A framework combining pseudo-excitation method and two-and-a-half-dimensional finite element method for random ground vibrations induced by high-speed trains

Abstract

A framework is developed in this article to predict the nonstationary random ground vibrations induced by high-speed trains, by combining the pseudo-excitation method with the two-and-a-half-dimensional finite element method. This development contains two steps. First, the power spectral density of the wheel–rail dynamic force is accurately obtained through the combination of the pseudo-excitation method and a vehicle–slab-track–ground theoretical model. Second, the nonstationary random ground vibrations are efficiently solved by combining the pseudo-excitation method and the two-and-a-half-dimensional finite element method, where the power spectral density of the wheel–rail dynamic force obtained in the former step is used to constitute the pseudo-loads. In the numerical examples, the accuracy and efficiency of the proposed approach are validated through the comparison to the fast three-dimensional random method for train–track–soil system developed previously. The results show that the proposed approach can predict the train-induced random ground vibrations with sufficient accuracy and has three-to-five times increase in efficiency in comparison to the fast three-dimensional random method.

Keywords

two-and-a-half-dimensional approach framework ground vibration high-speed railway random vibration track irregularity

Introduction

With the rapid development of high-speed railways, many propagation models for predicting ground vibration of high-speed train have been proposed (Xia et al., 2010). Due to the advance of computational tools, various numerical methods, such as finite element method (FEM) (Hung et al., 2001; Ma et al., 2019; Yang and Hung, 2001), boundary element method (BEM) (Andersen and Nielsen, 2003), and coupled FEM–BEM (Alves Costa et al., 2012; François et al., 2010; Galvín et al., 2010; Lombaert et al., 2014; Sheng et al., 2006) have been developed as supplements to analytical/semi-analytical method (Xia et al., 2010), field measurement method (Connolly et al., 2015), and empirical prediction (Verbraken et al., 2011) in predicting train-induced ground vibrations. A number of two-dimensional (2D) models (Hung et al., 2001; Metrikine and Vrouwenvelder, 2000) and three-dimensional (3D) models (Andersen and Nielsen, 2003; Galvín et al., 2010; Ma et al., 2016; Wang et al., 2018) have been used to predict ground vibration. However, the 2D models cannot account for wave propagation in the direction of the movement of the train. On the other hand, the 3D models require large computing resources and are limited for practical use. In many cases, it can be assumed that the ground and existing structures are uniform along the track. For this kind of engineering structure, many researchers have proposed and applied the so-called two-and-a-half-dimensional (2.5D) approach to solve the problem of ground vibrations caused by traffic, that is, to transform the problem into a series of 2D models according to the wave number along the track (Alves Costa et al., 2012; François et al., 2010; Ma et al., 2019; Sheng et al., 2006; Yang and Hung, 2001). Compared with the 2D and 3D approaches, the 2.5D approach has the advantages of better realism and computation efficiency (Yang et al., 2017).

The aforementioned studies deal mainly with deterministic dynamic response problems of ground subjected to moving loads, in which the moving loads are given in a prescribed form. Actually, because the ground vibrations are nonstationary random processes, random vibration approaches are required to reasonably simulate ground vibrations induced by trains moving over rails. Hunt (1991), Metrikine and Vrouwenvelder (2000), Sheng et al. (2004a), Lu et al. (2006), Si et al. (2016), and Li et al. (2019) investigated the stochastic dynamic responses of ground surface due to moving loads or moving trains. Unfortunately, these approaches are either based on analytical or semi-analytical methods, their applications are generally restricted to cases with simple geometries. In complex geological site, numerical methods are more suitable for dealing with such problems. However, two main difficulties arise when numerical methods are used. One is how to characterize the power spectral density (PSD) of the wheel–rail dynamic force. At present, there is no standard PSD data available for the wheel–rail dynamic force (Si et al., 2016). The other is how to obtain the random response of the coupled train–track–ground system efficiently. When the track–ground system is simulated by numerical methods, the computational cost grows exponentially with the number of degrees of freedom (DOFs) of the system. Based on the pseudo-excitation method (PEM) and the multifrontal method, Wang et al. (2018) proposed a fast algorithm to calculate the stochastic dynamic response of the coupled train–track–ground system. Although this algorithm improves the computational efficiency of the solution of random equations of the coupled system to a large extent, the algorithm relying on a 3D FEM model makes it uncompetitive in the stage of new line planning, because there are a lot of schemes that need to be compared. By using a 2.5D FEM–BEM model, Lombaert et al. (2014) studied the nonstationary random vibrations of track–ground system induced by moving trains. However, because a conventional random vibration method for a linear system with multiple excitations was used to calculate the autocorrelation function, the approach had relatively low efficiency.

As a conventional random vibration method, the Monte Carlo method (Rubinstein and Kroese, 2016) is widely used because of its simplicity. Generally, a series of responses due to different track irregularity samples are computed and the statistical characteristics of random responses are determined from them. However, due to the complexity of the train–track–ground system, the use of the Monte Carlo method is very time-consuming (Wang et al., 2018). Thus, a more efficient algorithm must be adopted to analyze the random vibration of the system. The PEM is an algorithm for calculating the response of linear time-varying systems under stationary or nonstationary random excitations with high efficiency and accuracy (Lin et al., 2001). At present, the PEM has been widely used in buildings (Huang et al., 2015), bridges (Zhang et al., 2014), train–bridge interactions (Zhai et al., 2019; Zhu et al., 2018b), and other engineering fields. Compared with the Monte Carlo method, the PEM improves the computational efficiency by one to two orders of magnitude.

