# Application of KLE-PEM for Random Dynamic Analysis of Nonlinear Train-Track-Bridge System

Authors: Lizhong Jiang; Xiang Liu; Tuo Zhou; Ping Xiang; Yuanjun Chen; Yulin Feng; Zhipeng Lai; Shanshan Cao
Journal: Shock and Vibration
 (2020)

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## Abstract

A nonlinear train-track-bridge system (TTBS) considering the random track irregularity and mass of train is discussed. Based on the Karhunen–Loéve theory, the track irregularity is expressed and input into the TTBS, and the result of random response is calculated using the point estimation method. Two cases are used to compare and validate the applicability of the proposed method, which show that the proposed method has a high precision and efficiency. Then, taking a 7-span bridge and a high-speed train as an example, the calculation results of random response of the nonlinear and linear wheel-rail model are compared, and the results show that for the bridge and rail response, the nonlinear and linear models are almost the same. Finally, comparing the calculated probability distribution results with the test results, it shows that the method can be applied to the prediction of actual response range.


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## Body

## 1. Introduction

Bridge structure occupies a large proportion in high-speed railway (HSR) lines [1, 2]. Because of inevitable errors in processing, pavement technology, environment, and other potential problems of track structure, random track irregularity occurs in a train-track-bridge system (TTBS) under excitation, affecting the stability and safety of train operation [3, 4]. When a train passes over a bridge, the bridge is affected by train vibration; simultaneously, the bridge vibration affects the train vibration [5–7]. The interaction between train and bridge vibrations leads to a coupling vibration effect; in this process, the effects on system vibration due to track irregularity cannot be ignored. Various rail irregularities are superimposed by random irregularities of different wavelengths, phases, and amplitudes [8]. This is a complex random process related to the railway line. Therefore, in dynamics analysis, a TTBS should be regarded as a random system. The track irregularity can be expressed by many methods including power spectrum statistics, Karhunen–Loéve expansion (KLE) [9], and others.In recent years, the random dynamics of the TTBS has attracted much attention. The most common calculation method for random dynamics is Monte Carlo simulation (MCS) [10, 11]. MCS requires abundant samples to obtain a convergence result in complex engineering problems, indicating that MCS is subjected to many limitations in practical applications. To obtain an efficient and accurate calculation method for the random dynamics of the TTBS, many studies have been conducted. For instance, the pseudo-excitation method was used for the random dynamics analysis of the train-track system and TTBS [12–16]. It is convenient to use the pseudo-excitation method for response power spectrum calculation in both nonstationary and stationary random excitation structural analysis with a high application value. The probability density evolution method (PDEM) was successfully applied in the analysis of train-track and train-bridge coupling dynamics considering track irregularity of randomness [17–21]. An efficient train-bridge random response method capable of global sensitivity analysis has been proposed in reference [22]. Wu and Law [23] used the stochastic finite element spectral method and regarded road irregularity as a random process. Jiang [24] used the point estimation method (PEM) to the stochastic TTBS and analyzed the sensitivity of random parameters. The PEM is a highly accurate and fast statistical moment calculation method, first proposed by Rosenblueth [25]. The core of the PEM is the use of the Gaussian integral method to calculate statistical moments.This paper is a further discussion of the previous works [24, 26, 27]. Hertz contact theory has been used for the calculation of normal contact force between a wheel and rail; the relationship between force and wheel/rail displacement is nonlinear. Under overloading and high-speed operation, strong wheel-rail dynamic effect results in the disruption of vertical dynamic load between the wheel and rail, severely affecting the bridge safety and passenger comfort as well as directly affecting the dynamic response of the bridge [28]. In the TTBS, the treatment of wheel-rail contact force as a nonlinear interaction can more truly reflect the dynamic situation; in addition, in some extreme track irregularity situations such as earthquake [29,30], settlement [31,32], creep, and rail damage [33,34], the wheel may jump from the rail; i.e., the always-close wheel/rail assumption is not suitable for use. These pose a challenge to the application of random vibration theory. Therefore, this paper proposes a nonlinear random dynamics calculation method suitable for the TTBS with stochastic track irregularity. In this method, the track irregularity is expressed by KLE, and then the system response is calculated using the PEM according to the characteristics of KLE. The difference between nonlinear and linear rail bridge systems is discussed as well.


## 2. Dynamic Model of Nonlinear TTBS

### 2.1. Wheel-Rail Contact

This study focused on the vertical movement of the TTBS; thus, only the vertical relationship of wheel-rail contact is considered. The normal force is the vertical wheel-rail force in this system, which is determined by compression and calculated using the famous Hertz nonlinear elastic contact theory, as follows:(1)Fci=1Gzwi−zri−zRoui1.5,zwi−zri−zRoui≤0,0,zwi−zri−zRoui>0,where zwi is the vertical displacement of the ith wheel set; zri is the vertical displacement of rail at the position of ith wheel set; zRoui is the track irregularity at the position of ith wheelset; and G is the contact constant. For the abrasion type tread, G = 3.86R−0.115 × 10−8(m/N2/3), and R is the radius of wheel rolling circle.Sometimes, the Hertz spring stiffness can be linearized as follows [35]:(2)Fci=−kHLzwi−zri−zRoui,zwi−zri−zRoui≤0,0,zwi−zri−zRoui>0,where kHL is the linearized contact stiffness of wheel-rail and can be calculated as follows:(3)kHL=23Gp01/3,where p0 is the static load of the wheel.


