Engineering Structures 33 (2011) 3546–3558

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Engineering Structures

journal homepage: www.elsevier.com/locate/engstruct

Random vibration of a train traversing a bridge subjected to travelingseismic wavesZhichao Zhang a,b, Yahui Zhang a,∗, Jiahao Lin a, Yan Zhao a, W.P. Howson c, F.W. Williams c

a State Key Laboratory of Structural Analysis for Industrial Equipment, Faculty of Vehicle Engineering and Mechanics, Dalian University of Technology, Dalian 116023, PR Chinab Locomotive & Car Research Institute, China Academy of Railway Sciences, Beijing 100000, PR Chinac Cardiff School of Engineering, Cardiff University, Cardiff CF24 3AA, Wales, UK

a r t i c l e i n f o

Article history:Received 18 October 2010Received in revised form14 July 2011Accepted 15 July 2011Available online 16 August 2011

Keywords:Train–bridge systemPseudo-excitation methodEarthquakeWave passage effectNon-stationary random vibrationTrack irregularity

a b s t r a c t

Non-stationary random vibration of 3D time-dependent train–bridge systems subjected to multi-pointearthquake excitations, including wave passage effect, is investigated using the pseudo-excitationmethod (PEM). The motion equation of such a system is established by coupling the train and bridgethrough wheel–rail contact relationships and accounting for the phase-lags between pier excitations.The horizontal and vertical earthquake excitations are both assumed to be uniformly modulated, fullycoherent random excitations with different phases, while the excitation due to track irregularitiesis assumed to be a 3D, fully coherent random excitation with velocity-dependent time lags. PEM isfirst proven to be applicable to such time-dependent systems, and is then used to transform therandom excitations into a series of deterministic pseudo excitations. By solving for the correspondingdeterministic pseudo responses, various non-stationary random responses, including the time-dependentpower spectral density functions (PSD) and standard deviations (SD), can be obtained easily. A case studyis then presented in which the China-Star high-speed train traverses a seven-span continuous bridgethat is being excited by an earthquake. The results show the effectiveness and accuracy of the proposedmethod by comparison with a Monte Carlo simulation. Additionally, the influences of seismic apparentwave velocity and train speed on the system random responses are discussed.

© 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The development of high-speed trains has led to more frequentuse of bridges or elevated railways. Hence the probability thatearthquakes occur while trains are crossing bridges has increasedconsiderably. The importance of this was highlighted during theNiigata Earthquake on 23 October 2004, when a Shinkansen high-speed train derailedwhile crossing an elevated bridge at 200 km/h.Since then, the dynamic analysis of train–bridge systems subjectedto seismic excitations has received much more attention [1–10].Matsumoto et al. [1] developed a vehicle/structure dynamicinteraction analysis program, known as DIASTARS, to study therunning safety of railway vehicles subjected to earthquakemotion.Yang and Wu [2] investigated the dynamic reliability of trains,initially static or traveling on bridges under different seismicconditions using a 3D train–track–bridge model.

∗ Corresponding address: State Key Laboratory of Structural Analysis forIndustrial Equipment, Department of Engineering Mechanics, Dalian Universityof Technology, Dalian 116023, PR China. Tel.: +86 411 84706337; fax: +86 41184708393.

E-mail address: zhangyh@dlut.edu.cn (Y. Zhang).

0141-0296/$ – see front matter© 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2011.07.018

This pioneering research has laid a good foundation for furtherwork. However, the dynamic interaction problem of train–bridgesystems under earthquakes has not been solved satisfactorily upuntil now because of its complexity. It is well known that forlong-span bridges subjected to earthquakes, it is very importantto account for the phase-lags between ground joints, i.e., theso-called wave passage effect. Doing so gives seismic responses,which may be quite different from those obtained by assuminguniform ground motion. Similarly, it is necessary to consider thewave passage effect in the seismic response analysis of coupledtrain–bridge systems. However, very little research has been donein this direction.

Yau and Fryba [3,4] analyzed the propagation effect of seismicwaves on the vibration of a suspension bridge subjected to theactions of moving loads and vertical support motions, but thedynamic responses and running safety of the train were notanalyzed. Xia et al. [5] studied the influence of seismic wavevelocity on the responses of vehicle–bridge systems and therunning safety of the train with the bridge quasi-static effectneglected.

It is known that both earthquakes and track irregularities arerandom. The random vibration analysis of such problems hasusually been performed bymeans of time-historymethods [1–10].

Z. Zhang et al. / Engineering Structures 33 (2011) 3546–3558 3547

Yau and Fryba [4,6] both used the modified Kobe earthquakerecords as input to compute the responses of the suspendedbridge, which was subjected to the simultaneous actions of themoving trains and the vertical motion of the earthquake. Xiaet al. [5] applied a seismic acceleration record from the 1952Kern County earthquake to investigate the influences of trainspeeds and seismic wave velocities on the dynamic responses ofvehicle–bridge systems. These time-history methods are not onlyrelatively simple in calculation, but also suitable for non-linearproblems. However, their precision relies heavily on the qualityof the samples, and the calculated results cannot fully reflect thecharacteristics of the system. On the other hand, although theconventional random vibration method [11,12] is an advancedprobabilistic analysis tool, it is computationally inefficient and toosophisticated for engineers to accept, especially for non-stationaryrandom vibration problems.

