Review ArticleRecent Research and Applications of NumericalSimulation for Dynamic Response of Long-Span BridgesSubjected to Multiple Loads
Zhiwei Chen1 and Bo Chen2
1 Department of Civil Engineering Xiamen University Xiamen Fujian 361005 China2 Key Laboratory of Roadway Bridge and Structural Engineering Wuhan University of Technology Wuhan 430070 China
Correspondence should be addressed to Zhiwei Chen cezhiweixmueducn
Received 19 March 2014 Accepted 21 April 2014 Published 21 May 2014
Academic Editor Ting-Hua Yi
Copyright copy 2014 Z Chen and B ChenThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Many long-span bridges have been built throughout the world in recent years but they are often subject to multiple types ofdynamic loads especially those located in wind-prone regions and carrying both trains and road vehicles To ensure the safetyand functionality of these bridges dynamic responses of long-span bridges are often required for bridge assessment Given thatthere are several limitations for the assessment based on field measurement of dynamic responses a promising approach is basedon numerical simulation technologies This paper provides a detailed review of key issues involved in dynamic response analysisof long-span multiload bridges based on numerical simulation technologies including dynamic interactions between runningtrains and bridge between running road vehicles and bridge and between wind and bridge and in the wind-vehicle-bridgecoupled systemThen a comprehensive review is conducted for engineering applications of newly developed numerical simulationtechnologies to safety assessment of long-span bridges such as assessment of fatigue damage and assessment under extreme eventsFinally the existing problems and promising research efforts for the numerical simulation technologies and their applications toassessment of long-span multiload bridges are explored
1 Introduction
Many long-span bridges have been built throughout theworld in the past few decades to meet the economic socialand recreational needs of communities Some of these bridgeshave main span lengths of more than 1000m (see Figure 1)such as the Akashi Kaikyo Bridge (1991m Japan 1998)the Xihoumen Bridge (1650m China 2009) the GreatBelt Bridge (1624m Denmark 1998) and the Run YangBridge (1490m China 2005) Some of them carry bothroad and rail traffic such as the Tsing Ma Bridge (1377mHong Kong 1997) the Minami Bisan-Seto Bridge (1100mJapan 1989) and the 25 de Abril Bridge (1013m Japan1966) Most of these bridges are located in wind-proneregions and long-span length makes them susceptible tostrong crosswinds Further the increases in traffic volumeand gross vehicle weight that accompany economic devel-opment significantly affect the local dynamic behavior of
such bridgesMost of long-span bridges aremultiload bridgessince they are simultaneously suffering combined effects ofmultiple dynamic loading such as railway highway andwindloading Multiload bridges play significant roles in the entiretransportation system and thus it is critically important toprotect such immense capital investments and ensure usercomfort and bridge safety
However the strength and integrity of these bridges willdecrease during the serviceability stage due to the degra-dation mechanisms induced by traffic wind temperaturecorrosion and environmental deterioration In order todetect the abnormal changes through nondestructive testing(NDT) technology or periodical evaluation a fundamentalbut critical step is to obtain dynamic responses at somecritical bridge locations The mostly concerned dynamicresponses of a multiload bridge may include global response(displacement velocity and acceleration) and local response(acceleration and stress) which are mainly induced by
Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 763810 17 pageshttpdxdoiorg1011552014763810
2 The Scientific World Journal
(a) Akashi-Kaikyo Bridge (b) Xihoumen Bridge
(c) Great Belt Bridge (d) Run Yang Bridge
Figure 1 Examples of long-span bridges
traditional live load (such as highway railway and windloading) or accidental live load (such as ship impact andearthquake) Structural intrinsic characteristics could beextracted from these dynamic responses (or vibration signals)to develop all sorts of vibration-based damage detection tech-niques A well-known family of them is based on structuraldynamic characteristics (such as frequencies mode shapesdamping ratios and strainmode shapes) and their derivatives[1ndash3] Some damage identification approaches were proposedbased on the dynamic responses of bridge structures undermoving vehicle loads [4ndash6] The dynamic responses of long-span bridges also could be used for structural assessmentfor example fatigue assessment at the critical locations overthe service history of the bridge [7ndash10] and assessment ofextreme events such as complex traffic congestion coupledwith moderate or even strong wind [11]
Over the past decades on-structure long-term structuralhealth monitoring systems (SHMSs) have been implementedon long-span bridges in Europe the United States CanadaJapan Korea China and other countries [12] They areinstalled in newly constructed bridges and existing bridgesfor monitoring structural behavior in real time evaluatingstructural performance under various loads and identifyingstructural damage or deterioration [13] To comprehen-sively understand the bridge performance dynamic bridgeresponses are important monitoring items of structuralhealth monitoring Global responses (such as displacement)are measured by GPS and accelerometers [14 15] and localresponses (such as strainstress) are normallymeasured in thecritical bridge components andwidely used for fatigue assess-ment [16] Although dynamic responses have been measuredfor those bridges installed with SHMSs condition evaluation
based on measurement still has some limitations (1) it isdifficult to identify all of the local critical locations and evenso it is uneconomical to monitor all critical locations in longterm (2) not every fatigue-critical location is suitable forsensor installation (3) it is difficult to obtain measurementdata in the extreme events (such as combination of trafficcongestion and strong wind) which rarely happen (4) itis hard to exactly predict the influence of possible trafficgrowths based on field measurement only Integrating withnumerical simulation technologies and field measurementsis an alternative approach which is able to overcome thelimitations of evaluation approaches only based on measure-ments The information on the concerned dynamic loadingsmeasured by the SHMS could be taken as inputs for thenumerical simulation and the computed dynamic responsescould be compared with the measured ones in the validation[17]
However numerical simulation of dynamic response ofa long-span multiload bridge is not an easy job becauseit requires a complex dynamic finite element model of thebridge including all important bridge components variousdynamic loading models for running trains running roadvehicles and high winds and interactive models betweenthe bridge and wind bridge and trains and bridge androad vehicles [17] This paper focuses on recent research andapplications of numerical simulation technology for dynamicresponse of long-span multiload bridges Firstly key issuesinvolved in dynamic response analysis of long-spanmultiloadbridges based on numerical simulation technologies arereviewed in Section 2 The applications of newly developednumerical simulation technologies to safety assessment oflong-span bridges are subsequently reviewed in Section 3
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y
xz Railway track
Bridge deck
Main cross-frame
(a)
Railway tracksOrthotropicdeck-plates (top)
Cross bracings
Cross bracings
Cross bracings(top centre)
Cross-frames(main and intermediate)
Longitudinaltrusses
(bottom outer)
Orthotropic deck-plates (bottom)
Corrugated sheets
bottom centre
(b)
Figure 2 Finite element model of suspended deck module (a) hybrid 3D bridge model [22] (b) full 3D model [25]
Finally the existing problems and promising research effortsfor the numerical simulation technologies and their appli-cations to assessment of long-span multiload bridges areexplored in Section 4
For themost complex situation a long-spanmultiload bridgewhich is located at a wind-prone region carries both railwayand highway traffic and thus the combined effect of runningtrains running road vehicles and wind is acting on thebridge Several key issues are involved in this complicatedsituation such as dynamic interaction between runningtrains and bridge dynamic interaction between running roadvehicles and bridge and dynamic interaction between windand bridge To give a comprehensive review the above threekey issues will be individually reviewed in Sections 21 to23 and then the dynamic interactions of wind-vehicle-bridgesystem as a whole are then reviewed in Section 24
21 Dynamic Interaction between Trains and Bridge
211 Modeling of a Cable-Supported Bridge In early researchin this area simplified bridge models were employed tostudy vehicle-bridge interactions For example a cable-stayedbridge was simulated as a beam resting on an elastic founda-tion by Meisenholder and Weidlinger [18] for the dynamicanalysis of cable-stayed guideways subject to track-levitatedvehicles moving at high speeds Mao [19] investigated theimpact factor of a cable-stayed bridge which was assumedto be formed of continuous elastic beams supported byintermediate elastic supports
More recently with the development of finite element(FE) technology it has become common practice to use acomputer software package to establish a finite elementmodel(FEM) of a cable-supported bridge This technology estab-lishes an accurate bridge model that takes into account thegeometric nonlinear behavior of a cable-supported bridgeTo make the bridge model close to the realistic bridgein terms of its dynamic properties the modal frequencies
and shapes determined by dynamic tests are used for fur-ther model validation or updating Many FEMs of cable-supported bridges have been established for the analysis oftrain-bridge interactions The Tsing Ma Suspension Bridgein Hong Kong can be used as an example to illustratethe various bridge models that have been established foranalysis The first generation of Tsing Ma Bridge model wasa spinal beam model [20] in which the hybrid steel deck wasrepresented by a single beam with equivalent cross-sectionalproperties two bridge towers made of reinforced concretethat were modeled by three-dimensional Timoshenko beamelements and cables and suspenders that were modeled bycable elements to account for geometric nonlinearity dueto cable tension The model was validated by comparing itwith measurements of the first 18 modal frequencies andshapes of the actual bridge Using this model Guo et al[21] predicted the dynamic displacement and accelerationresponses of coupled train and bridge systems in crosswindsHowever they modeled the bridge deck as a simplified spinebeam of equivalent sectional properties and were thus unableto capture the local stress and strain behavior of the bridgeA second-generation bridge model (hybrid 3-dimensionalmodel) was established to overcome this weakness [22] Themodeling work is based on the previous model developed byWong [23] In this model 15904 beam elements were usedto model the bridge deck to closely replicate the geometricdetails of the complicated deck in reality (see Figure 2(a))The railway beams and rails were modeled by beam elementsto allow the accurate computation of the contact forcesbetween the bridge and railway vehicle The deck-platescarrying the road vehicles were modeled by plate elementsto allow the accurate computation of the contact forces atthe contact points between the road surface and the vehicletires The bridge deck was modeled to closely replicate thegeometric details of the complicated deck which is requiredfor calculation of the action of the wind forces The bridgemodel was also updated using the first 18 measured naturalfrequencies and mode shapes Based on this model Xu et al[24] computed the stress and acceleration responses of localcritical components under running trains and Chen et al[17] computed dynamic stress response induced by railwayhighway and wind loading
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Mtij Jt120579ij
mwij1 120579wij1Ywij1
Y
U
120601ti2 120601ti1
Zti2k1i2 c
1i2
2di
kh1i ch1i2
kh2i2 ch2i2
Mti2
Yti2
2si
Yci
Mti1 Jt120601i1
Zti1 x
Zwij1
Bridge deck
Mci Jc120579i 120579ci
2bi Ycih1i
h2i
h3i
z2Bi
2ai
Ytij
Ywij1
120593ci
ZciMci Jc120593i
k2i2c2i2
Mti1
Jt120595i1Jt120595i2
Mci Jc120595i
Yci
120595ti2
Yti1
120595ti1120595ci
Mti2 Jt120601i2
kh2ij ch2ij
kh1ij ch1ij
k2ij c2ij
k1ij c1ij
Ztij
120579tij
Zwij1Jwij1
Figure 3 Dynamic model of a railway vehicle [51]
However the hybrid 3D model is still not fine enoughfor criticality analysis of bridge structures which requiresresults at strainstress level especially for some bridge detailsFor example the orthotropic decks (steel deck-plates sup-ported by U-shape troughs) were modeled by plate ele-ments with equivalent depths so that the measured resultsfrom strain gauges at the surfaces of deck-plates or U-shape troughs had no counterparts in computation resultsTherefore Duan et al [25] established the third-generationbridge model (full 3D model) for performance evaluationat stressstrain level (see Figure 2(b)) In this model themajor structural components were modeled in detail and theconnections and boundary conditions are modeled properlywhich results in about half million elements for the completebridge model The strainstress responses induced by atrain passing through the bridge were calculated by staticinfluence linemethod and comparedwithmeasured results inthe calibration
Although full 3D bridge model provides the possibilityfor exact stress analysis at the local components large com-putational efforts are needed for the refined section modelwith complicated structural details Li et al [26] proposed amultiscale FE modeling strategy for long-span bridges Theglobal structural analysis was carried out using the beamelement modeling method at the level of a meter The localdetailed hot-spot stress analysis was carried out using shellor solid elements at the level of a millimeter Based onthis model the global dynamic response of the bridge andlocal damage accumulation of two typical weld details ofthe bridge under traffic loading were numerically analyzedMultiscale FE modeling scheme was also proposed by Zhanget al [27] based on the equivalent orthotropic modelingmethod (EOMM) Bridge details withmultiple stiffenersweremodeled with shell elements using equivalent orthotropicmaterials Based on this model Zhang et al [10] computedthe dynamic stress responses of long-span bridges undercombined dynamic loads from winds and road vehicles
212 Modeling of Trains Previously running vehicles werecommonlymodeled as a series of moving forces either due tolimits on computational capacity or because it is easier to findthe analytical solutions inmany cases [28ndash37]This treatmentneglects the effect of interactions between the bridge andrunning vehicles For this reason the moving load model issuitable only for the case in which the mass of the vehicleis small relative to that of the bridge or when the vehicleresponse is not of interest [38] For cases in which the inertiaof the vehicle cannot be regarded as small a moving massmodel should be adopted instead [39ndash42] More recently theemergence of high-performance computers and advances incomputer technology hasmade it feasible tomore realisticallymodel the dynamic properties of the various components ofmoving vehicles [43ndash48]
In a more sophisticated railway vehicle model the sus-pension mechanisms are modeled by springs the dampingeffect of the suspension systems and air-cushion by dashpotsand the energy dissipating effect of the interleaf mechanismby frictional devices Using this technique a tractor-traileris represented as two discrete masses each of which issupported by two sets of springs and dashpots or frictionaldevices [38] To represent the various dynamic propertiesof railway vehicles vehicle models that contain dozens ofdegrees of freedom (DOFs) have been devised and used by[49ndash52] To investigate the dynamic interaction between along suspension bridge and running trains Xia et al [51]considered a train composed of a sequence of identicalrailway vehicles Each railway vehicle was assumed to consistof a rigid car body resting on front and rear bogies with eachbogie supported by two wheelsets (see Figure 3) Five DOFswere assigned to the car body and to each bogie to accountfor vertical lateral rolling yawing and pitching motions Incontrast only three DOFs were assigned to each wheelset toaccount for vertical lateral and rolling motions
Many vehicle models have been established for vehicle-bridge interaction analysis In most of these studies
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(a) (b) (c)
Car body
Connecting rigid-arm
Wheel
Wheel-rail contactConstrait
Bogie
Primary suspension
Secondary suspension
(d)
Figure 4 Finite element model of a railway vehicle (a) elevation view (b) side view (c) isometric view (d) model details [52]
the equations of motion of the vehicles were derivedanalytically However a great inconvenience of this methodis that the equations of motion of the whole vehicle-bridgesystem must be rederived if the vehicle type is changedFurthermore it is very difficult to derive the equation ofmotion for a complex vehicle model containing a largenumber of DOFs such as the articulated components of aTGV train with an 85-DOF dynamic system [53] Generalcommercial FE software has recently been adopted to makevehicle modeling more easily applicable for different vehicletypes [54] Li et al [55] described a four-step procedurefor modeling a four-axle railway vehicle by beam elements(1) the nodes and elements for the car body bogies andwheelsets respectively are defined by using beam elements sothat the spatial geometric configuration of each componentcan be built (see Figure 4) (2) sectional properties andmaterial properties are assigned to each beam element(3) rigid-arms and suspension units (systems) are used toconnect the three components (4) constraints are assignedto form a complete finite element model of the vehicle
213 Modeling of Rail Irregularities Track irregularities rep-resent an important source of excitation for bridges duringthe passage of railway vehicles Track irregularitiesmay occuras a result of initial installation errors the degradation ofsupport materials or the dislocation of track joints Fourgeometric parameters can be used to quantitatively describerail irregularities the vertical profile cross level alignmentand gauge [49 50 56] Vertical profile and cross level irreg-ularities chiefly influence the vertical vibrations of vehiclesand of the bridge whereas alignment gauge and crosslevel irregularities initiate horizontal transverse vibrationsof vehicles and the bridge and also the torsional movementof the bridge [57] Track irregularities may be periodic orrandom Random irregularities are due to wear clearancesubsidence and insufficient maintenance For engineeringapplications random irregularities can be approximatelyregarded as stationary and ergodic processes that can begenerated from measured results or simulated by numericalmethods Several numerical methods have been proposedfor the simulation of random rail irregularities such as
6 The Scientific World Journal
the trigonometry series white noise filtration autoregressive(AR) and power spectral density (PSD) sampling methodsAmong these methods the PSD sampling method has beenwidely adopted due to its high computational accuracy Thelateral and vertical irregularities could be all assumed tobe zero-mean stationary Gaussian random processes andexpressed through the inverse Fourier transformation of aPSD function [58]
119910119904(119909) =
119873
sum119896=1
radic2119878 (119891119896) Δ119891 cos (2120587119891
119896119909 + 120579119896) (1)
where 119878(119891) is the PSD function 119891119896= 119891l + (119896 minus 12)Δ119891
Δ119891 = (119891119906minus 119891l)119873 119891119906 and 119891119897 are the upper and lower cutoff
frequencies respectively and 120579119896is the random phase angle
uniformly distributed between 0 and 2120587 Rail irregularity inrailway engineering is commonly represented by a one-sidedPSD function
The PSD functions of rail irregularities have been devel-oped by different countries Based on the PSD functions ofrail irregularities developed by the Research Institute of theChina Railway Administration Zhai [59] expressed all railirregularities using the unified rational formula as follows
where 119891 = 1120578 (mminus1) is the spatial frequency in cyclem (120578 isthe wavelength) and119860 to119866 are the parameters recommendedby Zhai [59] specifically for vertical and lateral rail irregular-ities
214 Solution Methods The dynamic analysis of vehicle-bridge coupled system requires two sets of equations ofmotion for the bridge and vehicles respectively Thesedescribe the interaction or contact forces at the contactpoints of the two subsystems Because the contact pointsmove from time to time the system matrices are generallytime dependent and must be updated and factorized at eachtime step The various solution methods can be generalizedinto two groups according to whether or not an iterativeprocedure is needed at each time step
The first group ofmethods solves the equations of motionof a coupled vehicle-bridge system at each time step withoutiteration This approach has been widely used in coupledvehicle-bridge analysis [51 53 60ndash69] These methods havegood computational stability and are convenient for dealingwith vehicle-bridge interaction problems when the vehiclemodel is relatively simple The main disadvantage is thatthe equations of motion of the coupled system are timedependent and thus the characteristic matrices must bemodified at each time step In addition the equations ofmotion of the coupled vehicle-bridge system become verydifficult to determine if nonlinear wheel-rail contacts andnonlinear vehicle models are considered
The second group of methods solves the equations for thevehicles and bridge separately and requires an iterative pro-cess to obtain convergence for the displacements of the vehi-cles and bridge at all contact points Given that the conditions
of wheel-rail contact geometry and contact forces are rathercomplex a stable integration method adopting a small timeinterval is needed for obtaining the convergence of vehicleand bridge subsystems at the contact points in each time stepMany studies have applied this type of method to investigatevehicle-bridge interactions [70ndash76] The advantage of thesemethods is that the dynamic propertymatrices in the two setsof equations of motion remain constant which is convenientfor the consideration of nonlinear vehicle-bridge interactionsand nonlinear vehicle models [55] However in engineeringapplications the iterative convergence is a critical problemwith this type of method The low convergence rate andoccasional divergence of the solution have also been noted[77] Li et al [55] investigated the performance of theseiterative schemes using the Wilson-120579 method Newmark-120573method and an explicit integrationmethod proposed byZhai[59] and found that the latter gave amuch higher convergencerate than the former two methods
Most of the above methods solved the equations ofmotion of a coupled vehicle-bridge systemusing the nonjumpmodel which assumes that the moving vehicle travelingalong the bridge is always in contact with the rails nomatter what the sign is of the contact forces This is notalways true in view of the physics of the moving vehiclewhich simply sits on the upper surfaces of the rails Theinteraction forces between the moving vehicle and the bridgedepend on the motions of the vehicle the flexibility of thebridge and the track irregularities Li et al [55] utilizeda jump model to solve vehicle-bridge interaction problemusing a noniterative Runge-Kutta method and found thatthe acceleration responses of the car body using the wheel-jump model are smaller than those using the wheel nonjumpmodel when the vehicle speed exceeds 300 kmhr Antolin etal [78] proposed a nonlinear wheel-rail interaction modelwhich considers nonlinear wheel-rail contact forces in theinteraction as well as realistic wheel and rail profiles andapplied it for analysis of dynamic interaction between highspeed trains and bridges
22 Dynamic Interaction between Road Vehicles and BridgeSection 21 gave a detailed literature review of the dynamicinteractions between trains and bridges As there are somefundamental differences between trains and road vehiclesthis section reviews the modeling of road vehicles thesimulation of road vehicle flow and the modeling of roadsurface roughness
221 Modeling of Road Vehicles To analyze the dynamicinteraction between a bridge and running road vehicles amodel of road vehicles must be established A sophisticatedroad vehiclemodel is required tomake the simulation as real-istic as possible A road vehicle is modeled as a combinationof several rigid bodies each of which is connected by a set ofsprings and dashpots which model the elastic and dampingeffects of the tires and suspension systems respectivelyThereare various configurations of road vehicles such as a tractorand trailer with different axle spacing Road vehicle models
The Scientific World Journal 7
L11 L10 L9
L8
Ku5Cu5
Cl5Kl5
Zs5
Ku4Cu4
Cl4Kl4
Zs4
Ku3 Cu3
Cl3Kl3
Zs3
Ku2 Cu2
Cl2Kl2
Zs2
Ku1 Cu1
Cl1Kl1
Zs1
Ku1 Cu1
Cl1Kl1
Zs1
Ku6 Cu6
Cl6Kl6
Zs6
b1 b1
x yL1L2L3L4L5L6L7
Z3 1205793 Z2 Z1
Z1
1205792
1205791
zz
1206011
Figure 5 Dynamic model of a tractor-trailer [79]
zz
x y
h
2b1
ZZY
h1
120601
Kuz1 Kuz3Cuz1
Kuz1 Cuz1 Cuz3
Zs1Zs1
Zs3
Cuy1 Cuy3
Kuy1Kuy3
Ys1 Ys3
Klz1Klz1 Klz3Clz1 Clz3Clz1
Kuz2 Cuz2
Zs2
Klz2 Clz2
120579
L1L2
Kly1
Cly1
Kly3
Cly3
Figure 6 Dynamic model of a high-sided road vehicle [80]
that contain several DOFs have been devised for vehicle-bridge interaction analysis For example Guo and Xu [79]modeled a 17-DOF four-axle heavy tractor-trailer vehicle (seeFigure 5) to investigate the interaction between vehicles anda cable-stayed bridge A total of three DOFs were assignedto rigid bodies representing either the tractor or the trailerto account for vertical rolling and pitching motions Onlyone DOFwas assigned to the rigid body representing the axleset moving in the vertical direction Different vehicle modelsare adopted in wind-vehicle-bridge interaction analyses Xuand Guo [80] modeled a 13-DOF two-axle road vehicle (seeFigure 6) for the dynamic analysis of a coupled road vehicleand bridge system under turbulent wind Five DOFs wereassigned to the vehicle body with respect to its center ofgravity to account for vertical lateral rolling yawing andpitching motions and two DOFs were assigned to the frontand rear axle sets to account for motions in the vertical andlateral directionMoreDOFs are needed to account for lateralcrosswinds
222 Simulation of Road Vehicle Flow On long-span bridgesthere is a high probability of the simultaneous presence ofmultiple road vehicles including heavy trucks This maylead to larger amplitude stress responses and greater fatiguedamage of the local bridge components than would be thecase with only one road vehicle The simulation of roadvehicle flow is thus important in the analysis of the dynamicinteraction between road vehicles and bridges Rather simplepatterns of road vehicle flow have been assumed in most
vehicle-bridge coupled dynamic analyses [79 81 82] in whicheither one or several vehicles are distributed on the bridgein an assumed (usually uniform) pattern Obviously suchassumptions do not represent actual road traffic conditionsRecently Chen and Wu [83] modeled the stochastic trafficload for a long-span bridge based on the cellular automaton(CA) traffic flow simulation technique In this study theysimulated a complicated road vehicle flow on long-spanbridges in terms of vehicle number vehicle type combinationand driver operation characteristics such as lane changingacceleration or deceleration
223 Modeling of Road Surface Roughness Road surfaceroughness is an important factor that greatly affects vehicle-bridge interactions Paultre et al [84] pointed out that roadsurface or pavement roughness can significantly affect theimpact response of a bridge The roughness or surface profiledepends primarily on the workmanship involved in theconstruction of the pavement or roadway and how it is main-tained which although random in nature may contain someinherent frequencies [38] In most cases surface roughnesswhich is three-dimensional in reality is often approximatedby a two-dimensional profile To account for its randomnature the road profile can be modeled as a stationaryGaussian random process and derived using a certain powerspectral density function Other methods similar to this havebeen widely adopted by researchers studying vehicle-inducedbridge vibration [65 70 71 85ndash90] Dodds and Robson [91]developed power spectral density functions that were later
8 The Scientific World Journal
modified and used byWang and Huang [87] and Huang et al[92] This approach was also adopted by literatures [79 81] intheir dynamic analyses of coupled vehicle-bridge and wind-vehicle-bridge systems
23 Dynamic Interaction between Wind and Bridge Whena long-span cable-supported bridge is immersed in a givenflow field the bridge will be subject to mean and fluctuatingwind forces To simulate these forces a linear approximationof the time-averaged static and time-varying buffeting andself-excited force components must be formulated [93 94]As dynamic bridge responses are of concern in this study onlybuffeting and self-excited forces are considered and reviewedin this section
231 Buffeting Forces Buffeting action is a random vibrationcaused by turbulent wind that excites certain modes of vibra-tion across a bridge depending on the spectral distributionof the pressure vectors [95] Although the buffeting responsemay not lead to catastrophic failure it can lead to structuralfatigue and affect the safety of passing vehicles [96] Hencebuffeting analysis has received much attention in recentyears in research into the structural safety of bridges underturbulent wind action [81 95 97ndash102]
By assuming no interaction between buffeting forces andself-excited forces and using quasi-steady aerodynamic forcecoefficients the buffeting forces per unit span Feibf on the 119894thsection of a bridge deck can be expressed as [103]
bf are the buffeting drag lift andmoment respectively 119906
119894and 119908
119894are the horizontal and
vertical components respectively of fluctuating wind at the119894th section 120588 is the air density 119880
119894is the mean wind speed
at the 119894th section of the bridge deck 119861119894and 119871
119894are the width
and length of the 119894th bridge section 119862119863119894 119862119871119894 and 119862
119872119894are
the drag lift and moment coefficients respectively of the119894th bridge segment 1198621015840
119863119894= 119889119862
1198631198941198891205721015840 1198621015840
119871119894= 119889119862
1198711198941198891205721015840 and
1198621015840119872119894
= 1198891198621198721198941198891205721015840 1205721015840 is the angle of attack of a normal
wind incident on the horizontal plane of the deck and 120594119863bu
120594119863bw
120594119871bu
120594119871bw
120594119872bu
and 120594119872bw
are the aerodynamic transferfunctions between the fluctuating wind velocities and thebuffeting forces
It can be found from this equation that a series oftime histories of fluctuating wind velocity 119906
119894 119908119894119879 in the
longitudinal and vertical directions at various points alongthe bridge deck is needed to carry out a detailed buffetinganalysis To simulate the stochastic wind velocity field thefast spectral representation method proposed by Cao et al[104] that is based on the spectral representation methoddeveloped by Shinozuka and Jan [105] is often adopted Thismethod rests on the assumptions that (1) the bridge deck ishorizontal at the same elevation (2) the mean wind speedand wind spectra do not vary along the bridge deck and(3) the distance between any two successive points wherewind speeds are simulated is the same The time historiesof the along-wind component 119906(119905) and the upward windcomponent 119908(119905) at the jth point can be generated using thefollowing equations [104]
119906119895(119905) = radic2 (Δ120596)
119895
sum119898=1
119873119891
sum119896=1
radic119878119906119906(120596119898119896)
times 119866119895119898(120596119898119896) cos (120596
119898119896119905 + 120593119898119896)
(4a)
119908119895(119905) = radic2 (Δ120596)
119895
sum119898=1
119873119891
sum119896=1
radic119878119908119908(120596119898119896)
times 119866119895119898(120596119898119896) cos (120596
119898119896119905 + 120593119898119896)
(4b)
119866119895119898(120596) =
0 when 1 le 119895 lt 119898 le 119899
119862|119895minus119898| when 119897 = 1 119898 le 119895 le 119899
119862|119895minus119898|radic(1 minus 1198622) when 2 le 119898 le 119895 le 119899
(4c)
119862 = exp(minus120582120596119898119896Δ
2120587119880) Δ =
119871
119899119901minus 1
(4d)
120596119898119896= (119896 minus 1) Δ120596 +
119898
119899Δ120596 (119896 = 1 2 119873
119891) (4e)
where Δ120596 = 120596up119873119891 is the frequency interval between thespectral lines119873
119891is the total number of frequency intervals
120596up is the upper cutoff frequency n is the total number ofpoints at which wind speeds are simulated 119878uu and 119878ww arethe along-wind and vertical wind spectrum respectively 120593lmis a random variable that is uniformly distributed between 0and 2120587 L is the span length and 120582 is a parameter that usuallyfalls between 7 and 10
In reality the equivalent buffeting forces in (3) are actuallyassociated with the spatial distribution of the wind pressureson the surface of the bridge deck Ignoring the spatialdistribution or aerodynamic transfer function of the buffetingforces across the cross-section of the bridge deck may havea considerable impact on the accuracy of buffeting response
The Scientific World Journal 9
wi(t) UI + ui(t)
j
j + 1
k
Leibf Lkibf
Meibf
Deibf
Dkibf
ce
Figure 7 Buffeting wind pressures and buffeting forces at nodes [22]
predictions Furthermore the local structural behavior ofthe bridge deck associated with local stresses and strainswhich are prone to causing local damage cannot be predicteddirectly by the current approaches based on equivalentbuffeting forces In this regard Liu et al [22] proposed anapproach to consider the spatial distribution of buffetingforces on a bridge deck structure based on wind pressuredistributions from wind tunnel tests (see Figure 7)
232 Self-Excited Forces In addition to buffeting actionflutter instability caused by self-excited forces induced bywind-structure interactions is an important considerationin the design and construction of long-span suspensionbridges [96] because the additional energy injected into theoscillating structure by the aerodynamic forces increases themagnitude of vibration sometimes to catastrophic levels [95]The self-excited forces on a bridge deck are attributable tothe interactions between wind and the motion of the bridgeWhen the energy of motion extracted from the flow exceedsthe energy dissipated by the system through mechanicaldamping the magnitude of vibration can reach catastrophiclevels [106] Expressing self-excited forces in the form ofindicial functions was first suggested by Scanlan [94] Basedon the assumption that self-excited forces are generated in alinear fashion Lin and Yang [107] simplified the self-excitedforces acting on a bridge deck and expressed them in termsof convolution integrals between the bridge deck motion andthe impulse response functions
119891se119890119863(119905)
=1
21205881198802
int119905
minusinfin
[119868119863ℎ(119905 minus 120591) ℎ
119890(120591) + 119868
119863119902(119905 minus 120591) 119902
119890(120591)
+119861119868119863120579(119905 minus 120591) 120579
119890(120591)
] 119889120591
(5a)
119891se119890119871(119905)
=1
21205881198802
int119905
minusinfin
[119868119871ℎ(119905 minus 120591) ℎ
119890(120591) + 119868
119871119902(119905 minus 120591) 119902
119890(120591)
+119861119868119871120579(119905 minus 120591) 120579
is the impulse function of the self-excited forces in which120595 represents the corresponding force components and heqe and 120579119890 are the equivalent vertical lateral and torsionaldisplacements respectively at the center of elasticity of thebridge deck section The relationship between the aero-dynamic impulse functions and flutter derivatives can beobtained by taking the Fourier transform of (7) [98]
119868119863ℎ(120596) = 119870
2
(119875lowast
6+ 119894119875lowast
5) 119868
119863119902(120596) = 119870
2
(119875lowast
4+ 119894119875lowast
1)
119868119863120579(120596) = 119870
2
(119875lowast
3+ 119894119875lowast
2)
119868119871ℎ(120596) = 119870
2
(119867lowast
4+ 119894119867lowast
1) 119868
119871119902(120596) = 119870
2
(119867lowast
6+ 119894119867lowast
5)
119868119871120579(120596) = 119870
2
(119867lowast
3+ 119894119867lowast
2)
119868119872ℎ(120596) = 119870
2
(119860lowast
4+ 119894119860lowast
1) 119868
119872119902(120596) = 119870
2
(119860lowast
6+ 119894119860lowast
5)
119868119872120579(120596) = 119870
2
(119860lowast
3+ 119894119860lowast
2)
(6)
where the overbars denote the Fourier transform operationthe terms containing 119894 represent imaginary parts119875lowast
120595 119867lowast
120595 and
119860lowast120595(120595 = 1 2 6) are dimensionless flutter derivatives
obtained from wind tunnel tests 119870 = 120596119861119880 is the reducedfrequency and 120596 is the circular frequency of vibration
According to classical airfoil theory the impulse func-tions can reasonably be approximated by a rational function[108]
119868 (120596) = [1198621+ 1198941198622
2120587
]+
119898
sum119897=1
119862119897+2
41205872 + 1198942120587119889119897+2
]1198892119897+2
V2 + 41205872] (7)
where the value of 119898 determines the level of accuracy ofthe approximation 119862
1 1198622 119862119897+2
and 119889119897+2(119897 = 1 2 119898)
are the frequency independent coefficients and ] = 2120587119870
is the reduced mean wind velocity By equating the realand imaginary parts in the comparison of (8) and (9) therelationship between the dimensionless flutter derivativesand the coefficients 119862120595
1 119862120595
2 119862120595
119897+2 and 119889120595
119897+2(where 120595 = Dh
Dq 119863120579 Lh Lq 119871120579 Mh Mq 119872120579 and 119897 = 1 2 119898) canbe establishedThese coefficients are determined by using thenonlinear least-squares method to fit the measured flutter
10 The Scientific World Journal
derivatives at different reduced frequencies The expressionof the aerodynamic impulse functions in the time domaincan be obtained by taking the inverse Fourier transform ofthe impulse functions By substituting the related impulseresponse functions into (5b) the self-excited lift force at the119894th section of bridge deck can then be derived as
119891se119890119871119894
=1
21205881198861198802
119894119861119894119862119871120579
1119894120579119894(119905) + 119862
119871120579
2119894(119861119894
119880119894
) 120579119894(119905)
+ 119862119871120579
3119894(119861119894
119880119894
) 120579119894(119905) +
119898
sum119897=1
119862119871120579
119897+3119894
times int119905
minusinfin
120579119894(119905) exp[minus
119889119871120579119897+3119894119880119894
119861119894
(119905 minus 120591)] 119889120591
+1
21205881198861198802
119894119862119871ℎ
1119894ℎ119894(119905) + 119862
119871ℎ
2119894(119861119894
119880119894
) ℎ119894(119905)
+ 119862119871ℎ
3119894(119861119894
119880119894
) ℎ119894(119905) +
119898
sum119897=1
119862119871ℎ
119897+3119894
times int119905
minusinfin
ℎ119894(119905) exp[minus
119889119871ℎ119897+3119894119880119894
119861119894
(119905 minus 120591)] 119889120591
+1
21205881198861198802
119894
119862119871119902
1119894119902119894(119905) + 119862
119871119902
2119894(119861119894
119880119894
) 119902119894(119905)
+ 119862119871119902
3119894(119861119894
119880119894
) 119902119894(119905) +
119898
sum119897=1
119862119871119902
119897+3119894
times int119905
minusinfin
119902120595(119905) exp[
[
minus119889119871119901
119897+3119894119880119894
119861119894
(119905 minus 120591)]
]
119889120591
(8)
In practice the terms 1198621198711205793119894 119862119871ℎ3119894 and 119862119871119902
3119894 which are related to
additional aerodynamic masses are normally neglected andthe value of 119898 is often taken as 2 [101] Similar formulationsfor self-excited drag and moment can be derived withanalogous definitions The self-excited forces at the 119894th nodeof the bridge deck can thus be expressed as
Feise = EeiXei + GeiXei + Feise (9)
where
Xei =
0
ℎei119902ei120579ei0
0
Eei =1
21205881198802
119894
[[[[[[[[
[
0 0 0 0 0 0
0 119862119871ℎ
1119894119862119871119902
11198941198611198941198621198711205791119894
0 0
0 119862119863ℎ1119894
119862119863119902
11198941198611198941198621198631205791119894
0 0
0 119861119894119862119872ℎ1119894
119861119894119862119872119902
111989411986121198941198621198721205791119894
0 0
0 0 0 0 0 0
0 0 0 0 0 0
]]]]]]]]
]
Gei =1
21205881198802
119894119861119894
[[[[[[[[
[
0 0 0 0 0 0
0 119862119871ℎ
2119894119862119871119902
21198941198611198941198621198711205792119894
0 0
0 119862119863ℎ2119894
119862119863119902
21198941198611198941198621198631205792119894
0 0
0 119861119894119862119872ℎ2119894
119861119894119862119872119902
211989411986121198941198621198721205792119894
0 0
0 0 0 0 0 0
0 0 0 0 0 0
]]]]]]]]
]
Feise =
[[[[[[[
[
0
eise
119863eise
eise0
0
]]]]]]]
]
=
[[[[[[[[[[[[[[[[
[
02
sum119897=1
119862119871119902
l+3119894119881119871119902
l+3119894 +2
sum119897=1
119862119871ℎl+3119894119881119871ℎ
l+3119894 +2
sum119897=1
119862119871120579l+3119894119881119871120579
l+3119894
2
sum119897=1
119862119863119902
l+3119894119881119863119902
l+3119894 +2
sum119897=1
119862119863ℎl+3119894119881119863ℎ
l+3119894 +2
sum119897=1
119862119863120579l+3119894119881119863120579
l+3119894
2
sum119897=1
119862119872119902
l+3119894119881119872119902
l+3119894 +2
sum119897=1
119862119872ℎl+3119894119881119872ℎ
l+3119894 +2
sum119897=1
119862119872120579l+3119894119881119872120579
l+3119894
0
0
]]]]]]]]]]]]]]]]
]
(10)
where 119881120595l+3119894 (120595 = Dh Dq 119863120579 Lh Lq 119871120579 Mh Mq) are theconvolution integrations of the 119894th node and can be calculatedusing a recursive algorithm For example
(11)The self-excited forces expressed by (9) relate to the centerof elasticity of the 119894th deck section Hence the force modelmust be distributed to the nodal points of the section Adistribution based on the rigid body motion relationshipsbetween themotions at the nodal point and those at the centerof elasticity of the deck section [109] was applied by Liu etal [22] In this study by applying the virtual work principlethe self-excited forces at the center of elasticity of the givensection were distributed to all nodes (see Figure 8)
24 Dynamic Interactions in a Wind-Vehicle-Bridge SystemWhen trains and road vehicles are running on long-spanbridges under crosswinds complicated dynamic interac-tions occur among the trains road vehicles cable-supported
The Scientific World Journal 11
Ui h
p
120572
Lsec i
Msec i
Dsec i
Fsekiz
Fsekiy
ce
Figure 8 Self-excited forces at the centre of elasticity and at the nodes in the 119894th deck section [22]
bridge and windThe buffeting response of the bridge due tocrosswind is superimposed on the dynamic response of thebridge due to railway and road vehicles The large vibrationof the bridge will in turn considerably affect the safetyand ride comfort of the drivers of the road vehicles Thusthe dynamic responses of a coupled vehicle-bridge systemunder crosswinds are of great concern to both engineers andresearchers
Detailed reviews of the dynamic interactions betweentrains and bridges between road vehicles and bridges andbetween wind and bridges have been given in the previoussections However the interaction between wind and vehiclesmust also be taken into account in a coupled wind-vehicle-bridge analysis Many studies have investigated wind-vehicleinteractions in the past few decades Balzer [110] developeda theory to estimate the aerodynamic forces on a movingvehicle using Taylorrsquos hypothesis of ldquofrozen turbulencerdquo Forengineering applications Cooper [111] proposed the powerspectral density (PSD) square-root coherence functionphase-lag function and aerodynamic admittance functionto model the unsteady side forces on a moving vehicle andlaid down the foundations for investigating the effects ofwind on a moving vehicle in the frequency domain Bakerdeveloped a theoretical model that describes the dynamics ofvehicles in crosswinds in the time domain [112 113] whichwas later extended to include driver behavior [114] Baker[115 116] further investigated both the steady and unsteadyaerodynamic forces acting on a variety of vehicles and carriedout extensive studies of the interaction between aerodynamicforces and moving vehicles These approaches have all beenapplied in coupled vehicle-bridge analysis For example Xuet al [101] simulated the aerodynamic wind forces actingon running road vehicles using the quasi-steady approachand Xu and Ding [117] derived and simulated the steadyand unsteady aerodynamic forces acting on a moving railwayvehicle in crosswinds in the time domain
Based on these separate studies on the various typesof dynamic interactions among wind vehicles (trains orroad vehicles) and long-span bridges several researchersin the last decade have examined the wind-vehicle-bridgecoupled system as a whole For instance studies have beencarried out on coupled road vehicle and cable-stayed bridgesystems [81 82 118] and on coupled train and cable-supportedbridge systems in crosswinds [101 117 119ndash121] In therecent years several new advances have been made bothin numerical simulation technologies and in wind tunnelmeasurements Chen et al [17] proposed a wind-vehicle-bridge framework which enables considering the dynamiceffects induced by simultaneous actions of railway highway
and wind loading and it was applied to analyze dynamicstress of long suspension bridges Li et al [122] extended thewind-vehicle-bridge couple analysis to the case of two trainsmeeting on a long-span suspension bridge Chen and Wu[118] proposed a semideterministic analytical model whichis able to consider dynamic interactions between the bridgewind and stochastic ldquorealrdquo traffic Based on the wind tunneltests Dorigatti et al [123]measured crosswind loads on high-sided vehicles over long-span bridges taking three differentvehicles (van double deck bus and lorry) and two differentbridge deck configurations into consideration Zhu et al[124] investigated aerodynamic coefficients of road vehiclesby adopting different road vehicles types wind directionsand vehicle positions Li et al [122] studied the effects ofsudden changes of wind loads as the train passing througha bridge tower or two trains passing each other by usingthe wind tunnel test rig with moving train models Hanet al [125] developed an experimental setup for measuringthe aerodynamic characteristics of vehicles and the bridgein wind tunnel and then investigated the influences ofparameters adopted in the tests
3 Applications of Simulation Technology toBridge Assessment
After reviewing the key issues of numerical simulationsfor dynamic response of long-span multiload bridges thissection will review the engineering applications of the newlydeveloped technologies to safety assessment of long-spanbridges such as assessment of fatigue and assessment underextreme events
31 Assessment of FatigueDamage Steel structures are widelyused in long-span bridges Research by the ASCE [126] indi-cates that 80ndash90 of failures in steel structures are related tofatigue and fracture Several disasters resulting from fatigue-induced bridge failure have occurred in history For instance46 people died in the collapse of the Silver Bridge (USA1967) and 32 people lost their lives in the collapse of theSungsoo Grand Bridge (South Korea 1994) These disastersteach us that fatigue is an important aspect of the safety ofsteel bridges and action should be taken to prevent similarfatigue-induced bridge failures In the past few decadesfatigue assessment of steel bridges has attracted increasingattention from both governments and bridge engineers andrelevant provisions have been stipulated in several codes andstandards [127ndash130]
12 The Scientific World Journal
It has great advantages to evaluate fatigue damage of long-span bridges based on numerical simulation especially fora multiload bridge which suffers multiple types of dynamicloading such as railway highway andwind loadingDifferentfrom sudden structural damage fatigue damage accumulateswith load-induced dynamic stress (or stress fluctuation) overthe service life of a bridge lasting for more than 100 yearsThe increase in traffic volume and gross vehicle weight thataccompany economic development is very likely to happenin the long period Numerical simulation technology can bean ideal tool to study influences of traffic growth patternsto fatigue damage of bridge In addition slender long-span bridges built in wind-prone regions also suffer fromconsiderable wind induced vibration which appears withina wide range of wind speeds and lasts for almost the wholedesign life of the bridge Given the simultaneous presenceof multiple vehicles and wind it is necessary to considercombined effects of traffic loading (railway andor highwayloading) and wind loading in the fatigue assessment Sincemultiple loading is concerned in a long time period thereare a large number of loading combinations for multipleloading in different intensities It is almost unavailable forfield measurement to obtain such complete information butnumerical simulation could be a good choice to determinedynamic responses of a long-span bridge under multipleloading
A number of structural health monitoring systems(SHMSs) have been installed on numerous recently builtlong-span bridges and a variety of sensors are used for mon-itoring bridge loadings (eg traffic wind and earthquakes)and conditions (including global and local responses) toensure bridge safety and user comfort under in-service con-ditions Well-known examples include Tsing-Ma Bridge inHong Kong Akashi Kaikyo Bridge in Japan Binzhou YellowRiver Bridge in China and Jindo Bridge in Korea Integrationof numerical simulation technologies and measurement dataof structural health monitoring systems (SHMSs) installedon long-span bridges will make the fatigue assessment morereliable for several reasons (1) it is a perfect validation byusing field measurement data of the different types of loadingas input of numerical simulation and the measured dynamicresponses for comparison with the computed ones (2) a largenumber of measured loading data could be used to establishloading databases or probabilistic models of different loads