### 2.2. Dynamic Model of Train-Track-Bridge

As shown in Figure1, one carriage consists of one carbody, two bogies, and four wheelsets, which is a multifreedom mass-spring-dashpot system. In the model, mc, mt, and mw are the mass of carbody, bogie, and wheelset, respectively; Jc, Jt, and Jw are the moment of inertia of carbody, bogie, and wheelset, respectively; k1 and k2 are the stiffness of primary and second suspension spring, respectively; and c1 and c2 are the damping coefficient of primary and second suspension spring, respectively. Each carbody has two degrees of freedom (DOF) (zc, φc), each bogie has two DOFs (zt, φt) as well, and each wheel only has one DOF zw. Thus, each carriage has 10 DOFs. A dynamic equation of the train can be derived from the energy principle, as follows [36]:(4)MvXv..+CvXv.+KvXv=Fv.Figure 1
Train-track-bridge model.Elastic Bernoulli–Euler beams are utilized for rail, track slab, and bridge. Two steel rails are equivalent to one. The fasteners between the rail and rail plate are modeled as spring-dashpot systems, and the mortar layer between the track slab and girder is modeled as a continuous spring-damping system. The mass of fastener and CA mortar layer is neglected, and the structural damping of the rail and track slab is not considered. The damping of girders is assumed as Rayleigh damping. The entire track-bridge system can be modeled as a three-layer beam, the model can be established by FEM, and the dynamic equation can be written as follows:(5)MbXb..+CbXb.+KbXb=Fb,where Xb ,X˙b, and X¨b are the displacement, velocity, and acceleration of the bridge and Fb is the force vector of the bridge. The gravity of structure is neglected in Fb.The train and track-bridge system can be coupled using the wheel-rail relationship, and the system dynamic equation can be derived from equations (4) and (5), which can be written as follows:(6)MvMbX..vX..b+CvCbX.vX.b+KvKbXvXb=FvFb.The numerical time step-by-step integral method can be used to solve the system dynamic equation. Equation (1) or (2) shows that the calculation of force of wheel-rail is related to the structural response of the wheel; therefore, iteration is needed in the implicit algorithm or using the explicit algorithm [37].


## 2.1. Wheel-Rail Contact

This study focused on the vertical movement of the TTBS; thus, only the vertical relationship of wheel-rail contact is considered. The normal force is the vertical wheel-rail force in this system, which is determined by compression and calculated using the famous Hertz nonlinear elastic contact theory, as follows:(1)Fci=1Gzwi−zri−zRoui1.5,zwi−zri−zRoui≤0,0,zwi−zri−zRoui>0,where zwi is the vertical displacement of the ith wheel set; zri is the vertical displacement of rail at the position of ith wheel set; zRoui is the track irregularity at the position of ith wheelset; and G is the contact constant. For the abrasion type tread, G = 3.86R−0.115 × 10−8(m/N2/3), and R is the radius of wheel rolling circle.Sometimes, the Hertz spring stiffness can be linearized as follows [35]:(2)Fci=−kHLzwi−zri−zRoui,zwi−zri−zRoui≤0,0,zwi−zri−zRoui>0,where kHL is the linearized contact stiffness of wheel-rail and can be calculated as follows:(3)kHL=23Gp01/3,where p0 is the static load of the wheel.