PEM (the pseudo-excitation method) [13,14], which wasdeveloped from the field of bridge seismic resistance, is a highlyefficient and accurate random vibration algorithm. For long-span structures subjected to multi-point seismic excitations, itincludes both cross-correlation between participant modes andthe phase-lags between excitations. Recently, an investigationof the dynamic responses of train–bridge systems with andwithout the action of uniform earthquake excitations [15] hasshown that the standard deviations of system lateral randomvibrationswould bemuch stronger than thosewithout earthquakeexcitations. It was also shown that the PEM can execute the non-stationary random vibration analysis of 3D train–bridge systemssubjected to horizontal earthquakes efficiently and accurately. Thepresent paper extends PEM to investigate the random vibrationanalysis of train–bridge systems subjected to multi-point seismicexcitations. Firstly, through givenwheel–rail contact relationships,the motion equation of the train is combined with the generalizedcoordinate equation of the bridge when supports are excluded,with mode superposition adopted to reduce the volume ofcalculations. Secondly, the time-dependent motion equation ofthe train–bridge system subjected to multi-point earthquakeexcitations is established, based on the concept of pseudo-staticdisplacement. Thirdly, PEM is extended to such time-dependentsystems and is applied to transform the random excitations ofmulti-point earthquakes and track irregularities into a seriesof deterministic pseudo excitations. This not only simplifiesthe solution of the non-stationary random vibration equationsconsiderably, but also generates accurate numerical results.Finally, after generating the corresponding pseudo responses usingthe Newmark method, various random responses, including thetime-dependent PSD and SD, are obtained conveniently. In thenumerical examples, a China-Star high-speed train consistingof two locomotives and nine passenger cars is assumed totraverse a seven-span continuous bridge during an earthquakethat propagates in the subsoil from one end of the bridge tothe other. The non-stationary random responses were calculatedseparately by the presentmethod and theMonte Carlomethod andcomparison between these results justifies the effectiveness andaccuracy of the proposed method. Finally the influences of wavepassage effect and train speeds on the system random responsesare studied. All the abbreviations used in this paper have beenlisted in Table 1.

2. Model of the train–bridge system

The analytic model of the train–bridge system subjected tomulti-point seismic excitations can be regarded as a spatialdynamic system composed of the train subsystem and the bridgesubsystem, which interact with each other through the givenwheel–rail contact relationship. The system random excitationsthus comprise the non-uniform earthquake excitations imposedon the bridge piers together with the track irregularities betweenwheel and rail.

Table 1List of the abbreviations in the paper.

Abbreviations Meaning

DOF Degrees of freedomPEM The pseudo-excitation methodPSD Power spectral density functionsSD Standard deviations

Fig. 1. Dynamic interaction of vehicle and bridge.

2.1. Train model

As shown in Fig. 1, the train model consists of locomotivesand/or passenger coaches, which are each composed of a car body,two bogies, four wheel-pairs and the connections between them.All of the car bodies and bogies have 5 degrees of freedom (DOF),namely lateral, roll, yaw, vertical and pitch displacements,whereasthe wheel-pairs have only 3 DOF, namely lateral, roll and verticaldisplacements. Thus, the vehicle with four axles and two bogiesused in this paper has 27 DOF. All the parameters of the train canbe found in Fig. 1 or Refs. [7,16] and the symbols on the Figure aredefined in Table 2, or as needed. The motion equation of the traincan be expressed by the Lagrange equation as

MvvXv + CvvXv + KvvXv = Fv (1)

3548 Z. Zhang et al. / Engineering Structures 33 (2011) 3546–3558

Table 2Main parameters of the train used in this case study.

Item Unit Locomotive Passenger coach

Mass of car body (Mc ) kg 59,364 40,000Roll mass moment of car body (Jcθ ) kg m2 130,493 90,000Pitch mass moment of car body (Jcϕ ) kg m2 1,723,415 2,560,000Yaw mass moment of car body (Jcψ ) kg m2 1,796,565 2,560,000Mass of bogie (Mt ) kg 5631 2100Roll mass moment of bogie (Jtθ ) kg m2 2202 1701Pitch mass moment of bogie (Jtϕ ) kg m2 9488 2100Yaw mass moment of bogie (Jtψ ) kg m2 11,233 2100Mass of wheel-set (Mw) kg 1844 1950Roll mass moment of wheel-set (Jw) kg m2 1263 1248Vertical stiffness of 1st suspension system (per side) (kv1) kN/m 2400 600Lateral stiffness of 1st suspension system (per side) (kh1) kN/m 4878 4500Vertical stiffness of 2nd suspension system (per side) (kv2) kN/m 886 260Lateral stiffness of 2nd suspension system (per side) (kh2) kN/m 316 125Vertical damping of 1st suspension system (per side) (cv1 ) kN s/m 30 10Lateral damping of 1st suspension system (per side) (ch1 ) kN s/m 80 20Vertical damping of 2nd suspension system (per side) (cv2 ) kN s/m 45 60Lateral damping of 2nd suspension system (per side) (ch2 ) kN s/m 50 20Full length of a coach (Lv) m 21.290 25.500Half distance of two bogies (s) m 5.730 9.000Half distance of two wheel-sets (d) m 1.500 1.280Half span of the 1st suspension system (a) m 1.025 1.000Half span of the 2nd suspension system (b) m 1.025 1.000Half span of the wheel-set (B) m 0.7465 0.7465Lateral distance from the wheel-set to bridge center (e) m 2.500 2.500Height of car body above 2nd suspension system (h1) m 0.740 1.792Height of 2nd suspension system above bogie (h2) m 0.590 0.078Height of bogie above wheel-set (h3) m 0.015 0.042Height of wheel-set above bridge centroid (h4) m 1.8 1.8

in which Mvv , Cvv and Kvv are the mass, damping and stiffnessmatrices of the train, Xv is the corresponding displacement vector,and Fv is the vector of forces transmitted from the wheels to thebogies through the primary springs and dashpots. Details of thesematrices and vectors can be found in Refs. [7,16].

2.2. Bridge model

The bridge is supported at its ends and its piers are supportedat their lower ends. Themotion equation of the bridge, modeled by3D Bernoulli–Euler beam elements, can bewritten in the followingpartitioned form[Mbb Mbs

MTbs Mss

] Xb

Xs

+

[Cbb Cbs

CTbs Css

] Xb

Xs

+

[Kbb Kbs

KTbs Kss

] XbXs

=

FbFs

(2)

in which: Xs represents the support displacements of the bridge,i.e. those at its ends and at the bottoms of the piers; Xbrepresents all the other bridge displacements; superscript Tdenotes transpose; Mbb, Cbb and Kbb are the mass, damping andstiffness matrices of the bridge corresponding to Xb; Mss, Css andKss are the mass, damping and stiffness matrices of the bridgecorresponding to Xs; Mbs, Cbs and Kbs are the mass, damping andstiffness interaction matrices; Fs represents the forces exerted onthe structure by the ground and; Fb represents the forces exertedon the bridge by the running train, which has the form

Fb =

Nv−i=1

2−j=1

2−l=1

(FhijlRhijl + FvijlRvijl + Fθ ijlRθ ijl) (3)

where Fhijl, Fθ ijl and Fvijl are the lateral, torsional and vertical forcesgiven by the lth wheel in the jth bogie of the ith vehicle [7,16];and Rhijl, Rθ ijl and Rvijl are the decomposition-index vectors, whichdecompose the above forces in the element into nodal forces.

