Nonlinear random seismic analysis of 3D high-speed railway track-bridge system based on OpenSEES

Structures 24 (2020) 87–98

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Nonlinear random seismic analysis of 3D high-speed railway track-bridge system based on OpenSEES

T

⁎

Haiyan Lia,b, Zhiwu Yua,b, Jianfeng Maoa,b, , Lizhong Jianga,b a b

School of Civil Engineering, Central South University, Changsha 410075, China National Engineering Laboratory for High-speed Railway Construction, Changsha 410004, China

A R T I C LE I N FO

A B S T R A C T

Keywords: Random seismic response CRTS II slab ballastless track system High-speed railway Simple support bridge OpenSEES

A new nonlinear track-bridge random model was established based on OpenSEES, of which the random seismic responses is systematically studied by using probability density evolution method (PDEM). First, a three-dimensional detail nonlinear random dynamic model of the track-bridge coupled system in high-speed railway is established considering the earthquake as the random excitation. The dynamic model is composed with the rail, CRTS II slab ballastless track, the simple supported bridges, bearing, pier and the pile foundation, from which the random dynamic equation of the random model is formulated. In order to evaluate the seismic performance of track structures from a random perspective, the random seismic responses of high-speed railway bridge under random seismic excitations are investigated and analyzed by using the PDEM. Eventually, the Newmark β integration method and the double edge difference method of TVD format are adopted to obtain the random responses involving the probability functions, which can be translated to get the mean value and the standard deviation of responses. Thus, the abundant probability information of seismic responses of the high-speed railway bridge under random seismic excitations can be obtained. The PDF and CDF of the sliding layer are obtained; thus, the failure probability and reliability of different damage grades of sliding layer are obtained. Discussions and significant conclusions on the random dynamic responses are presented.

1. Introduction The dynamic interaction between the track and its supporting structures has turned into a significant issue with the rapid growth of high-speed railway transportation. Bridge is an important infrastructure widely used in high-speed railway as it can meet the high standard in train riding safety and train riding comfort [1,2]. As a result, the proportion of the bridges in high speed railway lines is greatly huge in China, e.g., 92% in Shanghai-Hangzhou high-speed railway, 80% in Beijing-Shanghai high-speed railway, et al. Earthquake is a serious natural disaster, which will cause the damage on hindering disaster relief work by affecting the safe evacuation of personnel and the transportation of materials to disaster areas. For example, the 8.0magnitude earthquake that happened in Wenchuan, Sichuan province, China on May 12, 2008, the 8.8-magnitude earthquake that happened in Chile on March 11, 2011, and the 9.0-magnitude earthquake that happened in Japan on March 11, 2011, and the recently 6.0-magnitude earthquake in Yibin, Sichuan Province in China. These earthquakes have caused much loss of life and property as well as disputing traffic due to its suddenness and randomness. Therefore, the larger of the

⁎

bridge proportion in high speed railway, the greater probability of bridge failure under the earthquake. Improve the seismic capacity of track-bridge coupled system is the first priority for engineers. On this basis, the study of seismic performance evaluation of high-speed railway track-bridge structure based on random dynamic response and probability analysis under earthquakes has great value. In recent years, various complex mechanical calculation models have been established to calculate the dynamic response of high-speed railway bridges. For example, the simple linear elastic beams or frame structure are often used to simulate the dynamic response of vehiclebridge system under general working condition, track irregularity or earthquakes [3–10]. However, so far, the detailed nonlinear finite element model considering the track-structure interaction for the seismic response calculating of high-speed railway bridges is quite limited. On the other hand, the track structure is often neglected or only ballasted track structure is mentioned in most of the existed models in publications. Since the track -bridge interaction has a great influence on the dynamic characteristics of track-bridge system in high-speed railway [11], it is necessary to consider the impact of track structure in future model and simulate its random seismic responses.

Corresponding author at: School of Civil Engineering, Central South University, Changsha 410075, China. E-mail addresses: [email protected] (H. Li), [email protected] (Z. Yu), [email protected] (J. Mao), [email protected] (L. Jiang).

https://doi.org/10.1016/j.istruc.2020.01.003 Received 21 October 2019; Accepted 4 January 2020 2352-0124/ © 2020 Institution of Structural Engineers. Published by Elsevier Ltd. All rights reserved.

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Fig. 1(a), and the 5# pier is especially emphasized in the study of this paper. Under each pier are the foundation system, which are 12 circular piles with a diameter of 1.0 m, are made of C30 concrete and Q235 steel bars. The CHSR adopts the technology of ballastless track with continuous welded rails due to the unique advantage comparing with the conventional non-welded track connected rail. In this bridge, the CRTS II ballastless track was adopted, which comprises base plates, track plates, rails and connecting members. The connecting members include sliding layers, shear cogging, CA layers, shear reinforcement, fasteners and lateral blocks as shown in Fig. 1(b) and (c). The base plate is made of C30 concrete, with a width of 2.95 m and a thickness of 0.19 m. The track plate is made of C55 concrete with a width of 2.55 m and a thickness of 0.2 m. The rail adopts CHN60 type. The sliding layer is arranged between the bridge deck and the base plate and consists of dual textiles and one membrane. The sliding layer near the fixed bearing is provided with the shear cogging, connecting the girder and the base plate. The CA layer is located between the base plate and the track plate. The shear bars are set in the CA layer near the beam end, connecting the track plate and the base plate to withstand the deformation caused by the rotation angle. The fasteners adopt the type of WJ-8C, and are set at intervals of 0.65 m.