## 2.2. Dynamic Model of Train-Track-Bridge

As shown in Figure1, one carriage consists of one carbody, two bogies, and four wheelsets, which is a multifreedom mass-spring-dashpot system. In the model, mc, mt, and mw are the mass of carbody, bogie, and wheelset, respectively; Jc, Jt, and Jw are the moment of inertia of carbody, bogie, and wheelset, respectively; k1 and k2 are the stiffness of primary and second suspension spring, respectively; and c1 and c2 are the damping coefficient of primary and second suspension spring, respectively. Each carbody has two degrees of freedom (DOF) (zc, φc), each bogie has two DOFs (zt, φt) as well, and each wheel only has one DOF zw. Thus, each carriage has 10 DOFs. A dynamic equation of the train can be derived from the energy principle, as follows [36]:(4)MvXv..+CvXv.+KvXv=Fv.Figure 1
Train-track-bridge model.Elastic Bernoulli–Euler beams are utilized for rail, track slab, and bridge. Two steel rails are equivalent to one. The fasteners between the rail and rail plate are modeled as spring-dashpot systems, and the mortar layer between the track slab and girder is modeled as a continuous spring-damping system. The mass of fastener and CA mortar layer is neglected, and the structural damping of the rail and track slab is not considered. The damping of girders is assumed as Rayleigh damping. The entire track-bridge system can be modeled as a three-layer beam, the model can be established by FEM, and the dynamic equation can be written as follows:(5)MbXb..+CbXb.+KbXb=Fb,where Xb ,X˙b, and X¨b are the displacement, velocity, and acceleration of the bridge and Fb is the force vector of the bridge. The gravity of structure is neglected in Fb.The train and track-bridge system can be coupled using the wheel-rail relationship, and the system dynamic equation can be derived from equations (4) and (5), which can be written as follows:(6)MvMbX..vX..b+CvCbX.vX.b+KvKbXvXb=FvFb.The numerical time step-by-step integral method can be used to solve the system dynamic equation. Equation (1) or (2) shows that the calculation of force of wheel-rail is related to the structural response of the wheel; therefore, iteration is needed in the implicit algorithm or using the explicit algorithm [37].


## 3. Karhunen–Loéve Expansion

### 3.1. Theory

It is assumed thatux,θ is a real-valued stochastic process, and u¯x is the mean value of ux,θ, and Cx1,x2is a covariance function. According to Mercer’s theorem, Cx1,x2 is bounded, symmetric, and positive and can be expanded as follows:(7)Cx1,x2=∑n=1∞λnφnx1φnx2,where λn and φnx are the eigenvalues and eigenfunctions of Cx1,x2, respectively. Equation (7) can be solved using the following equation:(8)∫DCx1,x2φnx1dx=λnφnx2.The eigenfunctionsφnx satisfy orthogonal property, and it can be written as follows:(9)∫Dφnx1φmx2dx=δnm,where δnm is the Kronecker delta. The stochastic process ux,θ can be expressed as follows:(10)ux,θ=u¯x,θ+∑n=1∞ξnθλnφnx.This is the KLE, whereξnθ is a set of uncorrelated random variables, which can also be written as follows:(11)ξnθ=1λn∫Du˜x,θφnxdx.Whenux,θ belongs to the Gaussian stochastic process, ξnθwill be a group of uncorrelated variables, obeying the standard normal distribution.In application, the firstM terms in equation (10) should be truncated, which can be expressed as follows:(12)ux,θ=u¯x,θ+∑n=1Mξnθλnφnx.The accuracy of representation of stochastic field depends onM.


### 3.2. Numerical Expression

It is assumed thatrx,θ is a spatial one-dimensional stochastic process; its length is L. Isometric discretization of stochastic processes is Δx, and the number of discrete points is n=L/Δx+1. After enough stochastic samples were obtained, the stochastic process rx,θ can be represented as follows.There is a sample set of stochasticRx,θ=r1x,θ;r2x,θ,…,rmx,θ, where m is the number of samples and rix,θ is one of the samples with order 1×n. The covariance matrix ΓR,R with order n×n can be obtained. Then, the corresponding eigenvalues and eigenfunctions of stochastic process can be calculated using the following equation:(13)ΓR,RΔxφi−λiφi=0,where λn and φnx are the eigenvalues and eigenfunctions of KLE, respectively, and φnx should be normalized.After the truncation at theMth order, the stochastic process can be represented by the KLE, which is denoted as follows:(14)rx,θ=r¯x,θ+∑n=1Mξnθλnφnx,where r¯x,θ is the expectation of Rx,θ.


## 3.1. Theory

It is assumed thatux,θ is a real-valued stochastic process, and u¯x is the mean value of ux,θ, and Cx1,x2is a covariance function. According to Mercer’s theorem, Cx1,x2 is bounded, symmetric, and positive and can be expanded as follows:(7)Cx1,x2=∑n=1∞λnφnx1φnx2,where λn and φnx are the eigenvalues and eigenfunctions of Cx1,x2, respectively. Equation (7) can be solved using the following equation:(8)∫DCx1,x2φnx1dx=λnφnx2.The eigenfunctionsφnx satisfy orthogonal property, and it can be written as follows:(9)∫Dφnx1φmx2dx=δnm,where δnm is the Kronecker delta. The stochastic process ux,θ can be expressed as follows:(10)ux,θ=u¯x,θ+∑n=1∞ξnθλnφnx.This is the KLE, whereξnθ is a set of uncorrelated random variables, which can also be written as follows:(11)ξnθ=1λn∫Du˜x,θφnxdx.Whenux,θ belongs to the Gaussian stochastic process, ξnθwill be a group of uncorrelated variables, obeying the standard normal distribution.In application, the firstM terms in equation (10) should be truncated, which can be expressed as follows:(12)ux,θ=u¯x,θ+∑n=1Mξnθλnφnx.The accuracy of representation of stochastic field depends onM.