The objective of this article is to present an efficient and robust framework for prediction of the nonstationary random vibrations of the ground induced by high-speed trains. The article is organized as follows. In the section “Framework for analysis of high-speed train-induced random ground vibration,” the framework for analysis of train-induced random ground vibration is recapitulated. In the section “PSD of the wheel–rail dynamic force based on the semi-analytical method and PEM,” the semi-analytical method and the PEM are effectively combined to derive the PSD of the wheel–rail dynamic force by employing the PSD of track irregularity. In the section “Random vibration analysis of the track–soil system by the PEM,” the nonstationary random vibration of the ground is determined according to the PEM and the 2.5D FEM. Finally, the correctness of the proposed random vibration approach and the accuracy and efficiency of the proposed approach are illustrated in the section “Comparison with 3D random method.”

Framework for analysis of high-speed train-induced random ground vibration

Ground surface condition has significant effect on the track response in comparison to such track response on a rigid foundation, especially for frequencies close to the cut-on frequency of the ground (Sheng et al., 2004b). Thus, it is important and necessary to consider the elasticity and energy radiation of the supporting ground. There are two main kinds of methods to simulate the track-ground system: semi-analytical and numerical. Semi-analytical methods model the railway infrastructure using simple elastodynamic elements (such as beams, plates, springs, dampers, and half-spaces). Usually, numerical models are based on the FEM or on a combination of the former with the BEM. Thus, the computational effects of semi-analytical models are significantly less than those associated to numerical approaches. However, numerical models are more accurate for the prediction of the behavior of the overall system.

In this article, a framework which combines the PEM with the semi-analytical method and the 2.5D FEM to quantify the effect of random track irregularity on high-speed train-induced ground vibrations is proposed. The PEM is a well-established algorithm to analyze the responses of linear time-dependent or independent systems under stationary or nonstationary random excitations (Lin et al., 2001). The semi-analytical method developed by Sheng et al. (2004a) and the 2.5D FEM developed by Yang and Hung (2001) are used to simulate the track–ground system.

The proposed framework includes two steps, as shown in Figure 1. The first step is to calculate the PSD of the wheel–rail dynamic force through the combination of the PEM and the semi-analytical method. The second step is to solve the nonstationary random responses of the track–soil system via the PEM and the 2.5D FEM. It is worth noting that a vehicle–ballast track–layered ground theoretical model was established by Sheng et al. (2004a). However, in this article, a vehicle–slab-track–layered ground theoretical model is developed for the ballastless track.

Figure 1.

Framework for analysis of high-speed train-induced random ground vibration.

PSD of the wheel–rail dynamic force based on the semi-analytical method and PEM

According to Figure 1, the PSD of the wheel–rail dynamic force should be obtained first. Usually, the relationship between the amplitude of the wheel–rail dynamic force and the amplitude of the track irregularity is represented by a yield equation, and the PSD of the wheel–rail dynamic force can be calculated by the PSD of the track irregularity in accordance with random vibration theory (Lombaert and Degrande, 2009). In the study by Sheng et al. (2004a), they established a vehicle–ballast-track–layered ground theoretical model that has also been used by other researchers (Sheng et al., 2004b, 2006). In this article, a vehicle–slab-track–layered ground theoretical model for the ballastless track is developed, as shown in Figure 2, where B denotes the contact width of railway and ground.

Figure 2.

Vehicle–slab-track–layered ground theoretical model.

Because this study does not focus on the dynamic response of the vehicle, the establishment of the vehicle–slab-track–layered ground theoretical model is used to obtain the PSD of the wheel–rail force. Based on the coordination condition of the displacement at wheel–rail contact points, the yield equation of the wheel–rail dynamic force can be expressed as (Sheng et al., 2004a)

(A_{W} + A_{R} + A_{Δ}) P (Ω) = Δ z (Ω)

(1)

where $A_{W}$ represents the receptance matrix of the vehicle at wheel–rail contact points, $A_{R}$ represents the receptance matrix of the track–soil system at wheel–rail contact points, $A_{Δ}$ represents the receptance matrix of the wheel–rail linear Hertzian contact. $Δ z (Ω)$ represents the amplitude of harmonic irregularity at wheel–rail contact points, $P (Ω)$ represents the amplitude of the wheel–rail dynamic forces. $Ω$ is the excitation angular frequency, rad/s. A detailed derivation of $A_{W}$ and $A_{Δ}$ can be found in the study by Sheng et al. (2004a), and the derivation of $A_{R}$ is shown as follows.

The rail, track slab, and concrete base are simulated by Euler beams and the elastic supports of the fasteners and the cement–asphalt–mortar (CAM) layer are simulated by linear spring–dampers uniformly distributed along the track. Thus, the governing equations for the slab-track system can be written as

\begin{matrix} E_{r} I_{r} \frac{\partial^{4} u_{r} (x, t)}{\partial x^{4}} + m_{r} \frac{\partial^{2} u_{r} (x, t)}{\partial t^{2}} + k_{p} [u_{r} (x, t) - u_{s} (x, t)] \\ + c_{p} [\frac{\partial u_{r} (x, t)}{\partial t} - \frac{\partial u_{s} (x, t)}{\partial t}] \\ = \sum_{h = 1}^{M} P_{h} δ (x - vt - α_{h}) \end{matrix}

(2)

\begin{matrix} E_{s} I_{s} \frac{\partial^{4} u_{s} (x, t)}{\partial x^{4}} + m_{s} \frac{\partial^{2} u_{s} (x, t)}{\partial t^{2}} - k_{p} [u_{r} (x, t) - u_{s} (x, t)] \\ - c_{p} [\frac{\partial u_{r} (x, t)}{\partial t} - \frac{\partial u_{s} (x, t)}{\partial t}] \\ + k_{q} [u_{s} (x, t) - u_{d} (x, t)] + c_{q} [\frac{\partial u_{s} (x, t)}{\partial t} - \frac{\partial u_{d} (x, t)}{\partial t}] = 0 \end{matrix}

