method, the high eﬃciency and the accuracy of this proposed model are validated. Numerical studies show that the ergodic properties of track irregularities on spectra, amplitudes, wavelengths, and phases should be taken into account in stochastic analysis of train-bridge interactions. Since the main contributive factors concerning diﬀerent dynamic indices are rather diﬀerent, diﬀerent failure modes possess no obvious or only weak correlations from the probabilistic perspective, and the ﬁrst-order reliability theory is suitable in achieving the system reliability.


Introduction
e accurate and reliable numerical estimation of rolling stocks passing over high-speed railway bridges is an important and cheap means to optimize the bridge design, improve the operation quality, and enhance the profitability of railway systems. However, the existence of uncertainties, which originate from the controllable or uncontrollable construction errors, environmental impact, and long-term natural evolution, seriously reduces the authority and persuasiveness of conventional deterministic numerical prediction. Against this backdrop, the dynamic assessment of train-bridge interaction system from the probabilistic perspective is being urgently sought.
For obtaining the full view of dynamic behaviors of train-bridge systems, much attention should be first given to the variability of track irregularities due to its great importance in the interaction. Until now, the statistically average power spectral density (PSD) is a common way to represent the track irregularities [1][2][3], based on which track irregularity sets with spatial randomness can be easily generated through some time-frequency transformation approaches [4,5]. However, the average track spectrum, which just represents such limited track profile deformations, is insufficient to describe the complete track-geometry variations in a railway line. In last decades, many scholars have focused on the modelling techniques of the track irregularity random field. For instance, Zhu et al. [6] considered track vertical profile and alignment irregularities as Gaussian process and studied the levelcrossing properties and peak statistics by comparing with the measurements. Perrin et al. [7][8][9] presented the track irregularity stochastic model by introducing approaches of Karhunen-Loéve expansion and polynomial chaos expansion, in which the statistical properties of this vectorvalued, non-Gaussian, and nonstationary track irregularity random field were properly considered and most importantly the dependencies of irregularities were taken into account. is technique was also applied into the French urban railway network [10]. Xu and Zhai [11,12] provided new insights into the probabilistic characteristics of track irregularity PSDs on the basis of the intrinsic features of track irregularities in the shape similarity of PSD functions and derived a spectral-based method to inversely and efficiently simulate the random track irregularities with ergodic properties. From the aforementioned studies, it can be seen that the methodologies for simulating random track irregularities are gradually advancing toward maturities.
In the last decades, research studies on the dynamic behaviors of track irregularity-induced train-bridge systems under have gradually been extended from the deterministic analysis to the probabilistic analysis. For example, Chatterjee et al. [13], Au et al. [14], Xia et al. [15], and Liu et al. [16,17] used multiple samples of track irregularities from one PSD to explore the effects of the location of irregularity amplitudes on system responses. By applying the pseudoexcitation method (PEM), researchers can efficiently study the random vibrations of train-bridge systems, where random irregularities were transformed into the superposition of a series of deterministic pseudoharmonic signals [18][19][20][21]. Rocha et al. [22,23] took the assumption that the track spectra are uniformly distributed and employed the Monte Carlo simulation and tail modelling technique to assess the track stability caused by excessive deck vibrations and running safety of trains for a short-span railway bridge. Subsequently, with the same assumption, they [24] tested the efficiencies of some typical sampling methods and compared their differences in estimating the probability of failure. Salcher et al. [25] also assumed that the track spectra are uniformly distributed and conducted an interesting work in the codebased reliability evaluation of train-bridge systems, in which the uncertainties of bridge, track, vehicle, track irregularity, and environmental impact are fully included. Mao et al. [26,27] used the probability density evolution method to perform random dynamic analysis of train-bridge systems subjected to random track irregularities with one PSD and system parameters. Jin et al. [28] applied the spectral method to obtain the random vibration of bridges due to a series of moving forces in frequency domain, and the hunting forces are modelled as a random process with the given PSD instead of harmonic forces. Cantero et al. [29] investigated the influences of wavelength ranges of track irregularities on train-bridge systems. In the above studies, abundant meaningful conclusions have been achieved. However, track irregularity PSDs were usually assumed to be uniformly distributed due to the scarcity of measurements, and therefore, the variability of track irregularity was not scientifically considered in the studies. Moreover, the majority of these investigations have simplified the wheel-rail interaction relationship to different levels for efficiency. is may lead to insufficient understandings. Currently, systematic work has rarely been reported in evaluating a relatively complete train-bridge system under random track irregularity from the probabilistic perspective. is paper aims to develop a probabilistic model to comprehensively assess the dynamic performance of trackbridge coupled systems under track random irregularities. First, a three-dimensional (3D) train-ballasted track-bridge model is introduced, in which the nonlinear wheel-rail contact relationship and the mechanics of different components are covered. Second, the approach in simulating the random field of track irregularities is presented, and the ergodic properties are entirely considered in the temporalspatial domain. en, the probability density evolution method (PDEM) is introduced to solve the probabilistic transmission between excitation inputs and response outputs. Finally, some significant aspects are investigated to reveal the random behaviors, including stochastic analysis, reliability analysis, and probabilistic relation analysis.