## 3.2. Numerical Expression

It is assumed thatrx,θ is a spatial one-dimensional stochastic process; its length is L. Isometric discretization of stochastic processes is Δx, and the number of discrete points is n=L/Δx+1. After enough stochastic samples were obtained, the stochastic process rx,θ can be represented as follows.There is a sample set of stochasticRx,θ=r1x,θ;r2x,θ,…,rmx,θ, where m is the number of samples and rix,θ is one of the samples with order 1×n. The covariance matrix ΓR,R with order n×n can be obtained. Then, the corresponding eigenvalues and eigenfunctions of stochastic process can be calculated using the following equation:(13)ΓR,RΔxφi−λiφi=0,where λn and φnx are the eigenvalues and eigenfunctions of KLE, respectively, and φnx should be normalized.After the truncation at theMth order, the stochastic process can be represented by the KLE, which is denoted as follows:(14)rx,θ=r¯x,θ+∑n=1Mξnθλnφnx,where r¯x,θ is the expectation of Rx,θ.


## 4. Point Estimation Method

It is assumed thatp(x) is a probability density function (PDF) of a continued random variable Y, and Y = gX. The expectation of random variable can be calculated using the following equation:(15)μ=EgX=∫−∞∞GXpXdX.The variance ofY can be calculated using the following formula:(16)σ2=∫−∞∞gX−μ2pXdX.When the function contains multiple random variables, it may be difficult to calculate the moments such as the central moments ofY = G(X). According to the dimension reduction method for multiple variables function proposed in [38], a n-dimensional variable function gX can be approximated using several s-dimensional variable function gsX, which can be written as follows:(17)gX≅gsX=∑i=0s−1iCn−s+i−1i∑k1<⋯<ks−iys−i,with ys−i=gc1,…,ck1−1,xk1,ck4+1,…,ckc−1−1,xkc−i,ckc−1+1,…,cn, where s<n and c=c1,c2,…,cn is the reference point.Whens = 1, the function is a one-variable dimensional reduction. Equation (17) can be converted into the following equation:(18)gX≅g1X=∑i=1ngiXi−n−1gc,with giXi=gc1,…,ci−1,Xi,ci+1,…,cn.By substituting equation (18) into equations (15) and (16), the expectation and variance of Y can be approximately expressed as follows:(19)μ=E∑i=1ngiXi−n−1gc≅∑i=1nEgiXi−n−1gc,(20)σ2≅E∑i=1ngi−μ2−n−1gc−μ2=∑i=1nEgiXi−μ2−n−1gc−μ2.When the random variable in functiongiXi obeys the standard Gaussian distribution, EgiXi and EgiXi−μ2 can be calculated approximately using the Gaussian–Hermite integration function, which can be expressed as follows:(21)EgiXi=∑l=1rwGH,lπgi2xGH,l,EgXi−μ2=∑l=1rwGH,lπgi2xGH,l−μ2,where r is the number of estimating points of Gaussian–Hermite integration and xGH,l and wGH,l are the abscissas and weights for Gaussian–Hermite integration, respectively; their detailed values with r = 3 are shown in Table 1.Table 1
Abscissas and weights for Gaussian–Hermite integration withr = 3 [39].
Point123xGH,l−1.2247401.22474wGH,l0.295411.181640.29541


## 5. Stochastic Dynamic Method for TTBS

Samples of track irregularity can be obtained from the actual railway line measurement or converted into spatial samples using power spectral density (PSD). After obtaining multiple spatial samples for irregularity, the eigenvalues and eigenfunctions expressed by the KLE of track irregularity can be derived. Assuming track irregularityz˜Rx,θ as a Gaussian random process, its KLE can be expressed as follows:(22)zx=z¯x+∑k=1mλkξkφkx,where z¯x,θ is the mean value of track irregularity and ξk is a set of distributed random variables independent of each other. Track irregularity is a stochastic process with zero mean value; thus, z¯x,θ=0. Equation (22) can be simplified as follows:(23)zx=z¯x+∑k=1mλkξkφkx.After obtaining an irregularity sample, it can be calculated as the irregularity amplitude in the model, i.e., the irregularity acts on the wheel-rail force of the train-bridge system as an external random excitation, and the corresponding response of the TTBS can be obtained. Therefore, the entire TTBS is a stochastic system withm random variables. According to the characteristics of equation (23), when the PEM is used for stochastic response calculation, zero can be selected as the reference point. In the calculation of stochastic system response through point estimation, the track irregularity sample corresponding to the lth Gaussian point of kth random variable can be expressed as follows:(24)z˜k,lx=2xGH,lλkφkx.Taking the displacement response of timet at some point of the bridge for instance, the corresponding response of the system R(k, l, t) can be obtained after calculating the track irregularity samples corresponding to each Gaussian point of each stochastic variable as the track irregularity samples of the system. The corresponding response of zero-irregularity sample is R0(t). The expectation and variance of response can be obtained by substituting all R(k, l, t) and R0 into equations (19) and (20):(25)Mean⋅t≈∑k=1m∑l=1rwGH,lπRk,l,t−m−1R0t,Var⋅t=∑n=1m∑l=1rwGH,lπRk,l,t−Mean⋅t2−m−1R0t−Mean⋅t2.Except the nonzero Gaussian point ofkth stochastic variable, other track irregularity samples are zero-amplitude irregularity, which is R(k, (r + 1)/2, t) = R0(t). Therefore, there is no need to repeat the calculation in the program. The standard deviation value of response can be calculated using the following equation:(26)Std.Dt=Var⋅t.