(3)

\begin{matrix} E_{d} I_{d} \frac{\partial^{4} u_{d} (x, t)}{\partial x^{4}} + m_{d} \frac{\partial^{2} u_{d} (x, t)}{\partial t^{2}} - k_{q} [u_{s} (x, t) - u_{d} (x, t)] \\ - c_{q} [\frac{\partial u_{s} (x, t)}{\partial t} - \frac{\partial u_{d} (x, t)}{\partial t}] \\ = - f_{d} (x, t) \end{matrix}

(4)

where $E$ , $I$ denote the elasticity modulus and horizontal moment of inertia, respectively; $u$ denotes the vertical displacement; $k$ , $c$ denote stiffness and damping, respectively; subscripts $r$ , $s$ , and $d$ denote rail, track slab, and concrete base, respectively; subscripts $p$ and $q$ represent the fasteners and CAM layer, respectively; $f_{d}$ denotes the load between the concrete base and the subgrade; $v$ denotes the speed of the train. $δ (\cdot)$ is the Dirac function; $M = 4 N$ , $N$ is the number of vehicles of the train. $P_{h}$ is the dynamic wheel–rail force of the $h$ -th wheelset, which is assumed to be a harmonic force with frequency $Ω$ . The positive direction of $P_{h}$ is upward. $α_{h}$ is the location of the $h$ -th wheelset at $t = 0$ .

The vibration of the subgrade system is considered in this article, which is simulated by linear mass–spring–dampers also uniformly distributed along the track. The governing equations of the subgrade system can be written as

\begin{matrix} \frac{m_{b}}{6} [2 \frac{\partial^{2} u_{d} (x, t)}{\partial t^{2}} + \frac{\partial^{2} u_{b} (x, t)}{\partial t^{2}}] + k_{b} [u_{d} (x, t) - u_{b} (x, t)] \\ + c_{b} [\frac{\partial u_{d} (x, t)}{\partial t} - \frac{\partial u_{b} (x, t)}{\partial t}] \\ = f_{d} (x, t) \end{matrix}

(5)

\begin{matrix} \frac{m_{b}}{6} [\frac{\partial^{2} u_{d} (x, t)}{\partial t^{2}} + 2 \frac{\partial^{2} u_{b} (x, t)}{\partial t^{2}}] - k_{b} [u_{d} (x, t) - u_{b} (x, t)] \\ - c_{b} [\frac{\partial u_{d} (x, t)}{\partial t} - \frac{\partial u_{b} (x, t)}{\partial t}] \\ = f_{b} (x, t) \end{matrix}

(6)

in which $b$ denotes the subgrade, $f_{b}$ is the load acting on the soil.

Compared with the ballast-track (Sheng et al., 1999), the governing equations of the slab-track (2)∼(4) add the terms of bending stiffness, $E_{s} I_{s} \frac{\partial^{4} u_{s} (x, t)}{\partial x^{4}}$ and $E_{d} I_{d} \frac{\partial^{4} u_{d} (x, t)}{\partial x^{4}}$ , on the track slab and concrete base, respectively. Using the method similar to the study by Sheng et al. (1999), the vertical displacement of the rail in the frequency–wavenumber domain can be obtained

U_{r} = \frac{\bar{P} (k_{x})}{[E_{r} I_{r} k_{x}^{4} - m_{r} ω^{2} + i ω c_{p} + k_{p} - \frac{{(k_{p} + i ω c_{p})}^{2}}{K_{11}}]}

(7)

where

\bar{P} (k_{x}) = \sum_{h = 1}^{M} P_{h} (Ω) e^{- i k_{x} α_{h}}

(8)

\begin{matrix} K_{11} = E_{s} I_{s} k_{x}^{4} - m_{s} ω^{2} + i ω (c_{p} + c_{q}) + k_{p} \\ + k_{q} - \frac{{(k_{q} + i ω c_{q})}^{2}}{K_{12}} \end{matrix}

(9)

\begin{matrix} K_{12} = E_{d} I_{d} k_{x}^{4} - (\frac{m_{b}}{3} + m_{d}) ω^{2} + i ω (c_{q} + c_{b}) \\ + k_{q} + k_{b} + \bar{H} (- m_{b} ω^{2} / 6 - k_{b} - i ω c_{b}) K_{13} \end{matrix}

(10)

K_{13} = \frac{m_{b} ω^{2} / 6 + i ω c_{b} + k_{b}}{(- m_{b} ω^{2} / 3 + k_{b} + i ω c_{b}) \bar{H} + 1}

(11)

where $i = \sqrt{- 1}$ , $k_{x}$ denotes the wavenumber in the $x$ -direction, rad/min; $ω = Ω - k_{x} v$ is the angular frequency of the system, rad/s. $\bar{H}$ is the transfer function of the layered ground and its derivation can be found in the study by Sheng et al. (1999). As a supplement, the Fourier transformed dynamic flexibility matrices of layered ground for the case when $k_{x} \neq 0$ are given in the Supplemental Appendix.

Suppose that the inverse Fourier transform of $U_{r}$ is ${\hat{U}}_{r}$ , the vertical displacement of the rail in the time–space domain can be written as

u_{r} (x, t) = {\hat{U}}_{r} (x - vt) e^{i Ω t}

(12)

Suppose that a vertical harmonic load $e^{i Ω t}$ moves along the rail at the speed of $v$ . When $t = 0$ , it is located at $x = 0$ , and the steady displacement of the rail is determined by

u_{r}^{e} (x, t) = {\hat{U}}_{r}^{e} (x - vt) e^{i Ω t}

(13)

Equation (13) means that the response of the track–soil system is harmonic and has the same frequency as that of the load when observed in a reference frame moving with the load. Therefore, the receptance of the $l$ -th wheel–rail contact point caused by a unit load acting at the $h$ -th wheel–rail contact point (both points move at speed $v$ ) can be expressed as

u_{r lh}^{e} = {\hat{U}}_{r}^{e} (α_{lh}) e^{i Ω t}

(14)

where $α_{lh} = α_{l} - α_{h}$ is the distance between the two contact points. When the $l$ -th contact point is ahead of the $h$ -th contact point, $α_{lh} > 0$ . The complex amplitudes of the displacements of the wheel–rail contact points on rail are given by

u_{r lh} = {\hat{U}}_{r}^{e} (α_{lh})