Since the seismic ground motion has the inherent feature of randomness, the seismic response is also random [12]. Therefore, conducting seismic response analysis from a random point of view is of great necessary. There have many studies on the deterministic dynamic analysis of track-bridge system formerly [13–17], considering only one or several ground motion samples, which is not enough to capture the random dynamic characteristics of the track-bridge coupling system. Previous deterministic dynamic analysis methods have not systematically revealed the random vibration of track and bridge systems, which has great limitations. Therefore, the more reasonable analysis method is to use the random vibration theory to describe the ground motion and analyze the seismic dynamic response of the structure. Based on stochastic vibration method, Li and Chen developed a generalized probability density evolution method (PDEM) [18–20], which provides a new way for the linear and nonlinear stochastic vibration analysis of complex engineering structures. Sun et al. [21] developed the expression of the second kind of random harmonic function of stochastic process and used it to generate random seismic waves. Yu and Mao [22–25] employed PDEM to analyze the random dynamic response of vehicle-bridge coupling system under random parameters and random track irregularity excitation. By comparing the results with those results of Monte Carlo Method (MCM), field measurement and classical results, the efficiency and accuracy of PDEM are verified, and some insightful conclusions are drawn. In present paper, PDEM is used to analyze random seismic response of a high-speed railway bridge. A refined three-dimensional (3D) trackgirder-pier interaction model, considering random ground motion and track system, is established to investigate the random vibration responses. The stochastic harmonic function (SHF) was used to generate representative random ground motions with probability characteristics, which is the way we considered the randomness of earthquake ground motions. On this basis, a series of seismic responses of track-bridge system are obtained by deterministic finite-element method, and the random seismic responses (PDFs of displacement) of high-speed railway bridge system are obtained by applying PDEM. Finally, the random vibration responses of the track-bridge system are analyzed and some useful conclusions are drawn. PDEM is capable of reliability analysis of high-speed railway bridge structure, which is of great significance for reliability analysis, design and reliability optimization of track-bridge structure in engineering practice. This study lays a foundation for the seismic reliability calculation of refined bridges based on probability density function and provides a new way for seismic analysis.

2.2. Finite element model of bridge system 2.2.1. Bridge girder, piers and piles The bridge girder is designed to be elastic as a capacity protection component. So, linear elastic beam-column element with equivalent section characteristics is adopted to model the superstructure, see Table 1. Each span of the bridge girder is discretized into 50 elements of identical lengths 0.65 m, which is the same as the interval between fasteners. The pier and pile foundation are simulated by a three-dimensional elastoplastic fiber element. By integrating the fiber characteristics over the pier cross-section, the nonlinear pier section characteristics are obtained. Considering the constraint effect of stirrups, those sections are divided into three parts, including cover concrete, core concrete and reinforcement, whose stress-strain curves are shown in Fig. 2. Because the fiber element accounts for material nonlinearity, geometry nonlinearity and bond slip effect of anchoring steel in joint, it is better than other simplified plastic hinge model. The piles are divided into several units with an interval of 1 m by the finite element models for the sake of simulating pile-soil interaction. According to M-method in the Chinese standard, each finite element is connected to the soil by three translational springs and three rotary springs, the spring value is assumed to be a constant.

2. Modelling of a detailed 3D track-bridge coupled system in OpenSEES In this section, a detailed three-dimensional dynamic coupled trackbridge system was established in OPENSEES, which considering trackstructure interaction for the seismic response of high-speed railway bridges to demonstrate the accuracy and efficiency of the PDEM.

2.2.2. Track-Bridge interaction and the bearings Fig. 1 shows a typical track plate system used for the CHSR prototype bridge. The track plate, base plate and rails are all designed to behave elastically as a capacity protected component. Therefore, they are modeled using linear elastic beam-column element and the crosssection parameters are listed in Table 1. The sliding layer is arranged between the bridge deck and the base plate and consists of dual textiles and one membrane. The sliding layer near the fixed bearing is provided with the shear cogging, connecting the girder and the base plate. The CA layer is located between the base plate and the track plate. The shear bars are set in the CA layer near the beam end, connecting the track plate and the base plate to withstand the deformation caused by the rotation angle. The rail is fixed to the track plate by means of fastener. Sliding layer, CA layer and fasteners allow for longitudinal slippery relative to the bridge girder along the bridge. The connection components, spherical steel bearings, sliding layers, CA layers and fasteners are simulated using a zero-length nonlinear connection element as shown in Table 2 and Fig. 3. The 3D nonlinear finite element model of CHSR bridge is

2.1. General information of track-bridge system The engineering prototype of this paper is a ballastless track prestressed concrete double-track simply supported girder bridge from Beijing to Xuzhou section of Beijing-Shanghai high-speed railway. Fig. 1(a) shows 10-span simply supported bridges on a high-speed railway. The bridge superstructure, a pre-stressed box girder, with the same span of 32 m, is made of C50 concrete, is 13.40 m wide at the top, 5.74 m wide at the bottom and 3.09 m deep from the top to bottom surface. Each girder end is supported by two bearings, the spherical steel bearings. The bearings are supported by pier columns made of C50 concrete and HRB335 steel bars. All of the piers have the identical height of 13.5 m from the top of the pile cap to the top surface of the pier head and the same rectangular section sizes of 5.434 m × 2.5 m. The number of piers from left to right is 0# to 10# as shown in 88

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3195 10 3195 10

10 3195 10 3195 10

10 3195 10 3195

1

0#

1#

2#

3#

4#

5#

2

2

1

7#

6#

8#

9#

10#

a) Elevation layout of high-speed railway bridge /cm

Rail

HRB 500 steel bar,diameter 28mm

CA layer Rigid foam board

Track plate Base plate

Shear nail Shear cogging 65

Shear cogging 19

Girder

100

19

19

100

19 19

100

19

270

Fixed bearing

Sliding bearing b) Schematic sketch of track and girder structure

c) a typical cross-section of track and girder structure Fig. 1. Schematic diagram of high-speed railway bridge. a) Elevation layout of high-speed railway bridge/cm. b) Schematic sketch of track and girder structure. c) A typical cross-section of track and girder structure.

and to assess its random seismic performance appropriately. Fig. 5(a) schematically shows the model of a single pier and its connection to the bridge deck, while Fig. 5(b) shows the model of a single bridge span and connection between the rails and the deck. The model takes into account various kinds of nonlinearity factors in the seismic response behavior of bridges, for example, material nonlinearities and geometric

established by using the OpenSEES program as shown in Fig. 4, which contains the following important contents, for example, the track system (base plate, track plate and rail), the bridge structure above the ground surface, the pile foundations and their section parameters are listed in Table 1. This FE model accounts for track structure interaction to simulate the random seismic response of the bridge and track system 89

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Table 1 Section parameters of elastic beam elements.