## 6. Numerical Simulation

Track irregularity can be measured using the railway line [9] and can also be obtained by converting the existing PSD function into spatial samples. Power spectrum simulation is used to obtain irregularity samples, where German low-interference PSD is used [12]. 10,000 irregularity samples with 1,000 meters are obtained using the trigonometric series method [8] and utilized as the initial sample. The results indicate that KLE truncation number 138 can obtain more than 98% of the probability information; therefore, the number of random variables is considered as 138.


### 6.1. Comparison and Validation

Case 1.
(a simple model). The nonlinear wheel-rail contact model may cause some trouble for the calculation of the stochastic TTBS; for example, the Hertz spring should be linearized in stochastic analysis [40]; besides, in the case of extreme track irregularity condition, the wheelset may jump, i.e., then there is no contact between the wheel and rail. To evaluate the applicability of the proposed method in the random TTBS with random track irregularity, a simple model is established, as shown in Figure 2. This model contains three DOFs, carbody-bogie system, wheel, and track. Besides gravity, the simple system is also excited by a random irregularity. When running to the third second, a 5 mm additional irregularity is suddenly added to force the wheel to jump to analyze the calculation accuracy of the proposed method after the wheel jump. The MCS method is used to compare and verify the accuracy of KLE-PEM. The mean and standard deviation values of displacement response of the wheel calculated using two methods are shown in Figure 3; it is observed that the dynamic response of the system enters the stable stage in two seconds. After being subjected to a sudden track irregularity excitation at the 3rd second, the wheel jumps from the track and then falls back to the track and returns to stability in about 0.2 s. In the entire process, the mean and standard deviation values of response calculated using the KLE-PEM are very close to those obtained by MCS, i.e., the proposed method can accurately calculate random responses with nonlinear wheel-rail contact even in the case of a jump, and it can be used to study extreme irregularities such as earthquake, settlement, creep, and rail damage in the future.Figure 2
Simple model of the train-track system.Figure 3
Comparison of displacement of the wheel.Case 2.
(comparison with MCS). The comparison results obtained with MCS are used to validate the accuracy of KLE-PEM in the nonlinear stochastic TTBS. In this case, only one carriage and one-span bridge are evaluated to save time, and the subgrade section is rigid. The calculation load of the TTBS for MCS is 5000. The number of KLE items after truncation is 138, and when using three nodes of Gaussian quadrature, the calculation load of the TTBS using KLE-PEM is 138 × 2 + 1 = 277. Wheel-rail contact force is calculated according to a nonlinear Hertz spring. The train parameters [41] are shown in Table 2, and the parameters of the bridge and rail system are shown in Table 3.
Figure4 shows the comparison of mean and standard deviation values and vertical displacement of rail fastener in the middle span calculated using KLE-PEM and MCS. It can be concluded that for the stochastic dynamic response of the track-bridge system, the results of KLE-PEM are almost consistent with the results of MCS. Figure 5 shows a comparison of the time-histories response of mean and standard deviation of the first wheel-rail force passing through the bridge. It can also be concluded that the results calculated using KLE-PEM are almost consistent with the results obtained using MCS.Table 2
Properties of train.
NotationUnitTrailer car/motor carmckg4.40 × 104/4.80 × 104mtkg2.40 × 103/3.20 × 103mwkg2.40 × 103/2.40 × 103Jckg·m22.70 × 106/2.70 × 106Jtkg·m22.20 × 103/7.20 × 103K1zN/m0.70 × 106/1.00 × 106K2zN/m3.00 × 105/4.00 × 105C1zN·s/m4.00 × 104/3.00 × 104C2zN·s/m5.00 × 104/4.00 × 104Lcm8.6875/8.6785Ltm1.25/1.25Lwwm4.90/4.90Table 3
Properties of the track-bridge.
NotationUnitItemValueErN/m2Young’s modulus of rail2.06 × 1011vr—Poisson’s ratio of rail0.3Irm4Mass moment of inertia of rail cross section2 × 3.217 × 10−5m¯rkg/mMass per unit length of rail2×60.64EsN/m2Young’s modulus of slab3.6 × 1010vs–Poisson’s ratio of slab0.2Ism4Mass moment of inertia of slab cross section1.6 × 10−3m¯skg/mMass per unit length of slab1.2 × 103krsN/mSpring stiffness of fastener2 × 5.0 × 107crsN·s/mDamping coefficient of fastener2 × 3.625 × 104k¯sbN/m2Spring stiffness of CA layer per unit length2 × 1.5 × 109c¯sbN·s/m2Damping coefficient of CA layer per unit length2 × 8.3 × 104EbN/m2Young’s modulus of girder3.45 × 1010vb—Poisson’s ratio of girder0.2Ibm4Mass moment of inertia of girder cross section12.744m¯bkg/mMass per unit length of bridge girder2.972 × 104ζb—Damping ratio of girder0.05Figure 4
Comparison of displacement of fastener at midspan.Figure 5
Comparison of wheel-rail contact force of first wheelset (double side).