(15)

Considering a vehicle with four wheelsets, the displacement amplitudes of the wheel–rail contact points can be written as

\begin{matrix} u_{R} (Ω) = {\begin{matrix} u_{R 1} (Ω) & u_{R 2} (Ω) & \begin{matrix} u_{R 3} (Ω) & u_{R 4} (Ω) \end{matrix} \end{matrix}}^{T} \\ = - A_{R} P (Ω) \end{matrix}

(16)

where

A_{R} = [\begin{matrix} u_{r 11} & u_{r 12} & \begin{matrix} u_{r 13} & u_{r 14} \end{matrix} \\ u_{r 21} & u_{r 22} & \begin{matrix} u_{r 23} & u_{r 24} \end{matrix} \\ \begin{matrix} u_{r 31} \\ u_{r 41} \end{matrix} & \begin{matrix} u_{r 32} \\ u_{r 42} \end{matrix} & \begin{matrix} \begin{matrix} u_{r 33} & u_{r 34} \end{matrix} \\ \begin{matrix} u_{r 43} & u_{r 44} \end{matrix} \end{matrix} \end{matrix}]

(17)

where $A_{R}$ is the receptance matrix of a single vehicle. When a train is composed of $N$ identical vehicles, the total number of wheel–rail contact points is $4 N$ , and the receptance matrix of the train system can be easily obtained.

According to the PEM, the pseudo-excitation of the system excited by the track irregularity can be expressed as (Lin et al., 2001)

Δ \tilde{z} (t) = [\begin{matrix} e^{- i Ω t_{1}} \\ e^{- i Ω t_{2}} \\ \begin{matrix} ⋮ \\ e^{- i Ω t_{M}} \end{matrix} \end{matrix}] \sqrt{S_{v} (Ω)} e^{i Ω t} = Δ \tilde{z} (Ω) e^{i Ω t}

(18)

where $t_{i} (i from 1 to M)$ is the time lag between the first wheelset and the $i$ -th wheelset, $S_{v} (Ω)$ is the PSD of the track irregularity, $Δ \tilde{z} (Ω)$ is the amplitude of the pseudo-excitation.

If we replace $Δ z (Ω)$ in equation (1) with $Δ \tilde{z} (Ω)$ , then

(A_{W} + A_{R} + A_{Δ}) \tilde{P} (Ω) = Δ \tilde{z} (Ω)

(19)

where $\tilde{P} (Ω)$ is the amplitude of the pseudo-wheel–rail dynamic force corresponding to the pseudo-excitation $Δ \tilde{z} (Ω)$ . Finally, the PSD of the wheel–rail dynamic force $S_{d} (Ω)$ can be expressed as

S_{d} (Ω) = \tilde{P} {(Ω)}^{*} \cdot \tilde{P} {(Ω)}^{T}

(20)

where $\tilde{P} (Ω)^{*}$ , $\tilde{P} (Ω)^{T}$ are the conjugate and transpose of $\tilde{P} (Ω)$ , respectively.

Random vibration analysis of the track–soil system by the PEM

2.5D FEM model of the track–soil system

In this section, we introduce a strategy for the modeling of the track–soil system. As shown in Figure 1, consider the Cartesian coordinate system $x = (x, y, z)$ with the origin at the center of the track and the $x$ -axis along the track direction and the z-axis pointing downward. Following the appropriate formulation of the 2.5D FEM, the dynamic equilibrium equation of the medium can be obtained by a variational method. Therefore, after we discretize the cross-section into 2.5D finite elements and apply the Fourier transform to the $x$ coordinate, the dynamic equilibrium can be described by the following equation

\begin{array}{l} [K_{1}^{global} + i k_{x} K_{2}^{global} + k_{x}^{2} K_{3}^{global} + k_{x}^{4} K_{4}^{global} - ω^{2} M^{global} + K_{5}^{global} (k_{x}, ω)] u (k_{x}, ω) \\ = F (k_{x}, ω) = F_{s} (k_{x}, ω) + F_{d} (k_{x}, ω) \end{array}

(21)

where $K_{i}^{global}$ ( $i$ = 1 to 4) are the stiffness matrices of the domain described by finite elements, $M^{global}$ is the mass matrix; $u$ is the vector of the node displacements; $F$ is the vector of external loads including two parts, that is, the deterministic load $(F_{s})$ caused by train weight, the random load $(F_{d})$ caused by vertical track irregularity; $K_{5}^{global}$ is the matrix that collects the impedance terms of the layered ground. Complementary information about the deduction of the aforementioned matrices can be found elsewhere (Bian et al., 2008). It should be mentioned that the track is modeled considering an equivalent continuous formulation in this study (Alves Costa et al., 2012). The adopted formulation is the most usual for this type of problem; nevertheless, the comparison between computed and experimental results proved to be satisfactory, mainly when the relevant frequencies were below 500 Hz (Knothe and Wu, 1998). Since the dominant frequencies for ground-borne vibration in buildings are in the range 1–80 Hz (Thompson et al., 2019), it is reasonable to adopt the continuous formulation.