Main girder

Sectional area/m2

Elastic modulus /kN/m2

Shear modulus /kN/m2

Torque /kN m

Inertia moment 1/m4

Inertia moment 2 /m4

9.06

3.45 × 107

1.44 × 107

2.26 × 101

1.10 × 101

9.48 × 101

107

107

10-3

Base plate

5.61 ×

Track plate

5.10 × 10-1

3.55 × 107

1.48 × 107

6.80 × 10-3

Rail

7.75 × 10-3

2.06 × 108

8.05 × 106

2.00 × 10-6

10-1

3.00 ×

1.25 ×

6.74 ×

nonlinearities. Each unit length for them is 0.65 m, which is the same as the interval between fasteners.

3.1. Ground motion samples based on stochastic harmonic function The random dynamic response of railway bridge system is affected by many factors, including random dynamic excitation (such as earthquake, wind load, track irregularity, etc.) and random system parameters (such as train parameters, structural parameters, etc.). Ground motion is considered as one of the primary factors that cause random vibration of track-bridge systems of high-speed railway. However, the vibration theory for a train-bridge system with random excitation and random structural parameters has not been systematically built yet. To this end, the primary objective of this study is to investigate the random vibration caused by random ground motions. With the numerical model, the first and significate procedure for seismic performance evaluation of track-bridge structures is to obtain a set of earthquake ground motions time histories. As described in Section 2.2, it is also a key step in solving PDEM equation. In the present study, the stochastic harmonic function (SHF) [26] was used to generate the representative acceleration time history of ground motion with probability characteristics. Generally speaking, according to the SHF the stochastic process of the ground motion samples can be expressed as:

i=1

dy /mm

Sliding layer CA layer Fastener Shear reinforcement Lateral block Fixed spherical steel bearing Sliding spherical steel bearing

6 41.5 15 22.5 453 5000 470

0.5 0.5 2 0.075 2 2 2

Friction force

dy

Fig. 3. Constitutive relation of zero length connection element.

Penzien model is taken as an example,

2S (ΩΘ, i )ΔΩΘ, i cos(ΩΘ, i + ϕΘ, i )

1 + 4ζ12 (ω/ ω1 )2 (1 + (ω/ ω1

) 2) 2

+

4ζ12 (ω/ ω1 )2

·

(ω/ ω2 )2 ·S0 (1 + (ω/ ω2 )2)2 + 4ζ22 (ω/ ω2 )2 (9)

(7)

where, ω denotes the white noise excitation of bedrock ground motion, and S0 is spectral intensity factor, ω1, ζ1, ω2 , ζ2 respectively denotes the filter parameters of the bottom layer and the surface layer, namely the circular frequency and damping ratio, which are determined according to the specification. For reference, all the earthquake ground motion parameters are determined by seismic zoning ad site grouping according to the Chinese seismic code, take the soil parameters of the ω1 = 17.95 rad·s−1, ζ1 = ζ2 = 0.72 , medium hard site, so − 1 ω2 = 1.795 rad·s [21]. The key point for generating X¨ g (Θ, χ ) is the definition of the representative discrete point sets of the random spatial frequencies ΩΘ, i

N

∑

Fy /kN

Relative displacement

where ΩΘ, i ∈ Θ and ϕΘ, i ∈ Θ, (i = 1, 2, ⋯N ) denote the random spatial frequency and phase of the ground motion respectively, and N is the number of random variables. A (ΩΘ, i) is an amplitude function related to the random frequencies ΩΘ, i , and A (ΩΘ, i) = 2S (ΩΘ, i )ΔΩΘ, i , S (ΩΘ, i) is the power spectrum density function of ground motions. So, Eq. (7) becomes,

X¨ g (Θ, χ ) =

5.00 × 10-6

Components

S (ω) =

i=1

2.76 × 10-1

Fy

N

∑ A (ΩΘ,i) cos(ΩΘ,i + ϕΘ,i)

4.06 × 10-1

1.70 × 10-3 3.20 × 10-5

Table 2 Parameters of zero length connection elements.

3. Solution by employ the probability density evolution method (PDEM)

X¨ g (Θ, χ ) =

1.69 × 10-3

(8)

The random process of ground motion power spectrum Clough-

Fig. 2. Models of pier and pile foundation. a) Stress-strain diagram of concrete. b) Stress-strain diagram of reinforcement. 90

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Rail

Fastener

Track plate

CA mortar layer

Base plate 1

3

2

0#

5

4

1#

2#

6

32m 7 8

3#

Sliding layer

32m 13 14

9 10 11 12 Fixed bearing Sliding bearing 4#

7#

6#

5#

15 16

17 18 19 20

9#

8#

10#

Fig. 4. Spatial finite element model of bridge.