### 6.2. Comparison between Linear and Nonlinear Simulation

A standard simply supported 7-span box girder with 32 m length per span is modeled to evaluate the effect of random track irregularity on the TTBS, and the difference between linear and nonlinear Hertz contacts is evaluated. A rigid subgrade is used for the simulation out of the bridge part. The treatment of boundary conditions between adjacent spans is shown in Figure1. The train uses four carriages ICE-3 train, and there are two motor cars at the head and tail of the train. To ensure the dynamic response stability before the train entering the bridge, the train starts running from 300 m away from the left side of the first span bridge. The operation speed of the train is 250 km/h. The TTBS considering linear Hertz and nonlinear Hertz is established.Figure6 shows the mean and standard deviation of midspan displacement response of fourth bridge span under the nonlinear and linear wheel-rail relationship, where the horizontal ordinate represents the time for the first wheelset entering the fourth bridge span. It can be concluded that the bridge dynamic responses under the two wheel-rail force models are similar. In addition, in the perspective of time-histories response trend, the mean value of time-histories response of the bridge has two peaks and three troughs. However, the standard deviation has only two peaks before and after, and the time of the first peak is inconsistent with the time of the first trough of the mean value.Figure 6
Vertical displacement of midspan of fourth span.The mean and standard deviation of time-histories response of vertical acceleration at the center of gravity of the first carbody are shown in Figure7, where the horizontal ordinate is the time taken for the first wheelset entering the first bridge span. It shows that the standard deviations of train acceleration obtained by the two wheel-rail force models are inconsistent. The standard deviation obtained by linear wheel-rail relationship is less than that obtained by nonlinear wheel-rail relationship in a relatively large time. In the entire process, the values of standard deviation obtained from the linear wheel-rail relationship are all around 6.7 × 10−2 m/s2, whereas the standard deviations obtained from the nonlinear relationship are relatively large at the initial stage of entering the bridge and then gradually decreases to 6.8 × 10−2 m/s2.Figure 7
Vertical acceleration at the center of gravity of first carbody.The mean and standard deviation of the time-histories response of vertical wheel-rail contact force of the first wheelset are shown in Figure8, where the horizontal ordinate is the time taken for the first wheelset entering the first bridge span. Because the subgrade section adopts rigid subgrade, there will be greater contact force at the moment when the wheelsets drive into and out of the bridge. Therefore, the wheel-rail force of the second span to the sixth span of the wheel set is selected for analysis; it can be seen that the mean values of contact force calculated by the nonlinear and linear model both oscillate about 20 kN along the axle load, and the magnitude of the contact force obtained from the nonlinear model is larger than that of the linear model. Regarding the standard deviations of wheel-rail force, it shows a significant difference between the two models at the initial stage when the train enters the bridge. Then, the difference gradually decreases, both of which oscillate between 2.5 kN and 4 kN. The standard deviations obtained from the linear wheel-rail relationship are less than that obtained from the nonlinear wheel-rail relationship in a relatively long time.Figure 8
Vertical contact force of first wheelset (double side).


### 6.3. Applied for Response Prediction

When it is necessary to obtain the response value of the bridge, the most accurate method is to install sensors in the field for actual measurement, but this method is very time-consuming and inconvenient. Therefore, the numerical simulation method is usually used to obtain the bridge response. However, if the deterministic analysis is used, the result will be different from the actual value because there are various randomness in the TTBS, such as track irregularity, train load, and so on. The results of probability range can be obtained by using the proposed stochastic analysis method. The accuracy of the probability distribution range obtained by KLE-PEM can be verified by comparing the test results in Ref. [42]. In this case, besides the stochastic track irregularity, there is also the random axle load, in which the train carbody mass is a normal distribution random variable with a coefficient of variation of 0.05. It was proved that the PEM can calculate the response of the TTBS with uncertain parameter precisely [24]; therefore, only one independent random variable needs to be added in KLE-PEM. (Mean ± 3 Std. D) are used to guarantee the probability range of results; the calculation results and the test data of maximum values of vertical displacement at bridge midspan are shown in Figure 9. It can be found that the test results are basically within the calculated probability distribution range, which means that the proposed method can be used to effectively predict the distribution range of response results.Figure 9
Calculation results and test data.