Now, the mode-superposition scheme can be used for thebridge subsystem to reduce the computational complexity.Assume that the first n modes of the bridge non-uniform partare written in a matrix as Qb and the corresponding generalizeddisplacement vector is Ub. Thus

Xb = QbUb. (4)

The required transformation matrix Q can now be constructedfromXbXs

=

[Qb 00 I

] UbXs

≡ Q

UbXs

. (5)

Substituting Q into Eq. (2) and pre-multiplying by QT gives[M′

BB MBsMsB Mss

] Ub

Xs

+

[C′

BB CBsCsB Css

] Ub

Xs

+

[K′

BB KBsKsB Kss

] UbXs

=

FBFs

. (6)

in which

M′

BB = QTbMbbQb; MBs = QT

bMbs;

MsB = MsbQb; C′

BB = QTbCbbQb;

CBs = QTbCbs; CsB = CsbQb

K′

BB = QTbKbbQb; KBs = QT

bKbs;

KsB = KsbQb; FB = QTbFb.

2.3. Earthquake model

The longitudinal degrees of freedomof trains are not consideredin the trainmodel and therefore groundmotion in the longitudinaldirection of the bridge is neglected.

As the earthquake duration time is comparable with thetime the train takes to cross the bridge, it is reasonable forthe earthquake to be considered as a non-stationary random

Z. Zhang et al. / Engineering Structures 33 (2011) 3546–3558 3549

process. Hence, it is assumed that the earthquake propagates in thelongitudinal direction of the bridge and that its vertical and lateralhorizontal components have the following uniformly modulatedevolutionary forms

xgh = ge(t)xh(t); xgv = ge(t)xv(t) (7)

where ge(t) is a uniform modulation function, taken as

ge(t) =

(t/tb)2 0 ≤ t < tb

1 tb ≤ t < tcexp(−c(t − tc)) t ≥ tc

(8)

in which c is the attenuation coefficient and tb and tc are the startand end times of the stationary main shock. xh(t) and xv(t) inEq. (7) are both zero-mean-valued stationary ground accelerationswith PSD values of Shxx(ω) and Svxx(ω).

The correlation between the lateral horizontal and verticalcomponents of the earthquake is taken into account and theircross-PSD functions are assumed to be

Shvxx (ω) = Svhxx (ω) = 0.6Shxx(ω)S

vxx(ω). (9)

2.4. Track irregularity model

The lateral, rotational and vertical irregularities of the track areall considered, i.e. ry(x), rθ (x) and rz(x), all of which are assumedto be zero-mean-valued stationary random processes with givenPSD of Sy(Ω), Sθ (Ω) and Sz(Ω) respectively, whereΩ is the spatialfrequency. IfV is the speedof the train, then the relationship x = Vtenables these track irregularities to be transformed from the spacedomain into the time domain as ry(t), rθ (t) and rz(t) and their PSDmatrix is expressed as

Srr(ω) = diagSy(Ω)/V Sθ (Ω)/V Sz(Ω)/V

= diag

Sy(ω) Sθ (ω) Sz(ω)

(10)

where ω = ΩV is the time frequency.

3. Equation of motion for train–bridge systems under multi-point earthquakes

The train and bridge subsystems are combined through thewheel–rail contact relationship, with the wheels assumed toalways remain in perfect contact with the track. Thus, it can bededuced asYwijlθwijlZwijl

=

Yb(xijl)+ h4iθb(xijl)+ ry(xijl)θb(xijl)+ rθ (xijl)Zb(xijl)+ eθb(xijl)+ rz(xijl)

(11)

where:Ywijl θwijl Zwijl

T is the vector of the ijl-th wheeldisplacements; Yb, θb, Zb represent the bridge displacements; xijlisthe distance from the left-hand end of the bridge to the locationof the ijl-th wheel; h4i is the vertical distance from the wheel axleto the bridge centroid and; e is the lateral distance from the trackcentroid to the bridge centroid.

The coupled equations of motion for the train–bridge systemcan be obtained by combining Eqs. (1) and (6) through (11). Thedetailed derivation given in Appendix shows that they can then berewritten in partitioned form by taking the train and bridge non-supporting part as a whole, as follows[MVB MVBs

MTVBs Mss

] XVB

Xs

+

[CVB CVBs

CTVBs Css

] XVB

Xs

+

[KVB KVBs

KTVBs Kss

] XVBXs

=

0Fs

+

Fr0

+

Fg0

(12)

where Fg represents the deterministic excitation due to gravityaction on the train and Fr is the vector of the random excitationdue to the track irregularities.

Using the superposition principle of linear systems, Eq. (12)can be solved separately for these different excitations Fs, Fr andFg . The deterministic responses caused by Fg , i.e. the mean valueresponses, can be computed easily by using a direct integrationmethod. The random equation of motion for the train–bridgesystem under multi-point earthquakes can be extracted fromEq. (12) as[MVB MVBs

MTVBs Mss

] Xe

VBXe

s

+

[CVB CVBs

CTVBs Css

] Xe

VBXe

s

+

[KVB KVBs

KTVBs Kss

] Xe

VBXe

s

=

0Fs

(13)

where the superscript e denotes that the random response iscaused by earthquake excitations. It should be noted that becausethe train is running on the bridge, all the matrices in Eq. (13) aretime-dependent.

The absolute displacement vector XeVB can be decomposed into

the two parts

XeVB = Xep

VB + XedVB (14)

where XepVB is the quasi-static displacement vector [16], which

satisfies

XepVB = −K−1

VB KVBsXes ≡ RVBs(t)Xe

s . (15)

Substituting Eqs. (14) and (15) into Eq. (13) gives

MVBXedVB + CVBXed

VB + KVBXedVB

= − (MVBRVBs + MVBs)Xes − (CVBRVBs + CVBs)Xe

s (16)

here, the difference from the common seismic analysis of long-span bridges is that all thematrices in Eq. (16) are time-dependent.