Introduction of Train-Track-Bridge Model
e train-track-bridge coupled system can be decomposed into three subsystems, namely, subsystems of the train, the track, and the bridge, which are spatially coupled by nonlinear wheel-rail interactions and track-bridge interactions, as shown in Figure 1. Without the loss of generality, the dynamic equations of motion for the coupled system can be written in the following form [31]: where M, C, and K are mass, damping, and stiffness submatrices, respectively; the subscripts "n," "t," and "b" represent the submodules of the vehicle, the track, and the bridge, respectively, and € X, _ X, X, and F are the vectors of acceleration, velocity, displacement, and force, respectively. e modelling methods are summarized in Tables 1 and 2.

Random Field of Track Irregularities
Rising from the manufacturing faultiness and environmental loads, e.g., cyclic wheel-rail interactions, material fatigue, and track settlement, track irregularities are characterized by high-dimensional randomicity in both spatial and temporal domains. Its spatial randomness behaves as the random distribution of amplitudes and phases along the track, and the temporal randomness is represented by the fluctuation of spectral lines, as shown in Figure 2. Xu and Zhai [11,12] pointed out that the ergodic properties of PSD probabilities, amplitudes, and wavelengths of track irregularities should be taken into account in the stochastic analysis of vehicle-track interactions and presented a track irregularity probabilistic model. However, being different from the repetitive feature of the train-track system, the dynamic response of train-bridge systems is dependent on the position of trains passing over the bridge. To completely consider the spatial randomness of track irregularities at every bridge position, the ergodic property of phases may also be considered in train-bridge systems.
In this paper, an extended approach is developed to characterize the ergodic properties of track irregularities on track spectrum, amplitudes, wavelengths, and phases based on the track irregularity probabilistic model proposed by Xu and Zhai [11,12]. e detailed modelling method is presented in the following sections.

PSD Probability Distribution.
e approach on the PSD probability distribution of track irregularities from big data of measurements proposed by Xu and Zhai [11,12] can be reconsidered as follows.
Let ℓ υ (kΔs) be a portion of track irregularities, where k � 1, 2, . . . , N; Δs is the discrete spacing; N is the total number of track irregularity segments; and υ, υ � υ 1 , υ 2 , υ 3 , υ 4 , represents the type of track irregularity, namely, vertical profile, alignment, cross level, and gauge, respectively. Define Z(·) as the PSD function operator, and then the PSD of ℓ υ (kΔs) can be conveniently derived by in which ω is the set of discrete frequency points with a total number of W. Out of the analytical convenience, a spectral density matrix with the order of N × W can be assembled by

Shock and Vibration
in which P is the discrete interval of spectral densities, Q is the total partition number, and int[·] is an operator used to round the number in the bracket to the nearest integer toward minus infinity.
Based on probability statistics, the probability density function (PDF) of P υ (ω d ) can be expressed as follows: in which ƛ(·) is the PDF operator. Owing to the shape similarity of PSD lines and probability equivalence, a cumulative probability index U υ ∈ [0, 1] is introduced to uniquely determine the value of spectral density over P υ (ω), namely, e statistically average spectrum can be easily derived, namely, As an illustration, Figure 3 plots the PDF of track vertical profile irregularity PSD against the cumulative probability index U υ . As recognized by the visual impression, the probability characteristic is definitely not uniformly distributed. e uniformly distributed assumption in previous studies [21][22][23][24][25] did not reflect the actual physical status. Meanwhile, it can be seen that the average track spectrum is approximately at the cumulative probability of 0.46.