## 6.1. Comparison and Validation

Case 1.
(a simple model). The nonlinear wheel-rail contact model may cause some trouble for the calculation of the stochastic TTBS; for example, the Hertz spring should be linearized in stochastic analysis [40]; besides, in the case of extreme track irregularity condition, the wheelset may jump, i.e., then there is no contact between the wheel and rail. To evaluate the applicability of the proposed method in the random TTBS with random track irregularity, a simple model is established, as shown in Figure 2. This model contains three DOFs, carbody-bogie system, wheel, and track. Besides gravity, the simple system is also excited by a random irregularity. When running to the third second, a 5 mm additional irregularity is suddenly added to force the wheel to jump to analyze the calculation accuracy of the proposed method after the wheel jump. The MCS method is used to compare and verify the accuracy of KLE-PEM. The mean and standard deviation values of displacement response of the wheel calculated using two methods are shown in Figure 3; it is observed that the dynamic response of the system enters the stable stage in two seconds. After being subjected to a sudden track irregularity excitation at the 3rd second, the wheel jumps from the track and then falls back to the track and returns to stability in about 0.2 s. In the entire process, the mean and standard deviation values of response calculated using the KLE-PEM are very close to those obtained by MCS, i.e., the proposed method can accurately calculate random responses with nonlinear wheel-rail contact even in the case of a jump, and it can be used to study extreme irregularities such as earthquake, settlement, creep, and rail damage in the future.Figure 2
Simple model of the train-track system.Figure 3
Comparison of displacement of the wheel.Case 2.
(comparison with MCS). The comparison results obtained with MCS are used to validate the accuracy of KLE-PEM in the nonlinear stochastic TTBS. In this case, only one carriage and one-span bridge are evaluated to save time, and the subgrade section is rigid. The calculation load of the TTBS for MCS is 5000. The number of KLE items after truncation is 138, and when using three nodes of Gaussian quadrature, the calculation load of the TTBS using KLE-PEM is 138 × 2 + 1 = 277. Wheel-rail contact force is calculated according to a nonlinear Hertz spring. The train parameters [41] are shown in Table 2, and the parameters of the bridge and rail system are shown in Table 3.
Figure4 shows the comparison of mean and standard deviation values and vertical displacement of rail fastener in the middle span calculated using KLE-PEM and MCS. It can be concluded that for the stochastic dynamic response of the track-bridge system, the results of KLE-PEM are almost consistent with the results of MCS. Figure 5 shows a comparison of the time-histories response of mean and standard deviation of the first wheel-rail force passing through the bridge. It can also be concluded that the results calculated using KLE-PEM are almost consistent with the results obtained using MCS.Table 2
Properties of train.
NotationUnitTrailer car/motor carmckg4.40 × 104/4.80 × 104mtkg2.40 × 103/3.20 × 103mwkg2.40 × 103/2.40 × 103Jckg·m22.70 × 106/2.70 × 106Jtkg·m22.20 × 103/7.20 × 103K1zN/m0.70 × 106/1.00 × 106K2zN/m3.00 × 105/4.00 × 105C1zN·s/m4.00 × 104/3.00 × 104C2zN·s/m5.00 × 104/4.00 × 104Lcm8.6875/8.6785Ltm1.25/1.25Lwwm4.90/4.90Table 3
Properties of the track-bridge.
NotationUnitItemValueErN/m2Young’s modulus of rail2.06 × 1011vr—Poisson’s ratio of rail0.3Irm4Mass moment of inertia of rail cross section2 × 3.217 × 10−5m¯rkg/mMass per unit length of rail2×60.64EsN/m2Young’s modulus of slab3.6 × 1010vs–Poisson’s ratio of slab0.2Ism4Mass moment of inertia of slab cross section1.6 × 10−3m¯skg/mMass per unit length of slab1.2 × 103krsN/mSpring stiffness of fastener2 × 5.0 × 107crsN·s/mDamping coefficient of fastener2 × 3.625 × 104k¯sbN/m2Spring stiffness of CA layer per unit length2 × 1.5 × 109c¯sbN·s/m2Damping coefficient of CA layer per unit length2 × 8.3 × 104EbN/m2Young’s modulus of girder3.45 × 1010vb—Poisson’s ratio of girder0.2Ibm4Mass moment of inertia of girder cross section12.744m¯bkg/mMass per unit length of bridge girder2.972 × 104ζb—Damping ratio of girder0.05Figure 4
Comparison of displacement of fastener at midspan.Figure 5
Comparison of wheel-rail contact force of first wheelset (double side).