where H = (H1, H2, ⋯, Hn )T , h = (h1, h2, ⋯, hn )T . More generally, any physical parameters in the structural system Eq. (2), such as displacement, velocity, acceleration, deformation, rotation, internal force, stress and strain of the control point, etc., all uniquely exist, and continuously depend on basic parameter Θ. Therefore, if the collection of physical parameters of interest are denoted as Z = (Z1, Z2, ⋯, Zm )T , so

and the random phase ϕΘ, i . It is necessary to select discrete representative points in the random vector space for the random seismic response probability density evolution analysis of bridge structure. Using number theory method (NTM) strategy for defining the representative point sets [27], same as the step1 of solution to the PDEM equation in Section 2.2. To generate a set of stochastic seismic ground motion samples, the representative sample points with 2 N-dimensional hypercube, is generated by sequence, j

(

j

)

χq, j = qH 2N + 1 − int qH 2N + 1 {q = 1, 2, ⋯nsel ; j = 1, 2, ⋯2N }

Z = HZ (Θ, t ), Z ̇ = hZ (Θ, t )

where hz = (h z1, h z 2, ⋯, h zm Obviously, in Eq. (3), it can be regarded as a random dynamical system, in which the randomness completely arises from Θ. For the extended system (Z , Θ) , if no random factors are added or disappeared, all the random factors are included, it is considered as a probabilistic conservative system. For convenience, the joint probability density function of (Z , Θ) is represented as p(z,Θ) (z , θ , t ) . According to the principle of conservation of probability, the generalized probability density evolution equation of the system is obtained as follows [18,19],

(10)

where χq, j ∈ {0, 1} denotes the discrete point of a 2 N-dimensional hypercube, H denotes a prime number, and nsel is the total number of the representative ground motion samples. Thus, the representative discrete points can be obtained by (p) (p) (p) ⎧ (Ωq, i)−1 = (Ωi - 1)−1 + [(Ωi - 1)−1 − (Ωi )−1] χq, i ⎨ ϕq, i = 2πχq, i ⎩

(11)

∂p(z,Θ) (z , θ , t )

where i = 1, 2, ⋯N , thus,

ξq = {Ωq,1, Ωq,2 , ⋯Ωq, N , ϕq,1, ϕq,2 , ⋯ϕq, N } ∈ Θ q = 1, 2, ⋯nsel

∂t (12)

pZ (Z , t ) =

∑ Zj̇ (θ, t )

∂p(z,Θ) (z , θ , t ) ∂z j

j=1

=0 (4)

∫ΩΘ p(z,Θ) (z, θ, t ) dθ

(5)

In Eq. (4), the dimensions of the PDEM equation m depend on those of the physical parameters being evaluated, are not related to the degree of freedom of the original physical system n , m could even be a one-dimensional equation. In order to evaluate the seismic performance of track-bridge system, the parameter we care about is the longitudinal displacement. Hence, the displacement time history was selected as a random process in the PDEM equation, and the following one-dimensional PDEM equation can be obtained:

3.2. Random dynamic equation of track-bridge coupled system The random dynamic equation of track-bridge system under earthquake excitation can be derived in a general form as follows:

∂p(D,Θ) (D , θ , t ) ∂t

(1)

+ Ḋ (θ , t )

∂p(D,Θ) (D , θ , t ) ∂Dj

=0

(6)

where D is the seismic time history response of the displacement of different components of track-bridge system.

where M , C , K respectively donates the mass, the damping and the stiffness matrix of n × n, order respectively. X¨ , Ẋ , X respectively donates the n-dimensional responses vectors of acceleration, velocity and displacement respectively. I is the n-dimensional unit vector. X¨ g (Θ, t ) is the random earthquake ground motion. Θ is a random vector that reflects the randomness of X¨ g . Obviously, the physical solutions of the Eq. (1) exist and continuously depend on the basic parameter Θ. For simplify, the solutions of the Eq. (1) can be written as

X = H (Θ, t ), Ẋ = h (Θ, t )

m

+

The bilateral difference method with the functionality of the total variation diminishing (TVD) format [18,19] is used to solve Eq. (4) and obtain the probabilistic function p(z,Θ) (z , θ , t ) . The joint probability density function (PDF) of random system response is given as follows

By substituting Eq. (11) into Eq. (8), a set of earthquake ground motions time histories are generated by SHF. In the present work, 300 samples of stochastic earthquake ground motions are prepared. The typical seismic acceleration sample time-history curve and its absolute acceleration response spectrum generated by the SHF method are shown in Fig. 6(a). The comparison of acceleration mean between the 300 samples and the target stochastic earthquake ground motion model are shown in Fig. 6(b), which verify the accuracy of the stochastic harmonic function (SHF) method and spectral representation in generating the representative random ground motion samples.

MX¨ + CẊ + KX = −MIX¨ g (Θ, t )

(3)

)T .

3.3. Solution for the random dynamic equation The track-bridge random dynamic system in Eq. (6) can be solved by following numerical method steps, as shown in Fig. 7: (1) In the random space Θ, the representative discrete points θq (q = 1, 2, ⋯, nsel ) are selected by numerical theory method (NTM) [27] and the corresponding probabilities are determined, nsel is the

(2) 91

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Fig. 5. Schematic diagram of the finite element model a) for a single bridge, b) for a single bridge span.

92

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0.3

0 -2 -4

0

2

4

6

8

10

12

14

16

18

20

Time(s)

Absolute Acceleration(m/s2 )

300 samples Target

2

15 10

Acceleration Mean(m/s2)

Acceleration(m/s2 )

4

5 0

0.2

0.1

0

-0.1

-0.2

0

1

2

3

4

5

-0.3

6

Tn(s)

0

2

4

6

8

10

12

14

16

18

20

Time(s)

Fig. 6. a) Typical seismic sample time curve and its acceleration response spectrum. b) Comparison of the acceleration mean between the 300 samples and target. Start

obtained on the basis of a series of representatively deterministic dynamic time history analysis. Equivalent extreme events can be used to construct a virtual stochastic process [28]. In order to obtain the result of interest, a virtual stochastic process can be constructed, then the process is substituted into the PDEM equation and solved to obtain the PDF of parameters of interest. For example, in the track-bridge system, the maximum displacement response of the rail under the earthquake excitation is meaningful, so the maximum displacement response of the rail constitutes a virtual stochastic process. by substituting this stochastic process into the PDEM equation and solving it, the random seismic performance of the track-bridge system can be obtained. As described in Section 2.2, various random seismic response can be calculated with the detailed procedures summarized in the solution flowchart in Fig. 7.