Time-Frequency Transformation.
Let T υ,U i (x) be a onedimensional stationary random process to characterize the PSD of track irregularity at cumulative probability U i , denoted by P υ,U i ′ (ω). As Shinozuka et al. [33] derived, T υ,U i (x) can be expressed by the following discrete integral form: cos ω k x du ω k + sin ω k x dv ω k , (8) in which ω k � kΔω where Δω is the frequency interval; u(ω) and v(ω) are two mutually orthogonal real processes with  orthogonal increments du(ω) and dv(ω); and du(ω k ) and dv(ω k ) can be defined as follows: with A k � (P υ,U i ′ (ω)Δω) 1/2 , in which X k , Y k is a group of orthogonal random variables and satisfies the conditions: where E[·] is the operator of mathematical expectation and δ jk is the Kronecker symbol. erefore, equation (8) can be approximately transformed into the sum of M terms: To further reduce the random parameters of T υ,U i (x) and improve the efficiency of stochastic analysis, a method expressing the random variables X k , Y k as random functions is developed by Liu et al. [34]. e procedure of realization can be expressed as follows: (1) Generate the independent random variables θ 1 , θ 2 , which are uniformly distributed within (0, t2π]. (2) Construct two sets of Gaussian, standard, orthonormal, and independent random variables, that is, where cas(·) � cos(·) + sin(·); Φ −1 (·) is the inverse function of standard normal distribution; and m � 1, 2, . . . , M. (3) Conduct a random mapping process, namely, X k , Y k ⟶ X k , Y k , and substitute it to equation (11).  With above theoretical derivations, the degree of random process T υ,U i (x) has been reduced from infinity to 2M and to 2, namely, θ 1 , θ 2 .
As an example, Figure 4(a) shows two track irregularity series derived from one spectral representation, and Figure 4(b) depicts the corresponding PSDs. e wavelengths are in the range of 0.5-120 m. It can be seen that the excellent agreement has been achieved between the simulating PSDs and the target PSD. Also, one can observe that the amplitudes in one position are significantly different due to the phase randomness. Figure 4(c) plots the mean values with 89 track irregularity samples obtained by the method in this paper. It can be seen that the mean values along the abscissa approach the ideal number, zero. e ergodic properties of track irregularities on amplitudes, wavelengths, and phases have been covered in this section.

Temporal-Spatial Track Random
Irregularity. By combining Sections 3.1 and 3.2, a comprehensive random field of track irregularity in the temporal and spatial domain has been constructed, in which the variables should be investigated in a synergetic and coupled manner, that is, Using some selecting point methods, such as Monte Carlo method (MCM), the number theoretical method [35], and Latin hypercube sampling (LHS) [36], the representative samples in the random field can be effectively generated.
For validation, comparisons are made of PDFs and CDFs obtained from the measured and simulated vertical profile irregularities. As can be seen from Figure 5, there is excellent agreement between them, which serves to illustrate the reliability of the proposed model.

Analysis Framework for System Probabilistic Analysis
In the following two sections, a PDEM developed by Li and Chen [37,38] is introduced to achieve the probabilistic transmissions between the system inputs and the response outputs. Besides, an analysis framework for the probabilistic assessment of train-bridge systems under random temporalspatial track irregularities will be presented in detail.

PDEM.
Correspondingly, the kinematic equations of the train-bridge system under random track irregularities can be transformed into the following form: where M s , C s , and K s are the mass, damping, and stiffness matrices of the dynamic system, respectively; X(t), _ X(t), and € X(t) are the state vectors of displacement, velocity, and acceleration, respectively; F s (Θ, t) is the load vectors including the random nonlinear wheel-rail interaction; and Θ is the random factors, herein, Θ � R. e solution of equation (14) is completely and continuously dependent on the random parameters Θ and can be expressed as a function of them [37,38]: en, the velocity of X(t), namely, the derivative of X(t) with respect to time, can be written as follows: in which h(·) is the derivative of H(·) with respect to time.
From the Lagrangian viewpoint, as long as random events neither appear nor disappear, the associated probabilities will remain constant; in other words, the probability will be preserved in the evolution process of the system [37,38]. Based on the principle of the preservation of probability, the joint PDF of the augmented vector (X(t), Θ) will follow the generalized probability density evolution function: under the initial condition: where δ(·) is the Dirac delta function, x 0 is the deterministic initial value, and p Θ (θ) is the joint probability of the random variables.
Equation (17) can be solved by a total variation diminishing (TVD) scheme [37,38] in conjunction with equation (15), and then the instantaneous PDF of X(t) can be obtained as follows: Once the PDF of dynamic indices is obtained, the reliability of which, R X (t), can be acquired consequently by in which x l and x u indicate the lower and upper bound of the dynamic index X(t), respectively.