In the recent years several researchers [7ndash10] appliedthe newly developed numerical simulation technologies tofatigue assessment of long-span bridges Chen et al [7]proposed a framework for fatigue analysis of a long-spansuspension bridge under railway highway and wind loadingby integrating computer simulation with SHMSs and itwas applied to evaluate fatigue damage of the Tsing MaSuspension Bridge over its design life as a case study Basedon this work Chen et al [8] proposed a framework forfatigue reliability analysis of long suspension bridges undermultiple loading inwhich the probabilisticmodels of railwayhighway and wind loading were established based on themeasurement data acquired from the SHMS of the TsingMa Bridge Wu et al [9] proposed a reliability-based fatigueapproach for slender long-span bridge and the combined
dynamic loading effects from wind and traffic as well asthe associated uncertainties were considered Based on theassumption that dynamic magnification related to vehicledynamics can be neglected in long suspension bridges Chenet al [8] established a framework for fatigue reliability anal-ysis To account for different types of long-span bridges withthe span length ranging from a few hundred to thousands ofmeters Zhang et al [10] proposed a more general frameworkwhich includes multiple random variables for the dynamicloads in a bridgersquos life cycle for the vehicle-bridge-winddynamic system such as road profile vehicle speed andwindvelocity and direction among other effects
32 Assessment under Extreme Events The aforementionedfatigue assessment mainly focuses on damage accumulationinduced by stress fluctuations under normal operationalcondition in a long-term period For long-span bridges inaddition to the normal operational conditions in which windspeeds are small ormoderate and traffic scenarios are normalthere are some extreme event conditions Extreme eventsmayinclude complex traffic congestion on the bridge coupledwith moderate or even strong wind [11] For example severetraffic congestions may be formed on the bridge as a resultof an evacuation or a partial blockage of driving lanes due totraffic accidents construction ormaintenance For hurricaneevacuations there are usually a lot of road vehicles passingthrough the bridge before the landfall of the hurricane whilethe wind speed may become pretty high already [131]
Although the excessive dynamic responses of the bridgesunder extreme events are rare it is also critical since itmay cause critical damage initiation or accumulation onsome local bridgemembers Furthermore the extreme events(eg heavy traffic) may even trigger the collapse of thewhole bridge by breaking the ldquoweakest linkrdquo especially whensome hidden damage or design flaw has not been detectedOne recent example is the Minnesota Bridge failure whichoccurred during rush hours with heavy traffic although trafficloads may not be the direct cause of failure For slender long-span bridges strong wind may also cause threats by workinginteractively with heavy traffic loads Therefore even thoughthe extreme cases associated with congested traffic andorwindy weather may be relatively rare and the durationscould be short it is still important for bridge engineers toappropriately look into these unusual extreme events duringstructural design and life-time management of these criticalinfrastructures [11]
The dynamic performance of long-span bridges undercombined actions of strong winds and running road vehicleshas been studied by many researchers in recent years [17 7981 82 132] Most of them studied bridge dynamic perfor-mance under road traffic inwhich only one or several vehiclesdistributed in an assumed (usually uniform) pattern on long-span bridges were considered Extreme events such as trafficcongestion coupled with strong wind were out of concernin those studies Recently Wu and Chen [11] conducteda research on the assessment of long-span bridges underextreme events which includes complex traffic congestioncoupled with moderate or even strong wind This study
The Scientific World Journal 13
applied the cellular automaton (CA) traffic model to thesimulation of the actual traffic flow through the bridgedefined representative scenarios for the extreme events andnumerically studied the bridge performance under thesepossible extreme events
4 Conclusions and Recommendations
Dynamic responses of long-span bridges are often requiredfor assessing the safety of these bridges and can be determinedby numerical simulation technologies This paper provides adetailed review of key issues involved in dynamic responseanalysis of long-span multiload bridges based on numericalsimulation including dynamic interactions between runningtrains and bridge between running road vehicles and bridgeand between wind and bridge and in the wind-vehicle-bridge coupled systemThen the review work was conductedfor engineering applications of newly developed numericalsimulation technologies to safety assessment of long-spanbridges such as assessment of fatigue damage and assessmentunder extreme event condition Although technologies fornumerical simulation of dynamic responses of long-spanmultiload bridge have achieved great advances in past fewdecades and successfully applied to several important bridgesit is still far from reach its maturity and enable to takeplace of traditional fieldmeasurementThe existing problemsand promising research efforts at least include the followingaspects
(1) After multiple types of dynamic interactions beingconsidered in the complex system computationalefficiency is a bottleneck problem for numericalsimulation of dynamic response of a long-span bridgeTypically when multiple loads are involved a largenumber of loading combinations for multiple load-ings must be considered in the assessment
(2) It is rather complex for the time-depending windloads acting on a long-span bridge and running vehi-cles especially for the case of rapid change of windloads such as a train passing through a bridge toweror two trains passing each other The aerodynamiccharacteristics of vehicles and the bridge under differ-ent loading scenarios can be determined through thewind tunnel testing and used in numerical simulationof dynamic responses of the bridge and vehicles
(3) It is a new trend to integrate numerical simulationtechnologies and measurement data of structuralhealth monitoring systems (SHMSs) installed onlong-span bridges whichmakes the safety assessmentof bridge structures more reliable Measured struc-tural responses could be used to validate numericalsimulation approach and measured loading infor-mation could be used for generating statistical orprobabilistic models of multiple loads
(4) It is important to study dynamic responses of bridgestructures under extreme events such as congestedtraffic coupled with windy weather which happens ina long-span bridge For the assessment under extreme
events using numerical simulation technologies sim-ulation of traffic flow and definition of representativescenarios of the extreme events are key issues
(5) It is necessary to consider the effects of typhoonwinds on the safety assessment of long-span bridgesin a reasonable way Few researches do this mostlybecause a probabilistic distribution of wind speedand direction specifically for typhoons is requiredfor assessment but there are insufficient measuredrecords to establish a reliable probabilistic typhoonwind model
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors wish to acknowledge the financial supportsfrom the National Natural Science Foundation of China(NSFC-51108395 and NSFC-51178366) the FundamentalResearch Funds for theCentral Universities (2012121032) andopen funding from Jiangsu Key Laboratory of EngineeringMechanics Special thanks go to the supervisor of the firstauthor Professor Y L XuHongKong PolytechnicUniversityfor his expert guidance and continuous support at all levelsthroughout his PhD study Sincere thanks should go tothe Highways Department of Hong Kong for providing theauthors with the field measurement data Any opinions andconcluding remarks presented in this paper are entirely thoseof the authors
References
[1] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[2] H Sohn C R Farrar N F Hunter and K Worden ldquoStructuralhealth monitoring using statistical pattern recognition tech-niquesrdquo Journal of Dynamic Systems Measurement and Controlvol 123 no 4 pp 706ndash711 2001
[3] W Fan and P Qiao ldquoVibration-based damage identificationmethods a review and comparative studyrdquo Structural HealthMonitoring vol 10 no 1 pp 83ndash111 2011
[4] X Q Zhu and S S Law ldquoDamage detection in simply supportedconcrete bridge structure under moving vehicular loadsrdquo Jour-nal of Vibration and Acoustics Transactions of the ASME vol129 no 1 pp 58ndash65 2007
[5] J Li and S S Law ldquoDamage identification of a target sub-structure with moving load excitationrdquoMechanical Systems andSignal Processing vol 30 pp 78ndash90 2012
[6] J Li S S Law and H Hao ldquoImproved damage identification inbridge structures subject tomoving loads numerical and exper-imental studiesrdquo International Journal of Mechanical Sciencesvol 74 pp 99ndash111 2013
14 The Scientific World Journal
[7] Z W Chen Y L Xu Y Xia Q Li and K Y Wong ldquoFatigueanalysis of long-span suspension bridges under multiple load-ing case studyrdquo Engineering Structures vol 33 no 12 pp 3246ndash3256 2011
[8] Z W Chen Y L Xu and X M Wang ldquoSHMS-based fatiguereliability analysis of multiloading suspension bridgesrdquo Journalof Structural Engineering-Asce vol 138 pp 299ndash307 2012
[9] J Wu S R Chen and J W van de Lindt ldquoFatigue assessmentof slender long-span bridges reliability approachrdquo Journal ofBridge Engineering vol 17 no 1 pp 47ndash57 2012
[10] W Zhang C S Cai and F Pan ldquoFatigue reliability assessmentfor long-span bridges under combined dynamic loads fromwinds and vehiclesrdquo Journal of Bridge Engineering vol 18 pp735ndash747 2013
[11] JWu and S R Chen ldquoProbabilistic dynamic behavior of a long-span bridge under extreme eventsrdquo Engineering Structures vol33 no 5 pp 1657ndash1665 2011
[12] J M Ko and Y Q Ni ldquoTechnology developments in structuralhealth monitoring of large-scale bridgesrdquo Engineering Struc-tures vol 27 no 12 pp 1715ndash1725 2005
[13] TH YiHN Li andHM Sun ldquoMulti-stage structural damagediagnosis method based on ldquoenergy-damagerdquo theoryrdquo SmartStructures and Systems vol 12 pp 345ndash361 2013
[14] T H Yi H N Li and M Gu ldquoFull-scale measurements ofdynamic response of suspension bridge subjected to environ-mental loads using GPS technologyrdquo Science China Technologi-cal Sciences vol 53 no 2 pp 469ndash479 2010
[15] T H Yi H N Li and M Gu ldquoExperimental assessmentof high-rate GPS receivers for deformation monitoring ofbridgerdquoMeasurement Journal of the InternationalMeasurementConfederation vol 46 pp 420ndash432 2013
[16] X W Ye Y Q Ni K Y Wong and J M Ko ldquoStatistical analysisof stress spectra for fatigue life assessment of steel bridges withstructural health monitoring datardquo Engineering Structures vol45 pp 166ndash176 2012
[17] Z W Chen Y L Xu Q Li and D J Wu ldquoDynamic stressanalysis of long suspension bridges under wind railway andhighway loadingsrdquo Journal of Bridge Engineering vol 16 no 3pp 383ndash391 2011
[18] S G Meisenholder and P Weidlinger ldquoDynamic interactionaspects of cable-stayed guideways for high speed ground trans-portationrdquoAmerican Society ofMechanical Engineers no 74 pp180ndash192 1974
[19] Q H Mao Research on the Highway Bridge Vibration Due toMoving Vehicles Tongji University Shang Hai China 1989
[20] Y L Xu J M Ko and Z Yu ldquoModal analysis of tower-cable system of Tsing Ma long suspension bridgerdquo EngineeringStructures vol 19 pp 857ndash867 1997
[21] W Guo H Xia and Y-L Xu ldquoDynamic response of a long spansuspension bridge and running safety of a train under windactionrdquo Frontiers of Architecture and Civil Engineering in Chinavol 1 no 1 pp 71ndash79 2007
[22] T T Liu Y L Xu W S Zhang K Y Wong H J Zhou and KW Y Chan ldquoBuffeting-induced stresses in a long suspensionbridge structural health monitoring oriented stress analysisrdquoWind and Structures An International Journal vol 12 no 6 pp479ndash504 2009
[23] K Y Wong ldquoStructural identification of Tsing Ma BridgerdquoTransactions Hong Kong Institution of Engineers vol 10 no 1pp 38ndash47 2003
[24] Y L XuQ Li D JWu and ZWChen ldquoStress and accelerationanalysis of coupled vehicle and long-span bridge systems usingthe mode superposition methodrdquo Engineering Structures vol32 no 5 pp 1356ndash1368 2010
[25] Y F Duan Y L Xu Q G Fei et al ldquoAdvanced finite elementmodel of Tsing Ma Bridge for structural health monitoringrdquoInternational Journal of Structural Stability and Dynamics vol11 no 2 pp 313ndash344 2011
[26] Z X Li T Q Zhou T H T Chan and Y Yu ldquoMulti-scalenumerical analysis on dynamic response and local damage inlong-span bridgesrdquo Engineering Structures vol 29 no 7 pp1507ndash1524 2007
[27] W Zhang C S Cai and F Pan ldquoFinite element modeling ofbridges with equivalent orthotropic material method for multi-scale dynamic loadsrdquo Engineering Structures vol 54 pp 82ndash932013
[28] S P Timoshenko ldquoOn the forced vibrations of bridgesrdquo Philo-sophical Magazine vol 6 no 257 pp 1018ndash1019 1922
[29] R S Ayre G Ford and L S Jacobsen ldquoTransverse vibration ofa two-span beam under the action of a moving constant forcerdquoJournal of Applied Mechanics vol 17 pp 1ndash12 1950
[30] R S Ayre andLS Jacobsen ldquoTransverse vibration of a two-spanbeam under the action of a moving alternating forcerdquo Journal ofApplied Mechanics vol 17 pp 283ndash290 1950
[31] L FrybaVibration of Solids and Structures underMoving LoadsASCE Press 1972
[32] J-S Wu and C-W Dai ldquoDynamic response of multispannonuniform beam due to moving loadsrdquo Journal of StructuralEngineering vol 113 no 3 pp 458ndash474 1987
[33] W Weaver S P Timoshenko and D H Young VibrationProblems in Engineering John Wiley amp Sons 1990
[34] N H Galdos D R Schelling and M A Sahin ldquoMethodologyfor impact factor of horizontally curved box bridgesrdquo Journal ofStructural Engineering vol 119 no 6 pp 1917ndash1934 1993
[35] J A Gbadeyan and S T Oni ldquoDynamic behaviour of beamsand rectangular plates under moving loadsrdquo Journal of Soundand Vibration vol 182 no 5 pp 677ndash695 1995
[36] D Y Zheng Y K Cheung F T K Au and Y S ChengldquoVibration of multi-span non-uniform beams under movingloads by using modified beam vibration functionsrdquo Journal ofSound and Vibration vol 212 no 3 pp 455ndash467 1998
[37] G V Rao ldquoLinear dynamics of an elastic beam under movingloadsrdquo Journal of Vibration and Acoustics Transactions of theASME vol 122 no 3 pp 281ndash289 2000
[38] Y B Yang J D Yau and Y S Wu Vehicle-Bridge InteractionDynamic with Applications to High-Speed Railways WorldScientific 2004
[39] E C Ting J Genin and J H Ginsberg ldquoA general algorithm formovingmass problemsrdquo Journal of Sound and Vibration vol 33no 1 pp 49ndash58 1974
[40] S Sadiku and H H E Leipholz ldquoOn the dynamics of elasticsystems with moving concentrated massesrdquo Ingenieur-Archivvol 57 no 3 pp 223ndash242 1987
[41] J E Akin and M Mofid ldquoNumerical solution for response ofbeamswithmovingmassrdquo Journal of Structural Engineering vol115 no 1 pp 120ndash131 1989
[42] M AMahmoud andM A Abou Zaid ldquoDynamic response of abeam with a crack subject to a moving massrdquo Journal of Soundand Vibration vol 256 no 4 pp 591ndash603 2002
[43] V K Garg Dynamics of Railway Vehicle Systems AcademicPress 1994
The Scientific World Journal 15
[44] T-LWang andDHuang ldquoCable-stayed bridge vibration due toroad surface roughnessrdquo Journal of Structural Engineering vol118 no 5 pp 1354ndash1374 1992
[45] Y-B Yang and B-H Lin ldquoVehicle-bridge interaction analysisby dynamic condensation methodrdquo Journal of Structural Engi-neering vol 121 no 11 pp 1636ndash1643 1995
[46] Y B Yang J D Yau and L C Hsu ldquoVibration of simple beamsdue to trainsmoving at high speedsrdquoEngineering Structures vol19 no 11 pp 936ndash943 1997
[47] B Tabarrok and E Esmailzadeh ldquoInduced vibration of bridgestransversed by moving vehiclesrdquo Transactions of the CanadianSociety for Mechanical Engineering B vol 24 no 1 pp 191ndash1982000
[48] C Liu T-L Wang and D Huang ldquoImpact study for multi-girder bridge based on correlated road roughnessrdquo StructuralEngineering and Mechanics vol 11 no 3 pp 259ndash272 2001
[49] K H Chu V K Garg and T L Wang ldquoImpact in railwayprestressed concrete bridgesrdquo Journal of Structural Engineeringvol 112 no 5 pp 1036ndash1051 1986
[50] T-LWang V K Garg and K-H Chu ldquoRailway bridgevehicleinteraction studies with new vehicle modelrdquo Journal of Struc-tural Engineering vol 117 no 7 pp 2099ndash2116 1991
[51] H Xia Y L Xu and T H T Chan ldquoDynamic interaction oflong suspension bridges with running trainsrdquo Journal of Soundand Vibration vol 237 no 2 pp 263ndash280 2000
[52] Q-L Zhang A Vrouwenvelder and J Wardenier ldquoNumericalsimulation of train-bridge interactive dynamicsrdquo Computersand Structures vol 79 no 10 pp 1059ndash1075 2001
[53] H Xia N Zhang and G de Roeck ldquoDynamic analysis of highspeed railway bridge under articulated trainsrdquo Computers andStructures vol 81 no 26-27 pp 2467ndash2478 2003
[54] G Diana F Cheli A Collina R Corradi and S MelzildquoThe development of a numerical model for railway vehiclescomfort assessment through comparison with experimentalmeasurementsrdquoVehicle SystemDynamics vol 38 no 3 pp 165ndash183 2002
[55] Q Li Y L Xu D J Wu and Z W Chen ldquoComputer-aided nonlinear vehicle-bridge interaction analysisrdquo Journal ofVibration and Control vol 16 pp 1791ndash1816 2010
[56] A Wiriyachai K H Chu and V K Garg ldquoBridge impact dueto wheel and track irregularitiesrdquo Journal of the EngineeringMechanics Division vol 108 no 4 pp 648ndash666 1982
[57] L Fryba Dynamics of Railway Bridges Inst of Civil Engineers1996
[58] D Huang and T-L Wang ldquoImpact analysis of cable-stayedbridgesrdquo Computers and Structures vol 43 no 5 pp 897ndash9081992
[59] W M Zhai Vehicle-Track Coupling Dynamics Chinese RailwayPress Beijing China 2007
[60] M Olsson ldquoFinite element modal co-ordinate analysis ofstructures subjected to moving loadsrdquo Journal of Sound andVibration vol 99 no 1 pp 1ndash12 1985
[61] Y-B Yang C-H Chang and J-D Yau ldquoAn element foranalysing vehicle-bridge systems considering vehiclersquos pitchingeffectrdquo International Journal for NumericalMethods in Engineer-ing vol 46 no 7 pp 1031ndash1047 1999
[62] Y B Yang and Y S Wu ldquoA versatile element for analyzingvehicle-bridge interaction responserdquo Engineering Structuresvol 23 no 5 pp 452ndash469 2001
[63] F T K Au J J Wang and Y K Cheung ldquoImpact study ofcable-stayed bridge under railway traffic using various modelsrdquo
Journal of Sound and Vibration vol 240 no 3 pp 447ndash4652001
[64] Y Q Sun andMDhanasekar ldquoA dynamicmodel for the verticalinteraction of the rail track and wagon systemrdquo InternationalJournal of Solids and Structures vol 39 no 5 pp 1337ndash13592002
[65] K Henchi M Fafard M Talbot and G Dhatt ldquoAn efficientalgorithm for dynamic analysis of bridges under moving vehi-cles using a coupledmodal and physical components approachrdquoJournal of Sound and Vibration vol 212 no 4 pp 663ndash6831998
[66] Y L Xu and L YWang ldquoAnalytical study of wind-rain-inducedcable vibration SDOFmodelrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 91 no 1-2 pp 27ndash40 2003
[67] B Biondi G Muscolino and A Sofi ldquoA substructure approachfor the dynamic analysis of train-track-bridge systemrdquoComput-ers and Structures vol 83 no 28ndash30 pp 2271ndash2281 2005
[68] J L Humar and A H Kashif ldquoDynamic response analysis ofslab-type bridgesrdquo Journal of Structural Engineering vol 121 no1 pp 48ndash62 1995
[69] P Lou and Q-Y Zeng ldquoFormulation of equations of verticalmotion for vehicle-track-bridge systemrdquo Journal of the ChinaRailway Society vol 26 no 5 p 71 2004
[70] O Coussy M Said and J-P van Hoove ldquoThe influence ofrandom surface irregularities on the dynamic response ofbridges under suspended moving loadsrdquo Journal of Sound andVibration vol 130 no 2 pp 313ndash320 1989
[71] E S Hwang and A S Nowak ldquoSimulation of dynamic loadfor bridgesrdquo Journal of Structural Engineering vol 117 pp 1413ndash1434 1991
[72] F Yang and G A Fonder ldquoAn iterative solution methodfor dynamic response of bridge-vehicles systemsrdquo EarthquakeEngineering and Structural Dynamics vol 25 pp 195ndash215 1996
[73] W Zhai and Z Cai ldquoDynamic interaction between a lumpedmass vehicle and a discretely supported continuous rail trackrdquoComputers and Structures vol 63 no 5 pp 987ndash997 1997
[74] W M Zhai and C B Cai ldquoTraintrackbridge dynamic inter-actions simulation and applicationsrdquo Vehicle System Dynamicsvol 37 pp 653ndash665 2003
[75] D Bruno F Greco and P Lonetti ldquoDynamic impact analysis oflong span cable-stayed bridges under moving loadsrdquo Engineer-ing Structures vol 30 no 4 pp 1160ndash1177 2008
[76] X D Song D J Wu and Q Li ldquoDynamic impact analysisof double-tower cable-stayed maglev bridges using a simplemodelrdquo Journal of Bridge Engineering vol 19 pp 34ndash43 2014
[77] Y-S Wu and Y-B Yang ldquoSteady-state response and ridingcomfort of trains moving over a series of simply supportedbridgesrdquoEngineering Structures vol 25 no 2 pp 251ndash265 2003
[78] P Antolin N Zhang J M Goicolea H Xia M A Astiz andJ Oliva ldquoConsideration of nonlinear wheel-rail contact forcesfor dynamic vehicle-bridge interaction in high-speed railwaysrdquoJournal of Sound and Vibration vol 332 no 5 pp 1231ndash12512013
[79] W H Guo and Y L Xu ldquoFully computerized approach to studycable-stayed bridge-vehicle interactionrdquo Journal of Sound andVibration vol 248 no 4 pp 745ndash761 2001
[80] Y L Xu and W H Guo ldquoDynamic behaviour of high-sidedroad vehicles subject to a sudden crosswind gustrdquo Wind andStructures vol 6 no 5 pp 325ndash346 2003
[81] Y L Xu and W H Guo ldquoDynamic analysis of coupled roadvehicle and cable-stayed bridge systems under turbulent windrdquoEngineering Structures vol 25 no 4 pp 473ndash486 2003
16 The Scientific World Journal
[82] C S Cai and S R Chen ldquoFramework of vehicle-bridge-winddynamic analysisrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 92 no 7-8 pp 579ndash607 2004
[83] S R Chen and J Wu ldquoModeling stochastic live load for long-span bridge based on microscopic traffic flow simulationrdquoComputers and Structures vol 89 no 9-10 pp 813ndash824 2011
[84] A V Paultre B Yang L A Bergman and C A Tan ldquoBridgedynamics and dynamic amplification factorsmdasha review ofanalytical and experimental findingsrdquoCanadian Journal of CivilEngineering vol 19 no 2 pp 260ndash278 1992
[85] H Honda Y Kajikawa and T Kobori ldquoSpectra of road surfaceroughness of bridgesrdquo Journal of the Structural Division vol 108pp 1956ndash1966 1982
[86] M J Inbanathan and M Wieland ldquoBridge vibrations dueto vehicle moving over rough surfacerdquo Journal of StructuralEngineering vol 113 no 9 pp 1994ndash2008 1987
[87] T-LWang andDHuang ldquoCable-stayed bridge vibration due toroad surface roughnessrdquo Journal of Structural Engineering vol118 no 5 pp 1354ndash1374 1992
[88] P K Chatterjee T K Datta and C S Surana ldquoVibrationsuspension bridges under vehicular movementrdquo Journal ofStructural Engineering vol 120 no 3 pp 681ndash703 1994
[89] D Chang and H Lee ldquoImpact factors for simple-span highwaygirder bridgesrdquo Journal of Structural Engineering vol 120 no 3pp 704ndash715 1994
[90] T-C Pan and J Li ldquoDynamic vehicle element method for tran-sient response of coupled vehicle-structure systemsrdquo Journal ofStructural Engineering vol 128 no 2 pp 214ndash223 2002
[91] C J Dodds and J D Robson ldquoThe description of road surfaceroughnessrdquo Journal of Sound Vibration vol 31 no 2 pp 175ndash183 1973
[92] D Huang T-L Wang and M Shahawy ldquoImpact studies ofmultigirder concrete bridgesrdquo Journal of Structural Engineeringvol 119 no 8 pp 2387ndash2402 1993
[93] A G Davenport ldquoBuffeting of a suspension bridge by stormwindrdquo Journal of Structural Division vol 88 pp 233ndash268 1962
[94] R H Scanlan ldquoThe action of flexible bridges under wind Iflutter theoryrdquo Journal of Sound and Vibration vol 60 no 2pp 187ndash199 1978
[95] Q Ding and P K K Lee ldquoComputer simulation of buffetingactions of suspension bridges under turbulentwindrdquoComputersand Structures vol 76 no 6 pp 787ndash797 2000
[96] V Boonyapinyo T Miyata and H Yamada ldquoAdvanced aerody-namic analysis of suspension bridges by state-space approachrdquoJournal of Structural Engineering vol 125 no 12 pp 1357ndash13661999
[97] Y-H Chen and C-Y Li ldquoDynamic response of elevated high-speed railwayrdquo Journal of Bridge Engineering vol 5 no 2 pp124ndash130 2000
[98] X Chen M Matsumoto and A Kareem ldquoTime domainflutter and buffeting response analysis of bridgesrdquo Journal ofEngineering Mechanics vol 126 no 1 pp 7ndash16 2000
[99] X Chen and A Kareem ldquoEquivalent static wind loads forbuffeting response of bridgesrdquo Journal of Structural Engineeringvol 127 no 12 pp 1467ndash1475 2001
[100] S R Chen and C S Cai ldquoEvolution of long-span bridgeresponse to wind-numerical simulation and discussionrdquo Com-puters and Structures vol 81 no 21 pp 2055ndash2066 2003
[101] Y L XuHXia andQ S Yan ldquoDynamic response of suspensionbridge to high wind and running trainrdquo Journal of BridgeEngineering vol 8 no 1 pp 46ndash55 2003
[102] A Guo Y L Xu and H Li ldquoDynamic performance of cable-stayed bridge tower with multi-stage pendulum mass damperunderwind excitations-II experimentrdquoEarthquake Engineeringand Engineering Vibration vol 6 no 4 pp 417ndash424 2007
[103] E Simiu and R H ScanlanWind Effects on Structures 1996[104] Y Cao H Xiang and Y Zhou ldquoSimulation of stochastic wind
velocity field on long-span bridgesrdquo Journal of EngineeringMechanics vol 126 no 1 pp 1ndash6 2000
[105] M Shinozuka and C-M Jan ldquoDigital simulation of randomprocesses and its applicationsrdquo Journal of Sound and Vibrationvol 25 no 1 pp 111ndash128 1972
[106] K M Shum Lateral and torsional vibration control of long spanbridge deck using novel tuned liquid column dampers [PhDthesis] Department of Civil and Structural Engineering TheHong Kong Polytechnic University 2004
[107] Y K Lin and J N Yang ldquoMultimode bridge response to windexcitationsrdquo Journal of EngineeringMechanics vol 109 no 2 pp586ndash603 1983
[108] Y S Lin Self-Excited Bridge Motion in Turbulent Wind 1978[109] D T Lau M S Cheung and S H Cheng ldquo3D flutter analysis
of bridges by spline finite-strip methodrdquo Journal of StructuralEngineering vol 126 no 10 pp 1246ndash1254 2000
[110] L A Balzer ldquoAtmospheric turbulence encountered by high-speed ground transport vehiclesrdquo Journal of Mechanical Engi-neering Science vol 19 pp 227ndash235 1977
[111] R K Cooper ldquoAtmospheric turbulence with respect to movingground vehiclesrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 17 no 2 pp 215ndash238 1984
[112] C J Baker ldquoA simplified analysis of various types of wind-induced road vehicle accidentsrdquo Journal of Wind Engineeringand Industrial Aerodynamics vol 22 no 1 pp 69ndash85 1986
[113] C J Baker ldquoMeasures to control vehicle movement at exposedsites during windy periodsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 25 no 2 pp 151ndash161 1987
[114] C J Baker ldquoHigh sided articulated road vehicles in strong crosswindsrdquo Journal of Wind Engineering and Industrial Aerodynam-ics vol 31 no 1 pp 67ndash85 1988
[115] C J Baker ldquoGround vehicles in high cross winds part I steadyaerodynamic forcesrdquo Journal of Fluids and Structures vol 5 no1 pp 69ndash90 1991
[116] C J Baker ldquoGround vehicles in high cross winds part IIunsteady aerodynamic forcesrdquo Journal of Fluids and Structuresvol 5 no 1 pp 91ndash111 1991
[117] Y L Xu and Q S Ding ldquoInteraction of railway vehicles withtrack in cross-windsrdquo Journal of Fluids and Structures vol 22no 3 pp 295ndash314 2006
[118] S R Chen and J Wu ldquoDynamic performance simulation oflong-span bridge under combined loads of stochastic traffic andwindrdquo Journal of Bridge Engineering vol 15 no 3 pp 219ndash2302010
[119] Y Li S Qiang H Liao and Y L Xu ldquoDynamics of wind-rail vehicle-bridge systemsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 93 no 6 pp 483ndash507 2005
[120] Y L Xu N Zhang and H Xia ldquoVibration of coupled trainand cable-stayed bridge systems in cross windsrdquo EngineeringStructures vol 26 no 10 pp 1389ndash1406 2004
[121] W W Guo Y L Xu H Xia W S Zhang and K M ShumldquoDynamic response of suspension bridge to typhoon and trainsII numerical resultsrdquo Journal of Structural Engineering vol 133no 1 pp 12ndash21 2007
The Scientific World Journal 17
[122] Y L Li H Y Xiang B Wang Y L Xu and S Z QiangldquoDynamic analysis of wind-vehicle-bridge coupling systemduring the meeting of two trainsrdquo Advances in StructuralEngineering vol 16 pp 1663ndash1670 2013
[123] F Dorigatti M Sterling D Rocchi et al ldquoWind tunnelmeasurements of crosswind loads on high sided vehicles overlong span bridgesrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 107 pp 214ndash224 2012
[124] L D Zhu L Li Y L Xu and Q Zhu ldquoWind tunnel investi-gations of aerodynamic coefficients of road vehicles on bridgedeckrdquo Journal of Fluids and Structures vol 30 pp 35ndash50 2012
[125] Y Han J X Hu C S Cai Z Q Chen and C G LildquoExperimental and numerical studies of aerodynamic forces onvehicles and bridgesrdquoWind and Structures vol 17 pp 163ndash1842013
[126] ASCE ldquoCommittee on fatigue and fracture reliability of thecommittee on structural safety and reliability of the structuraldivision fatigue reliability 1ndash4rdquo Journal of Structural Engineer-ing vol 108 pp 3ndash88 1982
[127] BS ldquoBS5400 part 10 code of practice for fatiguerdquo BritishStandard Institute 1980
[128] BS ldquoBS7608 code of practice for fatigue design and assessmentof steel structuresrdquo British Standard Institute 1993
[129] AASHTOGuide Specifications for Fatigue Evaluation of ExistingSteel Bridges 1990
[130] AASHTO Guide Manual for Condition Evaluation and Loadand Resistance Factor Rating (LRFR) of Highway Bridges 2003
[131] S R Chen C S Cai and B Wolshon ldquoFrom normal operationto evacuation Single-vehicle safety under adverse weathertopographic and operational conditionsrdquo Natural HazardsReview vol 10 no 2 pp 68ndash76 2009
[132] S R Chen and C S Cai ldquoEquivalent wheel load approach forslender cable-stayed bridge fatigue assessment under traffic andwind feasibility studyrdquo Journal of Bridge Engineering vol 12 no6 pp 755ndash764 2007
traditional live load (such as highway railway and windloading) or accidental live load (such as ship impact andearthquake) Structural intrinsic characteristics could beextracted from these dynamic responses (or vibration signals)to develop all sorts of vibration-based damage detection tech-niques A well-known family of them is based on structuraldynamic characteristics (such as frequencies mode shapesdamping ratios and strainmode shapes) and their derivatives[1ndash3] Some damage identification approaches were proposedbased on the dynamic responses of bridge structures undermoving vehicle loads [4ndash6] The dynamic responses of long-span bridges also could be used for structural assessmentfor example fatigue assessment at the critical locations overthe service history of the bridge [7ndash10] and assessment ofextreme events such as complex traffic congestion coupledwith moderate or even strong wind [11]
Over the past decades on-structure long-term structuralhealth monitoring systems (SHMSs) have been implementedon long-span bridges in Europe the United States CanadaJapan Korea China and other countries [12] They areinstalled in newly constructed bridges and existing bridgesfor monitoring structural behavior in real time evaluatingstructural performance under various loads and identifyingstructural damage or deterioration [13] To comprehen-sively understand the bridge performance dynamic bridgeresponses are important monitoring items of structuralhealth monitoring Global responses (such as displacement)are measured by GPS and accelerometers [14 15] and localresponses (such as strainstress) are normallymeasured in thecritical bridge components andwidely used for fatigue assess-ment [16] Although dynamic responses have been measuredfor those bridges installed with SHMSs condition evaluation
based on measurement still has some limitations (1) it isdifficult to identify all of the local critical locations and evenso it is uneconomical to monitor all critical locations in longterm (2) not every fatigue-critical location is suitable forsensor installation (3) it is difficult to obtain measurementdata in the extreme events (such as combination of trafficcongestion and strong wind) which rarely happen (4) itis hard to exactly predict the influence of possible trafficgrowths based on field measurement only Integrating withnumerical simulation technologies and field measurementsis an alternative approach which is able to overcome thelimitations of evaluation approaches only based on measure-ments The information on the concerned dynamic loadingsmeasured by the SHMS could be taken as inputs for thenumerical simulation and the computed dynamic responsescould be compared with the measured ones in the validation[17]
However numerical simulation of dynamic response ofa long-span multiload bridge is not an easy job becauseit requires a complex dynamic finite element model of thebridge including all important bridge components variousdynamic loading models for running trains running roadvehicles and high winds and interactive models betweenthe bridge and wind bridge and trains and bridge androad vehicles [17] This paper focuses on recent research andapplications of numerical simulation technology for dynamicresponse of long-span multiload bridges Firstly key issuesinvolved in dynamic response analysis of long-spanmultiloadbridges based on numerical simulation technologies arereviewed in Section 2 The applications of newly developednumerical simulation technologies to safety assessment oflong-span bridges are subsequently reviewed in Section 3
The Scientific World Journal 3
y
xz Railway track
Bridge deck
Main cross-frame
(a)
Railway tracksOrthotropicdeck-plates (top)
Cross bracings
Cross bracings
Cross bracings(top centre)
Cross-frames(main and intermediate)
Longitudinaltrusses
(bottom outer)
Orthotropic deck-plates (bottom)
Corrugated sheets
bottom centre
(b)
Figure 2 Finite element model of suspended deck module (a) hybrid 3D bridge model [22] (b) full 3D model [25]
Finally the existing problems and promising research effortsfor the numerical simulation technologies and their appli-cations to assessment of long-span multiload bridges areexplored in Section 4
For themost complex situation a long-spanmultiload bridgewhich is located at a wind-prone region carries both railwayand highway traffic and thus the combined effect of runningtrains running road vehicles and wind is acting on thebridge Several key issues are involved in this complicatedsituation such as dynamic interaction between runningtrains and bridge dynamic interaction between running roadvehicles and bridge and dynamic interaction between windand bridge To give a comprehensive review the above threekey issues will be individually reviewed in Sections 21 to23 and then the dynamic interactions of wind-vehicle-bridgesystem as a whole are then reviewed in Section 24
21 Dynamic Interaction between Trains and Bridge
211 Modeling of a Cable-Supported Bridge In early researchin this area simplified bridge models were employed tostudy vehicle-bridge interactions For example a cable-stayedbridge was simulated as a beam resting on an elastic founda-tion by Meisenholder and Weidlinger [18] for the dynamicanalysis of cable-stayed guideways subject to track-levitatedvehicles moving at high speeds Mao [19] investigated theimpact factor of a cable-stayed bridge which was assumedto be formed of continuous elastic beams supported byintermediate elastic supports
More recently with the development of finite element(FE) technology it has become common practice to use acomputer software package to establish a finite elementmodel(FEM) of a cable-supported bridge This technology estab-lishes an accurate bridge model that takes into account thegeometric nonlinear behavior of a cable-supported bridgeTo make the bridge model close to the realistic bridgein terms of its dynamic properties the modal frequencies
and shapes determined by dynamic tests are used for fur-ther model validation or updating Many FEMs of cable-supported bridges have been established for the analysis oftrain-bridge interactions The Tsing Ma Suspension Bridgein Hong Kong can be used as an example to illustratethe various bridge models that have been established foranalysis The first generation of Tsing Ma Bridge model wasa spinal beam model [20] in which the hybrid steel deck wasrepresented by a single beam with equivalent cross-sectionalproperties two bridge towers made of reinforced concretethat were modeled by three-dimensional Timoshenko beamelements and cables and suspenders that were modeled bycable elements to account for geometric nonlinearity dueto cable tension The model was validated by comparing itwith measurements of the first 18 modal frequencies andshapes of the actual bridge Using this model Guo et al[21] predicted the dynamic displacement and accelerationresponses of coupled train and bridge systems in crosswindsHowever they modeled the bridge deck as a simplified spinebeam of equivalent sectional properties and were thus unableto capture the local stress and strain behavior of the bridgeA second-generation bridge model (hybrid 3-dimensionalmodel) was established to overcome this weakness [22] Themodeling work is based on the previous model developed byWong [23] In this model 15904 beam elements were usedto model the bridge deck to closely replicate the geometricdetails of the complicated deck in reality (see Figure 2(a))The railway beams and rails were modeled by beam elementsto allow the accurate computation of the contact forcesbetween the bridge and railway vehicle The deck-platescarrying the road vehicles were modeled by plate elementsto allow the accurate computation of the contact forces atthe contact points between the road surface and the vehicletires The bridge deck was modeled to closely replicate thegeometric details of the complicated deck which is requiredfor calculation of the action of the wind forces The bridgemodel was also updated using the first 18 measured naturalfrequencies and mode shapes Based on this model Xu et al[24] computed the stress and acceleration responses of localcritical components under running trains and Chen et al[17] computed dynamic stress response induced by railwayhighway and wind loading
4 The Scientific World Journal
Mtij Jt120579ij
mwij1 120579wij1Ywij1
Y
U
120601ti2 120601ti1
Zti2k1i2 c
1i2
2di
kh1i ch1i2
kh2i2 ch2i2
Mti2
Yti2
2si
Yci
Mti1 Jt120601i1
Zti1 x
Zwij1
Bridge deck
Mci Jc120579i 120579ci
2bi Ycih1i
h2i
h3i
z2Bi
2ai
Ytij
Ywij1
120593ci
ZciMci Jc120593i
k2i2c2i2
Mti1
Jt120595i1Jt120595i2
Mci Jc120595i
Yci
120595ti2
Yti1
120595ti1120595ci
Mti2 Jt120601i2
kh2ij ch2ij
kh1ij ch1ij
k2ij c2ij
k1ij c1ij
Ztij
120579tij
Zwij1Jwij1
Figure 3 Dynamic model of a railway vehicle [51]
However the hybrid 3D model is still not fine enoughfor criticality analysis of bridge structures which requiresresults at strainstress level especially for some bridge detailsFor example the orthotropic decks (steel deck-plates sup-ported by U-shape troughs) were modeled by plate ele-ments with equivalent depths so that the measured resultsfrom strain gauges at the surfaces of deck-plates or U-shape troughs had no counterparts in computation resultsTherefore Duan et al [25] established the third-generationbridge model (full 3D model) for performance evaluationat stressstrain level (see Figure 2(b)) In this model themajor structural components were modeled in detail and theconnections and boundary conditions are modeled properlywhich results in about half million elements for the completebridge model The strainstress responses induced by atrain passing through the bridge were calculated by staticinfluence linemethod and comparedwithmeasured results inthe calibration
Although full 3D bridge model provides the possibilityfor exact stress analysis at the local components large com-putational efforts are needed for the refined section modelwith complicated structural details Li et al [26] proposed amultiscale FE modeling strategy for long-span bridges Theglobal structural analysis was carried out using the beamelement modeling method at the level of a meter The localdetailed hot-spot stress analysis was carried out using shellor solid elements at the level of a millimeter Based onthis model the global dynamic response of the bridge andlocal damage accumulation of two typical weld details ofthe bridge under traffic loading were numerically analyzedMultiscale FE modeling scheme was also proposed by Zhanget al [27] based on the equivalent orthotropic modelingmethod (EOMM) Bridge details withmultiple stiffenersweremodeled with shell elements using equivalent orthotropicmaterials Based on this model Zhang et al [10] computedthe dynamic stress responses of long-span bridges undercombined dynamic loads from winds and road vehicles
212 Modeling of Trains Previously running vehicles werecommonlymodeled as a series of moving forces either due tolimits on computational capacity or because it is easier to findthe analytical solutions inmany cases [28ndash37]This treatmentneglects the effect of interactions between the bridge andrunning vehicles For this reason the moving load model issuitable only for the case in which the mass of the vehicleis small relative to that of the bridge or when the vehicleresponse is not of interest [38] For cases in which the inertiaof the vehicle cannot be regarded as small a moving massmodel should be adopted instead [39ndash42] More recently theemergence of high-performance computers and advances incomputer technology hasmade it feasible tomore realisticallymodel the dynamic properties of the various components ofmoving vehicles [43ndash48]
In a more sophisticated railway vehicle model the sus-pension mechanisms are modeled by springs the dampingeffect of the suspension systems and air-cushion by dashpotsand the energy dissipating effect of the interleaf mechanismby frictional devices Using this technique a tractor-traileris represented as two discrete masses each of which issupported by two sets of springs and dashpots or frictionaldevices [38] To represent the various dynamic propertiesof railway vehicles vehicle models that contain dozens ofdegrees of freedom (DOFs) have been devised and used by[49ndash52] To investigate the dynamic interaction between along suspension bridge and running trains Xia et al [51]considered a train composed of a sequence of identicalrailway vehicles Each railway vehicle was assumed to consistof a rigid car body resting on front and rear bogies with eachbogie supported by two wheelsets (see Figure 3) Five DOFswere assigned to the car body and to each bogie to accountfor vertical lateral rolling yawing and pitching motions Incontrast only three DOFs were assigned to each wheelset toaccount for vertical lateral and rolling motions
Many vehicle models have been established for vehicle-bridge interaction analysis In most of these studies
The Scientific World Journal 5
(a) (b) (c)
Car body
Connecting rigid-arm
Wheel
Wheel-rail contactConstrait
Bogie
Primary suspension
Secondary suspension
(d)
Figure 4 Finite element model of a railway vehicle (a) elevation view (b) side view (c) isometric view (d) model details [52]
the equations of motion of the vehicles were derivedanalytically However a great inconvenience of this methodis that the equations of motion of the whole vehicle-bridgesystem must be rederived if the vehicle type is changedFurthermore it is very difficult to derive the equation ofmotion for a complex vehicle model containing a largenumber of DOFs such as the articulated components of aTGV train with an 85-DOF dynamic system [53] Generalcommercial FE software has recently been adopted to makevehicle modeling more easily applicable for different vehicletypes [54] Li et al [55] described a four-step procedurefor modeling a four-axle railway vehicle by beam elements(1) the nodes and elements for the car body bogies andwheelsets respectively are defined by using beam elements sothat the spatial geometric configuration of each componentcan be built (see Figure 4) (2) sectional properties andmaterial properties are assigned to each beam element(3) rigid-arms and suspension units (systems) are used toconnect the three components (4) constraints are assignedto form a complete finite element model of the vehicle
213 Modeling of Rail Irregularities Track irregularities rep-resent an important source of excitation for bridges duringthe passage of railway vehicles Track irregularitiesmay occuras a result of initial installation errors the degradation ofsupport materials or the dislocation of track joints Fourgeometric parameters can be used to quantitatively describerail irregularities the vertical profile cross level alignmentand gauge [49 50 56] Vertical profile and cross level irreg-ularities chiefly influence the vertical vibrations of vehiclesand of the bridge whereas alignment gauge and crosslevel irregularities initiate horizontal transverse vibrationsof vehicles and the bridge and also the torsional movementof the bridge [57] Track irregularities may be periodic orrandom Random irregularities are due to wear clearancesubsidence and insufficient maintenance For engineeringapplications random irregularities can be approximatelyregarded as stationary and ergodic processes that can begenerated from measured results or simulated by numericalmethods Several numerical methods have been proposedfor the simulation of random rail irregularities such as
6 The Scientific World Journal
the trigonometry series white noise filtration autoregressive(AR) and power spectral density (PSD) sampling methodsAmong these methods the PSD sampling method has beenwidely adopted due to its high computational accuracy Thelateral and vertical irregularities could be all assumed tobe zero-mean stationary Gaussian random processes andexpressed through the inverse Fourier transformation of aPSD function [58]
119910119904(119909) =
119873
sum119896=1
radic2119878 (119891119896) Δ119891 cos (2120587119891
119896119909 + 120579119896) (1)
where 119878(119891) is the PSD function 119891119896= 119891l + (119896 minus 12)Δ119891
Δ119891 = (119891119906minus 119891l)119873 119891119906 and 119891119897 are the upper and lower cutoff
frequencies respectively and 120579119896is the random phase angle
uniformly distributed between 0 and 2120587 Rail irregularity inrailway engineering is commonly represented by a one-sidedPSD function
The PSD functions of rail irregularities have been devel-oped by different countries Based on the PSD functions ofrail irregularities developed by the Research Institute of theChina Railway Administration Zhai [59] expressed all railirregularities using the unified rational formula as follows
where 119891 = 1120578 (mminus1) is the spatial frequency in cyclem (120578 isthe wavelength) and119860 to119866 are the parameters recommendedby Zhai [59] specifically for vertical and lateral rail irregular-ities
214 Solution Methods The dynamic analysis of vehicle-bridge coupled system requires two sets of equations ofmotion for the bridge and vehicles respectively Thesedescribe the interaction or contact forces at the contactpoints of the two subsystems Because the contact pointsmove from time to time the system matrices are generallytime dependent and must be updated and factorized at eachtime step The various solution methods can be generalizedinto two groups according to whether or not an iterativeprocedure is needed at each time step
The first group ofmethods solves the equations of motionof a coupled vehicle-bridge system at each time step withoutiteration This approach has been widely used in coupledvehicle-bridge analysis [51 53 60ndash69] These methods havegood computational stability and are convenient for dealingwith vehicle-bridge interaction problems when the vehiclemodel is relatively simple The main disadvantage is thatthe equations of motion of the coupled system are timedependent and thus the characteristic matrices must bemodified at each time step In addition the equations ofmotion of the coupled vehicle-bridge system become verydifficult to determine if nonlinear wheel-rail contacts andnonlinear vehicle models are considered
The second group of methods solves the equations for thevehicles and bridge separately and requires an iterative pro-cess to obtain convergence for the displacements of the vehi-cles and bridge at all contact points Given that the conditions
of wheel-rail contact geometry and contact forces are rathercomplex a stable integration method adopting a small timeinterval is needed for obtaining the convergence of vehicleand bridge subsystems at the contact points in each time stepMany studies have applied this type of method to investigatevehicle-bridge interactions [70ndash76] The advantage of thesemethods is that the dynamic propertymatrices in the two setsof equations of motion remain constant which is convenientfor the consideration of nonlinear vehicle-bridge interactionsand nonlinear vehicle models [55] However in engineeringapplications the iterative convergence is a critical problemwith this type of method The low convergence rate andoccasional divergence of the solution have also been noted[77] Li et al [55] investigated the performance of theseiterative schemes using the Wilson-120579 method Newmark-120573method and an explicit integrationmethod proposed byZhai[59] and found that the latter gave amuch higher convergencerate than the former two methods
Most of the above methods solved the equations ofmotion of a coupled vehicle-bridge systemusing the nonjumpmodel which assumes that the moving vehicle travelingalong the bridge is always in contact with the rails nomatter what the sign is of the contact forces This is notalways true in view of the physics of the moving vehiclewhich simply sits on the upper surfaces of the rails Theinteraction forces between the moving vehicle and the bridgedepend on the motions of the vehicle the flexibility of thebridge and the track irregularities Li et al [55] utilizeda jump model to solve vehicle-bridge interaction problemusing a noniterative Runge-Kutta method and found thatthe acceleration responses of the car body using the wheel-jump model are smaller than those using the wheel nonjumpmodel when the vehicle speed exceeds 300 kmhr Antolin etal [78] proposed a nonlinear wheel-rail interaction modelwhich considers nonlinear wheel-rail contact forces in theinteraction as well as realistic wheel and rail profiles andapplied it for analysis of dynamic interaction between highspeed trains and bridges
22 Dynamic Interaction between Road Vehicles and BridgeSection 21 gave a detailed literature review of the dynamicinteractions between trains and bridges As there are somefundamental differences between trains and road vehiclesthis section reviews the modeling of road vehicles thesimulation of road vehicle flow and the modeling of roadsurface roughness
221 Modeling of Road Vehicles To analyze the dynamicinteraction between a bridge and running road vehicles amodel of road vehicles must be established A sophisticatedroad vehiclemodel is required tomake the simulation as real-istic as possible A road vehicle is modeled as a combinationof several rigid bodies each of which is connected by a set ofsprings and dashpots which model the elastic and dampingeffects of the tires and suspension systems respectivelyThereare various configurations of road vehicles such as a tractorand trailer with different axle spacing Road vehicle models