Nonstationary random vibration analysis of the track–soil system

The mean value of the system response caused by train weight $F_{s}$ is deterministic and can be easily obtained by the conventional 2.5D method. The system response caused by random load $F_{d}$ is quantified by the proposed approach. Considering that the train travels along the track at a constant speed $v$ , the position of the $h$ -th axle depends on the time $t$ , that is, $x_{h} (t) = x_{h 0} + vt e_{x}$ , where $x_{h 0} = {x_{h 0}, y_{0}, z_{0}}^{T}$ is the position at $t = 0$ . Since the geometry and properties of the track and soil are assumed to be uniform along the direction of the track, the following expression of random displacement $u_{d} (x, t)$ at point $x = {x, y, z}^{T}$ are derived by using the Betti–Rayleigh reciprocity theorem (Lombaert et al., 2014)

u_{d} (x, t) = \sum_{h = 1}^{M} \int_{- \infty}^{t} H^{T} (x - x_{h 0} - n v τ, t - τ) f_{d h} (τ) d τ

(22)

where $f_{d h} (t)$ is the time history of the random load applied to the track by the $h$ -th axle, $H^{T} (x, t)$ is the 3×3 transfer function matrix, $n = {1, 0, 0}^{T}$ . The three columns of the matrix $H (x, t)$ represent the displacement at a point ${x, y, z}^{T}$ for an impulse load applied at the point ${0, y_{0}, z_{0}}^{T}$ in the $x$ -, $y$ -, $z$ -directions, respectively.

By combining the expectation operator $E [\cdot]$ with the Wiener–Khintchine transform (Lu et al., 2006), the correlation function of $u_{d}$ can be written as

\begin{matrix} R_{u} (x; t_{1}, t_{2}) = \\ \int_{- \infty}^{+ \infty} \sum_{h = 1}^{M} \sum_{l = 1}^{M} S_{d hl} (Ω) I^{*} (x; Ω, t_{1}) I^{T} (x; Ω, t_{2}) d Ω \end{matrix}

(23)

I (x; Ω, t) = \int_{- \infty}^{t} H^{T} (x - x_{0} - n v τ, t - τ) e^{i Ω τ} d τ

(24)

where $S_{d hl} (Ω)$ is the PSD function of the wheel–rail force. When $h = l$ , $S_{d hl} (Ω)$ represents the self-PSD function of $h$ -th wheelset; when $h \neq l$ , $S_{d hl} (Ω)$ represents the cross-PSD function between the $h$ -th and $l$ -th wheelset. As the vehicle moves along the track, its wheelsets are subjected to loads generated by the same track surface. Therefore, it can reasonably be assumed that if such wheelsets have very similar parameters, the loads will have identical PSD except for certain time lags. $I (x; Ω, t)$ is the response of the system under a harmonic load $e^{i Ω t}$ . Since the geometry and properties of the track and soil are assumed to be uniform along the track, $I (x; ω, t)$ can be effectively calculated in the wavenumber–frequency domain by means of double Fourier transform

\hat{I} (k_{x}; Ω, ω) = {\hat{H}}^{T} (k_{x}, ω) e^{i k_{x} x_{0}} δ (ω - k_{x} v)

(25)

where ${\hat{H}}^{T}$ represents the Fourier transform of $H^{T}$ , $k_{x} = (k_{x}, y, z)$ . The time-dependent variance can be obtained by setting $t_{1} = t_{2} = t$ in equation (23)

\begin{matrix} R_{u} (x; t) = σ_{u}^{2} (x; t) = \\ \int_{- \infty}^{+ \infty} \sum_{h = 1}^{M} \sum_{l = 1}^{M} S_{d hl} (Ω) I^{*} (x; Ω, t) I^{T} (x; Ω, t) d Ω \end{matrix}

(26)

Here, $σ_{u} (x; t)$ is the standard deviation vector of the displacements. The integrand function in equation (26) is the PSD of the response, which has nonstationary characteristics

S_{u} (x; Ω, t) = \sum_{h = 1}^{M} \sum_{l = 1}^{M} S_{d hl} (Ω) I^{*} (x; Ω, t) I^{T} (x; Ω, t)

(27)

Because the PSD functions of each wheelset are fully coherent except for certain time lags, the cross-PSD function between any two wheelsets can be expressed as

S_{d hl} (Ω) = S_{d} (Ω) e^{i Ω (t_{h} - t_{l})}

(28)

where $S_{d} (Ω)$ denotes the average of the PSD of all wheelsets of the train; $t_{h}$ , $t_{l}$ are the time lags between the $h$ -th, $l$ -th wheelsets and the first wheelset of the train, respectively. By substituting equation (28) into equation (27), we can obtain

\begin{matrix} S_{u} (x; Ω, t) = \sum_{h = 1}^{M} \sum_{l = 1}^{M} S_{d} (Ω) e^{i Ω (t_{h} - t_{l})} I^{*} (x; Ω, t) I^{T} (x; Ω, t) \\ = {[\sum_{h = 1}^{M} \sqrt{S_{d} (Ω)} e^{- i Ω t_{h}} I (x; Ω, t)]}^{*} \\ {[\sum_{h = 1}^{M} \sqrt{S_{d} (Ω)} e^{- i Ω t_{h}} I (x; Ω, t)]}^{T} \\ = {\tilde{I}}^{*} (x; Ω, t) {\tilde{I}}^{T} (x; Ω, t) \end{matrix}

(29)

where

\begin{matrix} \tilde{I} (x; Ω, t) = \int_{- \infty}^{t} \sum_{h = 1}^{M} \sqrt{S_{d} (Ω)} H^{T} \\ (x - x_{h 0} - n v τ, t - τ) e^{i Ω (t - t_{h})} d τ \\ = \int_{- \infty}^{t} H (x, t) {\tilde{f}}_{d} (x, Ω, t) d τ \end{matrix}

(30)

Obviously, if the pseudo-load ${\tilde{f}}_{d} (x, Ω, t)$ is constituted and applied to the track–soil model, the PSD of the random response can be obtained effectively according to equation (29).

When the PEM is used to solve the random vibration of the track–soil system caused by a train passing, a series of pseudo-loads should be formulated according to the geometric structure of the train. Figure 3 shows a train with a constant velocity $v$ consisting of $N$ cars with different car lengths $L_{j}$ , as well as the distance $b_{j}$ between two bogies and the distance $a_{j}$ between two wheelsets of each bogie.