Note that Eq. (16) cannot be reduced to the conventionalform [17] where Xe

s represents uniform ground displacements,since Eq. (16) implies that the damping forces are proportionalto the absolute velocity vector

XeT

VB XeTs

T . In order to avoidthis inconsistency, the damping forces should be assumed to beproportional to the relative velocity vector

XepT

VB 0T

in Eq. (16).This has been strictly proved in section 30.2.4 of Ref. [14]. Thus thefollowing equation can be obtained

MVBXedVB + CVBXed

VB + KVBXedVB

= −(MVBRVBs + MVBs)Xes = PXe

sXe

s . (17)

Compared with Eq. (16), the damping term on the right handside is neglected here. This approximation is reasonable, because ithas been pointed out by Clough and Penzien [17] that the dampingterms on the right hand side of this equation can be neglectedbecause they make little contribution to the effective load of arelatively lowly damped system.

By taking the ground acceleration vector Xes in non-uniform

form, Eq. (17) can be used to study the influence of the wavepassage effect on the random responses of the train–bridge system.

In addition, if only the random excitation of track irregularitiesis considered in Eq. (12), there will be no ground displacement, i.e.Xr

s = 0. Thus the random equation of motion for the train–bridgesystem due to track irregularities can be expressed as

MVBXrVB + CVBXr

VB + KVBXrVB = Fr (18)

where the superscript r denotes that the random response iscaused by the excitations due to track irregularity. In the nextsection, the random Eqs. (17) and (18) are solved using PEM.

3550 Z. Zhang et al. / Engineering Structures 33 (2011) 3546–3558

4. PEM for time-dependent train–bridge systems

Because it is a highly efficient and accurate algorithm forrandom vibration analysis, PEM has been established and usedsuccessfully in many time-independent systems with differentsingle-dimensional or multi-dimensional excitations [18–20].Here, PEM is extended to time-dependent systems and thenused to investigate the non-stationary random vibration of 3Dtrain–bridge systems under earthquakes with wave passage effectconsidered.

4.1. Time-dependent PEM for 2D asynchronous multi-point evolu-tionary random seismic excitations

Assume that phase-lags of groundmotion exist between bridgefoundations, so that the ground acceleration vector Xe

s in Eq. (17)can be expressed according to Eq. (7) as

Xes(t) = diag

G0e(t) G0e(t)

xTh(t) xTv(t)

T= Ge(t)xe(t) (19)

where

G0e(t) = diagge(t − t1) ge(t − t2) · · · ge(t − tN)

xh(t) =

xh(t − t1) xh(t − t2) · · · xh(t − tN)

T;

xv(t) =xv(t − t1) xv(t − t2) · · · xv(t − tN)

Tin which N is the number of bridge foundations. As any arbitraryPSD matrix must be Hermitian, the PSD matrix of the stationaryrandom processes xh(t) and xv(t), denoted by Se(ω), can bedecomposed into the form

Se(ω) =

[Shhxx (ω) Shvxx (ω)Svhxx (ω) Svvxx (ω)

]=

2−m=1

λemq∗

emqTem (20)

where λem and qem are the m-th eigenvalue and eigenvector ofSe(ω) and the superscript * represents the complex conjugate ofa matrix or vector.

The covariance matrix of xe(t) is

Exe(t1)xTe (t2)

=

[E

xh(t1)xTh(t2)

E

xh(t1)xTv(t2)

E

xv(t1)xTh(t2)

E

xv(t1)xTv(t2)

] (21)

whereE

xk(t1)xTl (t2)

=

xk(t − t1)xl(t − t1) xk(t − t1)xl(t − t2) · · · xk(t − t1)xl(t − tN )xk(t − t2)xl(t − t1) xk(t − t2)xl(t − t2) · · · xk(t − t2)xl(t − tN )...

.

.

.. . .

.

.

.xk(t − tN )xl(t − t1) xk(t − tN )xl(t − t2) · · · xk(t − tN )xl(t − tN )

in which k, l = h, v. Then using the Wiener–Khintchine theoremand letting τ = τ2 − τ1 yields

Exk(t1)xTl (t2)

=

∫+∞

−∞

V∗

e0VTe0S

klxx(ω)e

iωτdω (22)

where

Ve0 = [e−iωt1 e−iωt2 · · · e−iωtN ]T .

Substituting Eq. (22) into Eq. (21) gives

Exe(t1)xTe (t2)

=

∫+∞

−∞

[Ve0 00 Ve0

]∗

Se(ω)[Ve0 00 Ve0

]T

eiωτdω

=

∫+∞

−∞

V∗

eSe(ω)VTee

iωτdω. (23)

For the motion Eq. (17), any arbitrary response vector XedVB(t),

due to PXes(t)Xe

s(t), can be expressed by Duhamel integration as

XedVB(t) =

∫ t

0h(t − τ , τ )PXe

s(τ )Xe

s(τ )dτ (24)

where h(t − τ , τ ) contains the impulse response functions of thetime-variant system at time τ . The covariance matrix of Xed

VB(t) isthen

RXedVBX

edVB(t, t) = E

Xed

VB(t)XedVB

T (t)

=

∫ t

0

∫ t

0h(t − τ1, τ1)PXe

s(τ1)Ge(τ1)E

xe(t1)xTe (t2)

×GT

e (τ2)PTXes(τ2)hT (t − τ2, τ2)dτ1dτ2. (25)

Substituting Eq. (23) into Eq. (25) gives

RXedVBX

edVB(t, t) =

∫+∞

−∞

2−m=1

I∗em(ω, t)ITem(ω, t)d(ω) (26a)

Iem(ω, t) =

∫ t

0h(t − τ , τ )PXe

s(τ )Ge(τ )Ve

λemqemeiωτdτ . (26b)

It is known from Eq. (26b) that Iem(ω, t) is the response vectorcaused by the deterministic loading

√λemPXe

s(t)Ge(t)Veqemeiωt .