The Scientific World Journal 7
L11 L10 L9
L8
Ku5Cu5
Cl5Kl5
Zs5
Ku4Cu4
Cl4Kl4
Zs4
Ku3 Cu3
Cl3Kl3
Zs3
Ku2 Cu2
Cl2Kl2
Zs2
Ku1 Cu1
Cl1Kl1
Zs1
Ku1 Cu1
Cl1Kl1
Zs1
Ku6 Cu6
Cl6Kl6
Zs6
b1 b1
x yL1L2L3L4L5L6L7
Z3 1205793 Z2 Z1
Z1
1205792
1205791
zz
1206011
Figure 5 Dynamic model of a tractor-trailer [79]
zz
x y
h
2b1
ZZY
h1
120601
Kuz1 Kuz3Cuz1
Kuz1 Cuz1 Cuz3
Zs1Zs1
Zs3
Cuy1 Cuy3
Kuy1Kuy3
Ys1 Ys3
Klz1Klz1 Klz3Clz1 Clz3Clz1
Kuz2 Cuz2
Zs2
Klz2 Clz2
120579
L1L2
Kly1
Cly1
Kly3
Cly3
Figure 6 Dynamic model of a high-sided road vehicle [80]
that contain several DOFs have been devised for vehicle-bridge interaction analysis For example Guo and Xu [79]modeled a 17-DOF four-axle heavy tractor-trailer vehicle (seeFigure 5) to investigate the interaction between vehicles anda cable-stayed bridge A total of three DOFs were assignedto rigid bodies representing either the tractor or the trailerto account for vertical rolling and pitching motions Onlyone DOFwas assigned to the rigid body representing the axleset moving in the vertical direction Different vehicle modelsare adopted in wind-vehicle-bridge interaction analyses Xuand Guo [80] modeled a 13-DOF two-axle road vehicle (seeFigure 6) for the dynamic analysis of a coupled road vehicleand bridge system under turbulent wind Five DOFs wereassigned to the vehicle body with respect to its center ofgravity to account for vertical lateral rolling yawing andpitching motions and two DOFs were assigned to the frontand rear axle sets to account for motions in the vertical andlateral directionMoreDOFs are needed to account for lateralcrosswinds
222 Simulation of Road Vehicle Flow On long-span bridgesthere is a high probability of the simultaneous presence ofmultiple road vehicles including heavy trucks This maylead to larger amplitude stress responses and greater fatiguedamage of the local bridge components than would be thecase with only one road vehicle The simulation of roadvehicle flow is thus important in the analysis of the dynamicinteraction between road vehicles and bridges Rather simplepatterns of road vehicle flow have been assumed in most
vehicle-bridge coupled dynamic analyses [79 81 82] in whicheither one or several vehicles are distributed on the bridgein an assumed (usually uniform) pattern Obviously suchassumptions do not represent actual road traffic conditionsRecently Chen and Wu [83] modeled the stochastic trafficload for a long-span bridge based on the cellular automaton(CA) traffic flow simulation technique In this study theysimulated a complicated road vehicle flow on long-spanbridges in terms of vehicle number vehicle type combinationand driver operation characteristics such as lane changingacceleration or deceleration
223 Modeling of Road Surface Roughness Road surfaceroughness is an important factor that greatly affects vehicle-bridge interactions Paultre et al [84] pointed out that roadsurface or pavement roughness can significantly affect theimpact response of a bridge The roughness or surface profiledepends primarily on the workmanship involved in theconstruction of the pavement or roadway and how it is main-tained which although random in nature may contain someinherent frequencies [38] In most cases surface roughnesswhich is three-dimensional in reality is often approximatedby a two-dimensional profile To account for its randomnature the road profile can be modeled as a stationaryGaussian random process and derived using a certain powerspectral density function Other methods similar to this havebeen widely adopted by researchers studying vehicle-inducedbridge vibration [65 70 71 85ndash90] Dodds and Robson [91]developed power spectral density functions that were later
8 The Scientific World Journal
modified and used byWang and Huang [87] and Huang et al[92] This approach was also adopted by literatures [79 81] intheir dynamic analyses of coupled vehicle-bridge and wind-vehicle-bridge systems
23 Dynamic Interaction between Wind and Bridge Whena long-span cable-supported bridge is immersed in a givenflow field the bridge will be subject to mean and fluctuatingwind forces To simulate these forces a linear approximationof the time-averaged static and time-varying buffeting andself-excited force components must be formulated [93 94]As dynamic bridge responses are of concern in this study onlybuffeting and self-excited forces are considered and reviewedin this section
231 Buffeting Forces Buffeting action is a random vibrationcaused by turbulent wind that excites certain modes of vibra-tion across a bridge depending on the spectral distributionof the pressure vectors [95] Although the buffeting responsemay not lead to catastrophic failure it can lead to structuralfatigue and affect the safety of passing vehicles [96] Hencebuffeting analysis has received much attention in recentyears in research into the structural safety of bridges underturbulent wind action [81 95 97ndash102]
By assuming no interaction between buffeting forces andself-excited forces and using quasi-steady aerodynamic forcecoefficients the buffeting forces per unit span Feibf on the 119894thsection of a bridge deck can be expressed as [103]
bf are the buffeting drag lift andmoment respectively 119906
119894and 119908
119894are the horizontal and
vertical components respectively of fluctuating wind at the119894th section 120588 is the air density 119880
119894is the mean wind speed
at the 119894th section of the bridge deck 119861119894and 119871
119894are the width
and length of the 119894th bridge section 119862119863119894 119862119871119894 and 119862
119872119894are
the drag lift and moment coefficients respectively of the119894th bridge segment 1198621015840
119863119894= 119889119862
1198631198941198891205721015840 1198621015840
119871119894= 119889119862
1198711198941198891205721015840 and
1198621015840119872119894
= 1198891198621198721198941198891205721015840 1205721015840 is the angle of attack of a normal
wind incident on the horizontal plane of the deck and 120594119863bu
120594119863bw
120594119871bu
120594119871bw
120594119872bu
and 120594119872bw
are the aerodynamic transferfunctions between the fluctuating wind velocities and thebuffeting forces
It can be found from this equation that a series oftime histories of fluctuating wind velocity 119906
119894 119908119894119879 in the
longitudinal and vertical directions at various points alongthe bridge deck is needed to carry out a detailed buffetinganalysis To simulate the stochastic wind velocity field thefast spectral representation method proposed by Cao et al[104] that is based on the spectral representation methoddeveloped by Shinozuka and Jan [105] is often adopted Thismethod rests on the assumptions that (1) the bridge deck ishorizontal at the same elevation (2) the mean wind speedand wind spectra do not vary along the bridge deck and(3) the distance between any two successive points wherewind speeds are simulated is the same The time historiesof the along-wind component 119906(119905) and the upward windcomponent 119908(119905) at the jth point can be generated using thefollowing equations [104]
119906119895(119905) = radic2 (Δ120596)
119895
sum119898=1
119873119891
sum119896=1
radic119878119906119906(120596119898119896)
times 119866119895119898(120596119898119896) cos (120596
119898119896119905 + 120593119898119896)
(4a)
119908119895(119905) = radic2 (Δ120596)
119895
sum119898=1
119873119891
sum119896=1
radic119878119908119908(120596119898119896)
times 119866119895119898(120596119898119896) cos (120596
119898119896119905 + 120593119898119896)
(4b)
119866119895119898(120596) =
0 when 1 le 119895 lt 119898 le 119899
119862|119895minus119898| when 119897 = 1 119898 le 119895 le 119899
119862|119895minus119898|radic(1 minus 1198622) when 2 le 119898 le 119895 le 119899
(4c)
119862 = exp(minus120582120596119898119896Δ
2120587119880) Δ =
119871
119899119901minus 1
(4d)
120596119898119896= (119896 minus 1) Δ120596 +
119898
119899Δ120596 (119896 = 1 2 119873
119891) (4e)
where Δ120596 = 120596up119873119891 is the frequency interval between thespectral lines119873
119891is the total number of frequency intervals
120596up is the upper cutoff frequency n is the total number ofpoints at which wind speeds are simulated 119878uu and 119878ww arethe along-wind and vertical wind spectrum respectively 120593lmis a random variable that is uniformly distributed between 0and 2120587 L is the span length and 120582 is a parameter that usuallyfalls between 7 and 10
In reality the equivalent buffeting forces in (3) are actuallyassociated with the spatial distribution of the wind pressureson the surface of the bridge deck Ignoring the spatialdistribution or aerodynamic transfer function of the buffetingforces across the cross-section of the bridge deck may havea considerable impact on the accuracy of buffeting response
The Scientific World Journal 9
wi(t) UI + ui(t)
j
j + 1
k
Leibf Lkibf
Meibf
Deibf
Dkibf
ce
Figure 7 Buffeting wind pressures and buffeting forces at nodes [22]
predictions Furthermore the local structural behavior ofthe bridge deck associated with local stresses and strainswhich are prone to causing local damage cannot be predicteddirectly by the current approaches based on equivalentbuffeting forces In this regard Liu et al [22] proposed anapproach to consider the spatial distribution of buffetingforces on a bridge deck structure based on wind pressuredistributions from wind tunnel tests (see Figure 7)
232 Self-Excited Forces In addition to buffeting actionflutter instability caused by self-excited forces induced bywind-structure interactions is an important considerationin the design and construction of long-span suspensionbridges [96] because the additional energy injected into theoscillating structure by the aerodynamic forces increases themagnitude of vibration sometimes to catastrophic levels [95]The self-excited forces on a bridge deck are attributable tothe interactions between wind and the motion of the bridgeWhen the energy of motion extracted from the flow exceedsthe energy dissipated by the system through mechanicaldamping the magnitude of vibration can reach catastrophiclevels [106] Expressing self-excited forces in the form ofindicial functions was first suggested by Scanlan [94] Basedon the assumption that self-excited forces are generated in alinear fashion Lin and Yang [107] simplified the self-excitedforces acting on a bridge deck and expressed them in termsof convolution integrals between the bridge deck motion andthe impulse response functions
119891se119890119863(119905)
=1
21205881198802
int119905
minusinfin
[119868119863ℎ(119905 minus 120591) ℎ
119890(120591) + 119868
119863119902(119905 minus 120591) 119902
119890(120591)
+119861119868119863120579(119905 minus 120591) 120579
119890(120591)
] 119889120591
(5a)
119891se119890119871(119905)
=1
21205881198802
int119905
minusinfin
[119868119871ℎ(119905 minus 120591) ℎ
119890(120591) + 119868
119871119902(119905 minus 120591) 119902
119890(120591)
+119861119868119871120579(119905 minus 120591) 120579
is the impulse function of the self-excited forces in which120595 represents the corresponding force components and heqe and 120579119890 are the equivalent vertical lateral and torsionaldisplacements respectively at the center of elasticity of thebridge deck section The relationship between the aero-dynamic impulse functions and flutter derivatives can beobtained by taking the Fourier transform of (7) [98]
119868119863ℎ(120596) = 119870
2
(119875lowast
6+ 119894119875lowast
5) 119868
119863119902(120596) = 119870
2
(119875lowast
4+ 119894119875lowast
1)
119868119863120579(120596) = 119870
2
(119875lowast
3+ 119894119875lowast
2)
119868119871ℎ(120596) = 119870
2
(119867lowast
4+ 119894119867lowast
1) 119868
119871119902(120596) = 119870
2
(119867lowast
6+ 119894119867lowast
5)
119868119871120579(120596) = 119870
2
(119867lowast
3+ 119894119867lowast
2)
119868119872ℎ(120596) = 119870
2
(119860lowast
4+ 119894119860lowast
1) 119868
119872119902(120596) = 119870
2
(119860lowast
6+ 119894119860lowast
5)
119868119872120579(120596) = 119870
2
(119860lowast
3+ 119894119860lowast
2)
(6)
where the overbars denote the Fourier transform operationthe terms containing 119894 represent imaginary parts119875lowast
120595 119867lowast
120595 and
119860lowast120595(120595 = 1 2 6) are dimensionless flutter derivatives
obtained from wind tunnel tests 119870 = 120596119861119880 is the reducedfrequency and 120596 is the circular frequency of vibration
According to classical airfoil theory the impulse func-tions can reasonably be approximated by a rational function[108]
119868 (120596) = [1198621+ 1198941198622
2120587
]+
119898
sum119897=1
119862119897+2
41205872 + 1198942120587119889119897+2
]1198892119897+2
V2 + 41205872] (7)
where the value of 119898 determines the level of accuracy ofthe approximation 119862
1 1198622 119862119897+2
and 119889119897+2(119897 = 1 2 119898)
are the frequency independent coefficients and ] = 2120587119870
is the reduced mean wind velocity By equating the realand imaginary parts in the comparison of (8) and (9) therelationship between the dimensionless flutter derivativesand the coefficients 119862120595
1 119862120595
2 119862120595
119897+2 and 119889120595
119897+2(where 120595 = Dh
Dq 119863120579 Lh Lq 119871120579 Mh Mq 119872120579 and 119897 = 1 2 119898) canbe establishedThese coefficients are determined by using thenonlinear least-squares method to fit the measured flutter
10 The Scientific World Journal
derivatives at different reduced frequencies The expressionof the aerodynamic impulse functions in the time domaincan be obtained by taking the inverse Fourier transform ofthe impulse functions By substituting the related impulseresponse functions into (5b) the self-excited lift force at the119894th section of bridge deck can then be derived as
119891se119890119871119894
=1
21205881198861198802
119894119861119894119862119871120579
1119894120579119894(119905) + 119862
119871120579
2119894(119861119894
119880119894
) 120579119894(119905)
+ 119862119871120579
3119894(119861119894
119880119894
) 120579119894(119905) +
119898
sum119897=1
119862119871120579
119897+3119894
times int119905
minusinfin
120579119894(119905) exp[minus
119889119871120579119897+3119894119880119894
119861119894
(119905 minus 120591)] 119889120591
+1
21205881198861198802
119894119862119871ℎ
1119894ℎ119894(119905) + 119862
119871ℎ
2119894(119861119894
119880119894
) ℎ119894(119905)
+ 119862119871ℎ
3119894(119861119894
119880119894
) ℎ119894(119905) +
119898
sum119897=1
119862119871ℎ
119897+3119894
times int119905
minusinfin
ℎ119894(119905) exp[minus
119889119871ℎ119897+3119894119880119894
119861119894
(119905 minus 120591)] 119889120591
+1
21205881198861198802
119894
119862119871119902
1119894119902119894(119905) + 119862
119871119902
2119894(119861119894
119880119894
) 119902119894(119905)
+ 119862119871119902
3119894(119861119894
119880119894
) 119902119894(119905) +
119898
sum119897=1
119862119871119902
119897+3119894
times int119905
minusinfin
119902120595(119905) exp[
[
minus119889119871119901
119897+3119894119880119894
119861119894
(119905 minus 120591)]
]
119889120591
(8)
In practice the terms 1198621198711205793119894 119862119871ℎ3119894 and 119862119871119902
3119894 which are related to
additional aerodynamic masses are normally neglected andthe value of 119898 is often taken as 2 [101] Similar formulationsfor self-excited drag and moment can be derived withanalogous definitions The self-excited forces at the 119894th nodeof the bridge deck can thus be expressed as
Feise = EeiXei + GeiXei + Feise (9)
where
Xei =
0
ℎei119902ei120579ei0
0
Eei =1
21205881198802
119894
[[[[[[[[
[
0 0 0 0 0 0
0 119862119871ℎ
1119894119862119871119902
11198941198611198941198621198711205791119894
0 0
0 119862119863ℎ1119894
119862119863119902
11198941198611198941198621198631205791119894
0 0
0 119861119894119862119872ℎ1119894
119861119894119862119872119902
111989411986121198941198621198721205791119894
0 0
0 0 0 0 0 0
0 0 0 0 0 0
]]]]]]]]
]
Gei =1
21205881198802
119894119861119894
[[[[[[[[
[
0 0 0 0 0 0
0 119862119871ℎ
2119894119862119871119902
21198941198611198941198621198711205792119894
0 0
0 119862119863ℎ2119894
119862119863119902
21198941198611198941198621198631205792119894
0 0
0 119861119894119862119872ℎ2119894
119861119894119862119872119902
211989411986121198941198621198721205792119894
0 0
0 0 0 0 0 0
0 0 0 0 0 0
]]]]]]]]
]
Feise =
[[[[[[[
[
0
eise
119863eise
eise0
0
]]]]]]]
]
=
[[[[[[[[[[[[[[[[
[
02
sum119897=1
119862119871119902
l+3119894119881119871119902
l+3119894 +2
sum119897=1
119862119871ℎl+3119894119881119871ℎ
l+3119894 +2
sum119897=1
119862119871120579l+3119894119881119871120579
l+3119894
2
sum119897=1
119862119863119902
l+3119894119881119863119902
l+3119894 +2
sum119897=1
119862119863ℎl+3119894119881119863ℎ
l+3119894 +2
sum119897=1
119862119863120579l+3119894119881119863120579
l+3119894
2
sum119897=1
119862119872119902
l+3119894119881119872119902
l+3119894 +2
sum119897=1
119862119872ℎl+3119894119881119872ℎ
l+3119894 +2
sum119897=1
119862119872120579l+3119894119881119872120579
l+3119894
0
0
]]]]]]]]]]]]]]]]
]
(10)
where 119881120595l+3119894 (120595 = Dh Dq 119863120579 Lh Lq 119871120579 Mh Mq) are theconvolution integrations of the 119894th node and can be calculatedusing a recursive algorithm For example
(11)The self-excited forces expressed by (9) relate to the centerof elasticity of the 119894th deck section Hence the force modelmust be distributed to the nodal points of the section Adistribution based on the rigid body motion relationshipsbetween themotions at the nodal point and those at the centerof elasticity of the deck section [109] was applied by Liu etal [22] In this study by applying the virtual work principlethe self-excited forces at the center of elasticity of the givensection were distributed to all nodes (see Figure 8)
24 Dynamic Interactions in a Wind-Vehicle-Bridge SystemWhen trains and road vehicles are running on long-spanbridges under crosswinds complicated dynamic interac-tions occur among the trains road vehicles cable-supported
The Scientific World Journal 11
Ui h
p
120572
Lsec i
Msec i
Dsec i
Fsekiz
Fsekiy
ce
Figure 8 Self-excited forces at the centre of elasticity and at the nodes in the 119894th deck section [22]
bridge and windThe buffeting response of the bridge due tocrosswind is superimposed on the dynamic response of thebridge due to railway and road vehicles The large vibrationof the bridge will in turn considerably affect the safetyand ride comfort of the drivers of the road vehicles Thusthe dynamic responses of a coupled vehicle-bridge systemunder crosswinds are of great concern to both engineers andresearchers
Detailed reviews of the dynamic interactions betweentrains and bridges between road vehicles and bridges andbetween wind and bridges have been given in the previoussections However the interaction between wind and vehiclesmust also be taken into account in a coupled wind-vehicle-bridge analysis Many studies have investigated wind-vehicleinteractions in the past few decades Balzer [110] developeda theory to estimate the aerodynamic forces on a movingvehicle using Taylorrsquos hypothesis of ldquofrozen turbulencerdquo Forengineering applications Cooper [111] proposed the powerspectral density (PSD) square-root coherence functionphase-lag function and aerodynamic admittance functionto model the unsteady side forces on a moving vehicle andlaid down the foundations for investigating the effects ofwind on a moving vehicle in the frequency domain Bakerdeveloped a theoretical model that describes the dynamics ofvehicles in crosswinds in the time domain [112 113] whichwas later extended to include driver behavior [114] Baker[115 116] further investigated both the steady and unsteadyaerodynamic forces acting on a variety of vehicles and carriedout extensive studies of the interaction between aerodynamicforces and moving vehicles These approaches have all beenapplied in coupled vehicle-bridge analysis For example Xuet al [101] simulated the aerodynamic wind forces actingon running road vehicles using the quasi-steady approachand Xu and Ding [117] derived and simulated the steadyand unsteady aerodynamic forces acting on a moving railwayvehicle in crosswinds in the time domain
Based on these separate studies on the various typesof dynamic interactions among wind vehicles (trains orroad vehicles) and long-span bridges several researchersin the last decade have examined the wind-vehicle-bridgecoupled system as a whole For instance studies have beencarried out on coupled road vehicle and cable-stayed bridgesystems [81 82 118] and on coupled train and cable-supportedbridge systems in crosswinds [101 117 119ndash121] In therecent years several new advances have been made bothin numerical simulation technologies and in wind tunnelmeasurements Chen et al [17] proposed a wind-vehicle-bridge framework which enables considering the dynamiceffects induced by simultaneous actions of railway highway
and wind loading and it was applied to analyze dynamicstress of long suspension bridges Li et al [122] extended thewind-vehicle-bridge couple analysis to the case of two trainsmeeting on a long-span suspension bridge Chen and Wu[118] proposed a semideterministic analytical model whichis able to consider dynamic interactions between the bridgewind and stochastic ldquorealrdquo traffic Based on the wind tunneltests Dorigatti et al [123]measured crosswind loads on high-sided vehicles over long-span bridges taking three differentvehicles (van double deck bus and lorry) and two differentbridge deck configurations into consideration Zhu et al[124] investigated aerodynamic coefficients of road vehiclesby adopting different road vehicles types wind directionsand vehicle positions Li et al [122] studied the effects ofsudden changes of wind loads as the train passing througha bridge tower or two trains passing each other by usingthe wind tunnel test rig with moving train models Hanet al [125] developed an experimental setup for measuringthe aerodynamic characteristics of vehicles and the bridgein wind tunnel and then investigated the influences ofparameters adopted in the tests
3 Applications of Simulation Technology toBridge Assessment
After reviewing the key issues of numerical simulationsfor dynamic response of long-span multiload bridges thissection will review the engineering applications of the newlydeveloped technologies to safety assessment of long-spanbridges such as assessment of fatigue and assessment underextreme events
31 Assessment of FatigueDamage Steel structures are widelyused in long-span bridges Research by the ASCE [126] indi-cates that 80ndash90 of failures in steel structures are related tofatigue and fracture Several disasters resulting from fatigue-induced bridge failure have occurred in history For instance46 people died in the collapse of the Silver Bridge (USA1967) and 32 people lost their lives in the collapse of theSungsoo Grand Bridge (South Korea 1994) These disastersteach us that fatigue is an important aspect of the safety ofsteel bridges and action should be taken to prevent similarfatigue-induced bridge failures In the past few decadesfatigue assessment of steel bridges has attracted increasingattention from both governments and bridge engineers andrelevant provisions have been stipulated in several codes andstandards [127ndash130]
12 The Scientific World Journal
It has great advantages to evaluate fatigue damage of long-span bridges based on numerical simulation especially fora multiload bridge which suffers multiple types of dynamicloading such as railway highway andwind loadingDifferentfrom sudden structural damage fatigue damage accumulateswith load-induced dynamic stress (or stress fluctuation) overthe service life of a bridge lasting for more than 100 yearsThe increase in traffic volume and gross vehicle weight thataccompany economic development is very likely to happenin the long period Numerical simulation technology can bean ideal tool to study influences of traffic growth patternsto fatigue damage of bridge In addition slender long-span bridges built in wind-prone regions also suffer fromconsiderable wind induced vibration which appears withina wide range of wind speeds and lasts for almost the wholedesign life of the bridge Given the simultaneous presenceof multiple vehicles and wind it is necessary to considercombined effects of traffic loading (railway andor highwayloading) and wind loading in the fatigue assessment Sincemultiple loading is concerned in a long time period thereare a large number of loading combinations for multipleloading in different intensities It is almost unavailable forfield measurement to obtain such complete information butnumerical simulation could be a good choice to determinedynamic responses of a long-span bridge under multipleloading
A number of structural health monitoring systems(SHMSs) have been installed on numerous recently builtlong-span bridges and a variety of sensors are used for mon-itoring bridge loadings (eg traffic wind and earthquakes)and conditions (including global and local responses) toensure bridge safety and user comfort under in-service con-ditions Well-known examples include Tsing-Ma Bridge inHong Kong Akashi Kaikyo Bridge in Japan Binzhou YellowRiver Bridge in China and Jindo Bridge in Korea Integrationof numerical simulation technologies and measurement dataof structural health monitoring systems (SHMSs) installedon long-span bridges will make the fatigue assessment morereliable for several reasons (1) it is a perfect validation byusing field measurement data of the different types of loadingas input of numerical simulation and the measured dynamicresponses for comparison with the computed ones (2) a largenumber of measured loading data could be used to establishloading databases or probabilistic models of different loads
In the recent years several researchers [7ndash10] appliedthe newly developed numerical simulation technologies tofatigue assessment of long-span bridges Chen et al [7]proposed a framework for fatigue analysis of a long-spansuspension bridge under railway highway and wind loadingby integrating computer simulation with SHMSs and itwas applied to evaluate fatigue damage of the Tsing MaSuspension Bridge over its design life as a case study Basedon this work Chen et al [8] proposed a framework forfatigue reliability analysis of long suspension bridges undermultiple loading inwhich the probabilisticmodels of railwayhighway and wind loading were established based on themeasurement data acquired from the SHMS of the TsingMa Bridge Wu et al [9] proposed a reliability-based fatigueapproach for slender long-span bridge and the combined
dynamic loading effects from wind and traffic as well asthe associated uncertainties were considered Based on theassumption that dynamic magnification related to vehicledynamics can be neglected in long suspension bridges Chenet al [8] established a framework for fatigue reliability anal-ysis To account for different types of long-span bridges withthe span length ranging from a few hundred to thousands ofmeters Zhang et al [10] proposed a more general frameworkwhich includes multiple random variables for the dynamicloads in a bridgersquos life cycle for the vehicle-bridge-winddynamic system such as road profile vehicle speed andwindvelocity and direction among other effects
32 Assessment under Extreme Events The aforementionedfatigue assessment mainly focuses on damage accumulationinduced by stress fluctuations under normal operationalcondition in a long-term period For long-span bridges inaddition to the normal operational conditions in which windspeeds are small ormoderate and traffic scenarios are normalthere are some extreme event conditions Extreme eventsmayinclude complex traffic congestion on the bridge coupledwith moderate or even strong wind [11] For example severetraffic congestions may be formed on the bridge as a resultof an evacuation or a partial blockage of driving lanes due totraffic accidents construction ormaintenance For hurricaneevacuations there are usually a lot of road vehicles passingthrough the bridge before the landfall of the hurricane whilethe wind speed may become pretty high already [131]
Although the excessive dynamic responses of the bridgesunder extreme events are rare it is also critical since itmay cause critical damage initiation or accumulation onsome local bridgemembers Furthermore the extreme events(eg heavy traffic) may even trigger the collapse of thewhole bridge by breaking the ldquoweakest linkrdquo especially whensome hidden damage or design flaw has not been detectedOne recent example is the Minnesota Bridge failure whichoccurred during rush hours with heavy traffic although trafficloads may not be the direct cause of failure For slender long-span bridges strong wind may also cause threats by workinginteractively with heavy traffic loads Therefore even thoughthe extreme cases associated with congested traffic andorwindy weather may be relatively rare and the durationscould be short it is still important for bridge engineers toappropriately look into these unusual extreme events duringstructural design and life-time management of these criticalinfrastructures [11]
The dynamic performance of long-span bridges undercombined actions of strong winds and running road vehicleshas been studied by many researchers in recent years [17 7981 82 132] Most of them studied bridge dynamic perfor-mance under road traffic inwhich only one or several vehiclesdistributed in an assumed (usually uniform) pattern on long-span bridges were considered Extreme events such as trafficcongestion coupled with strong wind were out of concernin those studies Recently Wu and Chen [11] conducteda research on the assessment of long-span bridges underextreme events which includes complex traffic congestioncoupled with moderate or even strong wind This study
The Scientific World Journal 13
applied the cellular automaton (CA) traffic model to thesimulation of the actual traffic flow through the bridgedefined representative scenarios for the extreme events andnumerically studied the bridge performance under thesepossible extreme events
4 Conclusions and Recommendations
Dynamic responses of long-span bridges are often requiredfor assessing the safety of these bridges and can be determinedby numerical simulation technologies This paper provides adetailed review of key issues involved in dynamic responseanalysis of long-span multiload bridges based on numericalsimulation including dynamic interactions between runningtrains and bridge between running road vehicles and bridgeand between wind and bridge and in the wind-vehicle-bridge coupled systemThen the review work was conductedfor engineering applications of newly developed numericalsimulation technologies to safety assessment of long-spanbridges such as assessment of fatigue damage and assessmentunder extreme event condition Although technologies fornumerical simulation of dynamic responses of long-spanmultiload bridge have achieved great advances in past fewdecades and successfully applied to several important bridgesit is still far from reach its maturity and enable to takeplace of traditional fieldmeasurementThe existing problemsand promising research efforts at least include the followingaspects
(1) After multiple types of dynamic interactions beingconsidered in the complex system computationalefficiency is a bottleneck problem for numericalsimulation of dynamic response of a long-span bridgeTypically when multiple loads are involved a largenumber of loading combinations for multiple load-ings must be considered in the assessment
(2) It is rather complex for the time-depending windloads acting on a long-span bridge and running vehi-cles especially for the case of rapid change of windloads such as a train passing through a bridge toweror two trains passing each other The aerodynamiccharacteristics of vehicles and the bridge under differ-ent loading scenarios can be determined through thewind tunnel testing and used in numerical simulationof dynamic responses of the bridge and vehicles
(3) It is a new trend to integrate numerical simulationtechnologies and measurement data of structuralhealth monitoring systems (SHMSs) installed onlong-span bridges whichmakes the safety assessmentof bridge structures more reliable Measured struc-tural responses could be used to validate numericalsimulation approach and measured loading infor-mation could be used for generating statistical orprobabilistic models of multiple loads
(4) It is important to study dynamic responses of bridgestructures under extreme events such as congestedtraffic coupled with windy weather which happens ina long-span bridge For the assessment under extreme
events using numerical simulation technologies sim-ulation of traffic flow and definition of representativescenarios of the extreme events are key issues
(5) It is necessary to consider the effects of typhoonwinds on the safety assessment of long-span bridgesin a reasonable way Few researches do this mostlybecause a probabilistic distribution of wind speedand direction specifically for typhoons is requiredfor assessment but there are insufficient measuredrecords to establish a reliable probabilistic typhoonwind model
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors wish to acknowledge the financial supportsfrom the National Natural Science Foundation of China(NSFC-51108395 and NSFC-51178366) the FundamentalResearch Funds for theCentral Universities (2012121032) andopen funding from Jiangsu Key Laboratory of EngineeringMechanics Special thanks go to the supervisor of the firstauthor Professor Y L XuHongKong PolytechnicUniversityfor his expert guidance and continuous support at all levelsthroughout his PhD study Sincere thanks should go tothe Highways Department of Hong Kong for providing theauthors with the field measurement data Any opinions andconcluding remarks presented in this paper are entirely thoseof the authors
References
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[2] H Sohn C R Farrar N F Hunter and K Worden ldquoStructuralhealth monitoring using statistical pattern recognition tech-niquesrdquo Journal of Dynamic Systems Measurement and Controlvol 123 no 4 pp 706ndash711 2001
[3] W Fan and P Qiao ldquoVibration-based damage identificationmethods a review and comparative studyrdquo Structural HealthMonitoring vol 10 no 1 pp 83ndash111 2011
[4] X Q Zhu and S S Law ldquoDamage detection in simply supportedconcrete bridge structure under moving vehicular loadsrdquo Jour-nal of Vibration and Acoustics Transactions of the ASME vol129 no 1 pp 58ndash65 2007
[5] J Li and S S Law ldquoDamage identification of a target sub-structure with moving load excitationrdquoMechanical Systems andSignal Processing vol 30 pp 78ndash90 2012
[6] J Li S S Law and H Hao ldquoImproved damage identification inbridge structures subject tomoving loads numerical and exper-imental studiesrdquo International Journal of Mechanical Sciencesvol 74 pp 99ndash111 2013
14 The Scientific World Journal
[7] Z W Chen Y L Xu Y Xia Q Li and K Y Wong ldquoFatigueanalysis of long-span suspension bridges under multiple load-ing case studyrdquo Engineering Structures vol 33 no 12 pp 3246ndash3256 2011
[8] Z W Chen Y L Xu and X M Wang ldquoSHMS-based fatiguereliability analysis of multiloading suspension bridgesrdquo Journalof Structural Engineering-Asce vol 138 pp 299ndash307 2012
[9] J Wu S R Chen and J W van de Lindt ldquoFatigue assessmentof slender long-span bridges reliability approachrdquo Journal ofBridge Engineering vol 17 no 1 pp 47ndash57 2012
[10] W Zhang C S Cai and F Pan ldquoFatigue reliability assessmentfor long-span bridges under combined dynamic loads fromwinds and vehiclesrdquo Journal of Bridge Engineering vol 18 pp735ndash747 2013
[11] JWu and S R Chen ldquoProbabilistic dynamic behavior of a long-span bridge under extreme eventsrdquo Engineering Structures vol33 no 5 pp 1657ndash1665 2011
[12] J M Ko and Y Q Ni ldquoTechnology developments in structuralhealth monitoring of large-scale bridgesrdquo Engineering Struc-tures vol 27 no 12 pp 1715ndash1725 2005
[13] TH YiHN Li andHM Sun ldquoMulti-stage structural damagediagnosis method based on ldquoenergy-damagerdquo theoryrdquo SmartStructures and Systems vol 12 pp 345ndash361 2013
[14] T H Yi H N Li and M Gu ldquoFull-scale measurements ofdynamic response of suspension bridge subjected to environ-mental loads using GPS technologyrdquo Science China Technologi-cal Sciences vol 53 no 2 pp 469ndash479 2010
[15] T H Yi H N Li and M Gu ldquoExperimental assessmentof high-rate GPS receivers for deformation monitoring ofbridgerdquoMeasurement Journal of the InternationalMeasurementConfederation vol 46 pp 420ndash432 2013
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[21] W Guo H Xia and Y-L Xu ldquoDynamic response of a long spansuspension bridge and running safety of a train under windactionrdquo Frontiers of Architecture and Civil Engineering in Chinavol 1 no 1 pp 71ndash79 2007
[22] T T Liu Y L Xu W S Zhang K Y Wong H J Zhou and KW Y Chan ldquoBuffeting-induced stresses in a long suspensionbridge structural health monitoring oriented stress analysisrdquoWind and Structures An International Journal vol 12 no 6 pp479ndash504 2009
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[24] Y L XuQ Li D JWu and ZWChen ldquoStress and accelerationanalysis of coupled vehicle and long-span bridge systems usingthe mode superposition methodrdquo Engineering Structures vol32 no 5 pp 1356ndash1368 2010
[25] Y F Duan Y L Xu Q G Fei et al ldquoAdvanced finite elementmodel of Tsing Ma Bridge for structural health monitoringrdquoInternational Journal of Structural Stability and Dynamics vol11 no 2 pp 313ndash344 2011
[26] Z X Li T Q Zhou T H T Chan and Y Yu ldquoMulti-scalenumerical analysis on dynamic response and local damage inlong-span bridgesrdquo Engineering Structures vol 29 no 7 pp1507ndash1524 2007
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[34] N H Galdos D R Schelling and M A Sahin ldquoMethodologyfor impact factor of horizontally curved box bridgesrdquo Journal ofStructural Engineering vol 119 no 6 pp 1917ndash1934 1993
[35] J A Gbadeyan and S T Oni ldquoDynamic behaviour of beamsand rectangular plates under moving loadsrdquo Journal of Soundand Vibration vol 182 no 5 pp 677ndash695 1995
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[37] G V Rao ldquoLinear dynamics of an elastic beam under movingloadsrdquo Journal of Vibration and Acoustics Transactions of theASME vol 122 no 3 pp 281ndash289 2000
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[39] E C Ting J Genin and J H Ginsberg ldquoA general algorithm formovingmass problemsrdquo Journal of Sound and Vibration vol 33no 1 pp 49ndash58 1974
[40] S Sadiku and H H E Leipholz ldquoOn the dynamics of elasticsystems with moving concentrated massesrdquo Ingenieur-Archivvol 57 no 3 pp 223ndash242 1987
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The Scientific World Journal 15
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[47] B Tabarrok and E Esmailzadeh ldquoInduced vibration of bridgestransversed by moving vehiclesrdquo Transactions of the CanadianSociety for Mechanical Engineering B vol 24 no 1 pp 191ndash1982000
[48] C Liu T-L Wang and D Huang ldquoImpact study for multi-girder bridge based on correlated road roughnessrdquo StructuralEngineering and Mechanics vol 11 no 3 pp 259ndash272 2001
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[51] H Xia Y L Xu and T H T Chan ldquoDynamic interaction oflong suspension bridges with running trainsrdquo Journal of Soundand Vibration vol 237 no 2 pp 263ndash280 2000
[52] Q-L Zhang A Vrouwenvelder and J Wardenier ldquoNumericalsimulation of train-bridge interactive dynamicsrdquo Computersand Structures vol 79 no 10 pp 1059ndash1075 2001
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[54] G Diana F Cheli A Collina R Corradi and S MelzildquoThe development of a numerical model for railway vehiclescomfort assessment through comparison with experimentalmeasurementsrdquoVehicle SystemDynamics vol 38 no 3 pp 165ndash183 2002
[55] Q Li Y L Xu D J Wu and Z W Chen ldquoComputer-aided nonlinear vehicle-bridge interaction analysisrdquo Journal ofVibration and Control vol 16 pp 1791ndash1816 2010
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[98] X Chen M Matsumoto and A Kareem ldquoTime domainflutter and buffeting response analysis of bridgesrdquo Journal ofEngineering Mechanics vol 126 no 1 pp 7ndash16 2000
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[100] S R Chen and C S Cai ldquoEvolution of long-span bridgeresponse to wind-numerical simulation and discussionrdquo Com-puters and Structures vol 81 no 21 pp 2055ndash2066 2003
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[103] E Simiu and R H ScanlanWind Effects on Structures 1996[104] Y Cao H Xiang and Y Zhou ldquoSimulation of stochastic wind
velocity field on long-span bridgesrdquo Journal of EngineeringMechanics vol 126 no 1 pp 1ndash6 2000
[105] M Shinozuka and C-M Jan ldquoDigital simulation of randomprocesses and its applicationsrdquo Journal of Sound and Vibrationvol 25 no 1 pp 111ndash128 1972
[106] K M Shum Lateral and torsional vibration control of long spanbridge deck using novel tuned liquid column dampers [PhDthesis] Department of Civil and Structural Engineering TheHong Kong Polytechnic University 2004
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of bridges by spline finite-strip methodrdquo Journal of StructuralEngineering vol 126 no 10 pp 1246ndash1254 2000
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[117] Y L Xu and Q S Ding ldquoInteraction of railway vehicles withtrack in cross-windsrdquo Journal of Fluids and Structures vol 22no 3 pp 295ndash314 2006
[118] S R Chen and J Wu ldquoDynamic performance simulation oflong-span bridge under combined loads of stochastic traffic andwindrdquo Journal of Bridge Engineering vol 15 no 3 pp 219ndash2302010
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The Scientific World Journal 17
[122] Y L Li H Y Xiang B Wang Y L Xu and S Z QiangldquoDynamic analysis of wind-vehicle-bridge coupling systemduring the meeting of two trainsrdquo Advances in StructuralEngineering vol 16 pp 1663ndash1670 2013
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[125] Y Han J X Hu C S Cai Z Q Chen and C G LildquoExperimental and numerical studies of aerodynamic forces onvehicles and bridgesrdquoWind and Structures vol 17 pp 163ndash1842013
[126] ASCE ldquoCommittee on fatigue and fracture reliability of thecommittee on structural safety and reliability of the structuraldivision fatigue reliability 1ndash4rdquo Journal of Structural Engineer-ing vol 108 pp 3ndash88 1982
[127] BS ldquoBS5400 part 10 code of practice for fatiguerdquo BritishStandard Institute 1980
[128] BS ldquoBS7608 code of practice for fatigue design and assessmentof steel structuresrdquo British Standard Institute 1993
[129] AASHTOGuide Specifications for Fatigue Evaluation of ExistingSteel Bridges 1990
[130] AASHTO Guide Manual for Condition Evaluation and Loadand Resistance Factor Rating (LRFR) of Highway Bridges 2003
[131] S R Chen C S Cai and B Wolshon ldquoFrom normal operationto evacuation Single-vehicle safety under adverse weathertopographic and operational conditionsrdquo Natural HazardsReview vol 10 no 2 pp 68ndash76 2009
[132] S R Chen and C S Cai ldquoEquivalent wheel load approach forslender cable-stayed bridge fatigue assessment under traffic andwind feasibility studyrdquo Journal of Bridge Engineering vol 12 no6 pp 755ndash764 2007
Figure 2 Finite element model of suspended deck module (a) hybrid 3D bridge model [22] (b) full 3D model [25]
Finally the existing problems and promising research effortsfor the numerical simulation technologies and their appli-cations to assessment of long-span multiload bridges areexplored in Section 4
For themost complex situation a long-spanmultiload bridgewhich is located at a wind-prone region carries both railwayand highway traffic and thus the combined effect of runningtrains running road vehicles and wind is acting on thebridge Several key issues are involved in this complicatedsituation such as dynamic interaction between runningtrains and bridge dynamic interaction between running roadvehicles and bridge and dynamic interaction between windand bridge To give a comprehensive review the above threekey issues will be individually reviewed in Sections 21 to23 and then the dynamic interactions of wind-vehicle-bridgesystem as a whole are then reviewed in Section 24
21 Dynamic Interaction between Trains and Bridge
211 Modeling of a Cable-Supported Bridge In early researchin this area simplified bridge models were employed tostudy vehicle-bridge interactions For example a cable-stayedbridge was simulated as a beam resting on an elastic founda-tion by Meisenholder and Weidlinger [18] for the dynamicanalysis of cable-stayed guideways subject to track-levitatedvehicles moving at high speeds Mao [19] investigated theimpact factor of a cable-stayed bridge which was assumedto be formed of continuous elastic beams supported byintermediate elastic supports
More recently with the development of finite element(FE) technology it has become common practice to use acomputer software package to establish a finite elementmodel(FEM) of a cable-supported bridge This technology estab-lishes an accurate bridge model that takes into account thegeometric nonlinear behavior of a cable-supported bridgeTo make the bridge model close to the realistic bridgein terms of its dynamic properties the modal frequencies
and shapes determined by dynamic tests are used for fur-ther model validation or updating Many FEMs of cable-supported bridges have been established for the analysis oftrain-bridge interactions The Tsing Ma Suspension Bridgein Hong Kong can be used as an example to illustratethe various bridge models that have been established foranalysis The first generation of Tsing Ma Bridge model wasa spinal beam model [20] in which the hybrid steel deck wasrepresented by a single beam with equivalent cross-sectionalproperties two bridge towers made of reinforced concretethat were modeled by three-dimensional Timoshenko beamelements and cables and suspenders that were modeled bycable elements to account for geometric nonlinearity dueto cable tension The model was validated by comparing itwith measurements of the first 18 modal frequencies andshapes of the actual bridge Using this model Guo et al[21] predicted the dynamic displacement and accelerationresponses of coupled train and bridge systems in crosswindsHowever they modeled the bridge deck as a simplified spinebeam of equivalent sectional properties and were thus unableto capture the local stress and strain behavior of the bridgeA second-generation bridge model (hybrid 3-dimensionalmodel) was established to overcome this weakness [22] Themodeling work is based on the previous model developed byWong [23] In this model 15904 beam elements were usedto model the bridge deck to closely replicate the geometricdetails of the complicated deck in reality (see Figure 2(a))The railway beams and rails were modeled by beam elementsto allow the accurate computation of the contact forcesbetween the bridge and railway vehicle The deck-platescarrying the road vehicles were modeled by plate elementsto allow the accurate computation of the contact forces atthe contact points between the road surface and the vehicletires The bridge deck was modeled to closely replicate thegeometric details of the complicated deck which is requiredfor calculation of the action of the wind forces The bridgemodel was also updated using the first 18 measured naturalfrequencies and mode shapes Based on this model Xu et al[24] computed the stress and acceleration responses of localcritical components under running trains and Chen et al[17] computed dynamic stress response induced by railwayhighway and wind loading
4 The Scientific World Journal
Mtij Jt120579ij
mwij1 120579wij1Ywij1
Y
U
120601ti2 120601ti1
Zti2k1i2 c
1i2
2di
kh1i ch1i2
kh2i2 ch2i2
Mti2
Yti2
2si
Yci
Mti1 Jt120601i1
Zti1 x
Zwij1
Bridge deck
Mci Jc120579i 120579ci
2bi Ycih1i
h2i
h3i
z2Bi
2ai
Ytij
Ywij1
120593ci
ZciMci Jc120593i
k2i2c2i2
Mti1
Jt120595i1Jt120595i2
Mci Jc120595i
Yci
120595ti2
Yti1
120595ti1120595ci
Mti2 Jt120601i2
kh2ij ch2ij
kh1ij ch1ij
k2ij c2ij
k1ij c1ij
Ztij
120579tij
Zwij1Jwij1
Figure 3 Dynamic model of a railway vehicle [51]
However the hybrid 3D model is still not fine enoughfor criticality analysis of bridge structures which requiresresults at strainstress level especially for some bridge detailsFor example the orthotropic decks (steel deck-plates sup-ported by U-shape troughs) were modeled by plate ele-ments with equivalent depths so that the measured resultsfrom strain gauges at the surfaces of deck-plates or U-shape troughs had no counterparts in computation resultsTherefore Duan et al [25] established the third-generationbridge model (full 3D model) for performance evaluationat stressstrain level (see Figure 2(b)) In this model themajor structural components were modeled in detail and theconnections and boundary conditions are modeled properlywhich results in about half million elements for the completebridge model The strainstress responses induced by atrain passing through the bridge were calculated by staticinfluence linemethod and comparedwithmeasured results inthe calibration
Although full 3D bridge model provides the possibilityfor exact stress analysis at the local components large com-putational efforts are needed for the refined section modelwith complicated structural details Li et al [26] proposed amultiscale FE modeling strategy for long-span bridges Theglobal structural analysis was carried out using the beamelement modeling method at the level of a meter The localdetailed hot-spot stress analysis was carried out using shellor solid elements at the level of a millimeter Based onthis model the global dynamic response of the bridge andlocal damage accumulation of two typical weld details ofthe bridge under traffic loading were numerically analyzedMultiscale FE modeling scheme was also proposed by Zhanget al [27] based on the equivalent orthotropic modelingmethod (EOMM) Bridge details withmultiple stiffenersweremodeled with shell elements using equivalent orthotropicmaterials Based on this model Zhang et al [10] computedthe dynamic stress responses of long-span bridges undercombined dynamic loads from winds and road vehicles
212 Modeling of Trains Previously running vehicles werecommonlymodeled as a series of moving forces either due tolimits on computational capacity or because it is easier to findthe analytical solutions inmany cases [28ndash37]This treatmentneglects the effect of interactions between the bridge andrunning vehicles For this reason the moving load model issuitable only for the case in which the mass of the vehicleis small relative to that of the bridge or when the vehicleresponse is not of interest [38] For cases in which the inertiaof the vehicle cannot be regarded as small a moving massmodel should be adopted instead [39ndash42] More recently theemergence of high-performance computers and advances incomputer technology hasmade it feasible tomore realisticallymodel the dynamic properties of the various components ofmoving vehicles [43ndash48]