Choose appropriate ground motion power spectrum to produce Random ground motion samples

Build track-bridge dynamic model

The solution of deterministic dynamic system: Using a deterministic finite-element method to obtain a series of seismic response of track-bridge system

Solve the track-bridge random equation with generalized probability density evolution method

4. Proposal damage grades of track structure in high-speed railway Generally speaking, the performance objective of the structure is closely related to the damage grade. The structural damage is a direct factor that causes the damage of the structure’s function and collapse after the ground motion, so the displacement and damage index are often used to describe the damage grade of the structure. Each connecting members of the track structure in the high-speed railway track-bridge system will be damaged to varying degrees under the action of earthquakes. According to the research of other scholars, the damage grade of each component is quantitatively classified. The damage grade of each connecting member in the high-speed railway bridge system is divided into the following five, which are intact, mild, moderate, severe and complete damage. The seismic capacity is the four limit values among the five damage states, namely: intact to mild damage, mild to moderate damage, moderate to severe damage, and severe to complete damage. The main connecting components of the track structure include the sliding layer, the CA mortar layer and the fasteners as described in Section 2.2.2. The damage grades and corresponding damage index have not been fully studied before; so, it is difficult to accurately determine the value. Based on the ductile failure criterion and the constitutive relationship of each component, the limit values of damage grades are determined according to the practical experience. See Table 3 below for details.

Get the mean value and standard deviation of response and their probability density information

End

Fig. 7. Schematic illustration of the detailed procedures for the random seismic analysis of the coupled track-bridge system subjected to earthquake ground motions.

total number of discrete points. (2) Solving the deterministic dynamic system: For each given θq (q = 1, 2, ⋯, nsel ) , the dynamic equilibrium equation in Eq. (1) under the given earthquake action can be solved by time history dynamic analysis method, and a set of representatively physical quantities of seismic response can be obtained. (3) A series of seismic response (e.g. the displacement time histories) get in Step 2 are substituted into Eq. (6), which is solved using the bilateral difference method based on the total variation diminishing (TVD) [19]. Herein, the results obtain p(D,Θ) (D , θ , t )(q = 1, 2, ⋯, nsel ) . (4) By summing p(D,Θ) (D , θ , t )(q = 1, 2, ⋯, nsel ) , and the PDF value of displacement of the dynamic track-bridge system can be written as: nsel

p(D) (D , t ) =

∑ p(D,Θ) (D, θq, t ) q=1

5. Case study

(7)

The variability and reliability of PDEM are verified by a series of

As can be seen from the above steps, the solution of PDEM can be 93

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shown in the enlarged view of Fig. 8 (a) indicating by the black arrow, the maximum value of the mean value is about −0.281 mm, and the maximum result of one sample deterministic fastener result is −0.277 mm in reference [16], which is only 1.42% difference from the mean value, and practically in the midsection of the up and low extremum boundaries of the PDEM, which is very close to the mean value of −0.281 mm. The deterministic displacement results of the CA layers, the sliding layer are in good agreement with the PDEM result, these results can show that the method has good accuracy and reliability, which is not listed here because of the length. The probability density function (PDF) of the fastener longitudinal maximum displacement along the 5# span is given in Fig. 8. Due to random ground motion, the longitudinal vibration of the fastener displacement exhibits random vibration characteristics, which are directly reflected by the contour of the probability density evolution in Fig. 8(a), which illustrates that the PDF changes with position of the fastener. The probability density contour plot more closely resembles the flowing in a river, which indicates the liquidity of probability in the state space. The second-order statistics (mean value and the standard deviation) are calculated as well as other random methods. Fortunately, the contours of the probability density function are mostly located within the range of extreme values as shown in Fig. 8(a). This phenomenon indicates that use of the principle of ‘Mean ± 3Std’, which is based on the idea of the normal distribution, can accurately define the probability distribution range of the random bridge responses in the current study. The mean value is close to 0 mm at the ends of beams, then suddenly reaches the maximum value just above the bearing due to the seismic angle response near the beam ends, and then decreases to 0 mm in the middle of the span due to the compatibility of deformation. The mean value changes with position of the span. The maximum relative displacement of the fastener is located near the span end, for the left span end the displacement is in the range of 0.18 mm–0.25 mm, for the right span end the displacement is the range of −0.26 to −0.3 mm. The displacement at the right span end is larger than that at the left span end, which is due to the establishment of shear cogging at the right span end of the sliding layer, transferring more seismic energy to the upper layer. In the 3D position-varying process PDF as shown in Fig. 8(b), resembling a continuous mountain stretching into the distance, vividly shows a wavelike pattern that illustrate the physical characteristics of the displacement during earthquakes, the brighter the peak color, the greater the concentration of the probability density function observed. In order to describe the physical process more clearly, the probability density evolution functions of five typical position fasteners are shown in Fig. 8(c). Five typical positions are selected as shown in Fig. 8(c), from left to right, are the left span end, the 1/4 span, the midspan, the 3/4 span and the right span end of the 5th span fastener respectively. With the change of position from left to right, the probability density functions (PDF) begins at the initial level near the 0.6 mm at the left span end, increasingly changes to 0.12 mm at 1/4 span, then gradually decreases to the zero level at 1/2 span, afterwards it maintains the decreasing state and dropped to −0.15 mm at 3/4 spans, and finally rises to zero at the right span end. These demonstrate that the displacement develops and evolves over time and position, which also reveals the transfer processes of the probability statistics information. It is of great necessary to analyze the seismic response of the track-bridge system from a stochastic viewpoint. It can be seen from above that the maximum displacement of the fastener of one single span possibly occurs to 0.308 mm at the right span end. Comparing the maximum displacement of fasteners with the damage grade index of Section 4, the limit value of the fastener from intact state to mild damage is 2 mm, which is much larger than 0.308 mm. Consequently, fasteners are in a very healthy state and there is no damage at all.