Hence, the pseudo-excitation vector Fem(t) for this problem can beexpressed as

Fem(t) = PXes(t)

λemGe(t)Veqemeiωt = PXe

s(t) ˜Xe

sm (27)

where ˜Xesm =

√λemGe(t)Veqemeiωt is the vector of the pseudo

ground acceleration excitation. Clearly, the pseudo dynamicdisplacement vector of the train–bridge system is

XedVBm(ω, t) = Iem(ω, t). (28)

Meanwhile, substituting the vector of the correspondingpseudo ground displacement excitation obtained by ˜Xe

sm =

−˜Xesm/ω

2 into Eq. (15) gives the system pseudo quasi-staticdisplacement

XepVBm = −K−1

VB KVBsXesm ≡ RVBs(t)Xe

sm. (29)

Hence the pseudo absolute displacement response is

XeVBm(ω, t) = Xep

VBm(ω, t)+ XedVBm(ω, t). (30)

Based on PEM, its PSD matrix is obtained easily as

SXeVBX

eVB(ω, t) =

2−m=1

XeVBm

∗(ω, t)XeVBm

T (ω, t). (31)

4.2. Time-dependent PEM for 3D asynchronous multi-point randomexcitations due to track irregularities

As the PSDmatrix of the track irregularities ry(t), rθ (t) and rz(t)in Eq. (18) is Hermitian, it can be decomposed into the form

Srr(ω) = diagSy(ω) Sz(ω) Sθ (ω)

=

3−m=1

λrmq∗

rmqTrm (32)

where λrm and qrm are the m-th eigenvalue and eigenvector ofSrr(ω).

Based on random vibration theory, the PSD matrices of thetrack irregularities and their first and second derivatives have the

Z. Zhang et al. / Engineering Structures 33 (2011) 3546–3558 3551

relationships

Sr r(ω) = −Sr r(ω) = iωSrr(ω);Sr r(ω) = Sr r(ω) = −ω2Srr(ω)Sr r(ω) = −Sr r(ω) = iω3Srr(ω);

Sr r(ω) = ω2Srr(ω); Sr r(ω) = ω4Srr .(ω). (33)

For the motion Eq. (18), any arbitrary response vector XrVB(t),

due to Fr , can be expressed by the mean of Duhamel integration as

XrVB(t) =

∫ t

0h(t − τ , τ )(Pr1Γ r1(τ )+ Pr2Γ r2(τ )

+ Pr3Γ r3(τ ))dτ

=

∫ t

0h(t − τ , τ )

3−k=1

PrkΓ rk(τ )dτ . (34)

Hence, the covariance matrix of XrVB(t) is

RXrVBX

rVB(t, t) = E

Xr

VB(t)XrVB

T (t)

=

3−k=1

3−l=1

∫ t

0

∫ t

0h(t − τ1, τ1)PrkE

× (Γrk(τ1)ΓTrl (τ2))P

Trlh

T (t − τ2, τ2)dτ1dτ2. (35)

Using the Wiener–Khintchine theorem gives

E(Γ r1(τ1)ΓTr1(τ2)) =

∫+∞

−∞

V∗

r Srr(ω)VrTeiω(τ2−τ1)d(ω) (36)

where

Vr = diagVr0 Vr0 Vr0

;

Vr0 =e−iωt1 , e−iωt2 , . . . , e−iωtn

T.

In the same way,

E(Γ rk(τ1)ΓTrl(τ2)) =

∫+∞

−∞

V∗

r Srkrl(ω)VTr e

iω(t2−t1)d(ω);

(r1 = r; r2 = r; r3 = r; k, l = 1, 2, 3). (37)

Then substituting Eqs. (36) and (37) into Eq. (35) gives

RXrVBX

rVB(t, t) =

∫+∞

−∞

3−m=1

I∗rm(ω, t)ITrm(ω, t)d(ω) (38a)

Irm(ω, t) =

∫ t

0h(t − τ , τ )

λrm

× (Pr1 + iωPr2 − ω2Pr3)Vrqrmeiωτdτ . (38b)

Clearly, the integrand of Eq. (38a) is the PSD matrix of XrVB(t),

i.e.

SXrVBX

rVB(ω, t) =

3−m=1

I∗rm(ω, t)ITrm(ω, t). (39)

It is known from Eq. (38b) that Irm(ω, t) is the response causedby the deterministic loading

√λrm(Pr1 + iωPr2 − ω2Pr3)Vrqrmeiωt .

Hence, if the pseudo-excitation vector Frm(t) for this problem isexpressed as

Frm(t) =

λrm(Pr1 + iωPr2 − ω2Pr3)Vrqrmeiωt (40)

then, XrVBm(ω, t) = Irm(ω, t) will be the pseudo response due to

this pseudo excitation, while SXrVBX

rVB(ω, t) and RXr

VBXrVB(t, t)will be

the corresponding time-dependent PSD and covariance matrices,respectively.

After substituting the pseudo-excitations of Eqs. (27) and (40)into Eqs. (17) and (18) respectively, the corresponding pseudoresponses, Xe

VBm(ω, t) and XrVBm(ω, t), can be calculated by the

Newmark method. The corresponding PSD matrices, SXeVBX

eVB(ω, t)

and SXrVBX

rVB(ω, t), then follow easily from Eqs. (31) and (39). As

there is no correlation between the excitations of the earthquakeand the track irregularity, the PSD matrix of the whole systemrandom responses can be expressed as

SXVBXVB(ω, t) = SXeVBX

eVB(ω, t)+ SXr

VBXrVB(ω, t) (41)

and the SD of the random responses is then given by

σ2VB(t) =

∫+∞

−∞

SXVBXVB(ω, t)dω. (42)

5. Flowchart for random vibration analysis of train–bridgesystems

The above expositions of the behavior of 3D train–bridgesystems under the action of earthquakes and track irregularities,and of the multi-dimensional time-dependent PEM, are amplifiedby the flowchart for the random vibration analysis of train–bridgesystems given in Fig. 2.

6. Numerical examples

Consider the China-Star high-speed train traversing a uniformcontinuous concrete bridge. The bridge model consists of sevenspans with a length of 40 + 64 × 5 + 40 = 400 (m), as shownin Fig. 3. The parameters of its deck are: cross-section area A =

12.83m2, lateral bendingmoment of inertia Iz = 134.0m4, verticalbending moment of inertia Iy = 19.2 m4, torsional moment ofinertia Iρ = 51.9 m4, elastic modulus E = 2.5 × 1010 N/m2 andconcrete density ρ = 2500 kg/m3. The parameters of the bridgepiers are: height Hp = 16.0 m, cross-section area Ap = 6.2 m2,lateral bending moment of inertia Ipz = 28.7 m4, longitudinalbending moment of inertia Ipy = 2.4 m4 and torsional momentof inertia Ipρ = 10.17 m4. The bridge was assumed to have aRayleigh damping matrix C = 0.0108M + 0.1583K. The deck andpiers were discretized using 3D Bernoulli–Euler beam elements,with the deck sub-divided into 100 identical elements. The first60 modes of the bridge were included in the mode-superposition.The first vertical and lateral natural frequencies of the deck were11.42 rad/s (1.82 Hz) and 15.38 rad/s (2.45 Hz), respectively.