In a more sophisticated railway vehicle model the sus-pension mechanisms are modeled by springs the dampingeffect of the suspension systems and air-cushion by dashpotsand the energy dissipating effect of the interleaf mechanismby frictional devices Using this technique a tractor-traileris represented as two discrete masses each of which issupported by two sets of springs and dashpots or frictionaldevices [38] To represent the various dynamic propertiesof railway vehicles vehicle models that contain dozens ofdegrees of freedom (DOFs) have been devised and used by[49ndash52] To investigate the dynamic interaction between along suspension bridge and running trains Xia et al [51]considered a train composed of a sequence of identicalrailway vehicles Each railway vehicle was assumed to consistof a rigid car body resting on front and rear bogies with eachbogie supported by two wheelsets (see Figure 3) Five DOFswere assigned to the car body and to each bogie to accountfor vertical lateral rolling yawing and pitching motions Incontrast only three DOFs were assigned to each wheelset toaccount for vertical lateral and rolling motions
Many vehicle models have been established for vehicle-bridge interaction analysis In most of these studies
The Scientific World Journal 5
(a) (b) (c)
Car body
Connecting rigid-arm
Wheel
Wheel-rail contactConstrait
Bogie
Primary suspension
Secondary suspension
(d)
Figure 4 Finite element model of a railway vehicle (a) elevation view (b) side view (c) isometric view (d) model details [52]
the equations of motion of the vehicles were derivedanalytically However a great inconvenience of this methodis that the equations of motion of the whole vehicle-bridgesystem must be rederived if the vehicle type is changedFurthermore it is very difficult to derive the equation ofmotion for a complex vehicle model containing a largenumber of DOFs such as the articulated components of aTGV train with an 85-DOF dynamic system [53] Generalcommercial FE software has recently been adopted to makevehicle modeling more easily applicable for different vehicletypes [54] Li et al [55] described a four-step procedurefor modeling a four-axle railway vehicle by beam elements(1) the nodes and elements for the car body bogies andwheelsets respectively are defined by using beam elements sothat the spatial geometric configuration of each componentcan be built (see Figure 4) (2) sectional properties andmaterial properties are assigned to each beam element(3) rigid-arms and suspension units (systems) are used toconnect the three components (4) constraints are assignedto form a complete finite element model of the vehicle
213 Modeling of Rail Irregularities Track irregularities rep-resent an important source of excitation for bridges duringthe passage of railway vehicles Track irregularitiesmay occuras a result of initial installation errors the degradation ofsupport materials or the dislocation of track joints Fourgeometric parameters can be used to quantitatively describerail irregularities the vertical profile cross level alignmentand gauge [49 50 56] Vertical profile and cross level irreg-ularities chiefly influence the vertical vibrations of vehiclesand of the bridge whereas alignment gauge and crosslevel irregularities initiate horizontal transverse vibrationsof vehicles and the bridge and also the torsional movementof the bridge [57] Track irregularities may be periodic orrandom Random irregularities are due to wear clearancesubsidence and insufficient maintenance For engineeringapplications random irregularities can be approximatelyregarded as stationary and ergodic processes that can begenerated from measured results or simulated by numericalmethods Several numerical methods have been proposedfor the simulation of random rail irregularities such as
6 The Scientific World Journal
the trigonometry series white noise filtration autoregressive(AR) and power spectral density (PSD) sampling methodsAmong these methods the PSD sampling method has beenwidely adopted due to its high computational accuracy Thelateral and vertical irregularities could be all assumed tobe zero-mean stationary Gaussian random processes andexpressed through the inverse Fourier transformation of aPSD function [58]
119910119904(119909) =
119873
sum119896=1
radic2119878 (119891119896) Δ119891 cos (2120587119891
119896119909 + 120579119896) (1)
where 119878(119891) is the PSD function 119891119896= 119891l + (119896 minus 12)Δ119891
Δ119891 = (119891119906minus 119891l)119873 119891119906 and 119891119897 are the upper and lower cutoff
frequencies respectively and 120579119896is the random phase angle
uniformly distributed between 0 and 2120587 Rail irregularity inrailway engineering is commonly represented by a one-sidedPSD function
The PSD functions of rail irregularities have been devel-oped by different countries Based on the PSD functions ofrail irregularities developed by the Research Institute of theChina Railway Administration Zhai [59] expressed all railirregularities using the unified rational formula as follows
where 119891 = 1120578 (mminus1) is the spatial frequency in cyclem (120578 isthe wavelength) and119860 to119866 are the parameters recommendedby Zhai [59] specifically for vertical and lateral rail irregular-ities
214 Solution Methods The dynamic analysis of vehicle-bridge coupled system requires two sets of equations ofmotion for the bridge and vehicles respectively Thesedescribe the interaction or contact forces at the contactpoints of the two subsystems Because the contact pointsmove from time to time the system matrices are generallytime dependent and must be updated and factorized at eachtime step The various solution methods can be generalizedinto two groups according to whether or not an iterativeprocedure is needed at each time step
The first group ofmethods solves the equations of motionof a coupled vehicle-bridge system at each time step withoutiteration This approach has been widely used in coupledvehicle-bridge analysis [51 53 60ndash69] These methods havegood computational stability and are convenient for dealingwith vehicle-bridge interaction problems when the vehiclemodel is relatively simple The main disadvantage is thatthe equations of motion of the coupled system are timedependent and thus the characteristic matrices must bemodified at each time step In addition the equations ofmotion of the coupled vehicle-bridge system become verydifficult to determine if nonlinear wheel-rail contacts andnonlinear vehicle models are considered
The second group of methods solves the equations for thevehicles and bridge separately and requires an iterative pro-cess to obtain convergence for the displacements of the vehi-cles and bridge at all contact points Given that the conditions
of wheel-rail contact geometry and contact forces are rathercomplex a stable integration method adopting a small timeinterval is needed for obtaining the convergence of vehicleand bridge subsystems at the contact points in each time stepMany studies have applied this type of method to investigatevehicle-bridge interactions [70ndash76] The advantage of thesemethods is that the dynamic propertymatrices in the two setsof equations of motion remain constant which is convenientfor the consideration of nonlinear vehicle-bridge interactionsand nonlinear vehicle models [55] However in engineeringapplications the iterative convergence is a critical problemwith this type of method The low convergence rate andoccasional divergence of the solution have also been noted[77] Li et al [55] investigated the performance of theseiterative schemes using the Wilson-120579 method Newmark-120573method and an explicit integrationmethod proposed byZhai[59] and found that the latter gave amuch higher convergencerate than the former two methods
Most of the above methods solved the equations ofmotion of a coupled vehicle-bridge systemusing the nonjumpmodel which assumes that the moving vehicle travelingalong the bridge is always in contact with the rails nomatter what the sign is of the contact forces This is notalways true in view of the physics of the moving vehiclewhich simply sits on the upper surfaces of the rails Theinteraction forces between the moving vehicle and the bridgedepend on the motions of the vehicle the flexibility of thebridge and the track irregularities Li et al [55] utilizeda jump model to solve vehicle-bridge interaction problemusing a noniterative Runge-Kutta method and found thatthe acceleration responses of the car body using the wheel-jump model are smaller than those using the wheel nonjumpmodel when the vehicle speed exceeds 300 kmhr Antolin etal [78] proposed a nonlinear wheel-rail interaction modelwhich considers nonlinear wheel-rail contact forces in theinteraction as well as realistic wheel and rail profiles andapplied it for analysis of dynamic interaction between highspeed trains and bridges
22 Dynamic Interaction between Road Vehicles and BridgeSection 21 gave a detailed literature review of the dynamicinteractions between trains and bridges As there are somefundamental differences between trains and road vehiclesthis section reviews the modeling of road vehicles thesimulation of road vehicle flow and the modeling of roadsurface roughness
221 Modeling of Road Vehicles To analyze the dynamicinteraction between a bridge and running road vehicles amodel of road vehicles must be established A sophisticatedroad vehiclemodel is required tomake the simulation as real-istic as possible A road vehicle is modeled as a combinationof several rigid bodies each of which is connected by a set ofsprings and dashpots which model the elastic and dampingeffects of the tires and suspension systems respectivelyThereare various configurations of road vehicles such as a tractorand trailer with different axle spacing Road vehicle models
The Scientific World Journal 7
L11 L10 L9
L8
Ku5Cu5
Cl5Kl5
Zs5
Ku4Cu4
Cl4Kl4
Zs4
Ku3 Cu3
Cl3Kl3
Zs3
Ku2 Cu2
Cl2Kl2
Zs2
Ku1 Cu1
Cl1Kl1
Zs1
Ku1 Cu1
Cl1Kl1
Zs1
Ku6 Cu6
Cl6Kl6
Zs6
b1 b1
x yL1L2L3L4L5L6L7
Z3 1205793 Z2 Z1
Z1
1205792
1205791
zz
1206011
Figure 5 Dynamic model of a tractor-trailer [79]
zz
x y
h
2b1
ZZY
h1
120601
Kuz1 Kuz3Cuz1
Kuz1 Cuz1 Cuz3
Zs1Zs1
Zs3
Cuy1 Cuy3
Kuy1Kuy3
Ys1 Ys3
Klz1Klz1 Klz3Clz1 Clz3Clz1
Kuz2 Cuz2
Zs2
Klz2 Clz2
120579
L1L2
Kly1
Cly1
Kly3
Cly3
Figure 6 Dynamic model of a high-sided road vehicle [80]
that contain several DOFs have been devised for vehicle-bridge interaction analysis For example Guo and Xu [79]modeled a 17-DOF four-axle heavy tractor-trailer vehicle (seeFigure 5) to investigate the interaction between vehicles anda cable-stayed bridge A total of three DOFs were assignedto rigid bodies representing either the tractor or the trailerto account for vertical rolling and pitching motions Onlyone DOFwas assigned to the rigid body representing the axleset moving in the vertical direction Different vehicle modelsare adopted in wind-vehicle-bridge interaction analyses Xuand Guo [80] modeled a 13-DOF two-axle road vehicle (seeFigure 6) for the dynamic analysis of a coupled road vehicleand bridge system under turbulent wind Five DOFs wereassigned to the vehicle body with respect to its center ofgravity to account for vertical lateral rolling yawing andpitching motions and two DOFs were assigned to the frontand rear axle sets to account for motions in the vertical andlateral directionMoreDOFs are needed to account for lateralcrosswinds
222 Simulation of Road Vehicle Flow On long-span bridgesthere is a high probability of the simultaneous presence ofmultiple road vehicles including heavy trucks This maylead to larger amplitude stress responses and greater fatiguedamage of the local bridge components than would be thecase with only one road vehicle The simulation of roadvehicle flow is thus important in the analysis of the dynamicinteraction between road vehicles and bridges Rather simplepatterns of road vehicle flow have been assumed in most
vehicle-bridge coupled dynamic analyses [79 81 82] in whicheither one or several vehicles are distributed on the bridgein an assumed (usually uniform) pattern Obviously suchassumptions do not represent actual road traffic conditionsRecently Chen and Wu [83] modeled the stochastic trafficload for a long-span bridge based on the cellular automaton(CA) traffic flow simulation technique In this study theysimulated a complicated road vehicle flow on long-spanbridges in terms of vehicle number vehicle type combinationand driver operation characteristics such as lane changingacceleration or deceleration
223 Modeling of Road Surface Roughness Road surfaceroughness is an important factor that greatly affects vehicle-bridge interactions Paultre et al [84] pointed out that roadsurface or pavement roughness can significantly affect theimpact response of a bridge The roughness or surface profiledepends primarily on the workmanship involved in theconstruction of the pavement or roadway and how it is main-tained which although random in nature may contain someinherent frequencies [38] In most cases surface roughnesswhich is three-dimensional in reality is often approximatedby a two-dimensional profile To account for its randomnature the road profile can be modeled as a stationaryGaussian random process and derived using a certain powerspectral density function Other methods similar to this havebeen widely adopted by researchers studying vehicle-inducedbridge vibration [65 70 71 85ndash90] Dodds and Robson [91]developed power spectral density functions that were later
8 The Scientific World Journal
modified and used byWang and Huang [87] and Huang et al[92] This approach was also adopted by literatures [79 81] intheir dynamic analyses of coupled vehicle-bridge and wind-vehicle-bridge systems
23 Dynamic Interaction between Wind and Bridge Whena long-span cable-supported bridge is immersed in a givenflow field the bridge will be subject to mean and fluctuatingwind forces To simulate these forces a linear approximationof the time-averaged static and time-varying buffeting andself-excited force components must be formulated [93 94]As dynamic bridge responses are of concern in this study onlybuffeting and self-excited forces are considered and reviewedin this section
231 Buffeting Forces Buffeting action is a random vibrationcaused by turbulent wind that excites certain modes of vibra-tion across a bridge depending on the spectral distributionof the pressure vectors [95] Although the buffeting responsemay not lead to catastrophic failure it can lead to structuralfatigue and affect the safety of passing vehicles [96] Hencebuffeting analysis has received much attention in recentyears in research into the structural safety of bridges underturbulent wind action [81 95 97ndash102]
By assuming no interaction between buffeting forces andself-excited forces and using quasi-steady aerodynamic forcecoefficients the buffeting forces per unit span Feibf on the 119894thsection of a bridge deck can be expressed as [103]
bf are the buffeting drag lift andmoment respectively 119906
119894and 119908
119894are the horizontal and
vertical components respectively of fluctuating wind at the119894th section 120588 is the air density 119880
119894is the mean wind speed
at the 119894th section of the bridge deck 119861119894and 119871
119894are the width
and length of the 119894th bridge section 119862119863119894 119862119871119894 and 119862
119872119894are
the drag lift and moment coefficients respectively of the119894th bridge segment 1198621015840
119863119894= 119889119862
1198631198941198891205721015840 1198621015840
119871119894= 119889119862
1198711198941198891205721015840 and
1198621015840119872119894
= 1198891198621198721198941198891205721015840 1205721015840 is the angle of attack of a normal
wind incident on the horizontal plane of the deck and 120594119863bu
120594119863bw
120594119871bu
120594119871bw
120594119872bu
and 120594119872bw
are the aerodynamic transferfunctions between the fluctuating wind velocities and thebuffeting forces
It can be found from this equation that a series oftime histories of fluctuating wind velocity 119906
119894 119908119894119879 in the
longitudinal and vertical directions at various points alongthe bridge deck is needed to carry out a detailed buffetinganalysis To simulate the stochastic wind velocity field thefast spectral representation method proposed by Cao et al[104] that is based on the spectral representation methoddeveloped by Shinozuka and Jan [105] is often adopted Thismethod rests on the assumptions that (1) the bridge deck ishorizontal at the same elevation (2) the mean wind speedand wind spectra do not vary along the bridge deck and(3) the distance between any two successive points wherewind speeds are simulated is the same The time historiesof the along-wind component 119906(119905) and the upward windcomponent 119908(119905) at the jth point can be generated using thefollowing equations [104]
119906119895(119905) = radic2 (Δ120596)
119895
sum119898=1
119873119891
sum119896=1
radic119878119906119906(120596119898119896)
times 119866119895119898(120596119898119896) cos (120596
119898119896119905 + 120593119898119896)
(4a)
119908119895(119905) = radic2 (Δ120596)
119895
sum119898=1
119873119891
sum119896=1
radic119878119908119908(120596119898119896)
times 119866119895119898(120596119898119896) cos (120596
119898119896119905 + 120593119898119896)
(4b)
119866119895119898(120596) =
0 when 1 le 119895 lt 119898 le 119899
119862|119895minus119898| when 119897 = 1 119898 le 119895 le 119899
119862|119895minus119898|radic(1 minus 1198622) when 2 le 119898 le 119895 le 119899
(4c)
119862 = exp(minus120582120596119898119896Δ
2120587119880) Δ =
119871
119899119901minus 1
(4d)
120596119898119896= (119896 minus 1) Δ120596 +
119898
119899Δ120596 (119896 = 1 2 119873
119891) (4e)
where Δ120596 = 120596up119873119891 is the frequency interval between thespectral lines119873
119891is the total number of frequency intervals
120596up is the upper cutoff frequency n is the total number ofpoints at which wind speeds are simulated 119878uu and 119878ww arethe along-wind and vertical wind spectrum respectively 120593lmis a random variable that is uniformly distributed between 0and 2120587 L is the span length and 120582 is a parameter that usuallyfalls between 7 and 10
In reality the equivalent buffeting forces in (3) are actuallyassociated with the spatial distribution of the wind pressureson the surface of the bridge deck Ignoring the spatialdistribution or aerodynamic transfer function of the buffetingforces across the cross-section of the bridge deck may havea considerable impact on the accuracy of buffeting response
The Scientific World Journal 9
wi(t) UI + ui(t)
j
j + 1
k
Leibf Lkibf
Meibf
Deibf
Dkibf
ce
Figure 7 Buffeting wind pressures and buffeting forces at nodes [22]
predictions Furthermore the local structural behavior ofthe bridge deck associated with local stresses and strainswhich are prone to causing local damage cannot be predicteddirectly by the current approaches based on equivalentbuffeting forces In this regard Liu et al [22] proposed anapproach to consider the spatial distribution of buffetingforces on a bridge deck structure based on wind pressuredistributions from wind tunnel tests (see Figure 7)
232 Self-Excited Forces In addition to buffeting actionflutter instability caused by self-excited forces induced bywind-structure interactions is an important considerationin the design and construction of long-span suspensionbridges [96] because the additional energy injected into theoscillating structure by the aerodynamic forces increases themagnitude of vibration sometimes to catastrophic levels [95]The self-excited forces on a bridge deck are attributable tothe interactions between wind and the motion of the bridgeWhen the energy of motion extracted from the flow exceedsthe energy dissipated by the system through mechanicaldamping the magnitude of vibration can reach catastrophiclevels [106] Expressing self-excited forces in the form ofindicial functions was first suggested by Scanlan [94] Basedon the assumption that self-excited forces are generated in alinear fashion Lin and Yang [107] simplified the self-excitedforces acting on a bridge deck and expressed them in termsof convolution integrals between the bridge deck motion andthe impulse response functions
119891se119890119863(119905)
=1
21205881198802
int119905
minusinfin
[119868119863ℎ(119905 minus 120591) ℎ
119890(120591) + 119868
119863119902(119905 minus 120591) 119902
119890(120591)
+119861119868119863120579(119905 minus 120591) 120579
119890(120591)
] 119889120591
(5a)
119891se119890119871(119905)
=1
21205881198802
int119905
minusinfin
[119868119871ℎ(119905 minus 120591) ℎ
119890(120591) + 119868
119871119902(119905 minus 120591) 119902
119890(120591)
+119861119868119871120579(119905 minus 120591) 120579
is the impulse function of the self-excited forces in which120595 represents the corresponding force components and heqe and 120579119890 are the equivalent vertical lateral and torsionaldisplacements respectively at the center of elasticity of thebridge deck section The relationship between the aero-dynamic impulse functions and flutter derivatives can beobtained by taking the Fourier transform of (7) [98]
119868119863ℎ(120596) = 119870
2
(119875lowast
6+ 119894119875lowast
5) 119868
119863119902(120596) = 119870
2
(119875lowast
4+ 119894119875lowast
1)
119868119863120579(120596) = 119870
2
(119875lowast
3+ 119894119875lowast
2)
119868119871ℎ(120596) = 119870
2
(119867lowast
4+ 119894119867lowast
1) 119868
119871119902(120596) = 119870
2
(119867lowast
6+ 119894119867lowast
5)
119868119871120579(120596) = 119870
2
(119867lowast
3+ 119894119867lowast
2)
119868119872ℎ(120596) = 119870
2
(119860lowast
4+ 119894119860lowast
1) 119868
119872119902(120596) = 119870
2
(119860lowast
6+ 119894119860lowast
5)
119868119872120579(120596) = 119870
2
(119860lowast
3+ 119894119860lowast
2)
(6)
where the overbars denote the Fourier transform operationthe terms containing 119894 represent imaginary parts119875lowast
120595 119867lowast
120595 and
119860lowast120595(120595 = 1 2 6) are dimensionless flutter derivatives
obtained from wind tunnel tests 119870 = 120596119861119880 is the reducedfrequency and 120596 is the circular frequency of vibration
According to classical airfoil theory the impulse func-tions can reasonably be approximated by a rational function[108]
119868 (120596) = [1198621+ 1198941198622
2120587
]+
119898
sum119897=1
119862119897+2
41205872 + 1198942120587119889119897+2
]1198892119897+2
V2 + 41205872] (7)
where the value of 119898 determines the level of accuracy ofthe approximation 119862
1 1198622 119862119897+2
and 119889119897+2(119897 = 1 2 119898)
are the frequency independent coefficients and ] = 2120587119870
is the reduced mean wind velocity By equating the realand imaginary parts in the comparison of (8) and (9) therelationship between the dimensionless flutter derivativesand the coefficients 119862120595
1 119862120595
2 119862120595
119897+2 and 119889120595
119897+2(where 120595 = Dh
Dq 119863120579 Lh Lq 119871120579 Mh Mq 119872120579 and 119897 = 1 2 119898) canbe establishedThese coefficients are determined by using thenonlinear least-squares method to fit the measured flutter
10 The Scientific World Journal
derivatives at different reduced frequencies The expressionof the aerodynamic impulse functions in the time domaincan be obtained by taking the inverse Fourier transform ofthe impulse functions By substituting the related impulseresponse functions into (5b) the self-excited lift force at the119894th section of bridge deck can then be derived as
119891se119890119871119894
=1
21205881198861198802
119894119861119894119862119871120579
1119894120579119894(119905) + 119862
119871120579
2119894(119861119894
119880119894
) 120579119894(119905)
+ 119862119871120579
3119894(119861119894
119880119894
) 120579119894(119905) +
119898
sum119897=1
119862119871120579
119897+3119894
times int119905
minusinfin
120579119894(119905) exp[minus
119889119871120579119897+3119894119880119894
119861119894
(119905 minus 120591)] 119889120591
+1
21205881198861198802
119894119862119871ℎ
1119894ℎ119894(119905) + 119862
119871ℎ
2119894(119861119894
119880119894
) ℎ119894(119905)
+ 119862119871ℎ
3119894(119861119894
119880119894
) ℎ119894(119905) +
119898
sum119897=1
119862119871ℎ
119897+3119894
times int119905
minusinfin
ℎ119894(119905) exp[minus
119889119871ℎ119897+3119894119880119894
119861119894
(119905 minus 120591)] 119889120591
+1
21205881198861198802
119894
119862119871119902
1119894119902119894(119905) + 119862
119871119902
2119894(119861119894
119880119894
) 119902119894(119905)
+ 119862119871119902
3119894(119861119894
119880119894
) 119902119894(119905) +
119898
sum119897=1
119862119871119902
119897+3119894
times int119905
minusinfin
119902120595(119905) exp[
[
minus119889119871119901
119897+3119894119880119894
119861119894
(119905 minus 120591)]
]
119889120591
(8)
In practice the terms 1198621198711205793119894 119862119871ℎ3119894 and 119862119871119902
3119894 which are related to
additional aerodynamic masses are normally neglected andthe value of 119898 is often taken as 2 [101] Similar formulationsfor self-excited drag and moment can be derived withanalogous definitions The self-excited forces at the 119894th nodeof the bridge deck can thus be expressed as
Feise = EeiXei + GeiXei + Feise (9)
where
Xei =
0
ℎei119902ei120579ei0
0
Eei =1
21205881198802
119894
[[[[[[[[
[
0 0 0 0 0 0
0 119862119871ℎ
1119894119862119871119902
11198941198611198941198621198711205791119894
0 0
0 119862119863ℎ1119894
119862119863119902
11198941198611198941198621198631205791119894
0 0
0 119861119894119862119872ℎ1119894
119861119894119862119872119902
111989411986121198941198621198721205791119894
0 0
0 0 0 0 0 0
0 0 0 0 0 0
]]]]]]]]
]
Gei =1
21205881198802
119894119861119894
[[[[[[[[
[
0 0 0 0 0 0
0 119862119871ℎ
2119894119862119871119902
21198941198611198941198621198711205792119894
0 0
0 119862119863ℎ2119894
119862119863119902
21198941198611198941198621198631205792119894
0 0
0 119861119894119862119872ℎ2119894
119861119894119862119872119902
211989411986121198941198621198721205792119894
0 0
0 0 0 0 0 0
0 0 0 0 0 0
]]]]]]]]
]
Feise =
[[[[[[[
[
0
eise
119863eise
eise0
0
]]]]]]]
]
=
[[[[[[[[[[[[[[[[
[
02
sum119897=1
119862119871119902
l+3119894119881119871119902
l+3119894 +2
sum119897=1
119862119871ℎl+3119894119881119871ℎ
l+3119894 +2
sum119897=1
119862119871120579l+3119894119881119871120579
l+3119894
2
sum119897=1
119862119863119902
l+3119894119881119863119902
l+3119894 +2
sum119897=1
119862119863ℎl+3119894119881119863ℎ
l+3119894 +2
sum119897=1
119862119863120579l+3119894119881119863120579
l+3119894
2
sum119897=1
119862119872119902
l+3119894119881119872119902
l+3119894 +2
sum119897=1
119862119872ℎl+3119894119881119872ℎ
l+3119894 +2
sum119897=1
119862119872120579l+3119894119881119872120579
l+3119894
0
0
]]]]]]]]]]]]]]]]
]
(10)
where 119881120595l+3119894 (120595 = Dh Dq 119863120579 Lh Lq 119871120579 Mh Mq) are theconvolution integrations of the 119894th node and can be calculatedusing a recursive algorithm For example
(11)The self-excited forces expressed by (9) relate to the centerof elasticity of the 119894th deck section Hence the force modelmust be distributed to the nodal points of the section Adistribution based on the rigid body motion relationshipsbetween themotions at the nodal point and those at the centerof elasticity of the deck section [109] was applied by Liu etal [22] In this study by applying the virtual work principlethe self-excited forces at the center of elasticity of the givensection were distributed to all nodes (see Figure 8)
24 Dynamic Interactions in a Wind-Vehicle-Bridge SystemWhen trains and road vehicles are running on long-spanbridges under crosswinds complicated dynamic interac-tions occur among the trains road vehicles cable-supported
The Scientific World Journal 11
Ui h
p
120572
Lsec i
Msec i
Dsec i
Fsekiz
Fsekiy
ce
Figure 8 Self-excited forces at the centre of elasticity and at the nodes in the 119894th deck section [22]
bridge and windThe buffeting response of the bridge due tocrosswind is superimposed on the dynamic response of thebridge due to railway and road vehicles The large vibrationof the bridge will in turn considerably affect the safetyand ride comfort of the drivers of the road vehicles Thusthe dynamic responses of a coupled vehicle-bridge systemunder crosswinds are of great concern to both engineers andresearchers
Detailed reviews of the dynamic interactions betweentrains and bridges between road vehicles and bridges andbetween wind and bridges have been given in the previoussections However the interaction between wind and vehiclesmust also be taken into account in a coupled wind-vehicle-bridge analysis Many studies have investigated wind-vehicleinteractions in the past few decades Balzer [110] developeda theory to estimate the aerodynamic forces on a movingvehicle using Taylorrsquos hypothesis of ldquofrozen turbulencerdquo Forengineering applications Cooper [111] proposed the powerspectral density (PSD) square-root coherence functionphase-lag function and aerodynamic admittance functionto model the unsteady side forces on a moving vehicle andlaid down the foundations for investigating the effects ofwind on a moving vehicle in the frequency domain Bakerdeveloped a theoretical model that describes the dynamics ofvehicles in crosswinds in the time domain [112 113] whichwas later extended to include driver behavior [114] Baker[115 116] further investigated both the steady and unsteadyaerodynamic forces acting on a variety of vehicles and carriedout extensive studies of the interaction between aerodynamicforces and moving vehicles These approaches have all beenapplied in coupled vehicle-bridge analysis For example Xuet al [101] simulated the aerodynamic wind forces actingon running road vehicles using the quasi-steady approachand Xu and Ding [117] derived and simulated the steadyand unsteady aerodynamic forces acting on a moving railwayvehicle in crosswinds in the time domain
Based on these separate studies on the various typesof dynamic interactions among wind vehicles (trains orroad vehicles) and long-span bridges several researchersin the last decade have examined the wind-vehicle-bridgecoupled system as a whole For instance studies have beencarried out on coupled road vehicle and cable-stayed bridgesystems [81 82 118] and on coupled train and cable-supportedbridge systems in crosswinds [101 117 119ndash121] In therecent years several new advances have been made bothin numerical simulation technologies and in wind tunnelmeasurements Chen et al [17] proposed a wind-vehicle-bridge framework which enables considering the dynamiceffects induced by simultaneous actions of railway highway
and wind loading and it was applied to analyze dynamicstress of long suspension bridges Li et al [122] extended thewind-vehicle-bridge couple analysis to the case of two trainsmeeting on a long-span suspension bridge Chen and Wu[118] proposed a semideterministic analytical model whichis able to consider dynamic interactions between the bridgewind and stochastic ldquorealrdquo traffic Based on the wind tunneltests Dorigatti et al [123]measured crosswind loads on high-sided vehicles over long-span bridges taking three differentvehicles (van double deck bus and lorry) and two differentbridge deck configurations into consideration Zhu et al[124] investigated aerodynamic coefficients of road vehiclesby adopting different road vehicles types wind directionsand vehicle positions Li et al [122] studied the effects ofsudden changes of wind loads as the train passing througha bridge tower or two trains passing each other by usingthe wind tunnel test rig with moving train models Hanet al [125] developed an experimental setup for measuringthe aerodynamic characteristics of vehicles and the bridgein wind tunnel and then investigated the influences ofparameters adopted in the tests
3 Applications of Simulation Technology toBridge Assessment
After reviewing the key issues of numerical simulationsfor dynamic response of long-span multiload bridges thissection will review the engineering applications of the newlydeveloped technologies to safety assessment of long-spanbridges such as assessment of fatigue and assessment underextreme events
31 Assessment of FatigueDamage Steel structures are widelyused in long-span bridges Research by the ASCE [126] indi-cates that 80ndash90 of failures in steel structures are related tofatigue and fracture Several disasters resulting from fatigue-induced bridge failure have occurred in history For instance46 people died in the collapse of the Silver Bridge (USA1967) and 32 people lost their lives in the collapse of theSungsoo Grand Bridge (South Korea 1994) These disastersteach us that fatigue is an important aspect of the safety ofsteel bridges and action should be taken to prevent similarfatigue-induced bridge failures In the past few decadesfatigue assessment of steel bridges has attracted increasingattention from both governments and bridge engineers andrelevant provisions have been stipulated in several codes andstandards [127ndash130]
12 The Scientific World Journal
It has great advantages to evaluate fatigue damage of long-span bridges based on numerical simulation especially fora multiload bridge which suffers multiple types of dynamicloading such as railway highway andwind loadingDifferentfrom sudden structural damage fatigue damage accumulateswith load-induced dynamic stress (or stress fluctuation) overthe service life of a bridge lasting for more than 100 yearsThe increase in traffic volume and gross vehicle weight thataccompany economic development is very likely to happenin the long period Numerical simulation technology can bean ideal tool to study influences of traffic growth patternsto fatigue damage of bridge In addition slender long-span bridges built in wind-prone regions also suffer fromconsiderable wind induced vibration which appears withina wide range of wind speeds and lasts for almost the wholedesign life of the bridge Given the simultaneous presenceof multiple vehicles and wind it is necessary to considercombined effects of traffic loading (railway andor highwayloading) and wind loading in the fatigue assessment Sincemultiple loading is concerned in a long time period thereare a large number of loading combinations for multipleloading in different intensities It is almost unavailable forfield measurement to obtain such complete information butnumerical simulation could be a good choice to determinedynamic responses of a long-span bridge under multipleloading
A number of structural health monitoring systems(SHMSs) have been installed on numerous recently builtlong-span bridges and a variety of sensors are used for mon-itoring bridge loadings (eg traffic wind and earthquakes)and conditions (including global and local responses) toensure bridge safety and user comfort under in-service con-ditions Well-known examples include Tsing-Ma Bridge inHong Kong Akashi Kaikyo Bridge in Japan Binzhou YellowRiver Bridge in China and Jindo Bridge in Korea Integrationof numerical simulation technologies and measurement dataof structural health monitoring systems (SHMSs) installedon long-span bridges will make the fatigue assessment morereliable for several reasons (1) it is a perfect validation byusing field measurement data of the different types of loadingas input of numerical simulation and the measured dynamicresponses for comparison with the computed ones (2) a largenumber of measured loading data could be used to establishloading databases or probabilistic models of different loads
In the recent years several researchers [7ndash10] appliedthe newly developed numerical simulation technologies tofatigue assessment of long-span bridges Chen et al [7]proposed a framework for fatigue analysis of a long-spansuspension bridge under railway highway and wind loadingby integrating computer simulation with SHMSs and itwas applied to evaluate fatigue damage of the Tsing MaSuspension Bridge over its design life as a case study Basedon this work Chen et al [8] proposed a framework forfatigue reliability analysis of long suspension bridges undermultiple loading inwhich the probabilisticmodels of railwayhighway and wind loading were established based on themeasurement data acquired from the SHMS of the TsingMa Bridge Wu et al [9] proposed a reliability-based fatigueapproach for slender long-span bridge and the combined
dynamic loading effects from wind and traffic as well asthe associated uncertainties were considered Based on theassumption that dynamic magnification related to vehicledynamics can be neglected in long suspension bridges Chenet al [8] established a framework for fatigue reliability anal-ysis To account for different types of long-span bridges withthe span length ranging from a few hundred to thousands ofmeters Zhang et al [10] proposed a more general frameworkwhich includes multiple random variables for the dynamicloads in a bridgersquos life cycle for the vehicle-bridge-winddynamic system such as road profile vehicle speed andwindvelocity and direction among other effects
32 Assessment under Extreme Events The aforementionedfatigue assessment mainly focuses on damage accumulationinduced by stress fluctuations under normal operationalcondition in a long-term period For long-span bridges inaddition to the normal operational conditions in which windspeeds are small ormoderate and traffic scenarios are normalthere are some extreme event conditions Extreme eventsmayinclude complex traffic congestion on the bridge coupledwith moderate or even strong wind [11] For example severetraffic congestions may be formed on the bridge as a resultof an evacuation or a partial blockage of driving lanes due totraffic accidents construction ormaintenance For hurricaneevacuations there are usually a lot of road vehicles passingthrough the bridge before the landfall of the hurricane whilethe wind speed may become pretty high already [131]
Although the excessive dynamic responses of the bridgesunder extreme events are rare it is also critical since itmay cause critical damage initiation or accumulation onsome local bridgemembers Furthermore the extreme events(eg heavy traffic) may even trigger the collapse of thewhole bridge by breaking the ldquoweakest linkrdquo especially whensome hidden damage or design flaw has not been detectedOne recent example is the Minnesota Bridge failure whichoccurred during rush hours with heavy traffic although trafficloads may not be the direct cause of failure For slender long-span bridges strong wind may also cause threats by workinginteractively with heavy traffic loads Therefore even thoughthe extreme cases associated with congested traffic andorwindy weather may be relatively rare and the durationscould be short it is still important for bridge engineers toappropriately look into these unusual extreme events duringstructural design and life-time management of these criticalinfrastructures [11]
The dynamic performance of long-span bridges undercombined actions of strong winds and running road vehicleshas been studied by many researchers in recent years [17 7981 82 132] Most of them studied bridge dynamic perfor-mance under road traffic inwhich only one or several vehiclesdistributed in an assumed (usually uniform) pattern on long-span bridges were considered Extreme events such as trafficcongestion coupled with strong wind were out of concernin those studies Recently Wu and Chen [11] conducteda research on the assessment of long-span bridges underextreme events which includes complex traffic congestioncoupled with moderate or even strong wind This study
The Scientific World Journal 13
applied the cellular automaton (CA) traffic model to thesimulation of the actual traffic flow through the bridgedefined representative scenarios for the extreme events andnumerically studied the bridge performance under thesepossible extreme events
4 Conclusions and Recommendations
Dynamic responses of long-span bridges are often requiredfor assessing the safety of these bridges and can be determinedby numerical simulation technologies This paper provides adetailed review of key issues involved in dynamic responseanalysis of long-span multiload bridges based on numericalsimulation including dynamic interactions between runningtrains and bridge between running road vehicles and bridgeand between wind and bridge and in the wind-vehicle-bridge coupled systemThen the review work was conductedfor engineering applications of newly developed numericalsimulation technologies to safety assessment of long-spanbridges such as assessment of fatigue damage and assessmentunder extreme event condition Although technologies fornumerical simulation of dynamic responses of long-spanmultiload bridge have achieved great advances in past fewdecades and successfully applied to several important bridgesit is still far from reach its maturity and enable to takeplace of traditional fieldmeasurementThe existing problemsand promising research efforts at least include the followingaspects
(1) After multiple types of dynamic interactions beingconsidered in the complex system computationalefficiency is a bottleneck problem for numericalsimulation of dynamic response of a long-span bridgeTypically when multiple loads are involved a largenumber of loading combinations for multiple load-ings must be considered in the assessment
(2) It is rather complex for the time-depending windloads acting on a long-span bridge and running vehi-cles especially for the case of rapid change of windloads such as a train passing through a bridge toweror two trains passing each other The aerodynamiccharacteristics of vehicles and the bridge under differ-ent loading scenarios can be determined through thewind tunnel testing and used in numerical simulationof dynamic responses of the bridge and vehicles
(3) It is a new trend to integrate numerical simulationtechnologies and measurement data of structuralhealth monitoring systems (SHMSs) installed onlong-span bridges whichmakes the safety assessmentof bridge structures more reliable Measured struc-tural responses could be used to validate numericalsimulation approach and measured loading infor-mation could be used for generating statistical orprobabilistic models of multiple loads
(4) It is important to study dynamic responses of bridgestructures under extreme events such as congestedtraffic coupled with windy weather which happens ina long-span bridge For the assessment under extreme
events using numerical simulation technologies sim-ulation of traffic flow and definition of representativescenarios of the extreme events are key issues
(5) It is necessary to consider the effects of typhoonwinds on the safety assessment of long-span bridgesin a reasonable way Few researches do this mostlybecause a probabilistic distribution of wind speedand direction specifically for typhoons is requiredfor assessment but there are insufficient measuredrecords to establish a reliable probabilistic typhoonwind model
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors wish to acknowledge the financial supportsfrom the National Natural Science Foundation of China(NSFC-51108395 and NSFC-51178366) the FundamentalResearch Funds for theCentral Universities (2012121032) andopen funding from Jiangsu Key Laboratory of EngineeringMechanics Special thanks go to the supervisor of the firstauthor Professor Y L XuHongKong PolytechnicUniversityfor his expert guidance and continuous support at all levelsthroughout his PhD study Sincere thanks should go tothe Highways Department of Hong Kong for providing theauthors with the field measurement data Any opinions andconcluding remarks presented in this paper are entirely thoseof the authors
References
[1] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[2] H Sohn C R Farrar N F Hunter and K Worden ldquoStructuralhealth monitoring using statistical pattern recognition tech-niquesrdquo Journal of Dynamic Systems Measurement and Controlvol 123 no 4 pp 706ndash711 2001
[3] W Fan and P Qiao ldquoVibration-based damage identificationmethods a review and comparative studyrdquo Structural HealthMonitoring vol 10 no 1 pp 83ndash111 2011
[4] X Q Zhu and S S Law ldquoDamage detection in simply supportedconcrete bridge structure under moving vehicular loadsrdquo Jour-nal of Vibration and Acoustics Transactions of the ASME vol129 no 1 pp 58ndash65 2007
[5] J Li and S S Law ldquoDamage identification of a target sub-structure with moving load excitationrdquoMechanical Systems andSignal Processing vol 30 pp 78ndash90 2012
[6] J Li S S Law and H Hao ldquoImproved damage identification inbridge structures subject tomoving loads numerical and exper-imental studiesrdquo International Journal of Mechanical Sciencesvol 74 pp 99ndash111 2013
14 The Scientific World Journal
[7] Z W Chen Y L Xu Y Xia Q Li and K Y Wong ldquoFatigueanalysis of long-span suspension bridges under multiple load-ing case studyrdquo Engineering Structures vol 33 no 12 pp 3246ndash3256 2011
[8] Z W Chen Y L Xu and X M Wang ldquoSHMS-based fatiguereliability analysis of multiloading suspension bridgesrdquo Journalof Structural Engineering-Asce vol 138 pp 299ndash307 2012
[9] J Wu S R Chen and J W van de Lindt ldquoFatigue assessmentof slender long-span bridges reliability approachrdquo Journal ofBridge Engineering vol 17 no 1 pp 47ndash57 2012
[10] W Zhang C S Cai and F Pan ldquoFatigue reliability assessmentfor long-span bridges under combined dynamic loads fromwinds and vehiclesrdquo Journal of Bridge Engineering vol 18 pp735ndash747 2013
[11] JWu and S R Chen ldquoProbabilistic dynamic behavior of a long-span bridge under extreme eventsrdquo Engineering Structures vol33 no 5 pp 1657ndash1665 2011
[12] J M Ko and Y Q Ni ldquoTechnology developments in structuralhealth monitoring of large-scale bridgesrdquo Engineering Struc-tures vol 27 no 12 pp 1715ndash1725 2005
[13] TH YiHN Li andHM Sun ldquoMulti-stage structural damagediagnosis method based on ldquoenergy-damagerdquo theoryrdquo SmartStructures and Systems vol 12 pp 345ndash361 2013
[14] T H Yi H N Li and M Gu ldquoFull-scale measurements ofdynamic response of suspension bridge subjected to environ-mental loads using GPS technologyrdquo Science China Technologi-cal Sciences vol 53 no 2 pp 469ndash479 2010
[15] T H Yi H N Li and M Gu ldquoExperimental assessmentof high-rate GPS receivers for deformation monitoring ofbridgerdquoMeasurement Journal of the InternationalMeasurementConfederation vol 46 pp 420ndash432 2013
[16] X W Ye Y Q Ni K Y Wong and J M Ko ldquoStatistical analysisof stress spectra for fatigue life assessment of steel bridges withstructural health monitoring datardquo Engineering Structures vol45 pp 166ndash176 2012
[17] Z W Chen Y L Xu Q Li and D J Wu ldquoDynamic stressanalysis of long suspension bridges under wind railway andhighway loadingsrdquo Journal of Bridge Engineering vol 16 no 3pp 383ndash391 2011
[18] S G Meisenholder and P Weidlinger ldquoDynamic interactionaspects of cable-stayed guideways for high speed ground trans-portationrdquoAmerican Society ofMechanical Engineers no 74 pp180ndash192 1974
[19] Q H Mao Research on the Highway Bridge Vibration Due toMoving Vehicles Tongji University Shang Hai China 1989
[20] Y L Xu J M Ko and Z Yu ldquoModal analysis of tower-cable system of Tsing Ma long suspension bridgerdquo EngineeringStructures vol 19 pp 857ndash867 1997
[21] W Guo H Xia and Y-L Xu ldquoDynamic response of a long spansuspension bridge and running safety of a train under windactionrdquo Frontiers of Architecture and Civil Engineering in Chinavol 1 no 1 pp 71ndash79 2007
[22] T T Liu Y L Xu W S Zhang K Y Wong H J Zhou and KW Y Chan ldquoBuffeting-induced stresses in a long suspensionbridge structural health monitoring oriented stress analysisrdquoWind and Structures An International Journal vol 12 no 6 pp479ndash504 2009
[23] K Y Wong ldquoStructural identification of Tsing Ma BridgerdquoTransactions Hong Kong Institution of Engineers vol 10 no 1pp 38ndash47 2003
[24] Y L XuQ Li D JWu and ZWChen ldquoStress and accelerationanalysis of coupled vehicle and long-span bridge systems usingthe mode superposition methodrdquo Engineering Structures vol32 no 5 pp 1356ndash1368 2010
[25] Y F Duan Y L Xu Q G Fei et al ldquoAdvanced finite elementmodel of Tsing Ma Bridge for structural health monitoringrdquoInternational Journal of Structural Stability and Dynamics vol11 no 2 pp 313ndash344 2011
[26] Z X Li T Q Zhou T H T Chan and Y Yu ldquoMulti-scalenumerical analysis on dynamic response and local damage inlong-span bridgesrdquo Engineering Structures vol 29 no 7 pp1507ndash1524 2007
[27] W Zhang C S Cai and F Pan ldquoFinite element modeling ofbridges with equivalent orthotropic material method for multi-scale dynamic loadsrdquo Engineering Structures vol 54 pp 82ndash932013
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[29] R S Ayre G Ford and L S Jacobsen ldquoTransverse vibration ofa two-span beam under the action of a moving constant forcerdquoJournal of Applied Mechanics vol 17 pp 1ndash12 1950
[30] R S Ayre andLS Jacobsen ldquoTransverse vibration of a two-spanbeam under the action of a moving alternating forcerdquo Journal ofApplied Mechanics vol 17 pp 283ndash290 1950
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[32] J-S Wu and C-W Dai ldquoDynamic response of multispannonuniform beam due to moving loadsrdquo Journal of StructuralEngineering vol 113 no 3 pp 458ndash474 1987
[33] W Weaver S P Timoshenko and D H Young VibrationProblems in Engineering John Wiley amp Sons 1990
[34] N H Galdos D R Schelling and M A Sahin ldquoMethodologyfor impact factor of horizontally curved box bridgesrdquo Journal ofStructural Engineering vol 119 no 6 pp 1917ndash1934 1993
[35] J A Gbadeyan and S T Oni ldquoDynamic behaviour of beamsand rectangular plates under moving loadsrdquo Journal of Soundand Vibration vol 182 no 5 pp 677ndash695 1995
[36] D Y Zheng Y K Cheung F T K Au and Y S ChengldquoVibration of multi-span non-uniform beams under movingloads by using modified beam vibration functionsrdquo Journal ofSound and Vibration vol 212 no 3 pp 455ndash467 1998
[37] G V Rao ldquoLinear dynamics of an elastic beam under movingloadsrdquo Journal of Vibration and Acoustics Transactions of theASME vol 122 no 3 pp 281ndash289 2000
[38] Y B Yang J D Yau and Y S Wu Vehicle-Bridge InteractionDynamic with Applications to High-Speed Railways WorldScientific 2004
[39] E C Ting J Genin and J H Ginsberg ldquoA general algorithm formovingmass problemsrdquo Journal of Sound and Vibration vol 33no 1 pp 49ndash58 1974
[40] S Sadiku and H H E Leipholz ldquoOn the dynamics of elasticsystems with moving concentrated massesrdquo Ingenieur-Archivvol 57 no 3 pp 223ndash242 1987
[41] J E Akin and M Mofid ldquoNumerical solution for response ofbeamswithmovingmassrdquo Journal of Structural Engineering vol115 no 1 pp 120ndash131 1989
[42] M AMahmoud andM A Abou Zaid ldquoDynamic response of abeam with a crack subject to a moving massrdquo Journal of Soundand Vibration vol 256 no 4 pp 591ndash603 2002
[43] V K Garg Dynamics of Railway Vehicle Systems AcademicPress 1994
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[44] T-LWang andDHuang ldquoCable-stayed bridge vibration due toroad surface roughnessrdquo Journal of Structural Engineering vol118 no 5 pp 1354ndash1374 1992
[45] Y-B Yang and B-H Lin ldquoVehicle-bridge interaction analysisby dynamic condensation methodrdquo Journal of Structural Engi-neering vol 121 no 11 pp 1636ndash1643 1995
[46] Y B Yang J D Yau and L C Hsu ldquoVibration of simple beamsdue to trainsmoving at high speedsrdquoEngineering Structures vol19 no 11 pp 936ndash943 1997