Table 3 The limit value of damage grades of key components of high-speed railway bridge superstructure. Component

Fastener CA layer Sliding layer

Limit value of damage grades (Displacement) Intact to mild damage

Mild to moderate damage

Moderate to severe damage

Severe to complete damage

2 mm 0.5 mm 0.5 mm

3 mm 1 mm 1 mm

4 mm 1.5 mm 1.5 mm

5 mm 2 mm 2 mm

numerical examples, such as the results of MCM, the published results and the field test results. Details of verification can be found in the references [23]. Therefore, PDEM can be used for stochastic dynamic analysis of high-speed railway Track-Bridge system. Hence, the above track-bridge coupling model in Section 3 subjected to random ground motions is numerically simulated by using PDEM, which can transform the stochastic dynamic analysis into a series of deterministic seismic time-history analysis, and its results will be compared with the results got from the traditional deterministic method. The FE element model developed above is used to get a series of seismic response of trackbridge coupled model. The displacement response, which is easy to obtain, can directly reflects the safety, damage and even cracking degree of the system. Therefore, it is dependable performance index to measure the seismic performance of a track structure and bridge structure by collecting the displacement responses of each key component of the system. With a set of deterministic dynamic time history analysis, a series of time history displacement curves on the fasteners, CA layers, sliding layers, bearings and piers are obtained. These data are input into PDEM equation and solved by the bilateral difference method with TVD. On this basis, the random seismic response of the whole track-bridge coupled system are calculated. Due to the length of this article, the stochastic seismic response listed here are focus on the 5th span bridge that supported by piers 4# and 5#. Considering that the maximum displacement of fastener, sliding layer or CA layer of one span is not a time history, to obtain the PDF of the maximum displacement of one span, the virtual stochastic process was constructed by using maximum displacement of one span. Taking this virtual stochastic process as input into the PDEM equation gives the PDF of the maximum displacement of one span. Based on previous research, the seismic response that are of interest focus on the fifth span supported by pier 4# and pier 5#. The longitudinal responses of the system calculated by PDEM are presented in Figs. 8–10. For Figs. 8–10, in the (a) figures show the 3D probability density contours of the time–history curves of the fastener displacement, CA layer displacement, and sliding layer displacement. The curves for mean and standard deviation values are expressed in the form of ‘Mean ± 3Std’ to define the extreme displacement also as shown in the (a) figures. In (b) figures the probability density surfaces of the above responses are displayed. Because there is no probability loss enduring the whole process, the cross-sectional area of several representative time step is always equal to 1 which is shown in (c) figures. 5.1. Fastener Before the analysis, the results are compared and verified with the previous deterministic analysis results. Through the random ground motion, the dynamic response of the track-bridge displacement can be determined. The PDEM results are shown in Fig. 8 (a). The closer to the center of the probability distribution, the warmer the color and the higher the probability density value. Displacement results obtained from previous deterministic study [16] should fall within the scope of random results the PDEM on account of the same train-bridge model. As 94

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Fig. 8. The random displacement results of the fastener along the 5#span (a) The contour of probability density function (PDF) (b) 3D PDF (c) PDF at each section of one span.

5.2. CA layer

5.3. Sliding layer

Under the random ground motion, the probability density evolution contour, the second-order statistics (mean ± 3 Std) and the probability density function (PDF) surface of longitudinal displacement of the CA layer along the 5# span are calculated. The stochastic method is the same as that of fasteners in Section 4.1. The mean value and standard deviation values and the probability density contour plot of CA layer (Fig. 9(a)) show similar variation characteristics with those of fastener displacement. The probability density contour lays a foundation for further study of displacement reliability. The maximum relative displacement of the CA layer is located near the span end, for the left span end the displacement is in the range of 0.19 mm–0.25 mm, for the right span end the displacement is the range of −0.23 to −0.28 mm. Fig. 9(b) presents the 3-D probability density evolution surface for the CA layer displacement over the position of one CA layer span. It is clear that the PDF has a different distribution pattern in different position, e.g., the PDF evolves over position and has a mountain-peak shape. PDFs of several typical positions (Fig. 9(c)) also indicate the probability information changes with position, because of the coupling effect of the randomness and nonlinearity. It can be seen from above that the maximum displacement of the CA layer of one span is likely to reach 0.282 mm at the right span end. Comparing the maximum displacement of CA layer with the damage grade index of Section 4, the limit value of the CA layer from intact state to mild damage is 0.5 mm, which is larger than 0.282 mm. Consequently, CA layer are in a healthy state and there is no damage.