Analysis Framework.
Based on the work illustrated above, a framework shown in Figure 6 for analyzing the effects of track irregularities on train-track interactions can be constructed.

Applications
In the numerical examples, it is assumed that the train runs with a constant velocity of 300 km/h on a five-span simply supported concrete bridge with a length of 32 m for each span. is type of structural system is very common on Chinese high-speed railway lines.
e box girder has a e detailed parameters of the vehicles and tracks used in the calculation can be consulted in reference [3]. e random track irregularities from the simulations in Section 3 are used as the excitations.

Numerical Validation.
By postprocessing the responses of the dynamic indices using PDEM, the probability density surface of any arbitrary response index can be determined,

Shock and Vibration
irregularities. Compared to the mean values in Figure 8(b), there are no obvious distinctions between them. It is further illustrated that the random field model of track irregularities presented in this paper is a good characterization of the stationary random process with second-order statistics.

Stochastic Analysis.
In this section, two computational cases are set to quantitatively evaluate the influence of track random irregularities on system responses, namely, full PSD excitation and statistically average PSD excitation. Figure 9 shows the statistical results of some representative dynamic indices including the lateral acceleration of the carbody, the wheel-rail lateral force, and the vertical acceleration at the middle of the central span. From the figures, it can be seen that no matter for the full PSD excitation or average PSD excitation, both the lateral acceleration of the carbody and the wheel-rail lateral force behave as stationary random processes, approximatively; however, with regard to the vertical acceleration at the middle of the central span, the mean value and Std. D fluctuates with the time-varying characteristics of trains running across the bridge. Besides, it can be observed from Figure 9 that if only the average PSD is considered in the stochastic dynamical analysis, the Std. D of three dynamic indices are significantly smaller than those excited by full PSD excitation. It seems to be inconsistent with the probability distribution of track irregularities. It may be because that the system responses are not linear with the quality of track irregularities. e worse the track irregularity quality is, the larger the increment of system responses is.
In addition, these figures provide information about the statistically quantitative values. For example, the Std. D of wheel-rail lateral force is about 5 kN over time while its mean value is 1.5 kN. Apparently, the conventionally deterministic treatment is not sufficient to reveal the physical mechanisms of the random vibration of train-bridge systems or even erroneously evaluating the security.
In summary, the full excitation of track irregularities should be considered properly for evaluating the random behaviors of system components with higher precision.

Reliability Analysis of Dynamic Indices.
Based on the PDEM, the temporal reliabilities of arbitrary indices of the train-track-bridge system can be conveniently achieved. e same as stochastic analysis, the lateral acceleration of the carbody, the wheel-rail lateral force, and the vertical acceleration at the middle of the central span are selected as illustrations. Figures 10(a)-10(c) display the reliability curves versus time for the three indices, the safety threshold values of which are 0.6 m/s 2 , 10 + P 0 /3 kN, and 3.5 m/s 2 , respectively [39,40]. P 0 is the static axle load. As observed from the figures, their average reliabilities are around 98.7%, 99.9%, and 100%, respectively. Generally, the threshold of lateral acceleration of the carbody is represented as the serviceability limiting state to ensure the comfort of passengers, while the limits of wheel-rail lateral force and deck acceleration are set as the ultimate limiting state to prevent train derailment and guarantee the stability of interlocking of the ballast gravel, respectively. In accordance with the common sense, the serviceability limiting state has lower reliability than the ultimate limiting state. It is worth pointing out that the above limit values are valued based on the existing specification, and they already contain a high safety factor. erefore, the actual reliabilities are much higher than the values here.

System Failure Probability.
With the reliability results for each dynamic index above, the system failure probability can be further determined by considering the contributions from different dynamic indices exceeding their respective limiting values. Assuming the train-track-bridge coupled system as a series system, the first-order estimates of the upper and lower bounds on the probability of system failure can be expressed as follows [41,42]: with P Fi (t) � 1 − R X,i (t), where P Fi (t) denotes the probability of failure for i th index and is a function of time. For independent failure modes, the system failure probability can be represented as a function of the product of the mode survival probabilities, which provides the upper bound in equation (21). In cases where the failure modes are all fully dependent, the weakest failure mode will always be the most likely to fail, leading to the lower bound in equation (21). Based on the first-order reliability theory, the probability of failure for the train-bridge system under the random field of track irregularities can be achieved, as shown in Figure 11. Several failure modes listed in Table 3 are considered in the evaluation. It can be seen from the figure that the differences between the upper and lower bounds are pretty slight, which indicates that it is appropriate to employ the first-order bounds to construct the system failure probability represented by the conservative estimate from the upper bounds. e failure probability is very small, with an average value of 0.02, and general quality can be guaranteed for both running safety and ride comfort in this railway line. Furthermore, one would expect that using a single dynamic index (e.g., deck acceleration) would result in an underestimation of the train-track-bridge system's vulnerability.