## 6.2. Comparison between Linear and Nonlinear Simulation

A standard simply supported 7-span box girder with 32 m length per span is modeled to evaluate the effect of random track irregularity on the TTBS, and the difference between linear and nonlinear Hertz contacts is evaluated. A rigid subgrade is used for the simulation out of the bridge part. The treatment of boundary conditions between adjacent spans is shown in Figure1. The train uses four carriages ICE-3 train, and there are two motor cars at the head and tail of the train. To ensure the dynamic response stability before the train entering the bridge, the train starts running from 300 m away from the left side of the first span bridge. The operation speed of the train is 250 km/h. The TTBS considering linear Hertz and nonlinear Hertz is established.Figure6 shows the mean and standard deviation of midspan displacement response of fourth bridge span under the nonlinear and linear wheel-rail relationship, where the horizontal ordinate represents the time for the first wheelset entering the fourth bridge span. It can be concluded that the bridge dynamic responses under the two wheel-rail force models are similar. In addition, in the perspective of time-histories response trend, the mean value of time-histories response of the bridge has two peaks and three troughs. However, the standard deviation has only two peaks before and after, and the time of the first peak is inconsistent with the time of the first trough of the mean value.Figure 6
Vertical displacement of midspan of fourth span.The mean and standard deviation of time-histories response of vertical acceleration at the center of gravity of the first carbody are shown in Figure7, where the horizontal ordinate is the time taken for the first wheelset entering the first bridge span. It shows that the standard deviations of train acceleration obtained by the two wheel-rail force models are inconsistent. The standard deviation obtained by linear wheel-rail relationship is less than that obtained by nonlinear wheel-rail relationship in a relatively large time. In the entire process, the values of standard deviation obtained from the linear wheel-rail relationship are all around 6.7 × 10−2 m/s2, whereas the standard deviations obtained from the nonlinear relationship are relatively large at the initial stage of entering the bridge and then gradually decreases to 6.8 × 10−2 m/s2.Figure 7
Vertical acceleration at the center of gravity of first carbody.The mean and standard deviation of the time-histories response of vertical wheel-rail contact force of the first wheelset are shown in Figure8, where the horizontal ordinate is the time taken for the first wheelset entering the first bridge span. Because the subgrade section adopts rigid subgrade, there will be greater contact force at the moment when the wheelsets drive into and out of the bridge. Therefore, the wheel-rail force of the second span to the sixth span of the wheel set is selected for analysis; it can be seen that the mean values of contact force calculated by the nonlinear and linear model both oscillate about 20 kN along the axle load, and the magnitude of the contact force obtained from the nonlinear model is larger than that of the linear model. Regarding the standard deviations of wheel-rail force, it shows a significant difference between the two models at the initial stage when the train enters the bridge. Then, the difference gradually decreases, both of which oscillate between 2.5 kN and 4 kN. The standard deviations obtained from the linear wheel-rail relationship are less than that obtained from the nonlinear wheel-rail relationship in a relatively long time.Figure 8
Vertical contact force of first wheelset (double side).


## 6.3. Applied for Response Prediction

When it is necessary to obtain the response value of the bridge, the most accurate method is to install sensors in the field for actual measurement, but this method is very time-consuming and inconvenient. Therefore, the numerical simulation method is usually used to obtain the bridge response. However, if the deterministic analysis is used, the result will be different from the actual value because there are various randomness in the TTBS, such as track irregularity, train load, and so on. The results of probability range can be obtained by using the proposed stochastic analysis method. The accuracy of the probability distribution range obtained by KLE-PEM can be verified by comparing the test results in Ref. [42]. In this case, besides the stochastic track irregularity, there is also the random axle load, in which the train carbody mass is a normal distribution random variable with a coefficient of variation of 0.05. It was proved that the PEM can calculate the response of the TTBS with uncertain parameter precisely [24]; therefore, only one independent random variable needs to be added in KLE-PEM. (Mean ± 3 Std. D) are used to guarantee the probability range of results; the calculation results and the test data of maximum values of vertical displacement at bridge midspan are shown in Figure 9. It can be found that the test results are basically within the calculated probability distribution range, which means that the proposed method can be used to effectively predict the distribution range of response results.Figure 9
Calculation results and test data.


## 7. Conclusions

In this study, a stochastic analysis method KLE-PEM is introduced to analyze the random train-track-bridge dynamic model, and the difference between the nonlinear and linear Hertz contact model is discussed, and the proposed method compared with the actual measurement results mainly concluded as the following results:(1)
The proposed method can accurately calculate the random response after the wheel jumps from the extreme track irregularity.(2)
The mean and standard deviation of the stochastic dynamic system through KLE-PEM and MCS are approximate. Meanwhile, under the same accuracy, KLE-PEM is an order of magnitude higher than MCS.(3)
In the TTBS, nonlinear and linear wheel-rail contact models are compared. It can be concluded that there is almost no difference in the effect of these two models on the bridge response. The effects on train response are different at the initial stage when the train enters a bridge, but with the train operation, the difference decreases gradually.(4)
KLE-PEM can be used to effectively predict the distribution range of response.



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*Source: 8886737-2020-12-28.xml*