[47] B Tabarrok and E Esmailzadeh ldquoInduced vibration of bridgestransversed by moving vehiclesrdquo Transactions of the CanadianSociety for Mechanical Engineering B vol 24 no 1 pp 191ndash1982000
[48] C Liu T-L Wang and D Huang ldquoImpact study for multi-girder bridge based on correlated road roughnessrdquo StructuralEngineering and Mechanics vol 11 no 3 pp 259ndash272 2001
[49] K H Chu V K Garg and T L Wang ldquoImpact in railwayprestressed concrete bridgesrdquo Journal of Structural Engineeringvol 112 no 5 pp 1036ndash1051 1986
[50] T-LWang V K Garg and K-H Chu ldquoRailway bridgevehicleinteraction studies with new vehicle modelrdquo Journal of Struc-tural Engineering vol 117 no 7 pp 2099ndash2116 1991
[51] H Xia Y L Xu and T H T Chan ldquoDynamic interaction oflong suspension bridges with running trainsrdquo Journal of Soundand Vibration vol 237 no 2 pp 263ndash280 2000
[52] Q-L Zhang A Vrouwenvelder and J Wardenier ldquoNumericalsimulation of train-bridge interactive dynamicsrdquo Computersand Structures vol 79 no 10 pp 1059ndash1075 2001
[53] H Xia N Zhang and G de Roeck ldquoDynamic analysis of highspeed railway bridge under articulated trainsrdquo Computers andStructures vol 81 no 26-27 pp 2467ndash2478 2003
[54] G Diana F Cheli A Collina R Corradi and S MelzildquoThe development of a numerical model for railway vehiclescomfort assessment through comparison with experimentalmeasurementsrdquoVehicle SystemDynamics vol 38 no 3 pp 165ndash183 2002
[55] Q Li Y L Xu D J Wu and Z W Chen ldquoComputer-aided nonlinear vehicle-bridge interaction analysisrdquo Journal ofVibration and Control vol 16 pp 1791ndash1816 2010
[56] A Wiriyachai K H Chu and V K Garg ldquoBridge impact dueto wheel and track irregularitiesrdquo Journal of the EngineeringMechanics Division vol 108 no 4 pp 648ndash666 1982
[57] L Fryba Dynamics of Railway Bridges Inst of Civil Engineers1996
[58] D Huang and T-L Wang ldquoImpact analysis of cable-stayedbridgesrdquo Computers and Structures vol 43 no 5 pp 897ndash9081992
[59] W M Zhai Vehicle-Track Coupling Dynamics Chinese RailwayPress Beijing China 2007
[60] M Olsson ldquoFinite element modal co-ordinate analysis ofstructures subjected to moving loadsrdquo Journal of Sound andVibration vol 99 no 1 pp 1ndash12 1985
[61] Y-B Yang C-H Chang and J-D Yau ldquoAn element foranalysing vehicle-bridge systems considering vehiclersquos pitchingeffectrdquo International Journal for NumericalMethods in Engineer-ing vol 46 no 7 pp 1031ndash1047 1999
[62] Y B Yang and Y S Wu ldquoA versatile element for analyzingvehicle-bridge interaction responserdquo Engineering Structuresvol 23 no 5 pp 452ndash469 2001
[63] F T K Au J J Wang and Y K Cheung ldquoImpact study ofcable-stayed bridge under railway traffic using various modelsrdquo
Journal of Sound and Vibration vol 240 no 3 pp 447ndash4652001
[64] Y Q Sun andMDhanasekar ldquoA dynamicmodel for the verticalinteraction of the rail track and wagon systemrdquo InternationalJournal of Solids and Structures vol 39 no 5 pp 1337ndash13592002
[65] K Henchi M Fafard M Talbot and G Dhatt ldquoAn efficientalgorithm for dynamic analysis of bridges under moving vehi-cles using a coupledmodal and physical components approachrdquoJournal of Sound and Vibration vol 212 no 4 pp 663ndash6831998
[66] Y L Xu and L YWang ldquoAnalytical study of wind-rain-inducedcable vibration SDOFmodelrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 91 no 1-2 pp 27ndash40 2003
[67] B Biondi G Muscolino and A Sofi ldquoA substructure approachfor the dynamic analysis of train-track-bridge systemrdquoComput-ers and Structures vol 83 no 28ndash30 pp 2271ndash2281 2005
[68] J L Humar and A H Kashif ldquoDynamic response analysis ofslab-type bridgesrdquo Journal of Structural Engineering vol 121 no1 pp 48ndash62 1995
[69] P Lou and Q-Y Zeng ldquoFormulation of equations of verticalmotion for vehicle-track-bridge systemrdquo Journal of the ChinaRailway Society vol 26 no 5 p 71 2004
[70] O Coussy M Said and J-P van Hoove ldquoThe influence ofrandom surface irregularities on the dynamic response ofbridges under suspended moving loadsrdquo Journal of Sound andVibration vol 130 no 2 pp 313ndash320 1989
[71] E S Hwang and A S Nowak ldquoSimulation of dynamic loadfor bridgesrdquo Journal of Structural Engineering vol 117 pp 1413ndash1434 1991
[72] F Yang and G A Fonder ldquoAn iterative solution methodfor dynamic response of bridge-vehicles systemsrdquo EarthquakeEngineering and Structural Dynamics vol 25 pp 195ndash215 1996
[73] W Zhai and Z Cai ldquoDynamic interaction between a lumpedmass vehicle and a discretely supported continuous rail trackrdquoComputers and Structures vol 63 no 5 pp 987ndash997 1997
[74] W M Zhai and C B Cai ldquoTraintrackbridge dynamic inter-actions simulation and applicationsrdquo Vehicle System Dynamicsvol 37 pp 653ndash665 2003
[75] D Bruno F Greco and P Lonetti ldquoDynamic impact analysis oflong span cable-stayed bridges under moving loadsrdquo Engineer-ing Structures vol 30 no 4 pp 1160ndash1177 2008
[76] X D Song D J Wu and Q Li ldquoDynamic impact analysisof double-tower cable-stayed maglev bridges using a simplemodelrdquo Journal of Bridge Engineering vol 19 pp 34ndash43 2014
[77] Y-S Wu and Y-B Yang ldquoSteady-state response and ridingcomfort of trains moving over a series of simply supportedbridgesrdquoEngineering Structures vol 25 no 2 pp 251ndash265 2003
[78] P Antolin N Zhang J M Goicolea H Xia M A Astiz andJ Oliva ldquoConsideration of nonlinear wheel-rail contact forcesfor dynamic vehicle-bridge interaction in high-speed railwaysrdquoJournal of Sound and Vibration vol 332 no 5 pp 1231ndash12512013
[79] W H Guo and Y L Xu ldquoFully computerized approach to studycable-stayed bridge-vehicle interactionrdquo Journal of Sound andVibration vol 248 no 4 pp 745ndash761 2001
[80] Y L Xu and W H Guo ldquoDynamic behaviour of high-sidedroad vehicles subject to a sudden crosswind gustrdquo Wind andStructures vol 6 no 5 pp 325ndash346 2003
[81] Y L Xu and W H Guo ldquoDynamic analysis of coupled roadvehicle and cable-stayed bridge systems under turbulent windrdquoEngineering Structures vol 25 no 4 pp 473ndash486 2003
16 The Scientific World Journal
[82] C S Cai and S R Chen ldquoFramework of vehicle-bridge-winddynamic analysisrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 92 no 7-8 pp 579ndash607 2004
[83] S R Chen and J Wu ldquoModeling stochastic live load for long-span bridge based on microscopic traffic flow simulationrdquoComputers and Structures vol 89 no 9-10 pp 813ndash824 2011
[84] A V Paultre B Yang L A Bergman and C A Tan ldquoBridgedynamics and dynamic amplification factorsmdasha review ofanalytical and experimental findingsrdquoCanadian Journal of CivilEngineering vol 19 no 2 pp 260ndash278 1992
[85] H Honda Y Kajikawa and T Kobori ldquoSpectra of road surfaceroughness of bridgesrdquo Journal of the Structural Division vol 108pp 1956ndash1966 1982
[86] M J Inbanathan and M Wieland ldquoBridge vibrations dueto vehicle moving over rough surfacerdquo Journal of StructuralEngineering vol 113 no 9 pp 1994ndash2008 1987
[87] T-LWang andDHuang ldquoCable-stayed bridge vibration due toroad surface roughnessrdquo Journal of Structural Engineering vol118 no 5 pp 1354ndash1374 1992
[88] P K Chatterjee T K Datta and C S Surana ldquoVibrationsuspension bridges under vehicular movementrdquo Journal ofStructural Engineering vol 120 no 3 pp 681ndash703 1994
[89] D Chang and H Lee ldquoImpact factors for simple-span highwaygirder bridgesrdquo Journal of Structural Engineering vol 120 no 3pp 704ndash715 1994
[90] T-C Pan and J Li ldquoDynamic vehicle element method for tran-sient response of coupled vehicle-structure systemsrdquo Journal ofStructural Engineering vol 128 no 2 pp 214ndash223 2002
[91] C J Dodds and J D Robson ldquoThe description of road surfaceroughnessrdquo Journal of Sound Vibration vol 31 no 2 pp 175ndash183 1973
[92] D Huang T-L Wang and M Shahawy ldquoImpact studies ofmultigirder concrete bridgesrdquo Journal of Structural Engineeringvol 119 no 8 pp 2387ndash2402 1993
[93] A G Davenport ldquoBuffeting of a suspension bridge by stormwindrdquo Journal of Structural Division vol 88 pp 233ndash268 1962
[94] R H Scanlan ldquoThe action of flexible bridges under wind Iflutter theoryrdquo Journal of Sound and Vibration vol 60 no 2pp 187ndash199 1978
[95] Q Ding and P K K Lee ldquoComputer simulation of buffetingactions of suspension bridges under turbulentwindrdquoComputersand Structures vol 76 no 6 pp 787ndash797 2000
[96] V Boonyapinyo T Miyata and H Yamada ldquoAdvanced aerody-namic analysis of suspension bridges by state-space approachrdquoJournal of Structural Engineering vol 125 no 12 pp 1357ndash13661999
[97] Y-H Chen and C-Y Li ldquoDynamic response of elevated high-speed railwayrdquo Journal of Bridge Engineering vol 5 no 2 pp124ndash130 2000
[98] X Chen M Matsumoto and A Kareem ldquoTime domainflutter and buffeting response analysis of bridgesrdquo Journal ofEngineering Mechanics vol 126 no 1 pp 7ndash16 2000
[99] X Chen and A Kareem ldquoEquivalent static wind loads forbuffeting response of bridgesrdquo Journal of Structural Engineeringvol 127 no 12 pp 1467ndash1475 2001
[100] S R Chen and C S Cai ldquoEvolution of long-span bridgeresponse to wind-numerical simulation and discussionrdquo Com-puters and Structures vol 81 no 21 pp 2055ndash2066 2003
[101] Y L XuHXia andQ S Yan ldquoDynamic response of suspensionbridge to high wind and running trainrdquo Journal of BridgeEngineering vol 8 no 1 pp 46ndash55 2003
[102] A Guo Y L Xu and H Li ldquoDynamic performance of cable-stayed bridge tower with multi-stage pendulum mass damperunderwind excitations-II experimentrdquoEarthquake Engineeringand Engineering Vibration vol 6 no 4 pp 417ndash424 2007
[103] E Simiu and R H ScanlanWind Effects on Structures 1996[104] Y Cao H Xiang and Y Zhou ldquoSimulation of stochastic wind
velocity field on long-span bridgesrdquo Journal of EngineeringMechanics vol 126 no 1 pp 1ndash6 2000
[105] M Shinozuka and C-M Jan ldquoDigital simulation of randomprocesses and its applicationsrdquo Journal of Sound and Vibrationvol 25 no 1 pp 111ndash128 1972
[106] K M Shum Lateral and torsional vibration control of long spanbridge deck using novel tuned liquid column dampers [PhDthesis] Department of Civil and Structural Engineering TheHong Kong Polytechnic University 2004
[107] Y K Lin and J N Yang ldquoMultimode bridge response to windexcitationsrdquo Journal of EngineeringMechanics vol 109 no 2 pp586ndash603 1983
[108] Y S Lin Self-Excited Bridge Motion in Turbulent Wind 1978[109] D T Lau M S Cheung and S H Cheng ldquo3D flutter analysis
of bridges by spline finite-strip methodrdquo Journal of StructuralEngineering vol 126 no 10 pp 1246ndash1254 2000
[110] L A Balzer ldquoAtmospheric turbulence encountered by high-speed ground transport vehiclesrdquo Journal of Mechanical Engi-neering Science vol 19 pp 227ndash235 1977
[111] R K Cooper ldquoAtmospheric turbulence with respect to movingground vehiclesrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 17 no 2 pp 215ndash238 1984
[112] C J Baker ldquoA simplified analysis of various types of wind-induced road vehicle accidentsrdquo Journal of Wind Engineeringand Industrial Aerodynamics vol 22 no 1 pp 69ndash85 1986
[113] C J Baker ldquoMeasures to control vehicle movement at exposedsites during windy periodsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 25 no 2 pp 151ndash161 1987
[114] C J Baker ldquoHigh sided articulated road vehicles in strong crosswindsrdquo Journal of Wind Engineering and Industrial Aerodynam-ics vol 31 no 1 pp 67ndash85 1988
[115] C J Baker ldquoGround vehicles in high cross winds part I steadyaerodynamic forcesrdquo Journal of Fluids and Structures vol 5 no1 pp 69ndash90 1991
[116] C J Baker ldquoGround vehicles in high cross winds part IIunsteady aerodynamic forcesrdquo Journal of Fluids and Structuresvol 5 no 1 pp 91ndash111 1991
[117] Y L Xu and Q S Ding ldquoInteraction of railway vehicles withtrack in cross-windsrdquo Journal of Fluids and Structures vol 22no 3 pp 295ndash314 2006
[118] S R Chen and J Wu ldquoDynamic performance simulation oflong-span bridge under combined loads of stochastic traffic andwindrdquo Journal of Bridge Engineering vol 15 no 3 pp 219ndash2302010
[119] Y Li S Qiang H Liao and Y L Xu ldquoDynamics of wind-rail vehicle-bridge systemsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 93 no 6 pp 483ndash507 2005
[120] Y L Xu N Zhang and H Xia ldquoVibration of coupled trainand cable-stayed bridge systems in cross windsrdquo EngineeringStructures vol 26 no 10 pp 1389ndash1406 2004
[121] W W Guo Y L Xu H Xia W S Zhang and K M ShumldquoDynamic response of suspension bridge to typhoon and trainsII numerical resultsrdquo Journal of Structural Engineering vol 133no 1 pp 12ndash21 2007
The Scientific World Journal 17
[122] Y L Li H Y Xiang B Wang Y L Xu and S Z QiangldquoDynamic analysis of wind-vehicle-bridge coupling systemduring the meeting of two trainsrdquo Advances in StructuralEngineering vol 16 pp 1663ndash1670 2013
[123] F Dorigatti M Sterling D Rocchi et al ldquoWind tunnelmeasurements of crosswind loads on high sided vehicles overlong span bridgesrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 107 pp 214ndash224 2012
[124] L D Zhu L Li Y L Xu and Q Zhu ldquoWind tunnel investi-gations of aerodynamic coefficients of road vehicles on bridgedeckrdquo Journal of Fluids and Structures vol 30 pp 35ndash50 2012
[125] Y Han J X Hu C S Cai Z Q Chen and C G LildquoExperimental and numerical studies of aerodynamic forces onvehicles and bridgesrdquoWind and Structures vol 17 pp 163ndash1842013
[126] ASCE ldquoCommittee on fatigue and fracture reliability of thecommittee on structural safety and reliability of the structuraldivision fatigue reliability 1ndash4rdquo Journal of Structural Engineer-ing vol 108 pp 3ndash88 1982
[127] BS ldquoBS5400 part 10 code of practice for fatiguerdquo BritishStandard Institute 1980
[128] BS ldquoBS7608 code of practice for fatigue design and assessmentof steel structuresrdquo British Standard Institute 1993
[129] AASHTOGuide Specifications for Fatigue Evaluation of ExistingSteel Bridges 1990
[130] AASHTO Guide Manual for Condition Evaluation and Loadand Resistance Factor Rating (LRFR) of Highway Bridges 2003
[131] S R Chen C S Cai and B Wolshon ldquoFrom normal operationto evacuation Single-vehicle safety under adverse weathertopographic and operational conditionsrdquo Natural HazardsReview vol 10 no 2 pp 68ndash76 2009
[132] S R Chen and C S Cai ldquoEquivalent wheel load approach forslender cable-stayed bridge fatigue assessment under traffic andwind feasibility studyrdquo Journal of Bridge Engineering vol 12 no6 pp 755ndash764 2007
However the hybrid 3D model is still not fine enoughfor criticality analysis of bridge structures which requiresresults at strainstress level especially for some bridge detailsFor example the orthotropic decks (steel deck-plates sup-ported by U-shape troughs) were modeled by plate ele-ments with equivalent depths so that the measured resultsfrom strain gauges at the surfaces of deck-plates or U-shape troughs had no counterparts in computation resultsTherefore Duan et al [25] established the third-generationbridge model (full 3D model) for performance evaluationat stressstrain level (see Figure 2(b)) In this model themajor structural components were modeled in detail and theconnections and boundary conditions are modeled properlywhich results in about half million elements for the completebridge model The strainstress responses induced by atrain passing through the bridge were calculated by staticinfluence linemethod and comparedwithmeasured results inthe calibration
Although full 3D bridge model provides the possibilityfor exact stress analysis at the local components large com-putational efforts are needed for the refined section modelwith complicated structural details Li et al [26] proposed amultiscale FE modeling strategy for long-span bridges Theglobal structural analysis was carried out using the beamelement modeling method at the level of a meter The localdetailed hot-spot stress analysis was carried out using shellor solid elements at the level of a millimeter Based onthis model the global dynamic response of the bridge andlocal damage accumulation of two typical weld details ofthe bridge under traffic loading were numerically analyzedMultiscale FE modeling scheme was also proposed by Zhanget al [27] based on the equivalent orthotropic modelingmethod (EOMM) Bridge details withmultiple stiffenersweremodeled with shell elements using equivalent orthotropicmaterials Based on this model Zhang et al [10] computedthe dynamic stress responses of long-span bridges undercombined dynamic loads from winds and road vehicles
212 Modeling of Trains Previously running vehicles werecommonlymodeled as a series of moving forces either due tolimits on computational capacity or because it is easier to findthe analytical solutions inmany cases [28ndash37]This treatmentneglects the effect of interactions between the bridge andrunning vehicles For this reason the moving load model issuitable only for the case in which the mass of the vehicleis small relative to that of the bridge or when the vehicleresponse is not of interest [38] For cases in which the inertiaof the vehicle cannot be regarded as small a moving massmodel should be adopted instead [39ndash42] More recently theemergence of high-performance computers and advances incomputer technology hasmade it feasible tomore realisticallymodel the dynamic properties of the various components ofmoving vehicles [43ndash48]
In a more sophisticated railway vehicle model the sus-pension mechanisms are modeled by springs the dampingeffect of the suspension systems and air-cushion by dashpotsand the energy dissipating effect of the interleaf mechanismby frictional devices Using this technique a tractor-traileris represented as two discrete masses each of which issupported by two sets of springs and dashpots or frictionaldevices [38] To represent the various dynamic propertiesof railway vehicles vehicle models that contain dozens ofdegrees of freedom (DOFs) have been devised and used by[49ndash52] To investigate the dynamic interaction between along suspension bridge and running trains Xia et al [51]considered a train composed of a sequence of identicalrailway vehicles Each railway vehicle was assumed to consistof a rigid car body resting on front and rear bogies with eachbogie supported by two wheelsets (see Figure 3) Five DOFswere assigned to the car body and to each bogie to accountfor vertical lateral rolling yawing and pitching motions Incontrast only three DOFs were assigned to each wheelset toaccount for vertical lateral and rolling motions
Many vehicle models have been established for vehicle-bridge interaction analysis In most of these studies
The Scientific World Journal 5
(a) (b) (c)
Car body
Connecting rigid-arm
Wheel
Wheel-rail contactConstrait
Bogie
Primary suspension
Secondary suspension
(d)
Figure 4 Finite element model of a railway vehicle (a) elevation view (b) side view (c) isometric view (d) model details [52]
the equations of motion of the vehicles were derivedanalytically However a great inconvenience of this methodis that the equations of motion of the whole vehicle-bridgesystem must be rederived if the vehicle type is changedFurthermore it is very difficult to derive the equation ofmotion for a complex vehicle model containing a largenumber of DOFs such as the articulated components of aTGV train with an 85-DOF dynamic system [53] Generalcommercial FE software has recently been adopted to makevehicle modeling more easily applicable for different vehicletypes [54] Li et al [55] described a four-step procedurefor modeling a four-axle railway vehicle by beam elements(1) the nodes and elements for the car body bogies andwheelsets respectively are defined by using beam elements sothat the spatial geometric configuration of each componentcan be built (see Figure 4) (2) sectional properties andmaterial properties are assigned to each beam element(3) rigid-arms and suspension units (systems) are used toconnect the three components (4) constraints are assignedto form a complete finite element model of the vehicle
213 Modeling of Rail Irregularities Track irregularities rep-resent an important source of excitation for bridges duringthe passage of railway vehicles Track irregularitiesmay occuras a result of initial installation errors the degradation ofsupport materials or the dislocation of track joints Fourgeometric parameters can be used to quantitatively describerail irregularities the vertical profile cross level alignmentand gauge [49 50 56] Vertical profile and cross level irreg-ularities chiefly influence the vertical vibrations of vehiclesand of the bridge whereas alignment gauge and crosslevel irregularities initiate horizontal transverse vibrationsof vehicles and the bridge and also the torsional movementof the bridge [57] Track irregularities may be periodic orrandom Random irregularities are due to wear clearancesubsidence and insufficient maintenance For engineeringapplications random irregularities can be approximatelyregarded as stationary and ergodic processes that can begenerated from measured results or simulated by numericalmethods Several numerical methods have been proposedfor the simulation of random rail irregularities such as
6 The Scientific World Journal
the trigonometry series white noise filtration autoregressive(AR) and power spectral density (PSD) sampling methodsAmong these methods the PSD sampling method has beenwidely adopted due to its high computational accuracy Thelateral and vertical irregularities could be all assumed tobe zero-mean stationary Gaussian random processes andexpressed through the inverse Fourier transformation of aPSD function [58]
119910119904(119909) =
119873
sum119896=1
radic2119878 (119891119896) Δ119891 cos (2120587119891
119896119909 + 120579119896) (1)
where 119878(119891) is the PSD function 119891119896= 119891l + (119896 minus 12)Δ119891
Δ119891 = (119891119906minus 119891l)119873 119891119906 and 119891119897 are the upper and lower cutoff
frequencies respectively and 120579119896is the random phase angle
uniformly distributed between 0 and 2120587 Rail irregularity inrailway engineering is commonly represented by a one-sidedPSD function
The PSD functions of rail irregularities have been devel-oped by different countries Based on the PSD functions ofrail irregularities developed by the Research Institute of theChina Railway Administration Zhai [59] expressed all railirregularities using the unified rational formula as follows
where 119891 = 1120578 (mminus1) is the spatial frequency in cyclem (120578 isthe wavelength) and119860 to119866 are the parameters recommendedby Zhai [59] specifically for vertical and lateral rail irregular-ities
214 Solution Methods The dynamic analysis of vehicle-bridge coupled system requires two sets of equations ofmotion for the bridge and vehicles respectively Thesedescribe the interaction or contact forces at the contactpoints of the two subsystems Because the contact pointsmove from time to time the system matrices are generallytime dependent and must be updated and factorized at eachtime step The various solution methods can be generalizedinto two groups according to whether or not an iterativeprocedure is needed at each time step
The first group ofmethods solves the equations of motionof a coupled vehicle-bridge system at each time step withoutiteration This approach has been widely used in coupledvehicle-bridge analysis [51 53 60ndash69] These methods havegood computational stability and are convenient for dealingwith vehicle-bridge interaction problems when the vehiclemodel is relatively simple The main disadvantage is thatthe equations of motion of the coupled system are timedependent and thus the characteristic matrices must bemodified at each time step In addition the equations ofmotion of the coupled vehicle-bridge system become verydifficult to determine if nonlinear wheel-rail contacts andnonlinear vehicle models are considered
The second group of methods solves the equations for thevehicles and bridge separately and requires an iterative pro-cess to obtain convergence for the displacements of the vehi-cles and bridge at all contact points Given that the conditions
of wheel-rail contact geometry and contact forces are rathercomplex a stable integration method adopting a small timeinterval is needed for obtaining the convergence of vehicleand bridge subsystems at the contact points in each time stepMany studies have applied this type of method to investigatevehicle-bridge interactions [70ndash76] The advantage of thesemethods is that the dynamic propertymatrices in the two setsof equations of motion remain constant which is convenientfor the consideration of nonlinear vehicle-bridge interactionsand nonlinear vehicle models [55] However in engineeringapplications the iterative convergence is a critical problemwith this type of method The low convergence rate andoccasional divergence of the solution have also been noted[77] Li et al [55] investigated the performance of theseiterative schemes using the Wilson-120579 method Newmark-120573method and an explicit integrationmethod proposed byZhai[59] and found that the latter gave amuch higher convergencerate than the former two methods
Most of the above methods solved the equations ofmotion of a coupled vehicle-bridge systemusing the nonjumpmodel which assumes that the moving vehicle travelingalong the bridge is always in contact with the rails nomatter what the sign is of the contact forces This is notalways true in view of the physics of the moving vehiclewhich simply sits on the upper surfaces of the rails Theinteraction forces between the moving vehicle and the bridgedepend on the motions of the vehicle the flexibility of thebridge and the track irregularities Li et al [55] utilizeda jump model to solve vehicle-bridge interaction problemusing a noniterative Runge-Kutta method and found thatthe acceleration responses of the car body using the wheel-jump model are smaller than those using the wheel nonjumpmodel when the vehicle speed exceeds 300 kmhr Antolin etal [78] proposed a nonlinear wheel-rail interaction modelwhich considers nonlinear wheel-rail contact forces in theinteraction as well as realistic wheel and rail profiles andapplied it for analysis of dynamic interaction between highspeed trains and bridges
22 Dynamic Interaction between Road Vehicles and BridgeSection 21 gave a detailed literature review of the dynamicinteractions between trains and bridges As there are somefundamental differences between trains and road vehiclesthis section reviews the modeling of road vehicles thesimulation of road vehicle flow and the modeling of roadsurface roughness
221 Modeling of Road Vehicles To analyze the dynamicinteraction between a bridge and running road vehicles amodel of road vehicles must be established A sophisticatedroad vehiclemodel is required tomake the simulation as real-istic as possible A road vehicle is modeled as a combinationof several rigid bodies each of which is connected by a set ofsprings and dashpots which model the elastic and dampingeffects of the tires and suspension systems respectivelyThereare various configurations of road vehicles such as a tractorand trailer with different axle spacing Road vehicle models
The Scientific World Journal 7
L11 L10 L9
L8
Ku5Cu5
Cl5Kl5
Zs5
Ku4Cu4
Cl4Kl4
Zs4
Ku3 Cu3
Cl3Kl3
Zs3
Ku2 Cu2
Cl2Kl2
Zs2
Ku1 Cu1
Cl1Kl1
Zs1
Ku1 Cu1
Cl1Kl1
Zs1
Ku6 Cu6
Cl6Kl6
Zs6
b1 b1
x yL1L2L3L4L5L6L7
Z3 1205793 Z2 Z1
Z1
1205792
1205791
zz
1206011
Figure 5 Dynamic model of a tractor-trailer [79]
zz
x y
h
2b1
ZZY
h1
120601
Kuz1 Kuz3Cuz1
Kuz1 Cuz1 Cuz3
Zs1Zs1
Zs3
Cuy1 Cuy3
Kuy1Kuy3
Ys1 Ys3
Klz1Klz1 Klz3Clz1 Clz3Clz1
Kuz2 Cuz2
Zs2
Klz2 Clz2
120579
L1L2
Kly1
Cly1
Kly3
Cly3
Figure 6 Dynamic model of a high-sided road vehicle [80]
that contain several DOFs have been devised for vehicle-bridge interaction analysis For example Guo and Xu [79]modeled a 17-DOF four-axle heavy tractor-trailer vehicle (seeFigure 5) to investigate the interaction between vehicles anda cable-stayed bridge A total of three DOFs were assignedto rigid bodies representing either the tractor or the trailerto account for vertical rolling and pitching motions Onlyone DOFwas assigned to the rigid body representing the axleset moving in the vertical direction Different vehicle modelsare adopted in wind-vehicle-bridge interaction analyses Xuand Guo [80] modeled a 13-DOF two-axle road vehicle (seeFigure 6) for the dynamic analysis of a coupled road vehicleand bridge system under turbulent wind Five DOFs wereassigned to the vehicle body with respect to its center ofgravity to account for vertical lateral rolling yawing andpitching motions and two DOFs were assigned to the frontand rear axle sets to account for motions in the vertical andlateral directionMoreDOFs are needed to account for lateralcrosswinds
222 Simulation of Road Vehicle Flow On long-span bridgesthere is a high probability of the simultaneous presence ofmultiple road vehicles including heavy trucks This maylead to larger amplitude stress responses and greater fatiguedamage of the local bridge components than would be thecase with only one road vehicle The simulation of roadvehicle flow is thus important in the analysis of the dynamicinteraction between road vehicles and bridges Rather simplepatterns of road vehicle flow have been assumed in most
vehicle-bridge coupled dynamic analyses [79 81 82] in whicheither one or several vehicles are distributed on the bridgein an assumed (usually uniform) pattern Obviously suchassumptions do not represent actual road traffic conditionsRecently Chen and Wu [83] modeled the stochastic trafficload for a long-span bridge based on the cellular automaton(CA) traffic flow simulation technique In this study theysimulated a complicated road vehicle flow on long-spanbridges in terms of vehicle number vehicle type combinationand driver operation characteristics such as lane changingacceleration or deceleration
223 Modeling of Road Surface Roughness Road surfaceroughness is an important factor that greatly affects vehicle-bridge interactions Paultre et al [84] pointed out that roadsurface or pavement roughness can significantly affect theimpact response of a bridge The roughness or surface profiledepends primarily on the workmanship involved in theconstruction of the pavement or roadway and how it is main-tained which although random in nature may contain someinherent frequencies [38] In most cases surface roughnesswhich is three-dimensional in reality is often approximatedby a two-dimensional profile To account for its randomnature the road profile can be modeled as a stationaryGaussian random process and derived using a certain powerspectral density function Other methods similar to this havebeen widely adopted by researchers studying vehicle-inducedbridge vibration [65 70 71 85ndash90] Dodds and Robson [91]developed power spectral density functions that were later
8 The Scientific World Journal
modified and used byWang and Huang [87] and Huang et al[92] This approach was also adopted by literatures [79 81] intheir dynamic analyses of coupled vehicle-bridge and wind-vehicle-bridge systems
23 Dynamic Interaction between Wind and Bridge Whena long-span cable-supported bridge is immersed in a givenflow field the bridge will be subject to mean and fluctuatingwind forces To simulate these forces a linear approximationof the time-averaged static and time-varying buffeting andself-excited force components must be formulated [93 94]As dynamic bridge responses are of concern in this study onlybuffeting and self-excited forces are considered and reviewedin this section
231 Buffeting Forces Buffeting action is a random vibrationcaused by turbulent wind that excites certain modes of vibra-tion across a bridge depending on the spectral distributionof the pressure vectors [95] Although the buffeting responsemay not lead to catastrophic failure it can lead to structuralfatigue and affect the safety of passing vehicles [96] Hencebuffeting analysis has received much attention in recentyears in research into the structural safety of bridges underturbulent wind action [81 95 97ndash102]
By assuming no interaction between buffeting forces andself-excited forces and using quasi-steady aerodynamic forcecoefficients the buffeting forces per unit span Feibf on the 119894thsection of a bridge deck can be expressed as [103]
bf are the buffeting drag lift andmoment respectively 119906
119894and 119908
119894are the horizontal and
vertical components respectively of fluctuating wind at the119894th section 120588 is the air density 119880
119894is the mean wind speed
at the 119894th section of the bridge deck 119861119894and 119871
119894are the width
and length of the 119894th bridge section 119862119863119894 119862119871119894 and 119862
119872119894are
the drag lift and moment coefficients respectively of the119894th bridge segment 1198621015840
119863119894= 119889119862
1198631198941198891205721015840 1198621015840
119871119894= 119889119862
1198711198941198891205721015840 and
1198621015840119872119894
= 1198891198621198721198941198891205721015840 1205721015840 is the angle of attack of a normal
wind incident on the horizontal plane of the deck and 120594119863bu
120594119863bw
120594119871bu
120594119871bw
120594119872bu
and 120594119872bw
are the aerodynamic transferfunctions between the fluctuating wind velocities and thebuffeting forces
It can be found from this equation that a series oftime histories of fluctuating wind velocity 119906
119894 119908119894119879 in the
longitudinal and vertical directions at various points alongthe bridge deck is needed to carry out a detailed buffetinganalysis To simulate the stochastic wind velocity field thefast spectral representation method proposed by Cao et al[104] that is based on the spectral representation methoddeveloped by Shinozuka and Jan [105] is often adopted Thismethod rests on the assumptions that (1) the bridge deck ishorizontal at the same elevation (2) the mean wind speedand wind spectra do not vary along the bridge deck and(3) the distance between any two successive points wherewind speeds are simulated is the same The time historiesof the along-wind component 119906(119905) and the upward windcomponent 119908(119905) at the jth point can be generated using thefollowing equations [104]
119906119895(119905) = radic2 (Δ120596)
119895
sum119898=1
119873119891
sum119896=1
radic119878119906119906(120596119898119896)
times 119866119895119898(120596119898119896) cos (120596
119898119896119905 + 120593119898119896)
(4a)
119908119895(119905) = radic2 (Δ120596)
119895
sum119898=1
119873119891
sum119896=1
radic119878119908119908(120596119898119896)
times 119866119895119898(120596119898119896) cos (120596
119898119896119905 + 120593119898119896)
(4b)
119866119895119898(120596) =
0 when 1 le 119895 lt 119898 le 119899
119862|119895minus119898| when 119897 = 1 119898 le 119895 le 119899
119862|119895minus119898|radic(1 minus 1198622) when 2 le 119898 le 119895 le 119899
(4c)
119862 = exp(minus120582120596119898119896Δ
2120587119880) Δ =
119871
119899119901minus 1
(4d)
120596119898119896= (119896 minus 1) Δ120596 +
119898
119899Δ120596 (119896 = 1 2 119873
119891) (4e)
where Δ120596 = 120596up119873119891 is the frequency interval between thespectral lines119873
119891is the total number of frequency intervals
120596up is the upper cutoff frequency n is the total number ofpoints at which wind speeds are simulated 119878uu and 119878ww arethe along-wind and vertical wind spectrum respectively 120593lmis a random variable that is uniformly distributed between 0and 2120587 L is the span length and 120582 is a parameter that usuallyfalls between 7 and 10
In reality the equivalent buffeting forces in (3) are actuallyassociated with the spatial distribution of the wind pressureson the surface of the bridge deck Ignoring the spatialdistribution or aerodynamic transfer function of the buffetingforces across the cross-section of the bridge deck may havea considerable impact on the accuracy of buffeting response
The Scientific World Journal 9
wi(t) UI + ui(t)
j
j + 1
k
Leibf Lkibf
Meibf
Deibf
Dkibf
ce
Figure 7 Buffeting wind pressures and buffeting forces at nodes [22]
predictions Furthermore the local structural behavior ofthe bridge deck associated with local stresses and strainswhich are prone to causing local damage cannot be predicteddirectly by the current approaches based on equivalentbuffeting forces In this regard Liu et al [22] proposed anapproach to consider the spatial distribution of buffetingforces on a bridge deck structure based on wind pressuredistributions from wind tunnel tests (see Figure 7)
232 Self-Excited Forces In addition to buffeting actionflutter instability caused by self-excited forces induced bywind-structure interactions is an important considerationin the design and construction of long-span suspensionbridges [96] because the additional energy injected into theoscillating structure by the aerodynamic forces increases themagnitude of vibration sometimes to catastrophic levels [95]The self-excited forces on a bridge deck are attributable tothe interactions between wind and the motion of the bridgeWhen the energy of motion extracted from the flow exceedsthe energy dissipated by the system through mechanicaldamping the magnitude of vibration can reach catastrophiclevels [106] Expressing self-excited forces in the form ofindicial functions was first suggested by Scanlan [94] Basedon the assumption that self-excited forces are generated in alinear fashion Lin and Yang [107] simplified the self-excitedforces acting on a bridge deck and expressed them in termsof convolution integrals between the bridge deck motion andthe impulse response functions
119891se119890119863(119905)
=1
21205881198802
int119905
minusinfin
[119868119863ℎ(119905 minus 120591) ℎ
119890(120591) + 119868
119863119902(119905 minus 120591) 119902
119890(120591)
+119861119868119863120579(119905 minus 120591) 120579
119890(120591)
] 119889120591
(5a)
119891se119890119871(119905)
=1
21205881198802
int119905
minusinfin
[119868119871ℎ(119905 minus 120591) ℎ
119890(120591) + 119868
119871119902(119905 minus 120591) 119902
119890(120591)
+119861119868119871120579(119905 minus 120591) 120579
is the impulse function of the self-excited forces in which120595 represents the corresponding force components and heqe and 120579119890 are the equivalent vertical lateral and torsionaldisplacements respectively at the center of elasticity of thebridge deck section The relationship between the aero-dynamic impulse functions and flutter derivatives can beobtained by taking the Fourier transform of (7) [98]
119868119863ℎ(120596) = 119870
2
(119875lowast
6+ 119894119875lowast
5) 119868
119863119902(120596) = 119870
2
(119875lowast
4+ 119894119875lowast
1)
119868119863120579(120596) = 119870
2
(119875lowast
3+ 119894119875lowast
2)
119868119871ℎ(120596) = 119870
2
(119867lowast
4+ 119894119867lowast
1) 119868
119871119902(120596) = 119870
2
(119867lowast
6+ 119894119867lowast
5)
119868119871120579(120596) = 119870
2
(119867lowast
3+ 119894119867lowast
2)
119868119872ℎ(120596) = 119870
2
(119860lowast
4+ 119894119860lowast
1) 119868
119872119902(120596) = 119870
2
(119860lowast
6+ 119894119860lowast
5)
119868119872120579(120596) = 119870
2
(119860lowast
3+ 119894119860lowast
2)
(6)
where the overbars denote the Fourier transform operationthe terms containing 119894 represent imaginary parts119875lowast
120595 119867lowast
120595 and
119860lowast120595(120595 = 1 2 6) are dimensionless flutter derivatives
obtained from wind tunnel tests 119870 = 120596119861119880 is the reducedfrequency and 120596 is the circular frequency of vibration
According to classical airfoil theory the impulse func-tions can reasonably be approximated by a rational function[108]
119868 (120596) = [1198621+ 1198941198622
2120587
]+
119898
sum119897=1
119862119897+2
41205872 + 1198942120587119889119897+2
]1198892119897+2
V2 + 41205872] (7)
where the value of 119898 determines the level of accuracy ofthe approximation 119862
1 1198622 119862119897+2
and 119889119897+2(119897 = 1 2 119898)
are the frequency independent coefficients and ] = 2120587119870
is the reduced mean wind velocity By equating the realand imaginary parts in the comparison of (8) and (9) therelationship between the dimensionless flutter derivativesand the coefficients 119862120595
1 119862120595
2 119862120595
119897+2 and 119889120595
119897+2(where 120595 = Dh
Dq 119863120579 Lh Lq 119871120579 Mh Mq 119872120579 and 119897 = 1 2 119898) canbe establishedThese coefficients are determined by using thenonlinear least-squares method to fit the measured flutter
10 The Scientific World Journal
derivatives at different reduced frequencies The expressionof the aerodynamic impulse functions in the time domaincan be obtained by taking the inverse Fourier transform ofthe impulse functions By substituting the related impulseresponse functions into (5b) the self-excited lift force at the119894th section of bridge deck can then be derived as
119891se119890119871119894
=1
21205881198861198802
119894119861119894119862119871120579
1119894120579119894(119905) + 119862
119871120579
2119894(119861119894
119880119894
) 120579119894(119905)
+ 119862119871120579
3119894(119861119894
119880119894
) 120579119894(119905) +
119898
sum119897=1
119862119871120579
119897+3119894
times int119905
minusinfin
120579119894(119905) exp[minus
119889119871120579119897+3119894119880119894
119861119894
(119905 minus 120591)] 119889120591
+1
21205881198861198802
119894119862119871ℎ
1119894ℎ119894(119905) + 119862
119871ℎ
2119894(119861119894
119880119894
) ℎ119894(119905)
+ 119862119871ℎ
3119894(119861119894
119880119894
) ℎ119894(119905) +
119898
sum119897=1
119862119871ℎ
119897+3119894
times int119905
minusinfin
ℎ119894(119905) exp[minus
119889119871ℎ119897+3119894119880119894
119861119894
(119905 minus 120591)] 119889120591
+1
21205881198861198802
119894
119862119871119902
1119894119902119894(119905) + 119862
119871119902
2119894(119861119894
119880119894
) 119902119894(119905)
+ 119862119871119902
3119894(119861119894
119880119894
) 119902119894(119905) +
119898
sum119897=1
119862119871119902
119897+3119894
times int119905
minusinfin
119902120595(119905) exp[
[
minus119889119871119901
119897+3119894119880119894
119861119894
(119905 minus 120591)]
]
119889120591
(8)
In practice the terms 1198621198711205793119894 119862119871ℎ3119894 and 119862119871119902
3119894 which are related to
additional aerodynamic masses are normally neglected andthe value of 119898 is often taken as 2 [101] Similar formulationsfor self-excited drag and moment can be derived withanalogous definitions The self-excited forces at the 119894th nodeof the bridge deck can thus be expressed as
Feise = EeiXei + GeiXei + Feise (9)
where
Xei =
0
ℎei119902ei120579ei0
0
Eei =1
21205881198802
119894
[[[[[[[[
[
0 0 0 0 0 0
0 119862119871ℎ
1119894119862119871119902
11198941198611198941198621198711205791119894
0 0
0 119862119863ℎ1119894
119862119863119902
11198941198611198941198621198631205791119894
0 0
0 119861119894119862119872ℎ1119894
119861119894119862119872119902
111989411986121198941198621198721205791119894
0 0
0 0 0 0 0 0
0 0 0 0 0 0
]]]]]]]]
]
Gei =1
21205881198802
119894119861119894
[[[[[[[[
[
0 0 0 0 0 0
0 119862119871ℎ
2119894119862119871119902
21198941198611198941198621198711205792119894
0 0
0 119862119863ℎ2119894
119862119863119902
21198941198611198941198621198631205792119894
0 0
0 119861119894119862119872ℎ2119894
119861119894119862119872119902
211989411986121198941198621198721205792119894
0 0
0 0 0 0 0 0
0 0 0 0 0 0
]]]]]]]]
]
Feise =
[[[[[[[
[
0
eise
119863eise
eise0
0
]]]]]]]
]
=
[[[[[[[[[[[[[[[[
[
02
sum119897=1
119862119871119902
l+3119894119881119871119902
l+3119894 +2
sum119897=1
119862119871ℎl+3119894119881119871ℎ
l+3119894 +2
sum119897=1
119862119871120579l+3119894119881119871120579
l+3119894
2
sum119897=1
119862119863119902
l+3119894119881119863119902
l+3119894 +2
sum119897=1
119862119863ℎl+3119894119881119863ℎ
l+3119894 +2
sum119897=1
119862119863120579l+3119894119881119863120579
l+3119894
2
sum119897=1
119862119872119902
l+3119894119881119872119902
l+3119894 +2
sum119897=1
119862119872ℎl+3119894119881119872ℎ
l+3119894 +2
sum119897=1
119862119872120579l+3119894119881119872120579
l+3119894
0
0
]]]]]]]]]]]]]]]]
]
(10)
where 119881120595l+3119894 (120595 = Dh Dq 119863120579 Lh Lq 119871120579 Mh Mq) are theconvolution integrations of the 119894th node and can be calculatedusing a recursive algorithm For example
(11)The self-excited forces expressed by (9) relate to the centerof elasticity of the 119894th deck section Hence the force modelmust be distributed to the nodal points of the section Adistribution based on the rigid body motion relationshipsbetween themotions at the nodal point and those at the centerof elasticity of the deck section [109] was applied by Liu etal [22] In this study by applying the virtual work principlethe self-excited forces at the center of elasticity of the givensection were distributed to all nodes (see Figure 8)
24 Dynamic Interactions in a Wind-Vehicle-Bridge SystemWhen trains and road vehicles are running on long-spanbridges under crosswinds complicated dynamic interac-tions occur among the trains road vehicles cable-supported
The Scientific World Journal 11
Ui h
p
120572
Lsec i
Msec i
Dsec i
Fsekiz
Fsekiy
ce
Figure 8 Self-excited forces at the centre of elasticity and at the nodes in the 119894th deck section [22]
bridge and windThe buffeting response of the bridge due tocrosswind is superimposed on the dynamic response of thebridge due to railway and road vehicles The large vibrationof the bridge will in turn considerably affect the safetyand ride comfort of the drivers of the road vehicles Thusthe dynamic responses of a coupled vehicle-bridge systemunder crosswinds are of great concern to both engineers andresearchers
Detailed reviews of the dynamic interactions betweentrains and bridges between road vehicles and bridges andbetween wind and bridges have been given in the previoussections However the interaction between wind and vehiclesmust also be taken into account in a coupled wind-vehicle-bridge analysis Many studies have investigated wind-vehicleinteractions in the past few decades Balzer [110] developeda theory to estimate the aerodynamic forces on a movingvehicle using Taylorrsquos hypothesis of ldquofrozen turbulencerdquo Forengineering applications Cooper [111] proposed the powerspectral density (PSD) square-root coherence functionphase-lag function and aerodynamic admittance functionto model the unsteady side forces on a moving vehicle andlaid down the foundations for investigating the effects ofwind on a moving vehicle in the frequency domain Bakerdeveloped a theoretical model that describes the dynamics ofvehicles in crosswinds in the time domain [112 113] whichwas later extended to include driver behavior [114] Baker[115 116] further investigated both the steady and unsteadyaerodynamic forces acting on a variety of vehicles and carriedout extensive studies of the interaction between aerodynamicforces and moving vehicles These approaches have all beenapplied in coupled vehicle-bridge analysis For example Xuet al [101] simulated the aerodynamic wind forces actingon running road vehicles using the quasi-steady approachand Xu and Ding [117] derived and simulated the steadyand unsteady aerodynamic forces acting on a moving railwayvehicle in crosswinds in the time domain
Based on these separate studies on the various typesof dynamic interactions among wind vehicles (trains orroad vehicles) and long-span bridges several researchersin the last decade have examined the wind-vehicle-bridgecoupled system as a whole For instance studies have beencarried out on coupled road vehicle and cable-stayed bridgesystems [81 82 118] and on coupled train and cable-supportedbridge systems in crosswinds [101 117 119ndash121] In therecent years several new advances have been made bothin numerical simulation technologies and in wind tunnelmeasurements Chen et al [17] proposed a wind-vehicle-bridge framework which enables considering the dynamiceffects induced by simultaneous actions of railway highway
and wind loading and it was applied to analyze dynamicstress of long suspension bridges Li et al [122] extended thewind-vehicle-bridge couple analysis to the case of two trainsmeeting on a long-span suspension bridge Chen and Wu[118] proposed a semideterministic analytical model whichis able to consider dynamic interactions between the bridgewind and stochastic ldquorealrdquo traffic Based on the wind tunneltests Dorigatti et al [123]measured crosswind loads on high-sided vehicles over long-span bridges taking three differentvehicles (van double deck bus and lorry) and two differentbridge deck configurations into consideration Zhu et al[124] investigated aerodynamic coefficients of road vehiclesby adopting different road vehicles types wind directionsand vehicle positions Li et al [122] studied the effects ofsudden changes of wind loads as the train passing througha bridge tower or two trains passing each other by usingthe wind tunnel test rig with moving train models Hanet al [125] developed an experimental setup for measuringthe aerodynamic characteristics of vehicles and the bridgein wind tunnel and then investigated the influences ofparameters adopted in the tests
3 Applications of Simulation Technology toBridge Assessment
After reviewing the key issues of numerical simulationsfor dynamic response of long-span multiload bridges thissection will review the engineering applications of the newlydeveloped technologies to safety assessment of long-spanbridges such as assessment of fatigue and assessment underextreme events
31 Assessment of FatigueDamage Steel structures are widelyused in long-span bridges Research by the ASCE [126] indi-cates that 80ndash90 of failures in steel structures are related tofatigue and fracture Several disasters resulting from fatigue-induced bridge failure have occurred in history For instance46 people died in the collapse of the Silver Bridge (USA1967) and 32 people lost their lives in the collapse of theSungsoo Grand Bridge (South Korea 1994) These disastersteach us that fatigue is an important aspect of the safety ofsteel bridges and action should be taken to prevent similarfatigue-induced bridge failures In the past few decadesfatigue assessment of steel bridges has attracted increasingattention from both governments and bridge engineers andrelevant provisions have been stipulated in several codes andstandards [127ndash130]
12 The Scientific World Journal
It has great advantages to evaluate fatigue damage of long-span bridges based on numerical simulation especially fora multiload bridge which suffers multiple types of dynamicloading such as railway highway andwind loadingDifferentfrom sudden structural damage fatigue damage accumulateswith load-induced dynamic stress (or stress fluctuation) overthe service life of a bridge lasting for more than 100 yearsThe increase in traffic volume and gross vehicle weight thataccompany economic development is very likely to happenin the long period Numerical simulation technology can bean ideal tool to study influences of traffic growth patternsto fatigue damage of bridge In addition slender long-span bridges built in wind-prone regions also suffer fromconsiderable wind induced vibration which appears withina wide range of wind speeds and lasts for almost the wholedesign life of the bridge Given the simultaneous presenceof multiple vehicles and wind it is necessary to considercombined effects of traffic loading (railway andor highwayloading) and wind loading in the fatigue assessment Sincemultiple loading is concerned in a long time period thereare a large number of loading combinations for multipleloading in different intensities It is almost unavailable forfield measurement to obtain such complete information butnumerical simulation could be a good choice to determinedynamic responses of a long-span bridge under multipleloading