Figure 3.

Train loads model.

The total distribution function ${\tilde{f}}_{d} (x, Ω, t)$ of the pseudo-loads of the train can be expressed as

{\tilde{f}}_{d} (x, Ω, t) = \sum_{j = 1}^{N} {\tilde{f}}_{j} (x - x_{j 0} - vt, Ω)

(31)

Each element on the right-hand side represents the contribution of a single $j$ -th car. After Fourier transformation, the frequency–wavenumber domain representation of equation (31) is expressed as follows

{\tilde{f}}_{d}^{xt} (k_{x}, Ω, ω) = \frac{2 π}{v} δ (k_{x} v + Ω - ω) χ (k_{x}, Ω)

(32)

with

\begin{matrix} χ (k_{x}, Ω) = \sqrt{S_{d} (Ω)} \sum_{j = 1}^{N} \\ [1 + e^{- i a_{j} (k_{x} + Ω / v)} + e^{- i (a_{j} + b_{j}) (k_{x} + Ω / v)} + e^{- i (2 a_{j} + b_{j}) (k_{x} + Ω / v)}] \\ \exp [- i (k_{x} + Ω / v) \sum_{s = 0}^{n - 1} L_{s} - i L_{0} k_{x}] \end{matrix}

(33)

where $L_{0}$ is the distance to the reference position ahead of the first pseudo-load position; $t_{jm}$ is the time difference between the first axle and the $m$ -th axle of the $j$ -th car.

Combined with the pseudo-load function given in equation (32), the track–soil system governing equation under the pseudo-load can be expressed as

\begin{matrix} [K_{1}^{global} + i k_{x} K_{2}^{global} + k_{x}^{2} K_{3}^{global} + k_{x}^{4} K_{4}^{global} \\ - ω^{2} M^{global} + K_{5}^{global} (k_{x}, ω)] {\tilde{u}}_{d} = (k_{x}, Ω, ω) = Γ {\tilde{f}}_{d}^{xt} \end{matrix}

(34)

where ${\tilde{u}}_{d}$ represents the pseudo-displacement of the track–soil system in the frequency and wavenumber domains, respectively, and $Γ$ is the transformation matrix defining the DOF of the system under random excitation. When the pseudo-response of the system is obtained, the standard deviation of the random response can be easily obtained according to equation (26).

Comparison with 3D random method

Model and parameters

In this section, the accuracy and efficiency of the proposed approach are verified by comparing with the field measurement data (Bian et al., 2015) and the fast 3D random method for the train–track–soil coupling system (Wang et al., 2018). The tested site, carrying double ballastless tracks in an embankment section, is located on the Beijing to Shanghai high-speed railway line in China, and a detailed description can be found in the study by Bian et al. (2015). Based on the geometry and properties of the railway section, a 3D track–embankment–ground FEM model was established in the study by Wang et al. (2018). That model was used to verify the effectiveness of the fast 3D random method for train–track–soil system previously proposed by authors. In this article, a 2.5D track–embankment–ground FEM model in which the cross-section mesh is the same as that of the verified 3D model is established, as shown in Figure 4. Herein, the four-node quadrilateral elements are used to assemble the stiffness and mass matrices of the track–embankment–ground system, the Euler beams are used to represent the rails and the spring–dashpot elements are used to simulate the rail pads. To effectively control the mesh range of the model, the viscous-elastic boundary (Bian et al., 2008) is applied to the transverse and bottom boundaries. Regarding the element mesh sizes of the model, a detailed description can be found in the study by Wang et al. (2018).

Figure 4.

Cross-section of the 2.5D track–embankment–ground model (unit: m). CAM: cement–asphalt–mortar.

In this article, a CRH3-type high-speed train consisting of eight identical vehicles is adopted, and its parameters can be found in the study by Lei and Zhang (2011). The PSD of the track vertical profile irregularity of the Germany high-speed track spectrum of low irregularity is adopted. The wavelength coverage of track irregularity ranges from 1 to 100 m. The slab-track–ground theoretical model specified by the parameters in Table 1 is used to calculate the PSDs of the wheel–rail dynamic force. The track parameters are determined by the approach presented in the study by Ghangale et al. (2018), with the rail receptance obtained by the semi-analytical method and 2.5D FEM and the rail response matched at zero frequency to obtain equivalent track parameters. The ground parameters are determined by field testing in Shanghai (Feng et al., 2013). All numerical simulations in this section are simulated by a personal computer equipped with Intel E5-2620 central processing unit (CPU) and 64GB RAM.

Table 1.

Parameters of slab-track–layered ground theoretical model.

Notation	Item	Unit	Value
Slab track
$E I_{r}$	Bending stiffness of rail	N·m²	1.325×10⁷
$m_{r}$	Mass per unit length of rail	kg	120
$k_{p}$	Stiffness under rail pad	N/m	1.85×10⁸
$c_{p}$	Damping under rail pad	N·s/m	1.54×10⁵
$E I_{s}$	Bending stiffness of slab track	N·m²	6.63×10⁷
$m_{s}$	Mass per unit length of slab track	kg	1425
$k_{s}$	Stiffness per unit length of CAM layer	N/m	9×10⁸
$c_{s}$	Damping per unit length of CAM layer	N·s/m	8.3×10⁴
$E I_{d}$	Bending stiffness of concrete base	N·m²	2.093×10⁸
$m_{d}$	Mass per unit length of concrete base	Kg	2325
$m_{c}$	Mass per unit length of subgrade bed	Kg	19789
$k_{c}$	Stiffness per unit length of subgrade bed	N/m	3×10⁸
$c_{c}$	Damping per unit length of subgrade bed	N·s/m	2×10⁵
Soil
$H_{1}$	Depth of the first soil layer	M	3
$E_{1}$	Yang’s modulus of the first soil layer	MPa	100
$ρ_{1}$	Density of the first soil layer	kg/m³	1850
$γ_{1}$	Poisson ratio of the first soil layer	—	0.49
$β_{1}$	Damping ratio of the first soil layer	—	0.1
$E_{2}$	Yang’s modulus of the soil half-space	MPa	500
$ρ_{2}$	Density of the soil half-space	kg/m³	2000
$γ_{2}$	Poisson ratio of the soil half-space	—	0.4
$β_{2}$	Damping ratio of the soil half-space	—	0.1
$B$	Contact width of railway and ground	M	2.66

CAM: cement–asphalt–mortar.