By using the same stochastic calculation method as CA layer displacement, the probability density evolution contour, second-order statistics (mean ± 3 Std) and probability density function (PDF) surface of longitudinal displacement of sliding layer along 5th span under random ground motion are calculated. The probability density function (PDF) of the longitudinal maximum displacement of the sliding layer along the 5# span is shown in Fig. 10. Due to the randomness of the ground motion, the longitudinal vibration of sliding layer displacement also exhibits random vibration characteristics, which is directly reflected in the contour of probability density evolution in Fig. 10(a), indicating that PDF varies with the change of sliding layer position. The probability density contour map is closer to the flow in the river, which indicates the probability of liquidity in the state space. The mean value and Mean ± 3 Std are also calculated here, the Mean + 3 Std represents the upper boundary value, and the mean −3 Std represents the lower boundary value. Fortunately, the contours of the PDF are mostly in the extreme range, as shown in Fig. 10(a). This phenomenon shows that the probability distribution range of random bridge response in the current study can be accurately defined by using the 'Mean ± 3Std' principle based on normal distribution. The mean value of sliding layer displacement is close to 2.45 mm at the left span end of the 5#span beam, and then gradually decreases to 0 mm at the right span end due to the action of the shear cogging. The mean value varies with the change of span position. The one-sample deterministic result is almost in the middle of the upper and lower limits of the partial differential equation, which is very close to the mean value. Therefore, the proposed method proves to be accurate and reliable. The upper and lower limit values of the 95

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-0.262mm Mean value -0.263mm

(a) The contour of probability density function (PDF) Left Span End 1/4 Span 1/2 Span 3/4 Span Right Span End

PDF (1/mm)

40 30 20 10 0

-0.15 -0.1 -0.05

0

0.05 0.1 0.15 0.2 0.25

Displacement (mm)

(b) 3D PDF

(c) PDF at each section of one span

Fig. 9. The random displacement results of the CA layer along the 5#span (a) The contour of probability density function (PDF) (b) 3D PDF (c) PDF at each section of one span.

state, which is very detrimental to both structural safety and train operation safety. Therefore, it is necessary to carry out the damage probability and reliability analyze of the sliding layer. The maximum displacement of the sliding layer of one single span may be occurred at the location just near the 4# Pier, the relative displacement time histories of this location has been selected. Since the maximum value of relative displacement time histories is not a time history, using the equivalent extreme value event which is constructed by Li et al. [28] to generate a virtual stochastic process of the maximum value of relative displacement time histories. Just like described above, substitute the virtual stochastic process into the PDEM equation and solve the PDEM equation, the PDF of the equivalent extreme value event can be obtained as shown in Fig. 11(a). Right after seismic reliability information can be obtained based on the cumulative distribution function (CDF) by integrating the PDF of the equivalent extreme value event as shown in Fig. 11(b). The CDF based on the equivalent extreme value event of the relative displacement of sliding layer are obtained by using PDEM as shown in Fig. 11(b). According to the displacement index of the damage grades based on displacement proposed above in Section 4 to assess the seismic performance of sliding layer, the dynamic reliability of mild, moderate and severe are 0.8031, 1 and 1, respectively. The corresponding dynamic failure probability of mild, moderate and severe are 1 − 8031 = 0.1969, 1 − 1 = 0, and 1 − 1 = 0, respectively.

maximum relative displacement of the sliding layer are 2.4 mm–2.48 mm at the left span end and reduced to 0 mm at the right span end. Moreover, it can be known from the distribution of the PDF contour that the it has a larger probability value and a warmer color at the span ends on both sides, and a smaller probability value and the color is colder at the midspan. Fig. 10(b) shows the 3-D probability density evolution surface of the sliding layer displacement along one span. Obviously, PDFs have different distribution patterns at different locations of one span. The warmer the peak color, the greater the probability concentration. To clearer describe the physical process, the probability density evolution function of five typical positional sliding layers is shown in Fig. 10(c). Five typical positions are selected as shown in Fig. 10(c), from left to right, are the left span end, the 1/4 span, the mid span, the 3/4 span, and the right span end of the 5th span, respectively. As the position changes from left to right, the PDF begins at the initial state near the level of 2.45 mm at the left span end, then it decreases gradually to 2.0 mm at 1/4 span, and continues to decrease to 1.1 mm at 1/ 2 span. Then it keeps decreasing to 0.255 mm at 3/4 span, and finally to the zero at the end of the right span. This indicates that the displacement develops and evolves with time and position, and reveals the process of transmission of probability and statistics information. It is necessary to analyze the seismic response of the track-bridge system from a random perspective. From the discussion of above figures, it can be seen that the maximum displacement of the sliding layer could possibly reach 2.464 mm. Comparing the maximum displacement of sliding layer with the damage grade index in Section 4, in which the limit value of the sliding layer from severe damage to complete damage is 2 mm, less than 2.464 mm. Thus, the sliding layer may be in the complete damage a

6. Conclusion From the perspective of stochastic dynamics, the seismic performance of refined track-bridge coupled system is studied by employing PDEM and stochastic harmonic function. The second-order statistics 96

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Mean value -2.422mm -2.419mm

(a) The contour of probability density function (PDF) Left Span End 1/4 Span 1/2 Span 3/4 Span Right Span End

PDF (1/mm)

100 80 60 40 20 0

0

0.5

1

1.5

2

2.5

Displacement (mm)

(b) 3D PDF

(c) PDF at each section of one span

Fig. 10. The random displacement results of the sliding layer along the 5#span (a) The contour of probability density function (PDF) (b) 3D PDF (c) PDF at each section of one span.