A number of structural health monitoring systems(SHMSs) have been installed on numerous recently builtlong-span bridges and a variety of sensors are used for mon-itoring bridge loadings (eg traffic wind and earthquakes)and conditions (including global and local responses) toensure bridge safety and user comfort under in-service con-ditions Well-known examples include Tsing-Ma Bridge inHong Kong Akashi Kaikyo Bridge in Japan Binzhou YellowRiver Bridge in China and Jindo Bridge in Korea Integrationof numerical simulation technologies and measurement dataof structural health monitoring systems (SHMSs) installedon long-span bridges will make the fatigue assessment morereliable for several reasons (1) it is a perfect validation byusing field measurement data of the different types of loadingas input of numerical simulation and the measured dynamicresponses for comparison with the computed ones (2) a largenumber of measured loading data could be used to establishloading databases or probabilistic models of different loads
In the recent years several researchers [7ndash10] appliedthe newly developed numerical simulation technologies tofatigue assessment of long-span bridges Chen et al [7]proposed a framework for fatigue analysis of a long-spansuspension bridge under railway highway and wind loadingby integrating computer simulation with SHMSs and itwas applied to evaluate fatigue damage of the Tsing MaSuspension Bridge over its design life as a case study Basedon this work Chen et al [8] proposed a framework forfatigue reliability analysis of long suspension bridges undermultiple loading inwhich the probabilisticmodels of railwayhighway and wind loading were established based on themeasurement data acquired from the SHMS of the TsingMa Bridge Wu et al [9] proposed a reliability-based fatigueapproach for slender long-span bridge and the combined
dynamic loading effects from wind and traffic as well asthe associated uncertainties were considered Based on theassumption that dynamic magnification related to vehicledynamics can be neglected in long suspension bridges Chenet al [8] established a framework for fatigue reliability anal-ysis To account for different types of long-span bridges withthe span length ranging from a few hundred to thousands ofmeters Zhang et al [10] proposed a more general frameworkwhich includes multiple random variables for the dynamicloads in a bridgersquos life cycle for the vehicle-bridge-winddynamic system such as road profile vehicle speed andwindvelocity and direction among other effects
32 Assessment under Extreme Events The aforementionedfatigue assessment mainly focuses on damage accumulationinduced by stress fluctuations under normal operationalcondition in a long-term period For long-span bridges inaddition to the normal operational conditions in which windspeeds are small ormoderate and traffic scenarios are normalthere are some extreme event conditions Extreme eventsmayinclude complex traffic congestion on the bridge coupledwith moderate or even strong wind [11] For example severetraffic congestions may be formed on the bridge as a resultof an evacuation or a partial blockage of driving lanes due totraffic accidents construction ormaintenance For hurricaneevacuations there are usually a lot of road vehicles passingthrough the bridge before the landfall of the hurricane whilethe wind speed may become pretty high already [131]
Although the excessive dynamic responses of the bridgesunder extreme events are rare it is also critical since itmay cause critical damage initiation or accumulation onsome local bridgemembers Furthermore the extreme events(eg heavy traffic) may even trigger the collapse of thewhole bridge by breaking the ldquoweakest linkrdquo especially whensome hidden damage or design flaw has not been detectedOne recent example is the Minnesota Bridge failure whichoccurred during rush hours with heavy traffic although trafficloads may not be the direct cause of failure For slender long-span bridges strong wind may also cause threats by workinginteractively with heavy traffic loads Therefore even thoughthe extreme cases associated with congested traffic andorwindy weather may be relatively rare and the durationscould be short it is still important for bridge engineers toappropriately look into these unusual extreme events duringstructural design and life-time management of these criticalinfrastructures [11]
The dynamic performance of long-span bridges undercombined actions of strong winds and running road vehicleshas been studied by many researchers in recent years [17 7981 82 132] Most of them studied bridge dynamic perfor-mance under road traffic inwhich only one or several vehiclesdistributed in an assumed (usually uniform) pattern on long-span bridges were considered Extreme events such as trafficcongestion coupled with strong wind were out of concernin those studies Recently Wu and Chen [11] conducteda research on the assessment of long-span bridges underextreme events which includes complex traffic congestioncoupled with moderate or even strong wind This study
The Scientific World Journal 13
applied the cellular automaton (CA) traffic model to thesimulation of the actual traffic flow through the bridgedefined representative scenarios for the extreme events andnumerically studied the bridge performance under thesepossible extreme events
4 Conclusions and Recommendations
Dynamic responses of long-span bridges are often requiredfor assessing the safety of these bridges and can be determinedby numerical simulation technologies This paper provides adetailed review of key issues involved in dynamic responseanalysis of long-span multiload bridges based on numericalsimulation including dynamic interactions between runningtrains and bridge between running road vehicles and bridgeand between wind and bridge and in the wind-vehicle-bridge coupled systemThen the review work was conductedfor engineering applications of newly developed numericalsimulation technologies to safety assessment of long-spanbridges such as assessment of fatigue damage and assessmentunder extreme event condition Although technologies fornumerical simulation of dynamic responses of long-spanmultiload bridge have achieved great advances in past fewdecades and successfully applied to several important bridgesit is still far from reach its maturity and enable to takeplace of traditional fieldmeasurementThe existing problemsand promising research efforts at least include the followingaspects
(1) After multiple types of dynamic interactions beingconsidered in the complex system computationalefficiency is a bottleneck problem for numericalsimulation of dynamic response of a long-span bridgeTypically when multiple loads are involved a largenumber of loading combinations for multiple load-ings must be considered in the assessment
(2) It is rather complex for the time-depending windloads acting on a long-span bridge and running vehi-cles especially for the case of rapid change of windloads such as a train passing through a bridge toweror two trains passing each other The aerodynamiccharacteristics of vehicles and the bridge under differ-ent loading scenarios can be determined through thewind tunnel testing and used in numerical simulationof dynamic responses of the bridge and vehicles
(3) It is a new trend to integrate numerical simulationtechnologies and measurement data of structuralhealth monitoring systems (SHMSs) installed onlong-span bridges whichmakes the safety assessmentof bridge structures more reliable Measured struc-tural responses could be used to validate numericalsimulation approach and measured loading infor-mation could be used for generating statistical orprobabilistic models of multiple loads
(4) It is important to study dynamic responses of bridgestructures under extreme events such as congestedtraffic coupled with windy weather which happens ina long-span bridge For the assessment under extreme
events using numerical simulation technologies sim-ulation of traffic flow and definition of representativescenarios of the extreme events are key issues
(5) It is necessary to consider the effects of typhoonwinds on the safety assessment of long-span bridgesin a reasonable way Few researches do this mostlybecause a probabilistic distribution of wind speedand direction specifically for typhoons is requiredfor assessment but there are insufficient measuredrecords to establish a reliable probabilistic typhoonwind model
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors wish to acknowledge the financial supportsfrom the National Natural Science Foundation of China(NSFC-51108395 and NSFC-51178366) the FundamentalResearch Funds for theCentral Universities (2012121032) andopen funding from Jiangsu Key Laboratory of EngineeringMechanics Special thanks go to the supervisor of the firstauthor Professor Y L XuHongKong PolytechnicUniversityfor his expert guidance and continuous support at all levelsthroughout his PhD study Sincere thanks should go tothe Highways Department of Hong Kong for providing theauthors with the field measurement data Any opinions andconcluding remarks presented in this paper are entirely thoseof the authors
References
[1] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[2] H Sohn C R Farrar N F Hunter and K Worden ldquoStructuralhealth monitoring using statistical pattern recognition tech-niquesrdquo Journal of Dynamic Systems Measurement and Controlvol 123 no 4 pp 706ndash711 2001
[3] W Fan and P Qiao ldquoVibration-based damage identificationmethods a review and comparative studyrdquo Structural HealthMonitoring vol 10 no 1 pp 83ndash111 2011
[4] X Q Zhu and S S Law ldquoDamage detection in simply supportedconcrete bridge structure under moving vehicular loadsrdquo Jour-nal of Vibration and Acoustics Transactions of the ASME vol129 no 1 pp 58ndash65 2007
[5] J Li and S S Law ldquoDamage identification of a target sub-structure with moving load excitationrdquoMechanical Systems andSignal Processing vol 30 pp 78ndash90 2012
[6] J Li S S Law and H Hao ldquoImproved damage identification inbridge structures subject tomoving loads numerical and exper-imental studiesrdquo International Journal of Mechanical Sciencesvol 74 pp 99ndash111 2013
14 The Scientific World Journal
[7] Z W Chen Y L Xu Y Xia Q Li and K Y Wong ldquoFatigueanalysis of long-span suspension bridges under multiple load-ing case studyrdquo Engineering Structures vol 33 no 12 pp 3246ndash3256 2011
[8] Z W Chen Y L Xu and X M Wang ldquoSHMS-based fatiguereliability analysis of multiloading suspension bridgesrdquo Journalof Structural Engineering-Asce vol 138 pp 299ndash307 2012
[9] J Wu S R Chen and J W van de Lindt ldquoFatigue assessmentof slender long-span bridges reliability approachrdquo Journal ofBridge Engineering vol 17 no 1 pp 47ndash57 2012
[10] W Zhang C S Cai and F Pan ldquoFatigue reliability assessmentfor long-span bridges under combined dynamic loads fromwinds and vehiclesrdquo Journal of Bridge Engineering vol 18 pp735ndash747 2013
[11] JWu and S R Chen ldquoProbabilistic dynamic behavior of a long-span bridge under extreme eventsrdquo Engineering Structures vol33 no 5 pp 1657ndash1665 2011
[12] J M Ko and Y Q Ni ldquoTechnology developments in structuralhealth monitoring of large-scale bridgesrdquo Engineering Struc-tures vol 27 no 12 pp 1715ndash1725 2005
[13] TH YiHN Li andHM Sun ldquoMulti-stage structural damagediagnosis method based on ldquoenergy-damagerdquo theoryrdquo SmartStructures and Systems vol 12 pp 345ndash361 2013
[14] T H Yi H N Li and M Gu ldquoFull-scale measurements ofdynamic response of suspension bridge subjected to environ-mental loads using GPS technologyrdquo Science China Technologi-cal Sciences vol 53 no 2 pp 469ndash479 2010
[15] T H Yi H N Li and M Gu ldquoExperimental assessmentof high-rate GPS receivers for deformation monitoring ofbridgerdquoMeasurement Journal of the InternationalMeasurementConfederation vol 46 pp 420ndash432 2013
[16] X W Ye Y Q Ni K Y Wong and J M Ko ldquoStatistical analysisof stress spectra for fatigue life assessment of steel bridges withstructural health monitoring datardquo Engineering Structures vol45 pp 166ndash176 2012
[17] Z W Chen Y L Xu Q Li and D J Wu ldquoDynamic stressanalysis of long suspension bridges under wind railway andhighway loadingsrdquo Journal of Bridge Engineering vol 16 no 3pp 383ndash391 2011
[18] S G Meisenholder and P Weidlinger ldquoDynamic interactionaspects of cable-stayed guideways for high speed ground trans-portationrdquoAmerican Society ofMechanical Engineers no 74 pp180ndash192 1974
[19] Q H Mao Research on the Highway Bridge Vibration Due toMoving Vehicles Tongji University Shang Hai China 1989
[20] Y L Xu J M Ko and Z Yu ldquoModal analysis of tower-cable system of Tsing Ma long suspension bridgerdquo EngineeringStructures vol 19 pp 857ndash867 1997
[21] W Guo H Xia and Y-L Xu ldquoDynamic response of a long spansuspension bridge and running safety of a train under windactionrdquo Frontiers of Architecture and Civil Engineering in Chinavol 1 no 1 pp 71ndash79 2007
[22] T T Liu Y L Xu W S Zhang K Y Wong H J Zhou and KW Y Chan ldquoBuffeting-induced stresses in a long suspensionbridge structural health monitoring oriented stress analysisrdquoWind and Structures An International Journal vol 12 no 6 pp479ndash504 2009
[23] K Y Wong ldquoStructural identification of Tsing Ma BridgerdquoTransactions Hong Kong Institution of Engineers vol 10 no 1pp 38ndash47 2003
[24] Y L XuQ Li D JWu and ZWChen ldquoStress and accelerationanalysis of coupled vehicle and long-span bridge systems usingthe mode superposition methodrdquo Engineering Structures vol32 no 5 pp 1356ndash1368 2010
[25] Y F Duan Y L Xu Q G Fei et al ldquoAdvanced finite elementmodel of Tsing Ma Bridge for structural health monitoringrdquoInternational Journal of Structural Stability and Dynamics vol11 no 2 pp 313ndash344 2011
[26] Z X Li T Q Zhou T H T Chan and Y Yu ldquoMulti-scalenumerical analysis on dynamic response and local damage inlong-span bridgesrdquo Engineering Structures vol 29 no 7 pp1507ndash1524 2007
[27] W Zhang C S Cai and F Pan ldquoFinite element modeling ofbridges with equivalent orthotropic material method for multi-scale dynamic loadsrdquo Engineering Structures vol 54 pp 82ndash932013
[28] S P Timoshenko ldquoOn the forced vibrations of bridgesrdquo Philo-sophical Magazine vol 6 no 257 pp 1018ndash1019 1922
[29] R S Ayre G Ford and L S Jacobsen ldquoTransverse vibration ofa two-span beam under the action of a moving constant forcerdquoJournal of Applied Mechanics vol 17 pp 1ndash12 1950
[30] R S Ayre andLS Jacobsen ldquoTransverse vibration of a two-spanbeam under the action of a moving alternating forcerdquo Journal ofApplied Mechanics vol 17 pp 283ndash290 1950
[31] L FrybaVibration of Solids and Structures underMoving LoadsASCE Press 1972
[32] J-S Wu and C-W Dai ldquoDynamic response of multispannonuniform beam due to moving loadsrdquo Journal of StructuralEngineering vol 113 no 3 pp 458ndash474 1987
[33] W Weaver S P Timoshenko and D H Young VibrationProblems in Engineering John Wiley amp Sons 1990
[34] N H Galdos D R Schelling and M A Sahin ldquoMethodologyfor impact factor of horizontally curved box bridgesrdquo Journal ofStructural Engineering vol 119 no 6 pp 1917ndash1934 1993
[35] J A Gbadeyan and S T Oni ldquoDynamic behaviour of beamsand rectangular plates under moving loadsrdquo Journal of Soundand Vibration vol 182 no 5 pp 677ndash695 1995
[36] D Y Zheng Y K Cheung F T K Au and Y S ChengldquoVibration of multi-span non-uniform beams under movingloads by using modified beam vibration functionsrdquo Journal ofSound and Vibration vol 212 no 3 pp 455ndash467 1998
[37] G V Rao ldquoLinear dynamics of an elastic beam under movingloadsrdquo Journal of Vibration and Acoustics Transactions of theASME vol 122 no 3 pp 281ndash289 2000
[38] Y B Yang J D Yau and Y S Wu Vehicle-Bridge InteractionDynamic with Applications to High-Speed Railways WorldScientific 2004
[39] E C Ting J Genin and J H Ginsberg ldquoA general algorithm formovingmass problemsrdquo Journal of Sound and Vibration vol 33no 1 pp 49ndash58 1974
[40] S Sadiku and H H E Leipholz ldquoOn the dynamics of elasticsystems with moving concentrated massesrdquo Ingenieur-Archivvol 57 no 3 pp 223ndash242 1987
[41] J E Akin and M Mofid ldquoNumerical solution for response ofbeamswithmovingmassrdquo Journal of Structural Engineering vol115 no 1 pp 120ndash131 1989
[42] M AMahmoud andM A Abou Zaid ldquoDynamic response of abeam with a crack subject to a moving massrdquo Journal of Soundand Vibration vol 256 no 4 pp 591ndash603 2002
[43] V K Garg Dynamics of Railway Vehicle Systems AcademicPress 1994
The Scientific World Journal 15
[44] T-LWang andDHuang ldquoCable-stayed bridge vibration due toroad surface roughnessrdquo Journal of Structural Engineering vol118 no 5 pp 1354ndash1374 1992
[45] Y-B Yang and B-H Lin ldquoVehicle-bridge interaction analysisby dynamic condensation methodrdquo Journal of Structural Engi-neering vol 121 no 11 pp 1636ndash1643 1995
[46] Y B Yang J D Yau and L C Hsu ldquoVibration of simple beamsdue to trainsmoving at high speedsrdquoEngineering Structures vol19 no 11 pp 936ndash943 1997
[47] B Tabarrok and E Esmailzadeh ldquoInduced vibration of bridgestransversed by moving vehiclesrdquo Transactions of the CanadianSociety for Mechanical Engineering B vol 24 no 1 pp 191ndash1982000
[48] C Liu T-L Wang and D Huang ldquoImpact study for multi-girder bridge based on correlated road roughnessrdquo StructuralEngineering and Mechanics vol 11 no 3 pp 259ndash272 2001
[49] K H Chu V K Garg and T L Wang ldquoImpact in railwayprestressed concrete bridgesrdquo Journal of Structural Engineeringvol 112 no 5 pp 1036ndash1051 1986
[50] T-LWang V K Garg and K-H Chu ldquoRailway bridgevehicleinteraction studies with new vehicle modelrdquo Journal of Struc-tural Engineering vol 117 no 7 pp 2099ndash2116 1991
[51] H Xia Y L Xu and T H T Chan ldquoDynamic interaction oflong suspension bridges with running trainsrdquo Journal of Soundand Vibration vol 237 no 2 pp 263ndash280 2000
[52] Q-L Zhang A Vrouwenvelder and J Wardenier ldquoNumericalsimulation of train-bridge interactive dynamicsrdquo Computersand Structures vol 79 no 10 pp 1059ndash1075 2001
[53] H Xia N Zhang and G de Roeck ldquoDynamic analysis of highspeed railway bridge under articulated trainsrdquo Computers andStructures vol 81 no 26-27 pp 2467ndash2478 2003
[54] G Diana F Cheli A Collina R Corradi and S MelzildquoThe development of a numerical model for railway vehiclescomfort assessment through comparison with experimentalmeasurementsrdquoVehicle SystemDynamics vol 38 no 3 pp 165ndash183 2002
[55] Q Li Y L Xu D J Wu and Z W Chen ldquoComputer-aided nonlinear vehicle-bridge interaction analysisrdquo Journal ofVibration and Control vol 16 pp 1791ndash1816 2010
[56] A Wiriyachai K H Chu and V K Garg ldquoBridge impact dueto wheel and track irregularitiesrdquo Journal of the EngineeringMechanics Division vol 108 no 4 pp 648ndash666 1982
[57] L Fryba Dynamics of Railway Bridges Inst of Civil Engineers1996
[58] D Huang and T-L Wang ldquoImpact analysis of cable-stayedbridgesrdquo Computers and Structures vol 43 no 5 pp 897ndash9081992
[59] W M Zhai Vehicle-Track Coupling Dynamics Chinese RailwayPress Beijing China 2007
[60] M Olsson ldquoFinite element modal co-ordinate analysis ofstructures subjected to moving loadsrdquo Journal of Sound andVibration vol 99 no 1 pp 1ndash12 1985
[61] Y-B Yang C-H Chang and J-D Yau ldquoAn element foranalysing vehicle-bridge systems considering vehiclersquos pitchingeffectrdquo International Journal for NumericalMethods in Engineer-ing vol 46 no 7 pp 1031ndash1047 1999
[62] Y B Yang and Y S Wu ldquoA versatile element for analyzingvehicle-bridge interaction responserdquo Engineering Structuresvol 23 no 5 pp 452ndash469 2001
[63] F T K Au J J Wang and Y K Cheung ldquoImpact study ofcable-stayed bridge under railway traffic using various modelsrdquo
Journal of Sound and Vibration vol 240 no 3 pp 447ndash4652001
[64] Y Q Sun andMDhanasekar ldquoA dynamicmodel for the verticalinteraction of the rail track and wagon systemrdquo InternationalJournal of Solids and Structures vol 39 no 5 pp 1337ndash13592002
[65] K Henchi M Fafard M Talbot and G Dhatt ldquoAn efficientalgorithm for dynamic analysis of bridges under moving vehi-cles using a coupledmodal and physical components approachrdquoJournal of Sound and Vibration vol 212 no 4 pp 663ndash6831998
[66] Y L Xu and L YWang ldquoAnalytical study of wind-rain-inducedcable vibration SDOFmodelrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 91 no 1-2 pp 27ndash40 2003
[67] B Biondi G Muscolino and A Sofi ldquoA substructure approachfor the dynamic analysis of train-track-bridge systemrdquoComput-ers and Structures vol 83 no 28ndash30 pp 2271ndash2281 2005
[68] J L Humar and A H Kashif ldquoDynamic response analysis ofslab-type bridgesrdquo Journal of Structural Engineering vol 121 no1 pp 48ndash62 1995
[69] P Lou and Q-Y Zeng ldquoFormulation of equations of verticalmotion for vehicle-track-bridge systemrdquo Journal of the ChinaRailway Society vol 26 no 5 p 71 2004
[70] O Coussy M Said and J-P van Hoove ldquoThe influence ofrandom surface irregularities on the dynamic response ofbridges under suspended moving loadsrdquo Journal of Sound andVibration vol 130 no 2 pp 313ndash320 1989
[71] E S Hwang and A S Nowak ldquoSimulation of dynamic loadfor bridgesrdquo Journal of Structural Engineering vol 117 pp 1413ndash1434 1991
[72] F Yang and G A Fonder ldquoAn iterative solution methodfor dynamic response of bridge-vehicles systemsrdquo EarthquakeEngineering and Structural Dynamics vol 25 pp 195ndash215 1996
[73] W Zhai and Z Cai ldquoDynamic interaction between a lumpedmass vehicle and a discretely supported continuous rail trackrdquoComputers and Structures vol 63 no 5 pp 987ndash997 1997
[74] W M Zhai and C B Cai ldquoTraintrackbridge dynamic inter-actions simulation and applicationsrdquo Vehicle System Dynamicsvol 37 pp 653ndash665 2003
[75] D Bruno F Greco and P Lonetti ldquoDynamic impact analysis oflong span cable-stayed bridges under moving loadsrdquo Engineer-ing Structures vol 30 no 4 pp 1160ndash1177 2008
[76] X D Song D J Wu and Q Li ldquoDynamic impact analysisof double-tower cable-stayed maglev bridges using a simplemodelrdquo Journal of Bridge Engineering vol 19 pp 34ndash43 2014
[77] Y-S Wu and Y-B Yang ldquoSteady-state response and ridingcomfort of trains moving over a series of simply supportedbridgesrdquoEngineering Structures vol 25 no 2 pp 251ndash265 2003
[78] P Antolin N Zhang J M Goicolea H Xia M A Astiz andJ Oliva ldquoConsideration of nonlinear wheel-rail contact forcesfor dynamic vehicle-bridge interaction in high-speed railwaysrdquoJournal of Sound and Vibration vol 332 no 5 pp 1231ndash12512013
[79] W H Guo and Y L Xu ldquoFully computerized approach to studycable-stayed bridge-vehicle interactionrdquo Journal of Sound andVibration vol 248 no 4 pp 745ndash761 2001
[80] Y L Xu and W H Guo ldquoDynamic behaviour of high-sidedroad vehicles subject to a sudden crosswind gustrdquo Wind andStructures vol 6 no 5 pp 325ndash346 2003
[81] Y L Xu and W H Guo ldquoDynamic analysis of coupled roadvehicle and cable-stayed bridge systems under turbulent windrdquoEngineering Structures vol 25 no 4 pp 473ndash486 2003
16 The Scientific World Journal
[82] C S Cai and S R Chen ldquoFramework of vehicle-bridge-winddynamic analysisrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 92 no 7-8 pp 579ndash607 2004
[83] S R Chen and J Wu ldquoModeling stochastic live load for long-span bridge based on microscopic traffic flow simulationrdquoComputers and Structures vol 89 no 9-10 pp 813ndash824 2011
[84] A V Paultre B Yang L A Bergman and C A Tan ldquoBridgedynamics and dynamic amplification factorsmdasha review ofanalytical and experimental findingsrdquoCanadian Journal of CivilEngineering vol 19 no 2 pp 260ndash278 1992
[85] H Honda Y Kajikawa and T Kobori ldquoSpectra of road surfaceroughness of bridgesrdquo Journal of the Structural Division vol 108pp 1956ndash1966 1982
[86] M J Inbanathan and M Wieland ldquoBridge vibrations dueto vehicle moving over rough surfacerdquo Journal of StructuralEngineering vol 113 no 9 pp 1994ndash2008 1987
[87] T-LWang andDHuang ldquoCable-stayed bridge vibration due toroad surface roughnessrdquo Journal of Structural Engineering vol118 no 5 pp 1354ndash1374 1992
[88] P K Chatterjee T K Datta and C S Surana ldquoVibrationsuspension bridges under vehicular movementrdquo Journal ofStructural Engineering vol 120 no 3 pp 681ndash703 1994
[89] D Chang and H Lee ldquoImpact factors for simple-span highwaygirder bridgesrdquo Journal of Structural Engineering vol 120 no 3pp 704ndash715 1994
[90] T-C Pan and J Li ldquoDynamic vehicle element method for tran-sient response of coupled vehicle-structure systemsrdquo Journal ofStructural Engineering vol 128 no 2 pp 214ndash223 2002
[91] C J Dodds and J D Robson ldquoThe description of road surfaceroughnessrdquo Journal of Sound Vibration vol 31 no 2 pp 175ndash183 1973
[92] D Huang T-L Wang and M Shahawy ldquoImpact studies ofmultigirder concrete bridgesrdquo Journal of Structural Engineeringvol 119 no 8 pp 2387ndash2402 1993
[93] A G Davenport ldquoBuffeting of a suspension bridge by stormwindrdquo Journal of Structural Division vol 88 pp 233ndash268 1962
[94] R H Scanlan ldquoThe action of flexible bridges under wind Iflutter theoryrdquo Journal of Sound and Vibration vol 60 no 2pp 187ndash199 1978
[95] Q Ding and P K K Lee ldquoComputer simulation of buffetingactions of suspension bridges under turbulentwindrdquoComputersand Structures vol 76 no 6 pp 787ndash797 2000
[96] V Boonyapinyo T Miyata and H Yamada ldquoAdvanced aerody-namic analysis of suspension bridges by state-space approachrdquoJournal of Structural Engineering vol 125 no 12 pp 1357ndash13661999
[97] Y-H Chen and C-Y Li ldquoDynamic response of elevated high-speed railwayrdquo Journal of Bridge Engineering vol 5 no 2 pp124ndash130 2000
[98] X Chen M Matsumoto and A Kareem ldquoTime domainflutter and buffeting response analysis of bridgesrdquo Journal ofEngineering Mechanics vol 126 no 1 pp 7ndash16 2000
[99] X Chen and A Kareem ldquoEquivalent static wind loads forbuffeting response of bridgesrdquo Journal of Structural Engineeringvol 127 no 12 pp 1467ndash1475 2001
[100] S R Chen and C S Cai ldquoEvolution of long-span bridgeresponse to wind-numerical simulation and discussionrdquo Com-puters and Structures vol 81 no 21 pp 2055ndash2066 2003
[101] Y L XuHXia andQ S Yan ldquoDynamic response of suspensionbridge to high wind and running trainrdquo Journal of BridgeEngineering vol 8 no 1 pp 46ndash55 2003
[102] A Guo Y L Xu and H Li ldquoDynamic performance of cable-stayed bridge tower with multi-stage pendulum mass damperunderwind excitations-II experimentrdquoEarthquake Engineeringand Engineering Vibration vol 6 no 4 pp 417ndash424 2007
[103] E Simiu and R H ScanlanWind Effects on Structures 1996[104] Y Cao H Xiang and Y Zhou ldquoSimulation of stochastic wind
velocity field on long-span bridgesrdquo Journal of EngineeringMechanics vol 126 no 1 pp 1ndash6 2000
[105] M Shinozuka and C-M Jan ldquoDigital simulation of randomprocesses and its applicationsrdquo Journal of Sound and Vibrationvol 25 no 1 pp 111ndash128 1972
[106] K M Shum Lateral and torsional vibration control of long spanbridge deck using novel tuned liquid column dampers [PhDthesis] Department of Civil and Structural Engineering TheHong Kong Polytechnic University 2004
[107] Y K Lin and J N Yang ldquoMultimode bridge response to windexcitationsrdquo Journal of EngineeringMechanics vol 109 no 2 pp586ndash603 1983
[108] Y S Lin Self-Excited Bridge Motion in Turbulent Wind 1978[109] D T Lau M S Cheung and S H Cheng ldquo3D flutter analysis
of bridges by spline finite-strip methodrdquo Journal of StructuralEngineering vol 126 no 10 pp 1246ndash1254 2000
[110] L A Balzer ldquoAtmospheric turbulence encountered by high-speed ground transport vehiclesrdquo Journal of Mechanical Engi-neering Science vol 19 pp 227ndash235 1977
[111] R K Cooper ldquoAtmospheric turbulence with respect to movingground vehiclesrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 17 no 2 pp 215ndash238 1984
[112] C J Baker ldquoA simplified analysis of various types of wind-induced road vehicle accidentsrdquo Journal of Wind Engineeringand Industrial Aerodynamics vol 22 no 1 pp 69ndash85 1986
[113] C J Baker ldquoMeasures to control vehicle movement at exposedsites during windy periodsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 25 no 2 pp 151ndash161 1987
[114] C J Baker ldquoHigh sided articulated road vehicles in strong crosswindsrdquo Journal of Wind Engineering and Industrial Aerodynam-ics vol 31 no 1 pp 67ndash85 1988
[115] C J Baker ldquoGround vehicles in high cross winds part I steadyaerodynamic forcesrdquo Journal of Fluids and Structures vol 5 no1 pp 69ndash90 1991
[116] C J Baker ldquoGround vehicles in high cross winds part IIunsteady aerodynamic forcesrdquo Journal of Fluids and Structuresvol 5 no 1 pp 91ndash111 1991
[117] Y L Xu and Q S Ding ldquoInteraction of railway vehicles withtrack in cross-windsrdquo Journal of Fluids and Structures vol 22no 3 pp 295ndash314 2006
[118] S R Chen and J Wu ldquoDynamic performance simulation oflong-span bridge under combined loads of stochastic traffic andwindrdquo Journal of Bridge Engineering vol 15 no 3 pp 219ndash2302010
[119] Y Li S Qiang H Liao and Y L Xu ldquoDynamics of wind-rail vehicle-bridge systemsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 93 no 6 pp 483ndash507 2005
[120] Y L Xu N Zhang and H Xia ldquoVibration of coupled trainand cable-stayed bridge systems in cross windsrdquo EngineeringStructures vol 26 no 10 pp 1389ndash1406 2004
[121] W W Guo Y L Xu H Xia W S Zhang and K M ShumldquoDynamic response of suspension bridge to typhoon and trainsII numerical resultsrdquo Journal of Structural Engineering vol 133no 1 pp 12ndash21 2007
The Scientific World Journal 17
[122] Y L Li H Y Xiang B Wang Y L Xu and S Z QiangldquoDynamic analysis of wind-vehicle-bridge coupling systemduring the meeting of two trainsrdquo Advances in StructuralEngineering vol 16 pp 1663ndash1670 2013
[123] F Dorigatti M Sterling D Rocchi et al ldquoWind tunnelmeasurements of crosswind loads on high sided vehicles overlong span bridgesrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 107 pp 214ndash224 2012
[124] L D Zhu L Li Y L Xu and Q Zhu ldquoWind tunnel investi-gations of aerodynamic coefficients of road vehicles on bridgedeckrdquo Journal of Fluids and Structures vol 30 pp 35ndash50 2012
[125] Y Han J X Hu C S Cai Z Q Chen and C G LildquoExperimental and numerical studies of aerodynamic forces onvehicles and bridgesrdquoWind and Structures vol 17 pp 163ndash1842013
[126] ASCE ldquoCommittee on fatigue and fracture reliability of thecommittee on structural safety and reliability of the structuraldivision fatigue reliability 1ndash4rdquo Journal of Structural Engineer-ing vol 108 pp 3ndash88 1982
[127] BS ldquoBS5400 part 10 code of practice for fatiguerdquo BritishStandard Institute 1980
[128] BS ldquoBS7608 code of practice for fatigue design and assessmentof steel structuresrdquo British Standard Institute 1993
[129] AASHTOGuide Specifications for Fatigue Evaluation of ExistingSteel Bridges 1990
[130] AASHTO Guide Manual for Condition Evaluation and Loadand Resistance Factor Rating (LRFR) of Highway Bridges 2003
[131] S R Chen C S Cai and B Wolshon ldquoFrom normal operationto evacuation Single-vehicle safety under adverse weathertopographic and operational conditionsrdquo Natural HazardsReview vol 10 no 2 pp 68ndash76 2009
[132] S R Chen and C S Cai ldquoEquivalent wheel load approach forslender cable-stayed bridge fatigue assessment under traffic andwind feasibility studyrdquo Journal of Bridge Engineering vol 12 no6 pp 755ndash764 2007
Figure 4 Finite element model of a railway vehicle (a) elevation view (b) side view (c) isometric view (d) model details [52]
the equations of motion of the vehicles were derivedanalytically However a great inconvenience of this methodis that the equations of motion of the whole vehicle-bridgesystem must be rederived if the vehicle type is changedFurthermore it is very difficult to derive the equation ofmotion for a complex vehicle model containing a largenumber of DOFs such as the articulated components of aTGV train with an 85-DOF dynamic system [53] Generalcommercial FE software has recently been adopted to makevehicle modeling more easily applicable for different vehicletypes [54] Li et al [55] described a four-step procedurefor modeling a four-axle railway vehicle by beam elements(1) the nodes and elements for the car body bogies andwheelsets respectively are defined by using beam elements sothat the spatial geometric configuration of each componentcan be built (see Figure 4) (2) sectional properties andmaterial properties are assigned to each beam element(3) rigid-arms and suspension units (systems) are used toconnect the three components (4) constraints are assignedto form a complete finite element model of the vehicle
213 Modeling of Rail Irregularities Track irregularities rep-resent an important source of excitation for bridges duringthe passage of railway vehicles Track irregularitiesmay occuras a result of initial installation errors the degradation ofsupport materials or the dislocation of track joints Fourgeometric parameters can be used to quantitatively describerail irregularities the vertical profile cross level alignmentand gauge [49 50 56] Vertical profile and cross level irreg-ularities chiefly influence the vertical vibrations of vehiclesand of the bridge whereas alignment gauge and crosslevel irregularities initiate horizontal transverse vibrationsof vehicles and the bridge and also the torsional movementof the bridge [57] Track irregularities may be periodic orrandom Random irregularities are due to wear clearancesubsidence and insufficient maintenance For engineeringapplications random irregularities can be approximatelyregarded as stationary and ergodic processes that can begenerated from measured results or simulated by numericalmethods Several numerical methods have been proposedfor the simulation of random rail irregularities such as
6 The Scientific World Journal
the trigonometry series white noise filtration autoregressive(AR) and power spectral density (PSD) sampling methodsAmong these methods the PSD sampling method has beenwidely adopted due to its high computational accuracy Thelateral and vertical irregularities could be all assumed tobe zero-mean stationary Gaussian random processes andexpressed through the inverse Fourier transformation of aPSD function [58]
119910119904(119909) =
119873
sum119896=1
radic2119878 (119891119896) Δ119891 cos (2120587119891
119896119909 + 120579119896) (1)
where 119878(119891) is the PSD function 119891119896= 119891l + (119896 minus 12)Δ119891
Δ119891 = (119891119906minus 119891l)119873 119891119906 and 119891119897 are the upper and lower cutoff
frequencies respectively and 120579119896is the random phase angle
uniformly distributed between 0 and 2120587 Rail irregularity inrailway engineering is commonly represented by a one-sidedPSD function
The PSD functions of rail irregularities have been devel-oped by different countries Based on the PSD functions ofrail irregularities developed by the Research Institute of theChina Railway Administration Zhai [59] expressed all railirregularities using the unified rational formula as follows
where 119891 = 1120578 (mminus1) is the spatial frequency in cyclem (120578 isthe wavelength) and119860 to119866 are the parameters recommendedby Zhai [59] specifically for vertical and lateral rail irregular-ities
214 Solution Methods The dynamic analysis of vehicle-bridge coupled system requires two sets of equations ofmotion for the bridge and vehicles respectively Thesedescribe the interaction or contact forces at the contactpoints of the two subsystems Because the contact pointsmove from time to time the system matrices are generallytime dependent and must be updated and factorized at eachtime step The various solution methods can be generalizedinto two groups according to whether or not an iterativeprocedure is needed at each time step
The first group ofmethods solves the equations of motionof a coupled vehicle-bridge system at each time step withoutiteration This approach has been widely used in coupledvehicle-bridge analysis [51 53 60ndash69] These methods havegood computational stability and are convenient for dealingwith vehicle-bridge interaction problems when the vehiclemodel is relatively simple The main disadvantage is thatthe equations of motion of the coupled system are timedependent and thus the characteristic matrices must bemodified at each time step In addition the equations ofmotion of the coupled vehicle-bridge system become verydifficult to determine if nonlinear wheel-rail contacts andnonlinear vehicle models are considered
The second group of methods solves the equations for thevehicles and bridge separately and requires an iterative pro-cess to obtain convergence for the displacements of the vehi-cles and bridge at all contact points Given that the conditions
of wheel-rail contact geometry and contact forces are rathercomplex a stable integration method adopting a small timeinterval is needed for obtaining the convergence of vehicleand bridge subsystems at the contact points in each time stepMany studies have applied this type of method to investigatevehicle-bridge interactions [70ndash76] The advantage of thesemethods is that the dynamic propertymatrices in the two setsof equations of motion remain constant which is convenientfor the consideration of nonlinear vehicle-bridge interactionsand nonlinear vehicle models [55] However in engineeringapplications the iterative convergence is a critical problemwith this type of method The low convergence rate andoccasional divergence of the solution have also been noted[77] Li et al [55] investigated the performance of theseiterative schemes using the Wilson-120579 method Newmark-120573method and an explicit integrationmethod proposed byZhai[59] and found that the latter gave amuch higher convergencerate than the former two methods
Most of the above methods solved the equations ofmotion of a coupled vehicle-bridge systemusing the nonjumpmodel which assumes that the moving vehicle travelingalong the bridge is always in contact with the rails nomatter what the sign is of the contact forces This is notalways true in view of the physics of the moving vehiclewhich simply sits on the upper surfaces of the rails Theinteraction forces between the moving vehicle and the bridgedepend on the motions of the vehicle the flexibility of thebridge and the track irregularities Li et al [55] utilizeda jump model to solve vehicle-bridge interaction problemusing a noniterative Runge-Kutta method and found thatthe acceleration responses of the car body using the wheel-jump model are smaller than those using the wheel nonjumpmodel when the vehicle speed exceeds 300 kmhr Antolin etal [78] proposed a nonlinear wheel-rail interaction modelwhich considers nonlinear wheel-rail contact forces in theinteraction as well as realistic wheel and rail profiles andapplied it for analysis of dynamic interaction between highspeed trains and bridges
22 Dynamic Interaction between Road Vehicles and BridgeSection 21 gave a detailed literature review of the dynamicinteractions between trains and bridges As there are somefundamental differences between trains and road vehiclesthis section reviews the modeling of road vehicles thesimulation of road vehicle flow and the modeling of roadsurface roughness
221 Modeling of Road Vehicles To analyze the dynamicinteraction between a bridge and running road vehicles amodel of road vehicles must be established A sophisticatedroad vehiclemodel is required tomake the simulation as real-istic as possible A road vehicle is modeled as a combinationof several rigid bodies each of which is connected by a set ofsprings and dashpots which model the elastic and dampingeffects of the tires and suspension systems respectivelyThereare various configurations of road vehicles such as a tractorand trailer with different axle spacing Road vehicle models
The Scientific World Journal 7
L11 L10 L9
L8
Ku5Cu5
Cl5Kl5
Zs5
Ku4Cu4
Cl4Kl4
Zs4
Ku3 Cu3
Cl3Kl3
Zs3
Ku2 Cu2
Cl2Kl2
Zs2
Ku1 Cu1
Cl1Kl1
Zs1
Ku1 Cu1
Cl1Kl1
Zs1
Ku6 Cu6
Cl6Kl6
Zs6
b1 b1
x yL1L2L3L4L5L6L7
Z3 1205793 Z2 Z1
Z1
1205792
1205791
zz
1206011
Figure 5 Dynamic model of a tractor-trailer [79]
zz
x y
h
2b1
ZZY
h1
120601
Kuz1 Kuz3Cuz1
Kuz1 Cuz1 Cuz3
Zs1Zs1
Zs3
Cuy1 Cuy3
Kuy1Kuy3
Ys1 Ys3
Klz1Klz1 Klz3Clz1 Clz3Clz1
Kuz2 Cuz2
Zs2
Klz2 Clz2
120579
L1L2
Kly1
Cly1
Kly3
Cly3
Figure 6 Dynamic model of a high-sided road vehicle [80]
that contain several DOFs have been devised for vehicle-bridge interaction analysis For example Guo and Xu [79]modeled a 17-DOF four-axle heavy tractor-trailer vehicle (seeFigure 5) to investigate the interaction between vehicles anda cable-stayed bridge A total of three DOFs were assignedto rigid bodies representing either the tractor or the trailerto account for vertical rolling and pitching motions Onlyone DOFwas assigned to the rigid body representing the axleset moving in the vertical direction Different vehicle modelsare adopted in wind-vehicle-bridge interaction analyses Xuand Guo [80] modeled a 13-DOF two-axle road vehicle (seeFigure 6) for the dynamic analysis of a coupled road vehicleand bridge system under turbulent wind Five DOFs wereassigned to the vehicle body with respect to its center ofgravity to account for vertical lateral rolling yawing andpitching motions and two DOFs were assigned to the frontand rear axle sets to account for motions in the vertical andlateral directionMoreDOFs are needed to account for lateralcrosswinds
222 Simulation of Road Vehicle Flow On long-span bridgesthere is a high probability of the simultaneous presence ofmultiple road vehicles including heavy trucks This maylead to larger amplitude stress responses and greater fatiguedamage of the local bridge components than would be thecase with only one road vehicle The simulation of roadvehicle flow is thus important in the analysis of the dynamicinteraction between road vehicles and bridges Rather simplepatterns of road vehicle flow have been assumed in most
vehicle-bridge coupled dynamic analyses [79 81 82] in whicheither one or several vehicles are distributed on the bridgein an assumed (usually uniform) pattern Obviously suchassumptions do not represent actual road traffic conditionsRecently Chen and Wu [83] modeled the stochastic trafficload for a long-span bridge based on the cellular automaton(CA) traffic flow simulation technique In this study theysimulated a complicated road vehicle flow on long-spanbridges in terms of vehicle number vehicle type combinationand driver operation characteristics such as lane changingacceleration or deceleration
223 Modeling of Road Surface Roughness Road surfaceroughness is an important factor that greatly affects vehicle-bridge interactions Paultre et al [84] pointed out that roadsurface or pavement roughness can significantly affect theimpact response of a bridge The roughness or surface profiledepends primarily on the workmanship involved in theconstruction of the pavement or roadway and how it is main-tained which although random in nature may contain someinherent frequencies [38] In most cases surface roughnesswhich is three-dimensional in reality is often approximatedby a two-dimensional profile To account for its randomnature the road profile can be modeled as a stationaryGaussian random process and derived using a certain powerspectral density function Other methods similar to this havebeen widely adopted by researchers studying vehicle-inducedbridge vibration [65 70 71 85ndash90] Dodds and Robson [91]developed power spectral density functions that were later
8 The Scientific World Journal
modified and used byWang and Huang [87] and Huang et al[92] This approach was also adopted by literatures [79 81] intheir dynamic analyses of coupled vehicle-bridge and wind-vehicle-bridge systems
23 Dynamic Interaction between Wind and Bridge Whena long-span cable-supported bridge is immersed in a givenflow field the bridge will be subject to mean and fluctuatingwind forces To simulate these forces a linear approximationof the time-averaged static and time-varying buffeting andself-excited force components must be formulated [93 94]As dynamic bridge responses are of concern in this study onlybuffeting and self-excited forces are considered and reviewedin this section
231 Buffeting Forces Buffeting action is a random vibrationcaused by turbulent wind that excites certain modes of vibra-tion across a bridge depending on the spectral distributionof the pressure vectors [95] Although the buffeting responsemay not lead to catastrophic failure it can lead to structuralfatigue and affect the safety of passing vehicles [96] Hencebuffeting analysis has received much attention in recentyears in research into the structural safety of bridges underturbulent wind action [81 95 97ndash102]
By assuming no interaction between buffeting forces andself-excited forces and using quasi-steady aerodynamic forcecoefficients the buffeting forces per unit span Feibf on the 119894thsection of a bridge deck can be expressed as [103]
bf are the buffeting drag lift andmoment respectively 119906
119894and 119908
119894are the horizontal and
vertical components respectively of fluctuating wind at the119894th section 120588 is the air density 119880
119894is the mean wind speed
at the 119894th section of the bridge deck 119861119894and 119871
119894are the width
and length of the 119894th bridge section 119862119863119894 119862119871119894 and 119862
119872119894are
the drag lift and moment coefficients respectively of the119894th bridge segment 1198621015840
119863119894= 119889119862
1198631198941198891205721015840 1198621015840
119871119894= 119889119862
1198711198941198891205721015840 and
1198621015840119872119894
= 1198891198621198721198941198891205721015840 1205721015840 is the angle of attack of a normal
wind incident on the horizontal plane of the deck and 120594119863bu
120594119863bw
120594119871bu
120594119871bw
120594119872bu
and 120594119872bw
are the aerodynamic transferfunctions between the fluctuating wind velocities and thebuffeting forces
It can be found from this equation that a series oftime histories of fluctuating wind velocity 119906
119894 119908119894119879 in the
longitudinal and vertical directions at various points alongthe bridge deck is needed to carry out a detailed buffetinganalysis To simulate the stochastic wind velocity field thefast spectral representation method proposed by Cao et al[104] that is based on the spectral representation methoddeveloped by Shinozuka and Jan [105] is often adopted Thismethod rests on the assumptions that (1) the bridge deck ishorizontal at the same elevation (2) the mean wind speedand wind spectra do not vary along the bridge deck and(3) the distance between any two successive points wherewind speeds are simulated is the same The time historiesof the along-wind component 119906(119905) and the upward windcomponent 119908(119905) at the jth point can be generated using thefollowing equations [104]
119906119895(119905) = radic2 (Δ120596)
119895
sum119898=1
119873119891
sum119896=1
radic119878119906119906(120596119898119896)
times 119866119895119898(120596119898119896) cos (120596
119898119896119905 + 120593119898119896)
(4a)
119908119895(119905) = radic2 (Δ120596)
119895
sum119898=1
119873119891
sum119896=1
radic119878119908119908(120596119898119896)
times 119866119895119898(120596119898119896) cos (120596
119898119896119905 + 120593119898119896)
(4b)
119866119895119898(120596) =
0 when 1 le 119895 lt 119898 le 119899
119862|119895minus119898| when 119897 = 1 119898 le 119895 le 119899
119862|119895minus119898|radic(1 minus 1198622) when 2 le 119898 le 119895 le 119899
(4c)
119862 = exp(minus120582120596119898119896Δ
2120587119880) Δ =
119871
119899119901minus 1
(4d)
120596119898119896= (119896 minus 1) Δ120596 +
119898
119899Δ120596 (119896 = 1 2 119873
119891) (4e)
where Δ120596 = 120596up119873119891 is the frequency interval between thespectral lines119873
119891is the total number of frequency intervals
120596up is the upper cutoff frequency n is the total number ofpoints at which wind speeds are simulated 119878uu and 119878ww arethe along-wind and vertical wind spectrum respectively 120593lmis a random variable that is uniformly distributed between 0and 2120587 L is the span length and 120582 is a parameter that usuallyfalls between 7 and 10
In reality the equivalent buffeting forces in (3) are actuallyassociated with the spatial distribution of the wind pressureson the surface of the bridge deck Ignoring the spatialdistribution or aerodynamic transfer function of the buffetingforces across the cross-section of the bridge deck may havea considerable impact on the accuracy of buffeting response
The Scientific World Journal 9
wi(t) UI + ui(t)
j
j + 1
k
Leibf Lkibf
Meibf
Deibf
Dkibf
ce
Figure 7 Buffeting wind pressures and buffeting forces at nodes [22]
predictions Furthermore the local structural behavior ofthe bridge deck associated with local stresses and strainswhich are prone to causing local damage cannot be predicteddirectly by the current approaches based on equivalentbuffeting forces In this regard Liu et al [22] proposed anapproach to consider the spatial distribution of buffetingforces on a bridge deck structure based on wind pressuredistributions from wind tunnel tests (see Figure 7)
232 Self-Excited Forces In addition to buffeting actionflutter instability caused by self-excited forces induced bywind-structure interactions is an important considerationin the design and construction of long-span suspensionbridges [96] because the additional energy injected into theoscillating structure by the aerodynamic forces increases themagnitude of vibration sometimes to catastrophic levels [95]The self-excited forces on a bridge deck are attributable tothe interactions between wind and the motion of the bridgeWhen the energy of motion extracted from the flow exceedsthe energy dissipated by the system through mechanicaldamping the magnitude of vibration can reach catastrophiclevels [106] Expressing self-excited forces in the form ofindicial functions was first suggested by Scanlan [94] Basedon the assumption that self-excited forces are generated in alinear fashion Lin and Yang [107] simplified the self-excitedforces acting on a bridge deck and expressed them in termsof convolution integrals between the bridge deck motion andthe impulse response functions
119891se119890119863(119905)
=1
21205881198802
int119905
minusinfin
[119868119863ℎ(119905 minus 120591) ℎ
119890(120591) + 119868
119863119902(119905 minus 120591) 119902
119890(120591)
+119861119868119863120579(119905 minus 120591) 120579
119890(120591)
] 119889120591
(5a)
119891se119890119871(119905)
=1
21205881198802
int119905
minusinfin
[119868119871ℎ(119905 minus 120591) ℎ
119890(120591) + 119868
119871119902(119905 minus 120591) 119902
119890(120591)