Results, comparisons, and discussion

Before random vibration analysis, the reliability of the 2.5D track–ground–embankment FEM model is verified by comparisons with the 3D model results and field measurements. The 3D model results come from a prediction model that mixes the 2D train model and the 3D track–embankment–ground FEM model. The train is modeled as a 2D multi-rigid-body dynamic system with 10 DOFs for each vehicle. The track–embankment–ground system is modeled as a 3D model by a finite element analysis software package (Wang et al., 2018). Based on the wheel–rail linear Hertzian contact assumption (Ma et al., 2020), a time-dependent train–track–embankment–ground coupled system is established by means of the approach reported by Zhu et al. (2017, 2018a, 2019) and Gong et al. (2020). For each case of the 3D simulations, a time step Δt of 0.002 s is adopted. The field measurements come from a field test (Bian et al., 2015) in which a CRH3-type train traveled through the section at the speed of 316 km/h.

While the testing trains were running, the vibration responses of the track structure were subjected to real-time measurements. The measuring point of the field test was located on the concrete base, marked by “” in Figure 5. As vibration velocity can reflect the intensity of potential damage to structures, and the peak value has a direct relationship to the fatigue damage to buildings, the vibration velocity level was adopted in this study to assess the vibration. The vertical vibration velocity of the measuring point and its Fourier spectral analysis at the train speed of 316 km/h are shown in Figure 6. It can be seen from Figure 6(a) that the 2.5D model result agrees well with the field measurement. The absolute maximum values of the 2.5D model and field measurement are 0.0076 m/s and 0.0086 m/s, respectively. It also can be seen from Figure 6(a) that the 2.5D model result fits well with the 3D model result. The absolute maximum values of the 2.5D model and 3D model are 0.0076 m/s and 0.0075 m/s, respectively. Figure 6(b) also shows that there is a good match between the measurement and simulations of three main frequencies (3.5 Hz, 7.0 Hz, and 10.5 Hz). After comparisons with the field test results and 3D model results, it can be concluded that the 2.5D track–embankment–ground FEM model has good reliability and accuracy in predicting the train-induced vibrations.

Figure 5.

Schematic diagram of measuring point and observation points arrangement.

Figure 6.

Comparisons of measurements and simulations of the measuring point vertical velocity at train speed of 316 km/h: (a) time history; (b) frequency content.

To illustrate the accuracy and efficiency of the proposed approach, 3D analyses are performed in which the fast random method developed by the authors Wang et al. (2018) is used to calculate the random vibrations of the train–track–soil coupled system. The 3D FEM model established in the study by Wang et al. (2018) is used to calculate the random vibrations of the coupled system and the number of discrete points of the excitation frequency domain is equal to that in the 2.5D approach. For each frequency point, the transient analysis is used to obtain the pseudo-responses of the 3D model. Considering the train travels with a constant speed of 316 km/h, the PSDs of the wheel–rail dynamic force of the fourth wheelset of the second vehicle calculated by the 3D random method and the semi-analytical method are plotted in Figure 7. Figure 7(a) shows the mesh plot of the PSD of the wheel–rail dynamic force calculated by the 3D random method, which varies in the time and frequency domains. Figure 7(b) shows the PSDs of the wheel–rail dynamic force calculated by both methods with respect to frequency. It can be seen from Figure 7(a) that, due to the reality that the rails are discretely supported in the 3D FEM model, the PSD of the wheel–rail force varies in the time domain. It also can be seen from Figure 7(b) that the PSDs of the wheel–rail force calculated by both methods agree well. The domain vibration frequencies and peak values of the 3D random method and the semi-analytical method are 47.40 Hz and 3.34 kN²/s, 49.35 Hz and 3.37 kN²/s, respectively. The PSDs of the wheel–rail force of other wheelsets calculated by both methods also show good agreement.

Figure 7.

PSDs of the wheel–rail dynamic force of the last wheelset of the second vehicle at train speed of 316 km/h: (a) 3D random method; (b) 3D random method and semi-analytical method.

Another four observation points in the FEM model are selected for numerical simulations, their distance from the track centerline being 0 m, 7.5 m, 20 m, and 40 m, respectively, marked with “” in Figure 5. The standard deviation of the vertical vibration velocity of the observation points at (a) 0 m, (b) 7.5 m, (c) 20 m, and (d) 40 m from the track centerline at the train speed of 316 km/h are shown in Figure 8. It can be seen from Figure 8 that there are agreements between the proposed approach results and the 3D random method results. The maximum values of the proposed results and 3D results are 0.0026 m/s and 0.0028 m/s, 6.24×10⁻⁴ m/s and 5.83×10⁻⁴ m/s, 1.80×10⁻⁴ m/s and 1.70×10⁻⁴ m/s, 1.06×10⁻⁴ m/s and 1.09×10⁻⁴ m/s for the 0 m, 7.5 m, 20 m, and 40 m points, respectively. For the observation point at 0 m from the track centerline, the standard deviation of the vertical velocity calculated by the proposed approach has a lower amplitude than the 3D result. That is because the discretely supported rails in the 3D FEM model make the impact of train passage on the concrete base more obvious.

Figure 8.

Standard deviation of the vertical velocity of the observation points at (a) 0 m, (b) 7.5 m, (c) 20 m, and (d) 40 m from the track centerline at train speed of 316 km/h. 3D: three dimensional.