The China-Star high-speed train consists of one front locomo-tive, nine passenger cars and one rear locomotive. Their main pa-rameters are listed in Table 2. In order to simulate the motion ofthe train at the moment when it arrives at the bridge, it was as-sumed that the train starts to move from a location 50 m beforethe left-hand end of the bridge at a constant speed.

Assume that the earthquake wave propagates in the subsoilfrom one end of the bridge to the other while the train is traversingthe bridge. The lateral and vertical earthquake components havethe forms given in Eq. (7) and the parameters of the modulationfunctions are given in Eq. (8). The PSD functions take theClough–Penzien model [17]

Shxx(ω) =ω4

gh + 4ζ 2ghω

2ghω

2

(ω2gh − ω2)2 + 4ζ 2

ghω2ghω

2

×ω4

(ω2fh − ω2)2 + 4ζ 2

fhω2fhω

2S0h (43a)

Svxx(ω) =ω4

gv + 4ζ 2gvω

2gvω

2

(ω2gv − ω2)2 + 4ζ 2

gvω2gvω

2

×ω4

(ω2f v − ω2)2 + 4ζ 2

f vω2f vω

2S0v (43b)

3552 Z. Zhang et al. / Engineering Structures 33 (2011) 3546–3558

Fig. 2. Flowchart for the random vibration analysis of train–bridge systems subjected to earthquakes or track irregularities.

Fig. 3. Dynamic model of train–bridge system.

where: S0h and S0v are the spectral intensity factors; ζgh and ζgv arethe damping ratios of the soil; ωgh and ωgv are the predominantfrequencies of the soil and; ωfh and ωf v , ζfh and ζf v are the filterparameters, with

S0v = 0.218S0h; ζgv = ζgh; ωgv = 1.58ωgh

ζfh = ζgh; ωfh = 0.1ωgh ∼ 0.2ωgh;

ζf v = ζgv; ωf v = 0.1ωgv ∼ 0.2ωgv. (44)

Here, they take the values [21]: tb = 0.8 s, tc = 7 s, c = 0.35,ωgh = 17.95 (rad/s), ζgh = 0.72, and S0h = 3.64 × 10−4 (m2/s3).

In addition, the basic design acceleration of the ground motion istaken as 0.15 g, while the frequency region is ω ∈ [1, 80] rad/s.Meanwhile, the PSD functions of the three track irregularitiesin Eq. (10) use the German high-speed track spectrum of highirregularity, in which their spatial frequency lies between 0.01 ×

2π ∼ 0.4×2π (rad/m). The PSD function andmodulation functionof the above earthquake are both shown in Fig. 4.

It should be noted that in the following examples, the ‘bridgemidpoint’, ‘locomotive’ and ‘passenger’ mean, respectively, themidpoint of the fourth span for the seven-span bridge, the bodyof the first car and the body of the sixth car. The results displayed

Z. Zhang et al. / Engineering Structures 33 (2011) 3546–3558 3553

Fig. 4. Plots of the earthquake used in the numerical examples for: (a) PSD function of the Clough–Penzien model; (b) uniform modulation function.

Fig. 5. Comparison of the locomotive SD curves obtained by PEM and MC for: (a) lateral acceleration; (b) vertical acceleration.

in Figs. 5–10 all have time coordinates that start from the momentwhen the earthquake begins.

6.1. Verification by the Monte Carlo method

Assume that the seismic apparent wave velocity is ve =

1000 m/s and the train running speed is v = 180 km/h.Furthermore, it is assumed that the cross-correlation termsbetween the lateral and vertical random earthquake waves arenegligible in this example, i.e. Shvxx (ω) = Svhxx (ω) = 0 in Eq. (9).The random responses of the train–bridge system were calculatedseparately by the proposed method, marked as ‘‘PEM’’, and by theMonte Carlo method with 50 or 500 samples of earthquakes andtrack irregularities, marked as ‘‘MC’’. Fig. 5 shows the time-variantSD curves of the locomotive lateral and vertical accelerations,whileFig. 6 shows the time-variant SD curves for the lateral and verticaldisplacements of the bridge midpoint. It can be seen that as thenumber of samples increases, theMonte Carlomethod gives resultsthat are closer to those given by the present method. When 50samples were used, the maximum differences between the MonteCarlo method and the proposed method are, 28.8% and 31.7% forthe lateral and vertical acceleration SDof the locomotive, and 39.3%and 32.2% for the lateral and vertical displacement SD of the bridgemidpoint. However, when 500 samples were used, the differenceswere reduced to 10.7%, 6.9% and 9.9%, 7.9%, respectively. Thesedifferences would be further reduced with further increase of thesample number. It should be pointed out that the proposedmethodonly takes 48.7 CPU min on a Pentium (R) D computer with mainfrequency 3.2 Hz and memory 1 GB. This shows that by using thepresent method, the non-stationary random vibration analysis oftrain–bridge systems subjected to non-uniform seismic excitationscan now be performed accurately and efficiently for the firsttime.

6.2. Influence of the wave passage effectTo study the influence of wave passage effects on the random

responses of the train–bridge system, the seismic apparentwave velocity is assumed to be 600, 1200, 1800,+∞ (m/s),respectively. Here, ve = +∞ m/s means that no phase-lag existsbetween the bridge piers, i.e. the seismic wave passage effectis not considered for this case. The cross-correlation betweenthe lateral and vertical components of the seismic waves isgiven by Eq. (9) and the train speed is assumed to be v =

180 km/h. Figs. 7 and 8 show the resulting SD curves of thelateral and vertical accelerations for the locomotive and bridgemidpoint with different apparent wave velocities, respectively.It can be concluded that all the random response SD’s of thetrain–bridge system change significantly when considering thephase lags between the piers. When the apparent wave velocity is600m/s, themaximumdifferences in SD for the lateral and verticalaccelerations of the locomotive are −19.1% and −4.8% and thosefor the bridge midpoint are −32.3% and −38.9%, respectively.As the wave velocity increases to 1800 m/s, these discrepanciesreduce to −4.5%, −2.0% and −11.3%, −23.7%, respectively. Inaddition, it can also be seen that the influence of wave passageeffect on the bridge responses is more significant than its effecton the locomotive responses. The reason for this is that the seismicforces act directly on the bridge piers, but only transfer indirectlythrough the deck to the train.