1.0

7 6

0.8

PDF

5

0.8031

0.6

CDF

4 PDF

CDF Mild Moderate Severe

3

0.4

2 0.2

1 0 -1.5

-1.0

-0.5

0.0

0.5

1.0

0.0 -1.0

1.5

-0.5

0.0

0.5

1.0

1.5

2.0

Displacement

Displacement

(a)

(b) Fig. 11. PDF and CDF of the equivalent extreme value event.

can be found. It can be seen from the above random seismic response of track structure that the displacement of sliding layer is larger than that of fastener and CA layer, which is the weakest layer in track structure and functions to protect fastener and CA layer. Seismic isolation protecting the sliding layer studies can be performed in subsequent studies. (2) Based on the power spectrum model and random harmonic function, a set of representative acceleration time history of ground motion is generated, providing a large number of random ground

and probability information of each layer of the track structure are determined. The failure probability and reliability of the equivalent extreme value event has also been obtained to evaluate the seismic performance of sliding layer. The major conclusions are as follows: (1) In this paper, a refined track bridge model is established based on OpenSEES, and each layer of track structure is simulated in detail, thus the random seismic response of each layer of track structure under earthquake is calculated for the first time and the weak layer 97

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motions for the seismic evaluation of the track bridge system. (3) PDEM provides a basis for dynamic response, probability and reliability analysis mainly based on displacement aspect, namely deformation evaluation in the field of seismic performance evaluation of high-speed railway track system. PDEM provides an effective and feasible method to study seismic response and reliability analysis of railway bridge system from a random perspective.

seismic ground motion. Earthquake Eng Struct Dyn 2012;41(1):139–57. [9] Montenegro PA, Calçada R, Vila Pouca N, Tanabe M. Running safety assessment of trains moving over bridges subjected to moderate earthquakes. Earthquake Eng Struct Dyn 2016;45(3):483–504. [10] Yang YB, Wu YS. Dynamic stability of trains moving over bridges shaken by earthquakes. J Sound Vib 2002;258(1):65–94. [11] Yang YB, Yau JD, Wu YS. Vehicle bridge interaction dynamics. Singapore: World Scientific Pub Co Inc; 2004. [12] Housner GW. Characteristics of strong-motion earthquakes*. Bull Seismol Soc Am 1947;37(1):19–31. [13] Wei B, Wang P, Liu W, Yang M, Jiang L. The impact of the concave distribution of rolling friction coefficient on the seismic isolation performance of a spring-rolling system. Int J Non Linear Mech 2016;83:65–77. [14] Wei B, Wang P, Yang M, Jiang L. Seismic response of rolling isolation systems with concave friction distribution. J Earthquake Eng 2017;21(2):325–42. [15] Wei B, Zuo C, He X, Jiang L. Numerical investigation on scaling a pure friction isolation system for civil structures in shaking table model tests. Int J Non Linear Mech 2018;98:1–12. [16] Yu Z, Li H, Wei B, Jiang L, Mao J. Numerical analysis on longitudinal seismic responses of high-speed railway bridges isolated by friction pendulum bearings. J Vibroeng 2018;20(4):1748–60. [17] Yan B, Liu S, Pu H, Dai G, Cai X. Elastic-plastic seismic response of CRTS II slab ballastless track system on high-speed railway bridges. Sci China Technol Sci 2017;60(6):865–71. [18] Li J, Chen J. The principle of preservation of probability and the generalized density evolution equation. Struct Saf 2008;30(1):65–77. [19] Li J, Chen J. Stochastic dynamics of structures. John Wiley & Sons (Asia); 2009. [20] Li J, Chen J, Sun W, Peng Y. Advances of the probability density evolution method for nonlinear stochastic systems. Probab Eng Mech 2012;28:132–42. [21] Sun W, Chen J, Li J. Stochastic harmonic functions of second kind for spectral representations. J Tongji Univ (Nat Sci) 2011;10(39):1413–9. (in Chinese). [22] Yu Z, Mao J, Guo F, Guo W. Non-stationary random vibration analysis of a 3D train–bridge system using the probability density evolution method. J Sound Vib 2016;366:173–89. [23] Yu Z, Mao J. A stochastic dynamic model of train-track-bridge coupled system based on probability density evolution method. Appl Math Model 2018;59:205–32. [24] Yu Z, Mao J. Probability analysis of train-track-bridge interactions using a random wheel/rail contact model. Eng Struct 2017;144:120–38. [25] Mao J, Yu Z, Xiao Y, Jin C, Bai Y. Random dynamic analysis of a train-bridge coupled system involving random system parameters based on probability density evolution method. Probab Eng Mech 2016;46:48–61. [26] Chen J, Sun W, Li J, Xu J. Stochastic harmonic function representation of stochastic processes. J Appl Mech 2012:80(1). [27] Hua LK, Yuan W. Applications of number theory to numerical analysis. Berlin: Springer; 1981. [28] Li J, Chen J, Fan W. The equivalent extreme-value event and evaluation of the structural system reliability. Struct Saf 2007;29(2):112–31.

Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This work was supported by the Chinese Scholarship Council (Grant Nos. 201806370056) and the National Natural Science Foundation of China (Grant No. 51578549, 51708558, 51820105014). These financial supports are gratefully acknowledged. References [1] He XH, Wu T, Zou Y, Frank CY, Guo H, Yu Z. Recent developments of high speed railway bridges in China. Struct Infrastruct Eng 2017;13(12):1584–95. [2] Yan B, Dai G, Hu N. Recent development of design and construction of short span high-speed railway bridges in China. Eng Struct 2015;100:707–17. [3] Xia H, Zhang N. Dynamic analysis of railway bridge under high-speed trains. Comput Struct 2005;83(23–24):1891–901. [4] Yang Y, Wu Y. A versatile element for analyzing vehicle–bridge interaction response. Eng Struct 2001;23(5):452–69. [5] Zhai W, Xia H, Cai C, Gao M, Li X, Guo X, et al. High-speed train–track–bridge dynamic interactions – part I: theoretical model and numerical simulation. Int J Rail Transp 2013;1(1–2):3–24. [6] Lombaert G, Conte JP. Random vibration analysis of dynamic vehicle-bridge interaction due to road unevenness. J Eng Mech 2012;138(7):816–25. [7] Tan S, Yu Z, Shan Z, Mao J. Influences of train speed and concrete Young's modulus on random responses of a 3D train-track-girder-pier coupled system investigated by using PEM. Eur J Mech A Solids 2019;74:297–316. [8] Du XT, Xu YL, Xia H. Dynamic interaction of bridge–train system under non-uniform

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