Another five train speeds, 250 km/h, 300 km/h, 350 km/h, 400 km/h, and 450 km/h, are considered for calculation of the random responses of the track–embankment–ground system. To limit the computational cost of the 3D simulations, a CRH3-type train consisting of four identical vehicles is used in the analysis. For the proposed approach, the PSDs of the wheel–rail dynamic force that were used to constitute the pseudo-loads at different train speeds are plotted in Figure 9. As can be seen, the PSDs of the wheel–rail force increase rapidly at the train speeds of 400 km/h and 450 km/h. That is because these speeds are close to the critical speed of 420 km/h of the track–ground system (Bian et al., 2015).

Figure 9.

PSDs of the wheel–rail dynamic force used to constitute the pseudo-loads at different train speeds.

The attenuation of the maximum standard deviations of the vertical vibration velocity at train speeds of 250 km/h, 300 km/h, 350 km/h, 400 km/h, and 450 km/h is plotted in Figure 10. The horizontal axis represents the distance from the track centerline and the vertical axis represents the maximum standard deviation of vibration velocity, denoted by the vibration velocity level, unit dB

VL = 20 \lg (\frac{v_{e}}{v_{0}})

(35)

where $v_{e}$ is the maximum standard deviation of vertical vibration velocity and $v_{0}$ is the reference vibration velocity with the value of 1.0×10⁻⁸ m/s. The maximum velocity level (dB) of the observation points and the CPU time (h) at different train speeds are listed in Table 2, where “Prop.” and “3D” denote the proposed results and the 3D random method results, respectively. It can be seen from Figure 10 and Table 2 that the attenuation curves predicted by both methods fit well. The maximum differences between these two methods are 2.1 dB, 3.4 dB, 4.0 dB, 4.3 dB, and 4.6 dB for train speeds of 250 km/h, 300 km/h, 350 km/h, 400 km/h, and 450 km/h, respectively, and the corresponding relative errors are 2.6%, 4.4%, 4.4%, 5.4%, and 4.8%. In terms of computational efficiency, the proposed approach has three-to-five times increase compared with the 3D random method. For example, the CPU times required by the proposed approach and 3D method at the train speed of 350 km/h were 49.33 h and 188.75 h, respectively, in which the latter is 3.8 times that of the former.

Figure 10.

Attenuation of maximum standard deviation of the vertical vibration velocity at different train speeds: (a) 250 km/h; (b) 300 km/h; (c) 350 km/h; (d) 400 km/h; (e) 450 km/h. 3D: three dimensional.

Table 2.

Maximum velocity level (dB) of the observation points and CPU time (h) at different train speeds.

Train speed (km/h)	250		300		350		400		450
	3D	Prop.	3D	Prop.	3D	Prop.	3D	Prop.	3D	Prop.
D = 0 m (dB)	106.49	104.99	108.85	107.31	110.86	109.25	112.51	111.90	113.96	114.58
D = 7.5 m (dB)	92.47	93.00	94.90	96.12	96.16	96.72	96.27	99.12	96.80	100.44
D = 20 m (dB)	81.66	81.70	84.61	84.55	86.11	85.87	86.61	88.42	86.44	89.85
D = 40 m (dB)	78.31	77.85	79.38	79.33	81.14	81.79	80.51	82.54	82.48	82.99
CPU time (h)	247.81	49.40	213.94	49.31	188.75	49.33	167.60	49.37	154.30	49.30

3D: three dimensional; Prop: proposed.From the comparisons presented in this section, it can be confirmed that the proposed approach has sufficient accuracy and high efficiency in calculation of the nonstationary random ground vibrations induced by moving trains.

Conclusion

In this article, a framework is proposed that combines the PEM with the 2.5D FEM to quantify the effect of random track irregularity on high-speed train-induced ground vibrations. From the numerical results obtained in this work, the following conclusions are drawn:

From comparisons with the field measurement results obtained from the Beijing-Shanghai high-speed railway in China, the 2.5D track–embankment–ground prediction model established in this article is capable of describing the experimental data with sufficient accuracy.

Based on the combination of the PEM and the developed vehicle–slab-track–layered ground theoretical model, the PSD of the wheel–rail dynamic force can be efficiently obtained. The PSD of the wheel–rail dynamic force obtained by the proposed approach and the fully coupled train–track–soil model shows good agreement.

After the comparison to the fast 3D random method for train–track–soil system, it is confirmed that the proposed approach can predict the train-induced random ground vibrations with sufficient accuracy and has about three-to-five times increase in computation efficiency.

Supplemental Material

Supplemental_material – Supplemental material for A framework combining pseudo-excitation method and two-and-a-half-dimensional finite element method for random ground vibrations induced by high-speed trains

Supplemental material, Supplemental_material for A framework combining pseudo-excitation method and two-and-a-half-dimensional finite element method for random ground vibrations induced by high-speed trains by Lidong Wang, Zhihui Zhu, Pedro Alves Costa, Yu Bai, Zhiwu Yu and Yan Han in Advances in Structural Engineering

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 financially supported by (1) National Natural Science Foundation of China (Grant No. 51678576); National Key R&D Program of China (Grant No. 2017YFB1201204) and (2) Project POCI-01-0145-FEDER-007457—CONSTRUCT—Institute of R&D in Structures and Construction and funded by FEDER funds through COMPETE2020—Programa Operacional Competitividade e Internacionalização (POCI)—and by national funds through FCT—Fundação para a Ciência e a Tecnologia; Project POCI-01-0145-FEDER-029577-funded by FEDER funds through COMPETE2020—Programa Operacional Competitividade e Internacionalização (POCI) and by national funds (PIDDAC) through FCT/MCTES.

ORCID iD

Zhihui Zhu

Supplemental Material

Supplemental material for this article is available online.

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