Figs. 9 and 10 show the PSD plots of the lateral accelerationfor the locomotive and bridge midpoint locomotive, respectively,when ve = 600, 1200, 1800,+∞ (m/s). Fig. 9 shows thatthe first peaks of the PSD plots for the locomotive lateralacceleration all appear at 3.6 rad/s, which is the natural frequencyof the locomotive lateral vibration. However, the frequenciescorresponding to their second peaks are different from each other,which is consistentwith the corresponding peak frequencies of the

3554 Z. Zhang et al. / Engineering Structures 33 (2011) 3546–3558

Fig. 6. Comparison of the SD curves of the bridge midpoint obtained by PEM and MC for: (a) lateral displacement; (b) vertical displacement.

Fig. 7. The locomotive SD curves for different apparent wave velocities of: (a) lateral acceleration; (b) vertical acceleration.

Fig. 8. The SD curves of the bridge midpoint for different apparent wave velocities of: (a) lateral acceleration; (b) vertical acceleration.

lateral acceleration PSD plots of the bridge midpoint in Fig. 10.It can be seen from Fig. 9 that the maximum PSD of the lateralaccelerations of the bridge midpoint appear at nearly the first,third and fifth lateral natural frequencies of the bridge when ve =

600 m/s, i.e. 15.38 rad/s (2.45 Hz), 17.55 rad/s (2.79 Hz) and22.63 rad/s (3.60 Hz), respectively. As the apparent wave velocityincreases, the peak frequencies gradually get closer to the first andthird lateral natural frequencies when ve = 1200, 1800 (m/s)and finally converge on the first lateral natural frequency whenve = +∞. Hence, the conclusion that can be drawn is that thewave passage effect may reduce the peak values of the responsePSD for the case ve = +∞, but can enable the train–bridge systemto vibrate at higher natural frequencies.

In order to investigate the influence of seismic apparent wavevelocities on system random responses more deeply, the wavevelocity is now assumed to vary between 400 and 4000 m/s.Figs. 11 and 12 then show themaximumacceleration SD versus thewave velocity for the locomotive, passenger and bridge midpoint,

respectively. It can be seen that all the maximum acceleration SD’sof the locomotive and passenger increase gradually with seismicwave velocity, but those of the bridge midpoint do not alwaysincrease with the wave velocity. Fig. 12 shows that the max SDcurves of the lateral and vertical accelerations have peaks at thevelocities ve = 1000 m/s and ve = 600 m/s, respectively. This isbecause the variation of the seismic apparent wave velocity causesa corresponding change in the phase-lags between the bridgesupports.

6.3. Influence of train speed

The influence of train speed on the system random vibrationsis now considered by varying the train speed between 100 and420 km/h. Fig. 13 shows the distribution of the maximum SDfor the lateral acceleration of the locomotive and the verticalacceleration of the bridge midpoint versus train speed when ve =

600 m/s, for three different cases: the earthquake applied only,

Z. Zhang et al. / Engineering Structures 33 (2011) 3546–3558 3555

Fig. 9. The PSD plots of the locomotive lateral acceleration for different apparent wave velocities.

Fig. 10. The PSD plots of the lateral acceleration of the bridge midpoint for different apparent wave velocities.

the track irregularities applied only, and the earthquake and trackirregularities applied simultaneously. It can be seen that boththe curves of ‘Irregularity’ and ‘Earthquake + Irregularity’ varysignificantly with the increase of train speed, and their variationlaws are almost the same.However, the curves of ‘Earthquake’ havevery little variation with increase in train speed. Hence, the trainspeed has considerable influence on the system lateral randomvibrations caused by the excitations due to track irregularities,but has little effect on the random vibrations due to earthquakeexcitation.

In addition, Fig. 14 shows the distribution of the maximumacceleration SD versus train speed for the locomotive, passengercoach and bridge midpoint, when ve = 600, 1800, and +

∞ (m/s), with the seismic and track irregularities consideredsimultaneously. It can be seen from Fig. 14 that the max SD of thelateral and vertical acceleration for the passenger coach increasesquite rapidly with the train speed, while those for the bridgemidpoint also increase almost monotonically, but the incrementis not very large. However, the maximum SD of the locomotivelateral and vertical accelerations do not always increase with the

3556 Z. Zhang et al. / Engineering Structures 33 (2011) 3546–3558

Fig. 11. Variation with apparent wave velocity of the maximum SD for the locomotive and passenger coach for: (a) lateral accelerations; (b) vertical accelerations.

Fig. 12. Variation with apparent wave velocity of the maximum acceleration SD ofthe bridge midpoint.

train speed and even show a small decrease when the train speedincreases from 220 to 260 km/h. This is due to the fact that theinfluence of track irregularities on system random vibrations issignificant and hence changing the train speed will affect the PSDdistribution of track irregularity excitations versus frequency andthe corresponding time-lags between wheel excitations, which issimilar to the phenomena for wave passage effect studied above.

7. Conclusions

The non-stationary random vibration analysis of 3D train–bridge systems subjected to seismic excitations and includingthe wave passage effect has been solved by means of PEM.The equation of motion for time-dependent train–bridge systemsdue to multi-point seismic excitations has been establishedby combining the train and the bridge (excluding supports)as a coupled system through the wheel–rail contact relations.

The applicability of PEM to time-dependent systems has beenproven and then adopted to transform the seismic excitationsand track irregularities into a series of deterministic pseudoexcitations. In this way, various non-stationary random responses,including their time-dependent PSD and SD, can be obtainedconveniently. Numerical examples show that: (1) the presentmethod can execute the non-stationary random vibration analysisof 3D train–bridge systems due to multi-point seismic excitationsaccurately and very efficiently; (2) the wave passage effect ofearthquakes has significant influence on the random responsesof train–bridge systems and hence it is necessary to consider itsinfluence on coupled train and long-span bridge systems; (3) thetrain speed has little influence on the random responses caused byearthquakes, but the random responses due to track irregularitiesare significant.