Correlation Analysis.
Excited by the temporal-spatial track irregularities, there might be a specific probability correlation among different dynamic indices. To mutually evaluate and predict their dynamic responses, a probabilistic relation analysis is highly essential. e procedure for revealing the response relationships between arbitrary two indices is as follows.
e responses of the dynamic indices denoted by w i , i � 1, 2, . . . , n, can be represented as q wi (kΔt), k � 1, 2, ..., m, where m is the total number of discrete points and Δt is the time-domain integration interval for train-track-bridge interactions. For brevity, we take w x and w y as an example, where x, y ∈ [1, n]. To exclude the impact of data dimensions, the data are first normalized as where q wx,u and q wy,u are the upper bounds of q wx (kΔt) and q wy (kΔt), respectively. Meanwhile, q wx (kΔt) can be divided into α portions and expressed as follows: where α is the total partition number and q wx,l is the lower bound of q wx (kΔt).
For every response series of q wx (kΔt) and q wy (kΔt), it is clear that there is q wy (kΔt), k ∈ k, located at the ith interval [G i−1 , G i ]. Based on the methods of probability statistics [43], the PDF of q wy specified to the interval [G i−1 , G i ] can be written as p Ω x ,Ω y q wy |i � where f Ω x ,Ω y (q wx , q wy ) is the joint PDF of q wx and q wy and Ω x and Ω y denote the response amplitude range of q wx and q wy , respectively.
For instance, Figure 12 shows the probabilistic relationships among some dynamic indices:   (1) e relation between the lateral and vertical acceleration of carbody, as shown in Figure 12(a) (2) e relation between the lateral acceleration of the carbody and the wheel-rail lateral force, as shown in Figure 12(b) (3) e relation between the vertical acceleration of the carbody and the wheel-rail lateral force, as shown in Figure 12(c) Out of convenience, the lateral acceleration of the carbody, the vertical acceleration of the carbody, and the wheel-rail lateral force are abbreviated as "LACB," "VACB," and "WRLF," respectively. It can be seen from the figures that no significant probability flow occurs between LACB and VACB and between VACB and WRLF, while WRLF specified to the maximum probabilities is slightly increasing along with the increments of LACB. It is because that, in light of different dynamic indices, the sensitive factors are rather different. For example, the magnitude of VACB is mainly affected by vertical profile irregularities, but LACB is significantly influenced by alignment irregularities; although WRLF is one of the sources for LACB, the WRLF is rather sensitive to the short-wavelength irregularities while LACB is sensitive to the long-wavelength irregularities [44]. According to the theory of random process, this indicates that there are no obvious or only weak probabilistic correlations among them, and correspondingly, they can be considered as mutually independent processes. Based on this understanding, the failure modes for these two indices are individual and independent. e upper bound of system reliability determined in Section 5.3.2 is appropriate.

Conclusions
is paper introduces a 3D train-track-bridge coupled system with nonlinear wheel-rail interactions first. en, a probabilistic model is proposed to represent both spatial and spectral randomness of track irregularities. By combining these two models and PDEM, extensive studies are conducted on stochastic analysis, reliability analysis, and correlation analysis. e following remarks can be drawn from the results thereby obtained: (1) e full probability characteristics of track irregularities should be considered when assessing the random vibrations of train-bridge systems. (2) An accurate determination of the dynamic reliability of train-bridge systems has been achieved in a convenient manner. A full picture has been obtained of the PDF surfaces of the dynamic indices and of the dynamic reliability of the entire system versus time. (3) Since the main contributive factors with respect to different dynamic indices are rather different, such as track irregularity type and wavelength, different failure modes can be assumed as mutually independent processes from the probabilistic perspective.
Data Availability e track irregularity data used to support the findings of this study were supplied by Lifeng Xin under license and so cannot be made freely available. Requests for access to these data should be made to him, e-mail: sxxlf2010@163.com.

Conflicts of Interest
e authors declare that they have no conflicts of interest.