+119861119868119871120579(119905 minus 120591) 120579
is the impulse function of the self-excited forces in which120595 represents the corresponding force components and heqe and 120579119890 are the equivalent vertical lateral and torsionaldisplacements respectively at the center of elasticity of thebridge deck section The relationship between the aero-dynamic impulse functions and flutter derivatives can beobtained by taking the Fourier transform of (7) [98]
119868119863ℎ(120596) = 119870
2
(119875lowast
6+ 119894119875lowast
5) 119868
119863119902(120596) = 119870
2
(119875lowast
4+ 119894119875lowast
1)
119868119863120579(120596) = 119870
2
(119875lowast
3+ 119894119875lowast
2)
119868119871ℎ(120596) = 119870
2
(119867lowast
4+ 119894119867lowast
1) 119868
119871119902(120596) = 119870
2
(119867lowast
6+ 119894119867lowast
5)
119868119871120579(120596) = 119870
2
(119867lowast
3+ 119894119867lowast
2)
119868119872ℎ(120596) = 119870
2
(119860lowast
4+ 119894119860lowast
1) 119868
119872119902(120596) = 119870
2
(119860lowast
6+ 119894119860lowast
5)
119868119872120579(120596) = 119870
2
(119860lowast
3+ 119894119860lowast
2)
(6)
where the overbars denote the Fourier transform operationthe terms containing 119894 represent imaginary parts119875lowast
120595 119867lowast
120595 and
119860lowast120595(120595 = 1 2 6) are dimensionless flutter derivatives
obtained from wind tunnel tests 119870 = 120596119861119880 is the reducedfrequency and 120596 is the circular frequency of vibration
According to classical airfoil theory the impulse func-tions can reasonably be approximated by a rational function[108]
119868 (120596) = [1198621+ 1198941198622
2120587
]+
119898
sum119897=1
119862119897+2
41205872 + 1198942120587119889119897+2
]1198892119897+2
V2 + 41205872] (7)
where the value of 119898 determines the level of accuracy ofthe approximation 119862
1 1198622 119862119897+2
and 119889119897+2(119897 = 1 2 119898)
are the frequency independent coefficients and ] = 2120587119870
is the reduced mean wind velocity By equating the realand imaginary parts in the comparison of (8) and (9) therelationship between the dimensionless flutter derivativesand the coefficients 119862120595
1 119862120595
2 119862120595
119897+2 and 119889120595
119897+2(where 120595 = Dh
Dq 119863120579 Lh Lq 119871120579 Mh Mq 119872120579 and 119897 = 1 2 119898) canbe establishedThese coefficients are determined by using thenonlinear least-squares method to fit the measured flutter
10 The Scientific World Journal
derivatives at different reduced frequencies The expressionof the aerodynamic impulse functions in the time domaincan be obtained by taking the inverse Fourier transform ofthe impulse functions By substituting the related impulseresponse functions into (5b) the self-excited lift force at the119894th section of bridge deck can then be derived as
119891se119890119871119894
=1
21205881198861198802
119894119861119894119862119871120579
1119894120579119894(119905) + 119862
119871120579
2119894(119861119894
119880119894
) 120579119894(119905)
+ 119862119871120579
3119894(119861119894
119880119894
) 120579119894(119905) +
119898
sum119897=1
119862119871120579
119897+3119894
times int119905
minusinfin
120579119894(119905) exp[minus
119889119871120579119897+3119894119880119894
119861119894
(119905 minus 120591)] 119889120591
+1
21205881198861198802
119894119862119871ℎ
1119894ℎ119894(119905) + 119862
119871ℎ
2119894(119861119894
119880119894
) ℎ119894(119905)
+ 119862119871ℎ
3119894(119861119894
119880119894
) ℎ119894(119905) +
119898
sum119897=1
119862119871ℎ
119897+3119894
times int119905
minusinfin
ℎ119894(119905) exp[minus
119889119871ℎ119897+3119894119880119894
119861119894
(119905 minus 120591)] 119889120591
+1
21205881198861198802
119894
119862119871119902
1119894119902119894(119905) + 119862
119871119902
2119894(119861119894
119880119894
) 119902119894(119905)
+ 119862119871119902
3119894(119861119894
119880119894
) 119902119894(119905) +
119898
sum119897=1
119862119871119902
119897+3119894
times int119905
minusinfin
119902120595(119905) exp[
[
minus119889119871119901
119897+3119894119880119894
119861119894
(119905 minus 120591)]
]
119889120591
(8)
In practice the terms 1198621198711205793119894 119862119871ℎ3119894 and 119862119871119902
3119894 which are related to
additional aerodynamic masses are normally neglected andthe value of 119898 is often taken as 2 [101] Similar formulationsfor self-excited drag and moment can be derived withanalogous definitions The self-excited forces at the 119894th nodeof the bridge deck can thus be expressed as
Feise = EeiXei + GeiXei + Feise (9)
where
Xei =
0
ℎei119902ei120579ei0
0
Eei =1
21205881198802
119894
[[[[[[[[
[
0 0 0 0 0 0
0 119862119871ℎ
1119894119862119871119902
11198941198611198941198621198711205791119894
0 0
0 119862119863ℎ1119894
119862119863119902
11198941198611198941198621198631205791119894
0 0
0 119861119894119862119872ℎ1119894
119861119894119862119872119902
111989411986121198941198621198721205791119894
0 0
0 0 0 0 0 0
0 0 0 0 0 0
]]]]]]]]
]
Gei =1
21205881198802
119894119861119894
[[[[[[[[
[
0 0 0 0 0 0
0 119862119871ℎ
2119894119862119871119902
21198941198611198941198621198711205792119894
0 0
0 119862119863ℎ2119894
119862119863119902
21198941198611198941198621198631205792119894
0 0
0 119861119894119862119872ℎ2119894
119861119894119862119872119902
211989411986121198941198621198721205792119894
0 0
0 0 0 0 0 0
0 0 0 0 0 0
]]]]]]]]
]
Feise =
[[[[[[[
[
0
eise
119863eise
eise0
0
]]]]]]]
]
=
[[[[[[[[[[[[[[[[
[
02
sum119897=1
119862119871119902
l+3119894119881119871119902
l+3119894 +2
sum119897=1
119862119871ℎl+3119894119881119871ℎ
l+3119894 +2
sum119897=1
119862119871120579l+3119894119881119871120579
l+3119894
2
sum119897=1
119862119863119902
l+3119894119881119863119902
l+3119894 +2
sum119897=1
119862119863ℎl+3119894119881119863ℎ
l+3119894 +2
sum119897=1
119862119863120579l+3119894119881119863120579
l+3119894
2
sum119897=1
119862119872119902
l+3119894119881119872119902
l+3119894 +2
sum119897=1
119862119872ℎl+3119894119881119872ℎ
l+3119894 +2
sum119897=1
119862119872120579l+3119894119881119872120579
l+3119894
0
0
]]]]]]]]]]]]]]]]
]
(10)
where 119881120595l+3119894 (120595 = Dh Dq 119863120579 Lh Lq 119871120579 Mh Mq) are theconvolution integrations of the 119894th node and can be calculatedusing a recursive algorithm For example
(11)The self-excited forces expressed by (9) relate to the centerof elasticity of the 119894th deck section Hence the force modelmust be distributed to the nodal points of the section Adistribution based on the rigid body motion relationshipsbetween themotions at the nodal point and those at the centerof elasticity of the deck section [109] was applied by Liu etal [22] In this study by applying the virtual work principlethe self-excited forces at the center of elasticity of the givensection were distributed to all nodes (see Figure 8)
24 Dynamic Interactions in a Wind-Vehicle-Bridge SystemWhen trains and road vehicles are running on long-spanbridges under crosswinds complicated dynamic interac-tions occur among the trains road vehicles cable-supported
The Scientific World Journal 11
Ui h
p
120572
Lsec i
Msec i
Dsec i
Fsekiz
Fsekiy
ce
Figure 8 Self-excited forces at the centre of elasticity and at the nodes in the 119894th deck section [22]
bridge and windThe buffeting response of the bridge due tocrosswind is superimposed on the dynamic response of thebridge due to railway and road vehicles The large vibrationof the bridge will in turn considerably affect the safetyand ride comfort of the drivers of the road vehicles Thusthe dynamic responses of a coupled vehicle-bridge systemunder crosswinds are of great concern to both engineers andresearchers
Detailed reviews of the dynamic interactions betweentrains and bridges between road vehicles and bridges andbetween wind and bridges have been given in the previoussections However the interaction between wind and vehiclesmust also be taken into account in a coupled wind-vehicle-bridge analysis Many studies have investigated wind-vehicleinteractions in the past few decades Balzer [110] developeda theory to estimate the aerodynamic forces on a movingvehicle using Taylorrsquos hypothesis of ldquofrozen turbulencerdquo Forengineering applications Cooper [111] proposed the powerspectral density (PSD) square-root coherence functionphase-lag function and aerodynamic admittance functionto model the unsteady side forces on a moving vehicle andlaid down the foundations for investigating the effects ofwind on a moving vehicle in the frequency domain Bakerdeveloped a theoretical model that describes the dynamics ofvehicles in crosswinds in the time domain [112 113] whichwas later extended to include driver behavior [114] Baker[115 116] further investigated both the steady and unsteadyaerodynamic forces acting on a variety of vehicles and carriedout extensive studies of the interaction between aerodynamicforces and moving vehicles These approaches have all beenapplied in coupled vehicle-bridge analysis For example Xuet al [101] simulated the aerodynamic wind forces actingon running road vehicles using the quasi-steady approachand Xu and Ding [117] derived and simulated the steadyand unsteady aerodynamic forces acting on a moving railwayvehicle in crosswinds in the time domain
Based on these separate studies on the various typesof dynamic interactions among wind vehicles (trains orroad vehicles) and long-span bridges several researchersin the last decade have examined the wind-vehicle-bridgecoupled system as a whole For instance studies have beencarried out on coupled road vehicle and cable-stayed bridgesystems [81 82 118] and on coupled train and cable-supportedbridge systems in crosswinds [101 117 119ndash121] In therecent years several new advances have been made bothin numerical simulation technologies and in wind tunnelmeasurements Chen et al [17] proposed a wind-vehicle-bridge framework which enables considering the dynamiceffects induced by simultaneous actions of railway highway
and wind loading and it was applied to analyze dynamicstress of long suspension bridges Li et al [122] extended thewind-vehicle-bridge couple analysis to the case of two trainsmeeting on a long-span suspension bridge Chen and Wu[118] proposed a semideterministic analytical model whichis able to consider dynamic interactions between the bridgewind and stochastic ldquorealrdquo traffic Based on the wind tunneltests Dorigatti et al [123]measured crosswind loads on high-sided vehicles over long-span bridges taking three differentvehicles (van double deck bus and lorry) and two differentbridge deck configurations into consideration Zhu et al[124] investigated aerodynamic coefficients of road vehiclesby adopting different road vehicles types wind directionsand vehicle positions Li et al [122] studied the effects ofsudden changes of wind loads as the train passing througha bridge tower or two trains passing each other by usingthe wind tunnel test rig with moving train models Hanet al [125] developed an experimental setup for measuringthe aerodynamic characteristics of vehicles and the bridgein wind tunnel and then investigated the influences ofparameters adopted in the tests
3 Applications of Simulation Technology toBridge Assessment
After reviewing the key issues of numerical simulationsfor dynamic response of long-span multiload bridges thissection will review the engineering applications of the newlydeveloped technologies to safety assessment of long-spanbridges such as assessment of fatigue and assessment underextreme events
31 Assessment of FatigueDamage Steel structures are widelyused in long-span bridges Research by the ASCE [126] indi-cates that 80ndash90 of failures in steel structures are related tofatigue and fracture Several disasters resulting from fatigue-induced bridge failure have occurred in history For instance46 people died in the collapse of the Silver Bridge (USA1967) and 32 people lost their lives in the collapse of theSungsoo Grand Bridge (South Korea 1994) These disastersteach us that fatigue is an important aspect of the safety ofsteel bridges and action should be taken to prevent similarfatigue-induced bridge failures In the past few decadesfatigue assessment of steel bridges has attracted increasingattention from both governments and bridge engineers andrelevant provisions have been stipulated in several codes andstandards [127ndash130]
12 The Scientific World Journal
It has great advantages to evaluate fatigue damage of long-span bridges based on numerical simulation especially fora multiload bridge which suffers multiple types of dynamicloading such as railway highway andwind loadingDifferentfrom sudden structural damage fatigue damage accumulateswith load-induced dynamic stress (or stress fluctuation) overthe service life of a bridge lasting for more than 100 yearsThe increase in traffic volume and gross vehicle weight thataccompany economic development is very likely to happenin the long period Numerical simulation technology can bean ideal tool to study influences of traffic growth patternsto fatigue damage of bridge In addition slender long-span bridges built in wind-prone regions also suffer fromconsiderable wind induced vibration which appears withina wide range of wind speeds and lasts for almost the wholedesign life of the bridge Given the simultaneous presenceof multiple vehicles and wind it is necessary to considercombined effects of traffic loading (railway andor highwayloading) and wind loading in the fatigue assessment Sincemultiple loading is concerned in a long time period thereare a large number of loading combinations for multipleloading in different intensities It is almost unavailable forfield measurement to obtain such complete information butnumerical simulation could be a good choice to determinedynamic responses of a long-span bridge under multipleloading
A number of structural health monitoring systems(SHMSs) have been installed on numerous recently builtlong-span bridges and a variety of sensors are used for mon-itoring bridge loadings (eg traffic wind and earthquakes)and conditions (including global and local responses) toensure bridge safety and user comfort under in-service con-ditions Well-known examples include Tsing-Ma Bridge inHong Kong Akashi Kaikyo Bridge in Japan Binzhou YellowRiver Bridge in China and Jindo Bridge in Korea Integrationof numerical simulation technologies and measurement dataof structural health monitoring systems (SHMSs) installedon long-span bridges will make the fatigue assessment morereliable for several reasons (1) it is a perfect validation byusing field measurement data of the different types of loadingas input of numerical simulation and the measured dynamicresponses for comparison with the computed ones (2) a largenumber of measured loading data could be used to establishloading databases or probabilistic models of different loads
In the recent years several researchers [7ndash10] appliedthe newly developed numerical simulation technologies tofatigue assessment of long-span bridges Chen et al [7]proposed a framework for fatigue analysis of a long-spansuspension bridge under railway highway and wind loadingby integrating computer simulation with SHMSs and itwas applied to evaluate fatigue damage of the Tsing MaSuspension Bridge over its design life as a case study Basedon this work Chen et al [8] proposed a framework forfatigue reliability analysis of long suspension bridges undermultiple loading inwhich the probabilisticmodels of railwayhighway and wind loading were established based on themeasurement data acquired from the SHMS of the TsingMa Bridge Wu et al [9] proposed a reliability-based fatigueapproach for slender long-span bridge and the combined
dynamic loading effects from wind and traffic as well asthe associated uncertainties were considered Based on theassumption that dynamic magnification related to vehicledynamics can be neglected in long suspension bridges Chenet al [8] established a framework for fatigue reliability anal-ysis To account for different types of long-span bridges withthe span length ranging from a few hundred to thousands ofmeters Zhang et al [10] proposed a more general frameworkwhich includes multiple random variables for the dynamicloads in a bridgersquos life cycle for the vehicle-bridge-winddynamic system such as road profile vehicle speed andwindvelocity and direction among other effects
32 Assessment under Extreme Events The aforementionedfatigue assessment mainly focuses on damage accumulationinduced by stress fluctuations under normal operationalcondition in a long-term period For long-span bridges inaddition to the normal operational conditions in which windspeeds are small ormoderate and traffic scenarios are normalthere are some extreme event conditions Extreme eventsmayinclude complex traffic congestion on the bridge coupledwith moderate or even strong wind [11] For example severetraffic congestions may be formed on the bridge as a resultof an evacuation or a partial blockage of driving lanes due totraffic accidents construction ormaintenance For hurricaneevacuations there are usually a lot of road vehicles passingthrough the bridge before the landfall of the hurricane whilethe wind speed may become pretty high already [131]
Although the excessive dynamic responses of the bridgesunder extreme events are rare it is also critical since itmay cause critical damage initiation or accumulation onsome local bridgemembers Furthermore the extreme events(eg heavy traffic) may even trigger the collapse of thewhole bridge by breaking the ldquoweakest linkrdquo especially whensome hidden damage or design flaw has not been detectedOne recent example is the Minnesota Bridge failure whichoccurred during rush hours with heavy traffic although trafficloads may not be the direct cause of failure For slender long-span bridges strong wind may also cause threats by workinginteractively with heavy traffic loads Therefore even thoughthe extreme cases associated with congested traffic andorwindy weather may be relatively rare and the durationscould be short it is still important for bridge engineers toappropriately look into these unusual extreme events duringstructural design and life-time management of these criticalinfrastructures [11]
The dynamic performance of long-span bridges undercombined actions of strong winds and running road vehicleshas been studied by many researchers in recent years [17 7981 82 132] Most of them studied bridge dynamic perfor-mance under road traffic inwhich only one or several vehiclesdistributed in an assumed (usually uniform) pattern on long-span bridges were considered Extreme events such as trafficcongestion coupled with strong wind were out of concernin those studies Recently Wu and Chen [11] conducteda research on the assessment of long-span bridges underextreme events which includes complex traffic congestioncoupled with moderate or even strong wind This study
The Scientific World Journal 13
applied the cellular automaton (CA) traffic model to thesimulation of the actual traffic flow through the bridgedefined representative scenarios for the extreme events andnumerically studied the bridge performance under thesepossible extreme events
4 Conclusions and Recommendations
Dynamic responses of long-span bridges are often requiredfor assessing the safety of these bridges and can be determinedby numerical simulation technologies This paper provides adetailed review of key issues involved in dynamic responseanalysis of long-span multiload bridges based on numericalsimulation including dynamic interactions between runningtrains and bridge between running road vehicles and bridgeand between wind and bridge and in the wind-vehicle-bridge coupled systemThen the review work was conductedfor engineering applications of newly developed numericalsimulation technologies to safety assessment of long-spanbridges such as assessment of fatigue damage and assessmentunder extreme event condition Although technologies fornumerical simulation of dynamic responses of long-spanmultiload bridge have achieved great advances in past fewdecades and successfully applied to several important bridgesit is still far from reach its maturity and enable to takeplace of traditional fieldmeasurementThe existing problemsand promising research efforts at least include the followingaspects
(1) After multiple types of dynamic interactions beingconsidered in the complex system computationalefficiency is a bottleneck problem for numericalsimulation of dynamic response of a long-span bridgeTypically when multiple loads are involved a largenumber of loading combinations for multiple load-ings must be considered in the assessment
(2) It is rather complex for the time-depending windloads acting on a long-span bridge and running vehi-cles especially for the case of rapid change of windloads such as a train passing through a bridge toweror two trains passing each other The aerodynamiccharacteristics of vehicles and the bridge under differ-ent loading scenarios can be determined through thewind tunnel testing and used in numerical simulationof dynamic responses of the bridge and vehicles
(3) It is a new trend to integrate numerical simulationtechnologies and measurement data of structuralhealth monitoring systems (SHMSs) installed onlong-span bridges whichmakes the safety assessmentof bridge structures more reliable Measured struc-tural responses could be used to validate numericalsimulation approach and measured loading infor-mation could be used for generating statistical orprobabilistic models of multiple loads
(4) It is important to study dynamic responses of bridgestructures under extreme events such as congestedtraffic coupled with windy weather which happens ina long-span bridge For the assessment under extreme
events using numerical simulation technologies sim-ulation of traffic flow and definition of representativescenarios of the extreme events are key issues
(5) It is necessary to consider the effects of typhoonwinds on the safety assessment of long-span bridgesin a reasonable way Few researches do this mostlybecause a probabilistic distribution of wind speedand direction specifically for typhoons is requiredfor assessment but there are insufficient measuredrecords to establish a reliable probabilistic typhoonwind model
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors wish to acknowledge the financial supportsfrom the National Natural Science Foundation of China(NSFC-51108395 and NSFC-51178366) the FundamentalResearch Funds for theCentral Universities (2012121032) andopen funding from Jiangsu Key Laboratory of EngineeringMechanics Special thanks go to the supervisor of the firstauthor Professor Y L XuHongKong PolytechnicUniversityfor his expert guidance and continuous support at all levelsthroughout his PhD study Sincere thanks should go tothe Highways Department of Hong Kong for providing theauthors with the field measurement data Any opinions andconcluding remarks presented in this paper are entirely thoseof the authors
References
[1] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[2] H Sohn C R Farrar N F Hunter and K Worden ldquoStructuralhealth monitoring using statistical pattern recognition tech-niquesrdquo Journal of Dynamic Systems Measurement and Controlvol 123 no 4 pp 706ndash711 2001
[3] W Fan and P Qiao ldquoVibration-based damage identificationmethods a review and comparative studyrdquo Structural HealthMonitoring vol 10 no 1 pp 83ndash111 2011
[4] X Q Zhu and S S Law ldquoDamage detection in simply supportedconcrete bridge structure under moving vehicular loadsrdquo Jour-nal of Vibration and Acoustics Transactions of the ASME vol129 no 1 pp 58ndash65 2007
[5] J Li and S S Law ldquoDamage identification of a target sub-structure with moving load excitationrdquoMechanical Systems andSignal Processing vol 30 pp 78ndash90 2012
[6] J Li S S Law and H Hao ldquoImproved damage identification inbridge structures subject tomoving loads numerical and exper-imental studiesrdquo International Journal of Mechanical Sciencesvol 74 pp 99ndash111 2013
14 The Scientific World Journal
[7] Z W Chen Y L Xu Y Xia Q Li and K Y Wong ldquoFatigueanalysis of long-span suspension bridges under multiple load-ing case studyrdquo Engineering Structures vol 33 no 12 pp 3246ndash3256 2011
[8] Z W Chen Y L Xu and X M Wang ldquoSHMS-based fatiguereliability analysis of multiloading suspension bridgesrdquo Journalof Structural Engineering-Asce vol 138 pp 299ndash307 2012
[9] J Wu S R Chen and J W van de Lindt ldquoFatigue assessmentof slender long-span bridges reliability approachrdquo Journal ofBridge Engineering vol 17 no 1 pp 47ndash57 2012
[10] W Zhang C S Cai and F Pan ldquoFatigue reliability assessmentfor long-span bridges under combined dynamic loads fromwinds and vehiclesrdquo Journal of Bridge Engineering vol 18 pp735ndash747 2013
[11] JWu and S R Chen ldquoProbabilistic dynamic behavior of a long-span bridge under extreme eventsrdquo Engineering Structures vol33 no 5 pp 1657ndash1665 2011
[12] J M Ko and Y Q Ni ldquoTechnology developments in structuralhealth monitoring of large-scale bridgesrdquo Engineering Struc-tures vol 27 no 12 pp 1715ndash1725 2005
[13] TH YiHN Li andHM Sun ldquoMulti-stage structural damagediagnosis method based on ldquoenergy-damagerdquo theoryrdquo SmartStructures and Systems vol 12 pp 345ndash361 2013
[14] T H Yi H N Li and M Gu ldquoFull-scale measurements ofdynamic response of suspension bridge subjected to environ-mental loads using GPS technologyrdquo Science China Technologi-cal Sciences vol 53 no 2 pp 469ndash479 2010
[15] T H Yi H N Li and M Gu ldquoExperimental assessmentof high-rate GPS receivers for deformation monitoring ofbridgerdquoMeasurement Journal of the InternationalMeasurementConfederation vol 46 pp 420ndash432 2013
[16] X W Ye Y Q Ni K Y Wong and J M Ko ldquoStatistical analysisof stress spectra for fatigue life assessment of steel bridges withstructural health monitoring datardquo Engineering Structures vol45 pp 166ndash176 2012
[17] Z W Chen Y L Xu Q Li and D J Wu ldquoDynamic stressanalysis of long suspension bridges under wind railway andhighway loadingsrdquo Journal of Bridge Engineering vol 16 no 3pp 383ndash391 2011
[18] S G Meisenholder and P Weidlinger ldquoDynamic interactionaspects of cable-stayed guideways for high speed ground trans-portationrdquoAmerican Society ofMechanical Engineers no 74 pp180ndash192 1974
[19] Q H Mao Research on the Highway Bridge Vibration Due toMoving Vehicles Tongji University Shang Hai China 1989
[20] Y L Xu J M Ko and Z Yu ldquoModal analysis of tower-cable system of Tsing Ma long suspension bridgerdquo EngineeringStructures vol 19 pp 857ndash867 1997
[21] W Guo H Xia and Y-L Xu ldquoDynamic response of a long spansuspension bridge and running safety of a train under windactionrdquo Frontiers of Architecture and Civil Engineering in Chinavol 1 no 1 pp 71ndash79 2007
[22] T T Liu Y L Xu W S Zhang K Y Wong H J Zhou and KW Y Chan ldquoBuffeting-induced stresses in a long suspensionbridge structural health monitoring oriented stress analysisrdquoWind and Structures An International Journal vol 12 no 6 pp479ndash504 2009
[23] K Y Wong ldquoStructural identification of Tsing Ma BridgerdquoTransactions Hong Kong Institution of Engineers vol 10 no 1pp 38ndash47 2003
[24] Y L XuQ Li D JWu and ZWChen ldquoStress and accelerationanalysis of coupled vehicle and long-span bridge systems usingthe mode superposition methodrdquo Engineering Structures vol32 no 5 pp 1356ndash1368 2010
[25] Y F Duan Y L Xu Q G Fei et al ldquoAdvanced finite elementmodel of Tsing Ma Bridge for structural health monitoringrdquoInternational Journal of Structural Stability and Dynamics vol11 no 2 pp 313ndash344 2011
[26] Z X Li T Q Zhou T H T Chan and Y Yu ldquoMulti-scalenumerical analysis on dynamic response and local damage inlong-span bridgesrdquo Engineering Structures vol 29 no 7 pp1507ndash1524 2007
[27] W Zhang C S Cai and F Pan ldquoFinite element modeling ofbridges with equivalent orthotropic material method for multi-scale dynamic loadsrdquo Engineering Structures vol 54 pp 82ndash932013
[28] S P Timoshenko ldquoOn the forced vibrations of bridgesrdquo Philo-sophical Magazine vol 6 no 257 pp 1018ndash1019 1922
[29] R S Ayre G Ford and L S Jacobsen ldquoTransverse vibration ofa two-span beam under the action of a moving constant forcerdquoJournal of Applied Mechanics vol 17 pp 1ndash12 1950
[30] R S Ayre andLS Jacobsen ldquoTransverse vibration of a two-spanbeam under the action of a moving alternating forcerdquo Journal ofApplied Mechanics vol 17 pp 283ndash290 1950
[31] L FrybaVibration of Solids and Structures underMoving LoadsASCE Press 1972
[32] J-S Wu and C-W Dai ldquoDynamic response of multispannonuniform beam due to moving loadsrdquo Journal of StructuralEngineering vol 113 no 3 pp 458ndash474 1987
[33] W Weaver S P Timoshenko and D H Young VibrationProblems in Engineering John Wiley amp Sons 1990
[34] N H Galdos D R Schelling and M A Sahin ldquoMethodologyfor impact factor of horizontally curved box bridgesrdquo Journal ofStructural Engineering vol 119 no 6 pp 1917ndash1934 1993
[35] J A Gbadeyan and S T Oni ldquoDynamic behaviour of beamsand rectangular plates under moving loadsrdquo Journal of Soundand Vibration vol 182 no 5 pp 677ndash695 1995
[36] D Y Zheng Y K Cheung F T K Au and Y S ChengldquoVibration of multi-span non-uniform beams under movingloads by using modified beam vibration functionsrdquo Journal ofSound and Vibration vol 212 no 3 pp 455ndash467 1998
[37] G V Rao ldquoLinear dynamics of an elastic beam under movingloadsrdquo Journal of Vibration and Acoustics Transactions of theASME vol 122 no 3 pp 281ndash289 2000
[38] Y B Yang J D Yau and Y S Wu Vehicle-Bridge InteractionDynamic with Applications to High-Speed Railways WorldScientific 2004
[39] E C Ting J Genin and J H Ginsberg ldquoA general algorithm formovingmass problemsrdquo Journal of Sound and Vibration vol 33no 1 pp 49ndash58 1974
[40] S Sadiku and H H E Leipholz ldquoOn the dynamics of elasticsystems with moving concentrated massesrdquo Ingenieur-Archivvol 57 no 3 pp 223ndash242 1987
[41] J E Akin and M Mofid ldquoNumerical solution for response ofbeamswithmovingmassrdquo Journal of Structural Engineering vol115 no 1 pp 120ndash131 1989
[42] M AMahmoud andM A Abou Zaid ldquoDynamic response of abeam with a crack subject to a moving massrdquo Journal of Soundand Vibration vol 256 no 4 pp 591ndash603 2002
[43] V K Garg Dynamics of Railway Vehicle Systems AcademicPress 1994
The Scientific World Journal 15
[44] T-LWang andDHuang ldquoCable-stayed bridge vibration due toroad surface roughnessrdquo Journal of Structural Engineering vol118 no 5 pp 1354ndash1374 1992
[45] Y-B Yang and B-H Lin ldquoVehicle-bridge interaction analysisby dynamic condensation methodrdquo Journal of Structural Engi-neering vol 121 no 11 pp 1636ndash1643 1995
[46] Y B Yang J D Yau and L C Hsu ldquoVibration of simple beamsdue to trainsmoving at high speedsrdquoEngineering Structures vol19 no 11 pp 936ndash943 1997
[47] B Tabarrok and E Esmailzadeh ldquoInduced vibration of bridgestransversed by moving vehiclesrdquo Transactions of the CanadianSociety for Mechanical Engineering B vol 24 no 1 pp 191ndash1982000
[48] C Liu T-L Wang and D Huang ldquoImpact study for multi-girder bridge based on correlated road roughnessrdquo StructuralEngineering and Mechanics vol 11 no 3 pp 259ndash272 2001
[49] K H Chu V K Garg and T L Wang ldquoImpact in railwayprestressed concrete bridgesrdquo Journal of Structural Engineeringvol 112 no 5 pp 1036ndash1051 1986
[50] T-LWang V K Garg and K-H Chu ldquoRailway bridgevehicleinteraction studies with new vehicle modelrdquo Journal of Struc-tural Engineering vol 117 no 7 pp 2099ndash2116 1991
[51] H Xia Y L Xu and T H T Chan ldquoDynamic interaction oflong suspension bridges with running trainsrdquo Journal of Soundand Vibration vol 237 no 2 pp 263ndash280 2000
[52] Q-L Zhang A Vrouwenvelder and J Wardenier ldquoNumericalsimulation of train-bridge interactive dynamicsrdquo Computersand Structures vol 79 no 10 pp 1059ndash1075 2001
[53] H Xia N Zhang and G de Roeck ldquoDynamic analysis of highspeed railway bridge under articulated trainsrdquo Computers andStructures vol 81 no 26-27 pp 2467ndash2478 2003
[54] G Diana F Cheli A Collina R Corradi and S MelzildquoThe development of a numerical model for railway vehiclescomfort assessment through comparison with experimentalmeasurementsrdquoVehicle SystemDynamics vol 38 no 3 pp 165ndash183 2002
[55] Q Li Y L Xu D J Wu and Z W Chen ldquoComputer-aided nonlinear vehicle-bridge interaction analysisrdquo Journal ofVibration and Control vol 16 pp 1791ndash1816 2010
[56] A Wiriyachai K H Chu and V K Garg ldquoBridge impact dueto wheel and track irregularitiesrdquo Journal of the EngineeringMechanics Division vol 108 no 4 pp 648ndash666 1982
[57] L Fryba Dynamics of Railway Bridges Inst of Civil Engineers1996
[58] D Huang and T-L Wang ldquoImpact analysis of cable-stayedbridgesrdquo Computers and Structures vol 43 no 5 pp 897ndash9081992
[59] W M Zhai Vehicle-Track Coupling Dynamics Chinese RailwayPress Beijing China 2007
[60] M Olsson ldquoFinite element modal co-ordinate analysis ofstructures subjected to moving loadsrdquo Journal of Sound andVibration vol 99 no 1 pp 1ndash12 1985
[61] Y-B Yang C-H Chang and J-D Yau ldquoAn element foranalysing vehicle-bridge systems considering vehiclersquos pitchingeffectrdquo International Journal for NumericalMethods in Engineer-ing vol 46 no 7 pp 1031ndash1047 1999
[62] Y B Yang and Y S Wu ldquoA versatile element for analyzingvehicle-bridge interaction responserdquo Engineering Structuresvol 23 no 5 pp 452ndash469 2001
[63] F T K Au J J Wang and Y K Cheung ldquoImpact study ofcable-stayed bridge under railway traffic using various modelsrdquo
Journal of Sound and Vibration vol 240 no 3 pp 447ndash4652001
[64] Y Q Sun andMDhanasekar ldquoA dynamicmodel for the verticalinteraction of the rail track and wagon systemrdquo InternationalJournal of Solids and Structures vol 39 no 5 pp 1337ndash13592002
[65] K Henchi M Fafard M Talbot and G Dhatt ldquoAn efficientalgorithm for dynamic analysis of bridges under moving vehi-cles using a coupledmodal and physical components approachrdquoJournal of Sound and Vibration vol 212 no 4 pp 663ndash6831998
[66] Y L Xu and L YWang ldquoAnalytical study of wind-rain-inducedcable vibration SDOFmodelrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 91 no 1-2 pp 27ndash40 2003
[67] B Biondi G Muscolino and A Sofi ldquoA substructure approachfor the dynamic analysis of train-track-bridge systemrdquoComput-ers and Structures vol 83 no 28ndash30 pp 2271ndash2281 2005
[68] J L Humar and A H Kashif ldquoDynamic response analysis ofslab-type bridgesrdquo Journal of Structural Engineering vol 121 no1 pp 48ndash62 1995
[69] P Lou and Q-Y Zeng ldquoFormulation of equations of verticalmotion for vehicle-track-bridge systemrdquo Journal of the ChinaRailway Society vol 26 no 5 p 71 2004
[70] O Coussy M Said and J-P van Hoove ldquoThe influence ofrandom surface irregularities on the dynamic response ofbridges under suspended moving loadsrdquo Journal of Sound andVibration vol 130 no 2 pp 313ndash320 1989
[71] E S Hwang and A S Nowak ldquoSimulation of dynamic loadfor bridgesrdquo Journal of Structural Engineering vol 117 pp 1413ndash1434 1991
[72] F Yang and G A Fonder ldquoAn iterative solution methodfor dynamic response of bridge-vehicles systemsrdquo EarthquakeEngineering and Structural Dynamics vol 25 pp 195ndash215 1996
[73] W Zhai and Z Cai ldquoDynamic interaction between a lumpedmass vehicle and a discretely supported continuous rail trackrdquoComputers and Structures vol 63 no 5 pp 987ndash997 1997
[74] W M Zhai and C B Cai ldquoTraintrackbridge dynamic inter-actions simulation and applicationsrdquo Vehicle System Dynamicsvol 37 pp 653ndash665 2003
[75] D Bruno F Greco and P Lonetti ldquoDynamic impact analysis oflong span cable-stayed bridges under moving loadsrdquo Engineer-ing Structures vol 30 no 4 pp 1160ndash1177 2008
[76] X D Song D J Wu and Q Li ldquoDynamic impact analysisof double-tower cable-stayed maglev bridges using a simplemodelrdquo Journal of Bridge Engineering vol 19 pp 34ndash43 2014
[77] Y-S Wu and Y-B Yang ldquoSteady-state response and ridingcomfort of trains moving over a series of simply supportedbridgesrdquoEngineering Structures vol 25 no 2 pp 251ndash265 2003
[78] P Antolin N Zhang J M Goicolea H Xia M A Astiz andJ Oliva ldquoConsideration of nonlinear wheel-rail contact forcesfor dynamic vehicle-bridge interaction in high-speed railwaysrdquoJournal of Sound and Vibration vol 332 no 5 pp 1231ndash12512013
[79] W H Guo and Y L Xu ldquoFully computerized approach to studycable-stayed bridge-vehicle interactionrdquo Journal of Sound andVibration vol 248 no 4 pp 745ndash761 2001
[80] Y L Xu and W H Guo ldquoDynamic behaviour of high-sidedroad vehicles subject to a sudden crosswind gustrdquo Wind andStructures vol 6 no 5 pp 325ndash346 2003
[81] Y L Xu and W H Guo ldquoDynamic analysis of coupled roadvehicle and cable-stayed bridge systems under turbulent windrdquoEngineering Structures vol 25 no 4 pp 473ndash486 2003
16 The Scientific World Journal
[82] C S Cai and S R Chen ldquoFramework of vehicle-bridge-winddynamic analysisrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 92 no 7-8 pp 579ndash607 2004
[83] S R Chen and J Wu ldquoModeling stochastic live load for long-span bridge based on microscopic traffic flow simulationrdquoComputers and Structures vol 89 no 9-10 pp 813ndash824 2011
[84] A V Paultre B Yang L A Bergman and C A Tan ldquoBridgedynamics and dynamic amplification factorsmdasha review ofanalytical and experimental findingsrdquoCanadian Journal of CivilEngineering vol 19 no 2 pp 260ndash278 1992
[85] H Honda Y Kajikawa and T Kobori ldquoSpectra of road surfaceroughness of bridgesrdquo Journal of the Structural Division vol 108pp 1956ndash1966 1982
[86] M J Inbanathan and M Wieland ldquoBridge vibrations dueto vehicle moving over rough surfacerdquo Journal of StructuralEngineering vol 113 no 9 pp 1994ndash2008 1987
[87] T-LWang andDHuang ldquoCable-stayed bridge vibration due toroad surface roughnessrdquo Journal of Structural Engineering vol118 no 5 pp 1354ndash1374 1992
[88] P K Chatterjee T K Datta and C S Surana ldquoVibrationsuspension bridges under vehicular movementrdquo Journal ofStructural Engineering vol 120 no 3 pp 681ndash703 1994
[89] D Chang and H Lee ldquoImpact factors for simple-span highwaygirder bridgesrdquo Journal of Structural Engineering vol 120 no 3pp 704ndash715 1994
[90] T-C Pan and J Li ldquoDynamic vehicle element method for tran-sient response of coupled vehicle-structure systemsrdquo Journal ofStructural Engineering vol 128 no 2 pp 214ndash223 2002
[91] C J Dodds and J D Robson ldquoThe description of road surfaceroughnessrdquo Journal of Sound Vibration vol 31 no 2 pp 175ndash183 1973
[92] D Huang T-L Wang and M Shahawy ldquoImpact studies ofmultigirder concrete bridgesrdquo Journal of Structural Engineeringvol 119 no 8 pp 2387ndash2402 1993
[93] A G Davenport ldquoBuffeting of a suspension bridge by stormwindrdquo Journal of Structural Division vol 88 pp 233ndash268 1962
[94] R H Scanlan ldquoThe action of flexible bridges under wind Iflutter theoryrdquo Journal of Sound and Vibration vol 60 no 2pp 187ndash199 1978
[95] Q Ding and P K K Lee ldquoComputer simulation of buffetingactions of suspension bridges under turbulentwindrdquoComputersand Structures vol 76 no 6 pp 787ndash797 2000
[96] V Boonyapinyo T Miyata and H Yamada ldquoAdvanced aerody-namic analysis of suspension bridges by state-space approachrdquoJournal of Structural Engineering vol 125 no 12 pp 1357ndash13661999
[97] Y-H Chen and C-Y Li ldquoDynamic response of elevated high-speed railwayrdquo Journal of Bridge Engineering vol 5 no 2 pp124ndash130 2000
[98] X Chen M Matsumoto and A Kareem ldquoTime domainflutter and buffeting response analysis of bridgesrdquo Journal ofEngineering Mechanics vol 126 no 1 pp 7ndash16 2000
[99] X Chen and A Kareem ldquoEquivalent static wind loads forbuffeting response of bridgesrdquo Journal of Structural Engineeringvol 127 no 12 pp 1467ndash1475 2001
[100] S R Chen and C S Cai ldquoEvolution of long-span bridgeresponse to wind-numerical simulation and discussionrdquo Com-puters and Structures vol 81 no 21 pp 2055ndash2066 2003
[101] Y L XuHXia andQ S Yan ldquoDynamic response of suspensionbridge to high wind and running trainrdquo Journal of BridgeEngineering vol 8 no 1 pp 46ndash55 2003
[102] A Guo Y L Xu and H Li ldquoDynamic performance of cable-stayed bridge tower with multi-stage pendulum mass damperunderwind excitations-II experimentrdquoEarthquake Engineeringand Engineering Vibration vol 6 no 4 pp 417ndash424 2007
[103] E Simiu and R H ScanlanWind Effects on Structures 1996[104] Y Cao H Xiang and Y Zhou ldquoSimulation of stochastic wind
velocity field on long-span bridgesrdquo Journal of EngineeringMechanics vol 126 no 1 pp 1ndash6 2000
[105] M Shinozuka and C-M Jan ldquoDigital simulation of randomprocesses and its applicationsrdquo Journal of Sound and Vibrationvol 25 no 1 pp 111ndash128 1972
[106] K M Shum Lateral and torsional vibration control of long spanbridge deck using novel tuned liquid column dampers [PhDthesis] Department of Civil and Structural Engineering TheHong Kong Polytechnic University 2004
[107] Y K Lin and J N Yang ldquoMultimode bridge response to windexcitationsrdquo Journal of EngineeringMechanics vol 109 no 2 pp586ndash603 1983
[108] Y S Lin Self-Excited Bridge Motion in Turbulent Wind 1978[109] D T Lau M S Cheung and S H Cheng ldquo3D flutter analysis
of bridges by spline finite-strip methodrdquo Journal of StructuralEngineering vol 126 no 10 pp 1246ndash1254 2000
[110] L A Balzer ldquoAtmospheric turbulence encountered by high-speed ground transport vehiclesrdquo Journal of Mechanical Engi-neering Science vol 19 pp 227ndash235 1977
[111] R K Cooper ldquoAtmospheric turbulence with respect to movingground vehiclesrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 17 no 2 pp 215ndash238 1984
[112] C J Baker ldquoA simplified analysis of various types of wind-induced road vehicle accidentsrdquo Journal of Wind Engineeringand Industrial Aerodynamics vol 22 no 1 pp 69ndash85 1986
[113] C J Baker ldquoMeasures to control vehicle movement at exposedsites during windy periodsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 25 no 2 pp 151ndash161 1987
[114] C J Baker ldquoHigh sided articulated road vehicles in strong crosswindsrdquo Journal of Wind Engineering and Industrial Aerodynam-ics vol 31 no 1 pp 67ndash85 1988
[115] C J Baker ldquoGround vehicles in high cross winds part I steadyaerodynamic forcesrdquo Journal of Fluids and Structures vol 5 no1 pp 69ndash90 1991
[116] C J Baker ldquoGround vehicles in high cross winds part IIunsteady aerodynamic forcesrdquo Journal of Fluids and Structuresvol 5 no 1 pp 91ndash111 1991
[117] Y L Xu and Q S Ding ldquoInteraction of railway vehicles withtrack in cross-windsrdquo Journal of Fluids and Structures vol 22no 3 pp 295ndash314 2006
[118] S R Chen and J Wu ldquoDynamic performance simulation oflong-span bridge under combined loads of stochastic traffic andwindrdquo Journal of Bridge Engineering vol 15 no 3 pp 219ndash2302010
[119] Y Li S Qiang H Liao and Y L Xu ldquoDynamics of wind-rail vehicle-bridge systemsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 93 no 6 pp 483ndash507 2005
[120] Y L Xu N Zhang and H Xia ldquoVibration of coupled trainand cable-stayed bridge systems in cross windsrdquo EngineeringStructures vol 26 no 10 pp 1389ndash1406 2004
[121] W W Guo Y L Xu H Xia W S Zhang and K M ShumldquoDynamic response of suspension bridge to typhoon and trainsII numerical resultsrdquo Journal of Structural Engineering vol 133no 1 pp 12ndash21 2007
The Scientific World Journal 17
[122] Y L Li H Y Xiang B Wang Y L Xu and S Z QiangldquoDynamic analysis of wind-vehicle-bridge coupling systemduring the meeting of two trainsrdquo Advances in StructuralEngineering vol 16 pp 1663ndash1670 2013
[123] F Dorigatti M Sterling D Rocchi et al ldquoWind tunnelmeasurements of crosswind loads on high sided vehicles overlong span bridgesrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 107 pp 214ndash224 2012
[124] L D Zhu L Li Y L Xu and Q Zhu ldquoWind tunnel investi-gations of aerodynamic coefficients of road vehicles on bridgedeckrdquo Journal of Fluids and Structures vol 30 pp 35ndash50 2012
[125] Y Han J X Hu C S Cai Z Q Chen and C G LildquoExperimental and numerical studies of aerodynamic forces onvehicles and bridgesrdquoWind and Structures vol 17 pp 163ndash1842013
[126] ASCE ldquoCommittee on fatigue and fracture reliability of thecommittee on structural safety and reliability of the structuraldivision fatigue reliability 1ndash4rdquo Journal of Structural Engineer-ing vol 108 pp 3ndash88 1982
[127] BS ldquoBS5400 part 10 code of practice for fatiguerdquo BritishStandard Institute 1980
[128] BS ldquoBS7608 code of practice for fatigue design and assessmentof steel structuresrdquo British Standard Institute 1993
[129] AASHTOGuide Specifications for Fatigue Evaluation of ExistingSteel Bridges 1990
[130] AASHTO Guide Manual for Condition Evaluation and Loadand Resistance Factor Rating (LRFR) of Highway Bridges 2003
[131] S R Chen C S Cai and B Wolshon ldquoFrom normal operationto evacuation Single-vehicle safety under adverse weathertopographic and operational conditionsrdquo Natural HazardsReview vol 10 no 2 pp 68ndash76 2009
[132] S R Chen and C S Cai ldquoEquivalent wheel load approach forslender cable-stayed bridge fatigue assessment under traffic andwind feasibility studyrdquo Journal of Bridge Engineering vol 12 no6 pp 755ndash764 2007
the trigonometry series white noise filtration autoregressive(AR) and power spectral density (PSD) sampling methodsAmong these methods the PSD sampling method has beenwidely adopted due to its high computational accuracy Thelateral and vertical irregularities could be all assumed tobe zero-mean stationary Gaussian random processes andexpressed through the inverse Fourier transformation of aPSD function [58]
119910119904(119909) =
119873
sum119896=1
radic2119878 (119891119896) Δ119891 cos (2120587119891
119896119909 + 120579119896) (1)
where 119878(119891) is the PSD function 119891119896= 119891l + (119896 minus 12)Δ119891
Δ119891 = (119891119906minus 119891l)119873 119891119906 and 119891119897 are the upper and lower cutoff
frequencies respectively and 120579119896is the random phase angle
uniformly distributed between 0 and 2120587 Rail irregularity inrailway engineering is commonly represented by a one-sidedPSD function
The PSD functions of rail irregularities have been devel-oped by different countries Based on the PSD functions ofrail irregularities developed by the Research Institute of theChina Railway Administration Zhai [59] expressed all railirregularities using the unified rational formula as follows
where 119891 = 1120578 (mminus1) is the spatial frequency in cyclem (120578 isthe wavelength) and119860 to119866 are the parameters recommendedby Zhai [59] specifically for vertical and lateral rail irregular-ities
214 Solution Methods The dynamic analysis of vehicle-bridge coupled system requires two sets of equations ofmotion for the bridge and vehicles respectively Thesedescribe the interaction or contact forces at the contactpoints of the two subsystems Because the contact pointsmove from time to time the system matrices are generallytime dependent and must be updated and factorized at eachtime step The various solution methods can be generalizedinto two groups according to whether or not an iterativeprocedure is needed at each time step
The first group ofmethods solves the equations of motionof a coupled vehicle-bridge system at each time step withoutiteration This approach has been widely used in coupledvehicle-bridge analysis [51 53 60ndash69] These methods havegood computational stability and are convenient for dealingwith vehicle-bridge interaction problems when the vehiclemodel is relatively simple The main disadvantage is thatthe equations of motion of the coupled system are timedependent and thus the characteristic matrices must bemodified at each time step In addition the equations ofmotion of the coupled vehicle-bridge system become verydifficult to determine if nonlinear wheel-rail contacts andnonlinear vehicle models are considered
The second group of methods solves the equations for thevehicles and bridge separately and requires an iterative pro-cess to obtain convergence for the displacements of the vehi-cles and bridge at all contact points Given that the conditions
of wheel-rail contact geometry and contact forces are rathercomplex a stable integration method adopting a small timeinterval is needed for obtaining the convergence of vehicleand bridge subsystems at the contact points in each time stepMany studies have applied this type of method to investigatevehicle-bridge interactions [70ndash76] The advantage of thesemethods is that the dynamic propertymatrices in the two setsof equations of motion remain constant which is convenientfor the consideration of nonlinear vehicle-bridge interactionsand nonlinear vehicle models [55] However in engineeringapplications the iterative convergence is a critical problemwith this type of method The low convergence rate andoccasional divergence of the solution have also been noted[77] Li et al [55] investigated the performance of theseiterative schemes using the Wilson-120579 method Newmark-120573method and an explicit integrationmethod proposed byZhai[59] and found that the latter gave amuch higher convergencerate than the former two methods
Most of the above methods solved the equations ofmotion of a coupled vehicle-bridge systemusing the nonjumpmodel which assumes that the moving vehicle travelingalong the bridge is always in contact with the rails nomatter what the sign is of the contact forces This is notalways true in view of the physics of the moving vehiclewhich simply sits on the upper surfaces of the rails Theinteraction forces between the moving vehicle and the bridgedepend on the motions of the vehicle the flexibility of thebridge and the track irregularities Li et al [55] utilizeda jump model to solve vehicle-bridge interaction problemusing a noniterative Runge-Kutta method and found thatthe acceleration responses of the car body using the wheel-jump model are smaller than those using the wheel nonjumpmodel when the vehicle speed exceeds 300 kmhr Antolin etal [78] proposed a nonlinear wheel-rail interaction modelwhich considers nonlinear wheel-rail contact forces in theinteraction as well as realistic wheel and rail profiles andapplied it for analysis of dynamic interaction between highspeed trains and bridges
22 Dynamic Interaction between Road Vehicles and BridgeSection 21 gave a detailed literature review of the dynamicinteractions between trains and bridges As there are somefundamental differences between trains and road vehiclesthis section reviews the modeling of road vehicles thesimulation of road vehicle flow and the modeling of roadsurface roughness
221 Modeling of Road Vehicles To analyze the dynamicinteraction between a bridge and running road vehicles amodel of road vehicles must be established A sophisticatedroad vehiclemodel is required tomake the simulation as real-istic as possible A road vehicle is modeled as a combinationof several rigid bodies each of which is connected by a set ofsprings and dashpots which model the elastic and dampingeffects of the tires and suspension systems respectivelyThereare various configurations of road vehicles such as a tractorand trailer with different axle spacing Road vehicle models