It should be noted that the above conclusions were drawnfrom a single case-study, so more numerical computations may benecessary to confirm them. Additionally, the effects of incoherencyand spatially varying site effects are very important for theseismic analysis of long-span structures. PEMcanperfectly accountfor these effects [13]. In China, this method is very popular inpractical seismic analyses, e.g. for dams and bridges, and has alsobeen recommended by the Chinese government via the officialdocument ‘‘Guidelines for Seismic Design of Highway Bridges(2008)’’. Therefore it is intended to extend the present work tothe more general case by including the effects of incoherency andspatially varying site effects.

Acknowledgments

Support for this work from the National Natural ScienceFoundation of China (Nos. 10972048, 90815023), Chinese NationalScience and Technology support program ‘‘Improvement of vehiclebody structures of high-speed trains using the Pseudo Excitation

Fig. 13. Variation with train speed of the maximum SD for three different cases for: (a) locomotive lateral accelerations; (b) vertical accelerations of the bridge midpoint.

Z. Zhang et al. / Engineering Structures 33 (2011) 3546–3558 3557

Fig. 14. Variation with train speed of themaximum SD for: (a) locomotive lateral acceleration; (b) locomotive vertical acceleration; (c) passenger coach lateral acceleration;(d) passenger coach vertical acceleration; (e) bridge midpoint lateral acceleration; (f) bridge midpoint vertical acceleration.

Method’’ and from the Cardiff Advanced Chinese EngineeringCentre is gratefully acknowledged.

Appendix

Before explaining Eq. (12), it is necessary to define the symbolson Fig. 1 (noting that Table 2 gives more detail), as follows:subscripts i, j and l are as defined beneath Eq. (3); Y and Z are lateraland vertical displacements; θ , ϕ and ψ are rolling, pitching andyawing rotations (when used as subscripts they relate quantitiesto those rotations); J is rotatory inertia; M and m are mass and kand c are stiffness and damping, respectively; a, b, d, e, h and s arelinear dimensions; subscripts 1–4 are used on h to define the fourheights shown; subscripts c , t , w and b denote the vehicle body,bogie, wheel-set and bridge, respectively and; superscripts h and vdenote horizontal and vertical.

The mass stiffness sub-matrices are respectively expressed as

MVB =

[Mvv 00 MBB

];

MBB = M′

BB +

Nv−i=1

2−j=1

2−l=1

(mwijlR′

hijlR′

hijlT

+mwijlR′

vijlR′

vijlT

+ JwijlR′

θ ijlR′

θ ijlT )

KVB =

[Kvv KvBKTvB KBB

];

KBB = K′

BB +

Nv−i=1

2−j=1

2−l=1

(kh1ijR′

hijlR′

hijlT

+ kv1ijR′

vijlR′

vijlT

+ a2i kv1ijR

′

θ ijlR′

θ ijlT )

KvB =

KvB1KvB2...KvBNv

; KvBi =

0Ki

t1BKi

t2B

;

KitjB = −

kh1ijR

′

hijlT

a2i kv1ijR

′

θ ijlT

− h3ikh1ijR′

hijlT

ηjldikh1ijR′

hijlT

kv1ijR′

vijlT

ηjldikv1ijR′

vijlT

MVBs =

[0MBs

]; KVBs =

[0KBs

]; XVB =

XvUb

R′

hijl = QTb

Rhijl + h4iRθ ijl

;

R′

θ ijl = QTbRθ ijl; R′

vijl = QTb

Rvijl + eRθ ijl

.

3558 Z. Zhang et al. / Engineering Structures 33 (2011) 3546–3558

Each of the damping sub-matrices has the same form as thecorresponding stiffness sub-matrices and is obtained by simplyreplacing K by C and k by c.

The sub-vector Fr(t) is expressed as

Fr(t) = Pr1Γ r1(t)+ Pr2Γ r2(t)+ Pr3Γ r3(t)

Γr1(t) = Ustr (t); Γr2(t) = Ust

r (t); Γr3(t) = Ustr (t)

Ustr (t) =

rTy (t) rTθ (t) rTz (t)

T;

rm(t) =rm(t − t1) rm(t − t2) · · · rm(t − tn)

T;

(m = y, θ, z)

Pr1 =PvTr1 PbT

r1

T; Pr3 =

0 PbT

r3

Twhere Pr2 has the same form as Pr1 and is obtained by simplyreplacing k by c , and

Pvr1 =PvyTr1 PvθTr1 PvzTr1

T;

Pbr1 =

PbyTr1 PbθT

r1 PbzTr1

T;

Pbr3 =

PbyTr3 PbθT

r3 PbzTr3

TPvmr1 = diag

Pm1r1 Pm2

r1 · · · PmNvr1

;

Pbmr1 =

Pbm1r1 Pbm2

r1 · · · PbmNvr1

Pbmr3 =

Pbm1r3 Pbm2

r3 · · · PbmNvr3

; (m = y, θ, z)

Pmir1 = diag

0 Pmi1

r1 Pmi2r1

;

Pbmir1 =

Pbmi1r1 Pbmi2

r1

;

Pbmir3 =

Pbmi1r3 Pbmi2

r3

; (i = 1, 2, . . . ,Nv)

Pyijr1 = kh1ij

1 1−h3i −h3idi −di0 00 0

; Pθ ijr1 = kv1ij

0 0a2i a2i0 00 00 0

;

Pzijr1 = kv1ij

0 00 00 01 1di −di

; (j = 1, 2)

Pbyijr1 = −kh1ij

R′

hij1 R′hij2

; Pbθ ij

r1 = −a2i kv1ij

Rθ ij1 Rθ ij2

;

Pbzijr1 = −kv1ij

R′vij1 R′

vij2

Pbyijr3 = −mwijl

R′

hij1 R′hij2

; Pbθ ij

r1 = −JwijlRθ ij1 Rθ ij2

;

Pbzijr1 = −mwijl

R′vij1 R′

vij2

in which n is the number of train wheel-pairs; tq (q = 1, 2, . . . , n)is the time lag between the q-thwheel-pair and the firstwheel-pairwith t1 = 0.

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