The Scientific World Journal 7
L11 L10 L9
L8
Ku5Cu5
Cl5Kl5
Zs5
Ku4Cu4
Cl4Kl4
Zs4
Ku3 Cu3
Cl3Kl3
Zs3
Ku2 Cu2
Cl2Kl2
Zs2
Ku1 Cu1
Cl1Kl1
Zs1
Ku1 Cu1
Cl1Kl1
Zs1
Ku6 Cu6
Cl6Kl6
Zs6
b1 b1
x yL1L2L3L4L5L6L7
Z3 1205793 Z2 Z1
Z1
1205792
1205791
zz
1206011
Figure 5 Dynamic model of a tractor-trailer [79]
zz
x y
h
2b1
ZZY
h1
120601
Kuz1 Kuz3Cuz1
Kuz1 Cuz1 Cuz3
Zs1Zs1
Zs3
Cuy1 Cuy3
Kuy1Kuy3
Ys1 Ys3
Klz1Klz1 Klz3Clz1 Clz3Clz1
Kuz2 Cuz2
Zs2
Klz2 Clz2
120579
L1L2
Kly1
Cly1
Kly3
Cly3
Figure 6 Dynamic model of a high-sided road vehicle [80]
that contain several DOFs have been devised for vehicle-bridge interaction analysis For example Guo and Xu [79]modeled a 17-DOF four-axle heavy tractor-trailer vehicle (seeFigure 5) to investigate the interaction between vehicles anda cable-stayed bridge A total of three DOFs were assignedto rigid bodies representing either the tractor or the trailerto account for vertical rolling and pitching motions Onlyone DOFwas assigned to the rigid body representing the axleset moving in the vertical direction Different vehicle modelsare adopted in wind-vehicle-bridge interaction analyses Xuand Guo [80] modeled a 13-DOF two-axle road vehicle (seeFigure 6) for the dynamic analysis of a coupled road vehicleand bridge system under turbulent wind Five DOFs wereassigned to the vehicle body with respect to its center ofgravity to account for vertical lateral rolling yawing andpitching motions and two DOFs were assigned to the frontand rear axle sets to account for motions in the vertical andlateral directionMoreDOFs are needed to account for lateralcrosswinds
222 Simulation of Road Vehicle Flow On long-span bridgesthere is a high probability of the simultaneous presence ofmultiple road vehicles including heavy trucks This maylead to larger amplitude stress responses and greater fatiguedamage of the local bridge components than would be thecase with only one road vehicle The simulation of roadvehicle flow is thus important in the analysis of the dynamicinteraction between road vehicles and bridges Rather simplepatterns of road vehicle flow have been assumed in most
vehicle-bridge coupled dynamic analyses [79 81 82] in whicheither one or several vehicles are distributed on the bridgein an assumed (usually uniform) pattern Obviously suchassumptions do not represent actual road traffic conditionsRecently Chen and Wu [83] modeled the stochastic trafficload for a long-span bridge based on the cellular automaton(CA) traffic flow simulation technique In this study theysimulated a complicated road vehicle flow on long-spanbridges in terms of vehicle number vehicle type combinationand driver operation characteristics such as lane changingacceleration or deceleration
223 Modeling of Road Surface Roughness Road surfaceroughness is an important factor that greatly affects vehicle-bridge interactions Paultre et al [84] pointed out that roadsurface or pavement roughness can significantly affect theimpact response of a bridge The roughness or surface profiledepends primarily on the workmanship involved in theconstruction of the pavement or roadway and how it is main-tained which although random in nature may contain someinherent frequencies [38] In most cases surface roughnesswhich is three-dimensional in reality is often approximatedby a two-dimensional profile To account for its randomnature the road profile can be modeled as a stationaryGaussian random process and derived using a certain powerspectral density function Other methods similar to this havebeen widely adopted by researchers studying vehicle-inducedbridge vibration [65 70 71 85ndash90] Dodds and Robson [91]developed power spectral density functions that were later
8 The Scientific World Journal
modified and used byWang and Huang [87] and Huang et al[92] This approach was also adopted by literatures [79 81] intheir dynamic analyses of coupled vehicle-bridge and wind-vehicle-bridge systems
23 Dynamic Interaction between Wind and Bridge Whena long-span cable-supported bridge is immersed in a givenflow field the bridge will be subject to mean and fluctuatingwind forces To simulate these forces a linear approximationof the time-averaged static and time-varying buffeting andself-excited force components must be formulated [93 94]As dynamic bridge responses are of concern in this study onlybuffeting and self-excited forces are considered and reviewedin this section
231 Buffeting Forces Buffeting action is a random vibrationcaused by turbulent wind that excites certain modes of vibra-tion across a bridge depending on the spectral distributionof the pressure vectors [95] Although the buffeting responsemay not lead to catastrophic failure it can lead to structuralfatigue and affect the safety of passing vehicles [96] Hencebuffeting analysis has received much attention in recentyears in research into the structural safety of bridges underturbulent wind action [81 95 97ndash102]
By assuming no interaction between buffeting forces andself-excited forces and using quasi-steady aerodynamic forcecoefficients the buffeting forces per unit span Feibf on the 119894thsection of a bridge deck can be expressed as [103]
bf are the buffeting drag lift andmoment respectively 119906
119894and 119908
119894are the horizontal and
vertical components respectively of fluctuating wind at the119894th section 120588 is the air density 119880
119894is the mean wind speed
at the 119894th section of the bridge deck 119861119894and 119871
119894are the width
and length of the 119894th bridge section 119862119863119894 119862119871119894 and 119862
119872119894are
the drag lift and moment coefficients respectively of the119894th bridge segment 1198621015840
119863119894= 119889119862
1198631198941198891205721015840 1198621015840
119871119894= 119889119862
1198711198941198891205721015840 and
1198621015840119872119894
= 1198891198621198721198941198891205721015840 1205721015840 is the angle of attack of a normal
wind incident on the horizontal plane of the deck and 120594119863bu
120594119863bw
120594119871bu
120594119871bw
120594119872bu
and 120594119872bw
are the aerodynamic transferfunctions between the fluctuating wind velocities and thebuffeting forces
It can be found from this equation that a series oftime histories of fluctuating wind velocity 119906
119894 119908119894119879 in the
longitudinal and vertical directions at various points alongthe bridge deck is needed to carry out a detailed buffetinganalysis To simulate the stochastic wind velocity field thefast spectral representation method proposed by Cao et al[104] that is based on the spectral representation methoddeveloped by Shinozuka and Jan [105] is often adopted Thismethod rests on the assumptions that (1) the bridge deck ishorizontal at the same elevation (2) the mean wind speedand wind spectra do not vary along the bridge deck and(3) the distance between any two successive points wherewind speeds are simulated is the same The time historiesof the along-wind component 119906(119905) and the upward windcomponent 119908(119905) at the jth point can be generated using thefollowing equations [104]
119906119895(119905) = radic2 (Δ120596)
119895
sum119898=1
119873119891
sum119896=1
radic119878119906119906(120596119898119896)
times 119866119895119898(120596119898119896) cos (120596
119898119896119905 + 120593119898119896)
(4a)
119908119895(119905) = radic2 (Δ120596)
119895
sum119898=1
119873119891
sum119896=1
radic119878119908119908(120596119898119896)
times 119866119895119898(120596119898119896) cos (120596
119898119896119905 + 120593119898119896)
(4b)
119866119895119898(120596) =
0 when 1 le 119895 lt 119898 le 119899
119862|119895minus119898| when 119897 = 1 119898 le 119895 le 119899
119862|119895minus119898|radic(1 minus 1198622) when 2 le 119898 le 119895 le 119899
(4c)
119862 = exp(minus120582120596119898119896Δ
2120587119880) Δ =
119871
119899119901minus 1
(4d)
120596119898119896= (119896 minus 1) Δ120596 +
119898
119899Δ120596 (119896 = 1 2 119873
119891) (4e)
where Δ120596 = 120596up119873119891 is the frequency interval between thespectral lines119873
119891is the total number of frequency intervals
120596up is the upper cutoff frequency n is the total number ofpoints at which wind speeds are simulated 119878uu and 119878ww arethe along-wind and vertical wind spectrum respectively 120593lmis a random variable that is uniformly distributed between 0and 2120587 L is the span length and 120582 is a parameter that usuallyfalls between 7 and 10
In reality the equivalent buffeting forces in (3) are actuallyassociated with the spatial distribution of the wind pressureson the surface of the bridge deck Ignoring the spatialdistribution or aerodynamic transfer function of the buffetingforces across the cross-section of the bridge deck may havea considerable impact on the accuracy of buffeting response
The Scientific World Journal 9
wi(t) UI + ui(t)
j
j + 1
k
Leibf Lkibf
Meibf
Deibf
Dkibf
ce
Figure 7 Buffeting wind pressures and buffeting forces at nodes [22]
predictions Furthermore the local structural behavior ofthe bridge deck associated with local stresses and strainswhich are prone to causing local damage cannot be predicteddirectly by the current approaches based on equivalentbuffeting forces In this regard Liu et al [22] proposed anapproach to consider the spatial distribution of buffetingforces on a bridge deck structure based on wind pressuredistributions from wind tunnel tests (see Figure 7)
232 Self-Excited Forces In addition to buffeting actionflutter instability caused by self-excited forces induced bywind-structure interactions is an important considerationin the design and construction of long-span suspensionbridges [96] because the additional energy injected into theoscillating structure by the aerodynamic forces increases themagnitude of vibration sometimes to catastrophic levels [95]The self-excited forces on a bridge deck are attributable tothe interactions between wind and the motion of the bridgeWhen the energy of motion extracted from the flow exceedsthe energy dissipated by the system through mechanicaldamping the magnitude of vibration can reach catastrophiclevels [106] Expressing self-excited forces in the form ofindicial functions was first suggested by Scanlan [94] Basedon the assumption that self-excited forces are generated in alinear fashion Lin and Yang [107] simplified the self-excitedforces acting on a bridge deck and expressed them in termsof convolution integrals between the bridge deck motion andthe impulse response functions
119891se119890119863(119905)
=1
21205881198802
int119905
minusinfin
[119868119863ℎ(119905 minus 120591) ℎ
119890(120591) + 119868
119863119902(119905 minus 120591) 119902
119890(120591)
+119861119868119863120579(119905 minus 120591) 120579
119890(120591)
] 119889120591
(5a)
119891se119890119871(119905)
=1
21205881198802
int119905
minusinfin
[119868119871ℎ(119905 minus 120591) ℎ
119890(120591) + 119868
119871119902(119905 minus 120591) 119902
119890(120591)
+119861119868119871120579(119905 minus 120591) 120579
is the impulse function of the self-excited forces in which120595 represents the corresponding force components and heqe and 120579119890 are the equivalent vertical lateral and torsionaldisplacements respectively at the center of elasticity of thebridge deck section The relationship between the aero-dynamic impulse functions and flutter derivatives can beobtained by taking the Fourier transform of (7) [98]
119868119863ℎ(120596) = 119870
2
(119875lowast
6+ 119894119875lowast
5) 119868
119863119902(120596) = 119870
2
(119875lowast
4+ 119894119875lowast
1)
119868119863120579(120596) = 119870
2
(119875lowast
3+ 119894119875lowast
2)
119868119871ℎ(120596) = 119870
2
(119867lowast
4+ 119894119867lowast
1) 119868
119871119902(120596) = 119870
2
(119867lowast
6+ 119894119867lowast
5)
119868119871120579(120596) = 119870
2
(119867lowast
3+ 119894119867lowast
2)
119868119872ℎ(120596) = 119870
2
(119860lowast
4+ 119894119860lowast
1) 119868
119872119902(120596) = 119870
2
(119860lowast
6+ 119894119860lowast
5)
119868119872120579(120596) = 119870
2
(119860lowast
3+ 119894119860lowast
2)
(6)
where the overbars denote the Fourier transform operationthe terms containing 119894 represent imaginary parts119875lowast
120595 119867lowast
120595 and
119860lowast120595(120595 = 1 2 6) are dimensionless flutter derivatives
obtained from wind tunnel tests 119870 = 120596119861119880 is the reducedfrequency and 120596 is the circular frequency of vibration
According to classical airfoil theory the impulse func-tions can reasonably be approximated by a rational function[108]
119868 (120596) = [1198621+ 1198941198622
2120587
]+
119898
sum119897=1
119862119897+2
41205872 + 1198942120587119889119897+2
]1198892119897+2
V2 + 41205872] (7)
where the value of 119898 determines the level of accuracy ofthe approximation 119862
1 1198622 119862119897+2
and 119889119897+2(119897 = 1 2 119898)
are the frequency independent coefficients and ] = 2120587119870
is the reduced mean wind velocity By equating the realand imaginary parts in the comparison of (8) and (9) therelationship between the dimensionless flutter derivativesand the coefficients 119862120595
1 119862120595
2 119862120595
119897+2 and 119889120595
119897+2(where 120595 = Dh
Dq 119863120579 Lh Lq 119871120579 Mh Mq 119872120579 and 119897 = 1 2 119898) canbe establishedThese coefficients are determined by using thenonlinear least-squares method to fit the measured flutter
10 The Scientific World Journal
derivatives at different reduced frequencies The expressionof the aerodynamic impulse functions in the time domaincan be obtained by taking the inverse Fourier transform ofthe impulse functions By substituting the related impulseresponse functions into (5b) the self-excited lift force at the119894th section of bridge deck can then be derived as
119891se119890119871119894
=1
21205881198861198802
119894119861119894119862119871120579
1119894120579119894(119905) + 119862
119871120579
2119894(119861119894
119880119894
) 120579119894(119905)
+ 119862119871120579
3119894(119861119894
119880119894
) 120579119894(119905) +
119898
sum119897=1
119862119871120579
119897+3119894
times int119905
minusinfin
120579119894(119905) exp[minus
119889119871120579119897+3119894119880119894
119861119894
(119905 minus 120591)] 119889120591
+1
21205881198861198802
119894119862119871ℎ
1119894ℎ119894(119905) + 119862
119871ℎ
2119894(119861119894
119880119894
) ℎ119894(119905)
+ 119862119871ℎ
3119894(119861119894
119880119894
) ℎ119894(119905) +
119898
sum119897=1
119862119871ℎ
119897+3119894
times int119905
minusinfin
ℎ119894(119905) exp[minus
119889119871ℎ119897+3119894119880119894
119861119894
(119905 minus 120591)] 119889120591
+1
21205881198861198802
119894
119862119871119902
1119894119902119894(119905) + 119862
119871119902
2119894(119861119894
119880119894
) 119902119894(119905)
+ 119862119871119902
3119894(119861119894
119880119894
) 119902119894(119905) +
119898
sum119897=1
119862119871119902
119897+3119894
times int119905
minusinfin
119902120595(119905) exp[
[
minus119889119871119901
119897+3119894119880119894
119861119894
(119905 minus 120591)]
]
119889120591
(8)
In practice the terms 1198621198711205793119894 119862119871ℎ3119894 and 119862119871119902
3119894 which are related to
additional aerodynamic masses are normally neglected andthe value of 119898 is often taken as 2 [101] Similar formulationsfor self-excited drag and moment can be derived withanalogous definitions The self-excited forces at the 119894th nodeof the bridge deck can thus be expressed as
Feise = EeiXei + GeiXei + Feise (9)
where
Xei =
0
ℎei119902ei120579ei0
0
Eei =1
21205881198802
119894
[[[[[[[[
[
0 0 0 0 0 0
0 119862119871ℎ
1119894119862119871119902
11198941198611198941198621198711205791119894
0 0
0 119862119863ℎ1119894
119862119863119902
11198941198611198941198621198631205791119894
0 0
0 119861119894119862119872ℎ1119894
119861119894119862119872119902
111989411986121198941198621198721205791119894
0 0
0 0 0 0 0 0
0 0 0 0 0 0
]]]]]]]]
]
Gei =1
21205881198802
119894119861119894
[[[[[[[[
[
0 0 0 0 0 0
0 119862119871ℎ
2119894119862119871119902
21198941198611198941198621198711205792119894
0 0
0 119862119863ℎ2119894
119862119863119902
21198941198611198941198621198631205792119894
0 0
0 119861119894119862119872ℎ2119894
119861119894119862119872119902
211989411986121198941198621198721205792119894
0 0
0 0 0 0 0 0
0 0 0 0 0 0
]]]]]]]]
]
Feise =
[[[[[[[
[
0
eise
119863eise
eise0
0
]]]]]]]
]
=
[[[[[[[[[[[[[[[[
[
02
sum119897=1
119862119871119902
l+3119894119881119871119902
l+3119894 +2
sum119897=1
119862119871ℎl+3119894119881119871ℎ
l+3119894 +2
sum119897=1
119862119871120579l+3119894119881119871120579
l+3119894
2
sum119897=1
119862119863119902
l+3119894119881119863119902
l+3119894 +2
sum119897=1
119862119863ℎl+3119894119881119863ℎ
l+3119894 +2
sum119897=1
119862119863120579l+3119894119881119863120579
l+3119894
2
sum119897=1
119862119872119902
l+3119894119881119872119902
l+3119894 +2
sum119897=1
119862119872ℎl+3119894119881119872ℎ
l+3119894 +2
sum119897=1
119862119872120579l+3119894119881119872120579
l+3119894
0
0
]]]]]]]]]]]]]]]]
]
(10)
where 119881120595l+3119894 (120595 = Dh Dq 119863120579 Lh Lq 119871120579 Mh Mq) are theconvolution integrations of the 119894th node and can be calculatedusing a recursive algorithm For example
(11)The self-excited forces expressed by (9) relate to the centerof elasticity of the 119894th deck section Hence the force modelmust be distributed to the nodal points of the section Adistribution based on the rigid body motion relationshipsbetween themotions at the nodal point and those at the centerof elasticity of the deck section [109] was applied by Liu etal [22] In this study by applying the virtual work principlethe self-excited forces at the center of elasticity of the givensection were distributed to all nodes (see Figure 8)
24 Dynamic Interactions in a Wind-Vehicle-Bridge SystemWhen trains and road vehicles are running on long-spanbridges under crosswinds complicated dynamic interac-tions occur among the trains road vehicles cable-supported
The Scientific World Journal 11
Ui h
p
120572
Lsec i
Msec i
Dsec i
Fsekiz
Fsekiy
ce
Figure 8 Self-excited forces at the centre of elasticity and at the nodes in the 119894th deck section [22]
bridge and windThe buffeting response of the bridge due tocrosswind is superimposed on the dynamic response of thebridge due to railway and road vehicles The large vibrationof the bridge will in turn considerably affect the safetyand ride comfort of the drivers of the road vehicles Thusthe dynamic responses of a coupled vehicle-bridge systemunder crosswinds are of great concern to both engineers andresearchers
Detailed reviews of the dynamic interactions betweentrains and bridges between road vehicles and bridges andbetween wind and bridges have been given in the previoussections However the interaction between wind and vehiclesmust also be taken into account in a coupled wind-vehicle-bridge analysis Many studies have investigated wind-vehicleinteractions in the past few decades Balzer [110] developeda theory to estimate the aerodynamic forces on a movingvehicle using Taylorrsquos hypothesis of ldquofrozen turbulencerdquo Forengineering applications Cooper [111] proposed the powerspectral density (PSD) square-root coherence functionphase-lag function and aerodynamic admittance functionto model the unsteady side forces on a moving vehicle andlaid down the foundations for investigating the effects ofwind on a moving vehicle in the frequency domain Bakerdeveloped a theoretical model that describes the dynamics ofvehicles in crosswinds in the time domain [112 113] whichwas later extended to include driver behavior [114] Baker[115 116] further investigated both the steady and unsteadyaerodynamic forces acting on a variety of vehicles and carriedout extensive studies of the interaction between aerodynamicforces and moving vehicles These approaches have all beenapplied in coupled vehicle-bridge analysis For example Xuet al [101] simulated the aerodynamic wind forces actingon running road vehicles using the quasi-steady approachand Xu and Ding [117] derived and simulated the steadyand unsteady aerodynamic forces acting on a moving railwayvehicle in crosswinds in the time domain
Based on these separate studies on the various typesof dynamic interactions among wind vehicles (trains orroad vehicles) and long-span bridges several researchersin the last decade have examined the wind-vehicle-bridgecoupled system as a whole For instance studies have beencarried out on coupled road vehicle and cable-stayed bridgesystems [81 82 118] and on coupled train and cable-supportedbridge systems in crosswinds [101 117 119ndash121] In therecent years several new advances have been made bothin numerical simulation technologies and in wind tunnelmeasurements Chen et al [17] proposed a wind-vehicle-bridge framework which enables considering the dynamiceffects induced by simultaneous actions of railway highway
and wind loading and it was applied to analyze dynamicstress of long suspension bridges Li et al [122] extended thewind-vehicle-bridge couple analysis to the case of two trainsmeeting on a long-span suspension bridge Chen and Wu[118] proposed a semideterministic analytical model whichis able to consider dynamic interactions between the bridgewind and stochastic ldquorealrdquo traffic Based on the wind tunneltests Dorigatti et al [123]measured crosswind loads on high-sided vehicles over long-span bridges taking three differentvehicles (van double deck bus and lorry) and two differentbridge deck configurations into consideration Zhu et al[124] investigated aerodynamic coefficients of road vehiclesby adopting different road vehicles types wind directionsand vehicle positions Li et al [122] studied the effects ofsudden changes of wind loads as the train passing througha bridge tower or two trains passing each other by usingthe wind tunnel test rig with moving train models Hanet al [125] developed an experimental setup for measuringthe aerodynamic characteristics of vehicles and the bridgein wind tunnel and then investigated the influences ofparameters adopted in the tests
3 Applications of Simulation Technology toBridge Assessment
After reviewing the key issues of numerical simulationsfor dynamic response of long-span multiload bridges thissection will review the engineering applications of the newlydeveloped technologies to safety assessment of long-spanbridges such as assessment of fatigue and assessment underextreme events
31 Assessment of FatigueDamage Steel structures are widelyused in long-span bridges Research by the ASCE [126] indi-cates that 80ndash90 of failures in steel structures are related tofatigue and fracture Several disasters resulting from fatigue-induced bridge failure have occurred in history For instance46 people died in the collapse of the Silver Bridge (USA1967) and 32 people lost their lives in the collapse of theSungsoo Grand Bridge (South Korea 1994) These disastersteach us that fatigue is an important aspect of the safety ofsteel bridges and action should be taken to prevent similarfatigue-induced bridge failures In the past few decadesfatigue assessment of steel bridges has attracted increasingattention from both governments and bridge engineers andrelevant provisions have been stipulated in several codes andstandards [127ndash130]
12 The Scientific World Journal
It has great advantages to evaluate fatigue damage of long-span bridges based on numerical simulation especially fora multiload bridge which suffers multiple types of dynamicloading such as railway highway andwind loadingDifferentfrom sudden structural damage fatigue damage accumulateswith load-induced dynamic stress (or stress fluctuation) overthe service life of a bridge lasting for more than 100 yearsThe increase in traffic volume and gross vehicle weight thataccompany economic development is very likely to happenin the long period Numerical simulation technology can bean ideal tool to study influences of traffic growth patternsto fatigue damage of bridge In addition slender long-span bridges built in wind-prone regions also suffer fromconsiderable wind induced vibration which appears withina wide range of wind speeds and lasts for almost the wholedesign life of the bridge Given the simultaneous presenceof multiple vehicles and wind it is necessary to considercombined effects of traffic loading (railway andor highwayloading) and wind loading in the fatigue assessment Sincemultiple loading is concerned in a long time period thereare a large number of loading combinations for multipleloading in different intensities It is almost unavailable forfield measurement to obtain such complete information butnumerical simulation could be a good choice to determinedynamic responses of a long-span bridge under multipleloading
A number of structural health monitoring systems(SHMSs) have been installed on numerous recently builtlong-span bridges and a variety of sensors are used for mon-itoring bridge loadings (eg traffic wind and earthquakes)and conditions (including global and local responses) toensure bridge safety and user comfort under in-service con-ditions Well-known examples include Tsing-Ma Bridge inHong Kong Akashi Kaikyo Bridge in Japan Binzhou YellowRiver Bridge in China and Jindo Bridge in Korea Integrationof numerical simulation technologies and measurement dataof structural health monitoring systems (SHMSs) installedon long-span bridges will make the fatigue assessment morereliable for several reasons (1) it is a perfect validation byusing field measurement data of the different types of loadingas input of numerical simulation and the measured dynamicresponses for comparison with the computed ones (2) a largenumber of measured loading data could be used to establishloading databases or probabilistic models of different loads
In the recent years several researchers [7ndash10] appliedthe newly developed numerical simulation technologies tofatigue assessment of long-span bridges Chen et al [7]proposed a framework for fatigue analysis of a long-spansuspension bridge under railway highway and wind loadingby integrating computer simulation with SHMSs and itwas applied to evaluate fatigue damage of the Tsing MaSuspension Bridge over its design life as a case study Basedon this work Chen et al [8] proposed a framework forfatigue reliability analysis of long suspension bridges undermultiple loading inwhich the probabilisticmodels of railwayhighway and wind loading were established based on themeasurement data acquired from the SHMS of the TsingMa Bridge Wu et al [9] proposed a reliability-based fatigueapproach for slender long-span bridge and the combined
dynamic loading effects from wind and traffic as well asthe associated uncertainties were considered Based on theassumption that dynamic magnification related to vehicledynamics can be neglected in long suspension bridges Chenet al [8] established a framework for fatigue reliability anal-ysis To account for different types of long-span bridges withthe span length ranging from a few hundred to thousands ofmeters Zhang et al [10] proposed a more general frameworkwhich includes multiple random variables for the dynamicloads in a bridgersquos life cycle for the vehicle-bridge-winddynamic system such as road profile vehicle speed andwindvelocity and direction among other effects
32 Assessment under Extreme Events The aforementionedfatigue assessment mainly focuses on damage accumulationinduced by stress fluctuations under normal operationalcondition in a long-term period For long-span bridges inaddition to the normal operational conditions in which windspeeds are small ormoderate and traffic scenarios are normalthere are some extreme event conditions Extreme eventsmayinclude complex traffic congestion on the bridge coupledwith moderate or even strong wind [11] For example severetraffic congestions may be formed on the bridge as a resultof an evacuation or a partial blockage of driving lanes due totraffic accidents construction ormaintenance For hurricaneevacuations there are usually a lot of road vehicles passingthrough the bridge before the landfall of the hurricane whilethe wind speed may become pretty high already [131]
Although the excessive dynamic responses of the bridgesunder extreme events are rare it is also critical since itmay cause critical damage initiation or accumulation onsome local bridgemembers Furthermore the extreme events(eg heavy traffic) may even trigger the collapse of thewhole bridge by breaking the ldquoweakest linkrdquo especially whensome hidden damage or design flaw has not been detectedOne recent example is the Minnesota Bridge failure whichoccurred during rush hours with heavy traffic although trafficloads may not be the direct cause of failure For slender long-span bridges strong wind may also cause threats by workinginteractively with heavy traffic loads Therefore even thoughthe extreme cases associated with congested traffic andorwindy weather may be relatively rare and the durationscould be short it is still important for bridge engineers toappropriately look into these unusual extreme events duringstructural design and life-time management of these criticalinfrastructures [11]
The dynamic performance of long-span bridges undercombined actions of strong winds and running road vehicleshas been studied by many researchers in recent years [17 7981 82 132] Most of them studied bridge dynamic perfor-mance under road traffic inwhich only one or several vehiclesdistributed in an assumed (usually uniform) pattern on long-span bridges were considered Extreme events such as trafficcongestion coupled with strong wind were out of concernin those studies Recently Wu and Chen [11] conducteda research on the assessment of long-span bridges underextreme events which includes complex traffic congestioncoupled with moderate or even strong wind This study
The Scientific World Journal 13
applied the cellular automaton (CA) traffic model to thesimulation of the actual traffic flow through the bridgedefined representative scenarios for the extreme events andnumerically studied the bridge performance under thesepossible extreme events
4 Conclusions and Recommendations
Dynamic responses of long-span bridges are often requiredfor assessing the safety of these bridges and can be determinedby numerical simulation technologies This paper provides adetailed review of key issues involved in dynamic responseanalysis of long-span multiload bridges based on numericalsimulation including dynamic interactions between runningtrains and bridge between running road vehicles and bridgeand between wind and bridge and in the wind-vehicle-bridge coupled systemThen the review work was conductedfor engineering applications of newly developed numericalsimulation technologies to safety assessment of long-spanbridges such as assessment of fatigue damage and assessmentunder extreme event condition Although technologies fornumerical simulation of dynamic responses of long-spanmultiload bridge have achieved great advances in past fewdecades and successfully applied to several important bridgesit is still far from reach its maturity and enable to takeplace of traditional fieldmeasurementThe existing problemsand promising research efforts at least include the followingaspects
(1) After multiple types of dynamic interactions beingconsidered in the complex system computationalefficiency is a bottleneck problem for numericalsimulation of dynamic response of a long-span bridgeTypically when multiple loads are involved a largenumber of loading combinations for multiple load-ings must be considered in the assessment
(2) It is rather complex for the time-depending windloads acting on a long-span bridge and running vehi-cles especially for the case of rapid change of windloads such as a train passing through a bridge toweror two trains passing each other The aerodynamiccharacteristics of vehicles and the bridge under differ-ent loading scenarios can be determined through thewind tunnel testing and used in numerical simulationof dynamic responses of the bridge and vehicles
(3) It is a new trend to integrate numerical simulationtechnologies and measurement data of structuralhealth monitoring systems (SHMSs) installed onlong-span bridges whichmakes the safety assessmentof bridge structures more reliable Measured struc-tural responses could be used to validate numericalsimulation approach and measured loading infor-mation could be used for generating statistical orprobabilistic models of multiple loads
(4) It is important to study dynamic responses of bridgestructures under extreme events such as congestedtraffic coupled with windy weather which happens ina long-span bridge For the assessment under extreme
events using numerical simulation technologies sim-ulation of traffic flow and definition of representativescenarios of the extreme events are key issues
(5) It is necessary to consider the effects of typhoonwinds on the safety assessment of long-span bridgesin a reasonable way Few researches do this mostlybecause a probabilistic distribution of wind speedand direction specifically for typhoons is requiredfor assessment but there are insufficient measuredrecords to establish a reliable probabilistic typhoonwind model
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors wish to acknowledge the financial supportsfrom the National Natural Science Foundation of China(NSFC-51108395 and NSFC-51178366) the FundamentalResearch Funds for theCentral Universities (2012121032) andopen funding from Jiangsu Key Laboratory of EngineeringMechanics Special thanks go to the supervisor of the firstauthor Professor Y L XuHongKong PolytechnicUniversityfor his expert guidance and continuous support at all levelsthroughout his PhD study Sincere thanks should go tothe Highways Department of Hong Kong for providing theauthors with the field measurement data Any opinions andconcluding remarks presented in this paper are entirely thoseof the authors
References
[1] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[2] H Sohn C R Farrar N F Hunter and K Worden ldquoStructuralhealth monitoring using statistical pattern recognition tech-niquesrdquo Journal of Dynamic Systems Measurement and Controlvol 123 no 4 pp 706ndash711 2001
[3] W Fan and P Qiao ldquoVibration-based damage identificationmethods a review and comparative studyrdquo Structural HealthMonitoring vol 10 no 1 pp 83ndash111 2011
[4] X Q Zhu and S S Law ldquoDamage detection in simply supportedconcrete bridge structure under moving vehicular loadsrdquo Jour-nal of Vibration and Acoustics Transactions of the ASME vol129 no 1 pp 58ndash65 2007
[5] J Li and S S Law ldquoDamage identification of a target sub-structure with moving load excitationrdquoMechanical Systems andSignal Processing vol 30 pp 78ndash90 2012
[6] J Li S S Law and H Hao ldquoImproved damage identification inbridge structures subject tomoving loads numerical and exper-imental studiesrdquo International Journal of Mechanical Sciencesvol 74 pp 99ndash111 2013
14 The Scientific World Journal
[7] Z W Chen Y L Xu Y Xia Q Li and K Y Wong ldquoFatigueanalysis of long-span suspension bridges under multiple load-ing case studyrdquo Engineering Structures vol 33 no 12 pp 3246ndash3256 2011
[8] Z W Chen Y L Xu and X M Wang ldquoSHMS-based fatiguereliability analysis of multiloading suspension bridgesrdquo Journalof Structural Engineering-Asce vol 138 pp 299ndash307 2012
[9] J Wu S R Chen and J W van de Lindt ldquoFatigue assessmentof slender long-span bridges reliability approachrdquo Journal ofBridge Engineering vol 17 no 1 pp 47ndash57 2012
[10] W Zhang C S Cai and F Pan ldquoFatigue reliability assessmentfor long-span bridges under combined dynamic loads fromwinds and vehiclesrdquo Journal of Bridge Engineering vol 18 pp735ndash747 2013
[11] JWu and S R Chen ldquoProbabilistic dynamic behavior of a long-span bridge under extreme eventsrdquo Engineering Structures vol33 no 5 pp 1657ndash1665 2011
[12] J M Ko and Y Q Ni ldquoTechnology developments in structuralhealth monitoring of large-scale bridgesrdquo Engineering Struc-tures vol 27 no 12 pp 1715ndash1725 2005
[13] TH YiHN Li andHM Sun ldquoMulti-stage structural damagediagnosis method based on ldquoenergy-damagerdquo theoryrdquo SmartStructures and Systems vol 12 pp 345ndash361 2013
[14] T H Yi H N Li and M Gu ldquoFull-scale measurements ofdynamic response of suspension bridge subjected to environ-mental loads using GPS technologyrdquo Science China Technologi-cal Sciences vol 53 no 2 pp 469ndash479 2010
[15] T H Yi H N Li and M Gu ldquoExperimental assessmentof high-rate GPS receivers for deformation monitoring ofbridgerdquoMeasurement Journal of the InternationalMeasurementConfederation vol 46 pp 420ndash432 2013
[16] X W Ye Y Q Ni K Y Wong and J M Ko ldquoStatistical analysisof stress spectra for fatigue life assessment of steel bridges withstructural health monitoring datardquo Engineering Structures vol45 pp 166ndash176 2012
[17] Z W Chen Y L Xu Q Li and D J Wu ldquoDynamic stressanalysis of long suspension bridges under wind railway andhighway loadingsrdquo Journal of Bridge Engineering vol 16 no 3pp 383ndash391 2011
[18] S G Meisenholder and P Weidlinger ldquoDynamic interactionaspects of cable-stayed guideways for high speed ground trans-portationrdquoAmerican Society ofMechanical Engineers no 74 pp180ndash192 1974
[19] Q H Mao Research on the Highway Bridge Vibration Due toMoving Vehicles Tongji University Shang Hai China 1989
[20] Y L Xu J M Ko and Z Yu ldquoModal analysis of tower-cable system of Tsing Ma long suspension bridgerdquo EngineeringStructures vol 19 pp 857ndash867 1997
[21] W Guo H Xia and Y-L Xu ldquoDynamic response of a long spansuspension bridge and running safety of a train under windactionrdquo Frontiers of Architecture and Civil Engineering in Chinavol 1 no 1 pp 71ndash79 2007
[22] T T Liu Y L Xu W S Zhang K Y Wong H J Zhou and KW Y Chan ldquoBuffeting-induced stresses in a long suspensionbridge structural health monitoring oriented stress analysisrdquoWind and Structures An International Journal vol 12 no 6 pp479ndash504 2009
[23] K Y Wong ldquoStructural identification of Tsing Ma BridgerdquoTransactions Hong Kong Institution of Engineers vol 10 no 1pp 38ndash47 2003
[24] Y L XuQ Li D JWu and ZWChen ldquoStress and accelerationanalysis of coupled vehicle and long-span bridge systems usingthe mode superposition methodrdquo Engineering Structures vol32 no 5 pp 1356ndash1368 2010
[25] Y F Duan Y L Xu Q G Fei et al ldquoAdvanced finite elementmodel of Tsing Ma Bridge for structural health monitoringrdquoInternational Journal of Structural Stability and Dynamics vol11 no 2 pp 313ndash344 2011
[26] Z X Li T Q Zhou T H T Chan and Y Yu ldquoMulti-scalenumerical analysis on dynamic response and local damage inlong-span bridgesrdquo Engineering Structures vol 29 no 7 pp1507ndash1524 2007
[27] W Zhang C S Cai and F Pan ldquoFinite element modeling ofbridges with equivalent orthotropic material method for multi-scale dynamic loadsrdquo Engineering Structures vol 54 pp 82ndash932013
[28] S P Timoshenko ldquoOn the forced vibrations of bridgesrdquo Philo-sophical Magazine vol 6 no 257 pp 1018ndash1019 1922
[29] R S Ayre G Ford and L S Jacobsen ldquoTransverse vibration ofa two-span beam under the action of a moving constant forcerdquoJournal of Applied Mechanics vol 17 pp 1ndash12 1950
[30] R S Ayre andLS Jacobsen ldquoTransverse vibration of a two-spanbeam under the action of a moving alternating forcerdquo Journal ofApplied Mechanics vol 17 pp 283ndash290 1950
[31] L FrybaVibration of Solids and Structures underMoving LoadsASCE Press 1972
[32] J-S Wu and C-W Dai ldquoDynamic response of multispannonuniform beam due to moving loadsrdquo Journal of StructuralEngineering vol 113 no 3 pp 458ndash474 1987
[33] W Weaver S P Timoshenko and D H Young VibrationProblems in Engineering John Wiley amp Sons 1990
[34] N H Galdos D R Schelling and M A Sahin ldquoMethodologyfor impact factor of horizontally curved box bridgesrdquo Journal ofStructural Engineering vol 119 no 6 pp 1917ndash1934 1993
[35] J A Gbadeyan and S T Oni ldquoDynamic behaviour of beamsand rectangular plates under moving loadsrdquo Journal of Soundand Vibration vol 182 no 5 pp 677ndash695 1995
[36] D Y Zheng Y K Cheung F T K Au and Y S ChengldquoVibration of multi-span non-uniform beams under movingloads by using modified beam vibration functionsrdquo Journal ofSound and Vibration vol 212 no 3 pp 455ndash467 1998
[37] G V Rao ldquoLinear dynamics of an elastic beam under movingloadsrdquo Journal of Vibration and Acoustics Transactions of theASME vol 122 no 3 pp 281ndash289 2000
[38] Y B Yang J D Yau and Y S Wu Vehicle-Bridge InteractionDynamic with Applications to High-Speed Railways WorldScientific 2004
[39] E C Ting J Genin and J H Ginsberg ldquoA general algorithm formovingmass problemsrdquo Journal of Sound and Vibration vol 33no 1 pp 49ndash58 1974
[40] S Sadiku and H H E Leipholz ldquoOn the dynamics of elasticsystems with moving concentrated massesrdquo Ingenieur-Archivvol 57 no 3 pp 223ndash242 1987
[41] J E Akin and M Mofid ldquoNumerical solution for response ofbeamswithmovingmassrdquo Journal of Structural Engineering vol115 no 1 pp 120ndash131 1989
[42] M AMahmoud andM A Abou Zaid ldquoDynamic response of abeam with a crack subject to a moving massrdquo Journal of Soundand Vibration vol 256 no 4 pp 591ndash603 2002
[43] V K Garg Dynamics of Railway Vehicle Systems AcademicPress 1994
The Scientific World Journal 15
[44] T-LWang andDHuang ldquoCable-stayed bridge vibration due toroad surface roughnessrdquo Journal of Structural Engineering vol118 no 5 pp 1354ndash1374 1992
[45] Y-B Yang and B-H Lin ldquoVehicle-bridge interaction analysisby dynamic condensation methodrdquo Journal of Structural Engi-neering vol 121 no 11 pp 1636ndash1643 1995
[46] Y B Yang J D Yau and L C Hsu ldquoVibration of simple beamsdue to trainsmoving at high speedsrdquoEngineering Structures vol19 no 11 pp 936ndash943 1997
[47] B Tabarrok and E Esmailzadeh ldquoInduced vibration of bridgestransversed by moving vehiclesrdquo Transactions of the CanadianSociety for Mechanical Engineering B vol 24 no 1 pp 191ndash1982000
[48] C Liu T-L Wang and D Huang ldquoImpact study for multi-girder bridge based on correlated road roughnessrdquo StructuralEngineering and Mechanics vol 11 no 3 pp 259ndash272 2001
[49] K H Chu V K Garg and T L Wang ldquoImpact in railwayprestressed concrete bridgesrdquo Journal of Structural Engineeringvol 112 no 5 pp 1036ndash1051 1986
[50] T-LWang V K Garg and K-H Chu ldquoRailway bridgevehicleinteraction studies with new vehicle modelrdquo Journal of Struc-tural Engineering vol 117 no 7 pp 2099ndash2116 1991
[51] H Xia Y L Xu and T H T Chan ldquoDynamic interaction oflong suspension bridges with running trainsrdquo Journal of Soundand Vibration vol 237 no 2 pp 263ndash280 2000
[52] Q-L Zhang A Vrouwenvelder and J Wardenier ldquoNumericalsimulation of train-bridge interactive dynamicsrdquo Computersand Structures vol 79 no 10 pp 1059ndash1075 2001
[53] H Xia N Zhang and G de Roeck ldquoDynamic analysis of highspeed railway bridge under articulated trainsrdquo Computers andStructures vol 81 no 26-27 pp 2467ndash2478 2003
[54] G Diana F Cheli A Collina R Corradi and S MelzildquoThe development of a numerical model for railway vehiclescomfort assessment through comparison with experimentalmeasurementsrdquoVehicle SystemDynamics vol 38 no 3 pp 165ndash183 2002
[55] Q Li Y L Xu D J Wu and Z W Chen ldquoComputer-aided nonlinear vehicle-bridge interaction analysisrdquo Journal ofVibration and Control vol 16 pp 1791ndash1816 2010
[56] A Wiriyachai K H Chu and V K Garg ldquoBridge impact dueto wheel and track irregularitiesrdquo Journal of the EngineeringMechanics Division vol 108 no 4 pp 648ndash666 1982
[57] L Fryba Dynamics of Railway Bridges Inst of Civil Engineers1996
[58] D Huang and T-L Wang ldquoImpact analysis of cable-stayedbridgesrdquo Computers and Structures vol 43 no 5 pp 897ndash9081992
[59] W M Zhai Vehicle-Track Coupling Dynamics Chinese RailwayPress Beijing China 2007
[60] M Olsson ldquoFinite element modal co-ordinate analysis ofstructures subjected to moving loadsrdquo Journal of Sound andVibration vol 99 no 1 pp 1ndash12 1985
[61] Y-B Yang C-H Chang and J-D Yau ldquoAn element foranalysing vehicle-bridge systems considering vehiclersquos pitchingeffectrdquo International Journal for NumericalMethods in Engineer-ing vol 46 no 7 pp 1031ndash1047 1999
[62] Y B Yang and Y S Wu ldquoA versatile element for analyzingvehicle-bridge interaction responserdquo Engineering Structuresvol 23 no 5 pp 452ndash469 2001
[63] F T K Au J J Wang and Y K Cheung ldquoImpact study ofcable-stayed bridge under railway traffic using various modelsrdquo
Journal of Sound and Vibration vol 240 no 3 pp 447ndash4652001
[64] Y Q Sun andMDhanasekar ldquoA dynamicmodel for the verticalinteraction of the rail track and wagon systemrdquo InternationalJournal of Solids and Structures vol 39 no 5 pp 1337ndash13592002
[65] K Henchi M Fafard M Talbot and G Dhatt ldquoAn efficientalgorithm for dynamic analysis of bridges under moving vehi-cles using a coupledmodal and physical components approachrdquoJournal of Sound and Vibration vol 212 no 4 pp 663ndash6831998
[66] Y L Xu and L YWang ldquoAnalytical study of wind-rain-inducedcable vibration SDOFmodelrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 91 no 1-2 pp 27ndash40 2003
[67] B Biondi G Muscolino and A Sofi ldquoA substructure approachfor the dynamic analysis of train-track-bridge systemrdquoComput-ers and Structures vol 83 no 28ndash30 pp 2271ndash2281 2005
[68] J L Humar and A H Kashif ldquoDynamic response analysis ofslab-type bridgesrdquo Journal of Structural Engineering vol 121 no1 pp 48ndash62 1995
[69] P Lou and Q-Y Zeng ldquoFormulation of equations of verticalmotion for vehicle-track-bridge systemrdquo Journal of the ChinaRailway Society vol 26 no 5 p 71 2004
[70] O Coussy M Said and J-P van Hoove ldquoThe influence ofrandom surface irregularities on the dynamic response ofbridges under suspended moving loadsrdquo Journal of Sound andVibration vol 130 no 2 pp 313ndash320 1989
[71] E S Hwang and A S Nowak ldquoSimulation of dynamic loadfor bridgesrdquo Journal of Structural Engineering vol 117 pp 1413ndash1434 1991
[72] F Yang and G A Fonder ldquoAn iterative solution methodfor dynamic response of bridge-vehicles systemsrdquo EarthquakeEngineering and Structural Dynamics vol 25 pp 195ndash215 1996
[73] W Zhai and Z Cai ldquoDynamic interaction between a lumpedmass vehicle and a discretely supported continuous rail trackrdquoComputers and Structures vol 63 no 5 pp 987ndash997 1997
[74] W M Zhai and C B Cai ldquoTraintrackbridge dynamic inter-actions simulation and applicationsrdquo Vehicle System Dynamicsvol 37 pp 653ndash665 2003
[75] D Bruno F Greco and P Lonetti ldquoDynamic impact analysis oflong span cable-stayed bridges under moving loadsrdquo Engineer-ing Structures vol 30 no 4 pp 1160ndash1177 2008
[76] X D Song D J Wu and Q Li ldquoDynamic impact analysisof double-tower cable-stayed maglev bridges using a simplemodelrdquo Journal of Bridge Engineering vol 19 pp 34ndash43 2014
[77] Y-S Wu and Y-B Yang ldquoSteady-state response and ridingcomfort of trains moving over a series of simply supportedbridgesrdquoEngineering Structures vol 25 no 2 pp 251ndash265 2003
[78] P Antolin N Zhang J M Goicolea H Xia M A Astiz andJ Oliva ldquoConsideration of nonlinear wheel-rail contact forcesfor dynamic vehicle-bridge interaction in high-speed railwaysrdquoJournal of Sound and Vibration vol 332 no 5 pp 1231ndash12512013
[79] W H Guo and Y L Xu ldquoFully computerized approach to studycable-stayed bridge-vehicle interactionrdquo Journal of Sound andVibration vol 248 no 4 pp 745ndash761 2001
[80] Y L Xu and W H Guo ldquoDynamic behaviour of high-sidedroad vehicles subject to a sudden crosswind gustrdquo Wind andStructures vol 6 no 5 pp 325ndash346 2003
[81] Y L Xu and W H Guo ldquoDynamic analysis of coupled roadvehicle and cable-stayed bridge systems under turbulent windrdquoEngineering Structures vol 25 no 4 pp 473ndash486 2003
16 The Scientific World Journal
[82] C S Cai and S R Chen ldquoFramework of vehicle-bridge-winddynamic analysisrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 92 no 7-8 pp 579ndash607 2004
[83] S R Chen and J Wu ldquoModeling stochastic live load for long-span bridge based on microscopic traffic flow simulationrdquoComputers and Structures vol 89 no 9-10 pp 813ndash824 2011
[84] A V Paultre B Yang L A Bergman and C A Tan ldquoBridgedynamics and dynamic amplification factorsmdasha review ofanalytical and experimental findingsrdquoCanadian Journal of CivilEngineering vol 19 no 2 pp 260ndash278 1992
[85] H Honda Y Kajikawa and T Kobori ldquoSpectra of road surfaceroughness of bridgesrdquo Journal of the Structural Division vol 108pp 1956ndash1966 1982
[86] M J Inbanathan and M Wieland ldquoBridge vibrations dueto vehicle moving over rough surfacerdquo Journal of StructuralEngineering vol 113 no 9 pp 1994ndash2008 1987
[87] T-LWang andDHuang ldquoCable-stayed bridge vibration due toroad surface roughnessrdquo Journal of Structural Engineering vol118 no 5 pp 1354ndash1374 1992
[88] P K Chatterjee T K Datta and C S Surana ldquoVibrationsuspension bridges under vehicular movementrdquo Journal ofStructural Engineering vol 120 no 3 pp 681ndash703 1994
[89] D Chang and H Lee ldquoImpact factors for simple-span highwaygirder bridgesrdquo Journal of Structural Engineering vol 120 no 3pp 704ndash715 1994
[90] T-C Pan and J Li ldquoDynamic vehicle element method for tran-sient response of coupled vehicle-structure systemsrdquo Journal ofStructural Engineering vol 128 no 2 pp 214ndash223 2002
[91] C J Dodds and J D Robson ldquoThe description of road surfaceroughnessrdquo Journal of Sound Vibration vol 31 no 2 pp 175ndash183 1973
[92] D Huang T-L Wang and M Shahawy ldquoImpact studies ofmultigirder concrete bridgesrdquo Journal of Structural Engineeringvol 119 no 8 pp 2387ndash2402 1993
[93] A G Davenport ldquoBuffeting of a suspension bridge by stormwindrdquo Journal of Structural Division vol 88 pp 233ndash268 1962
[94] R H Scanlan ldquoThe action of flexible bridges under wind Iflutter theoryrdquo Journal of Sound and Vibration vol 60 no 2pp 187ndash199 1978
[95] Q Ding and P K K Lee ldquoComputer simulation of buffetingactions of suspension bridges under turbulentwindrdquoComputersand Structures vol 76 no 6 pp 787ndash797 2000
[96] V Boonyapinyo T Miyata and H Yamada ldquoAdvanced aerody-namic analysis of suspension bridges by state-space approachrdquoJournal of Structural Engineering vol 125 no 12 pp 1357ndash13661999
[97] Y-H Chen and C-Y Li ldquoDynamic response of elevated high-speed railwayrdquo Journal of Bridge Engineering vol 5 no 2 pp124ndash130 2000
[98] X Chen M Matsumoto and A Kareem ldquoTime domainflutter and buffeting response analysis of bridgesrdquo Journal ofEngineering Mechanics vol 126 no 1 pp 7ndash16 2000
[99] X Chen and A Kareem ldquoEquivalent static wind loads forbuffeting response of bridgesrdquo Journal of Structural Engineeringvol 127 no 12 pp 1467ndash1475 2001
[100] S R Chen and C S Cai ldquoEvolution of long-span bridgeresponse to wind-numerical simulation and discussionrdquo Com-puters and Structures vol 81 no 21 pp 2055ndash2066 2003
[101] Y L XuHXia andQ S Yan ldquoDynamic response of suspensionbridge to high wind and running trainrdquo Journal of BridgeEngineering vol 8 no 1 pp 46ndash55 2003
[102] A Guo Y L Xu and H Li ldquoDynamic performance of cable-stayed bridge tower with multi-stage pendulum mass damperunderwind excitations-II experimentrdquoEarthquake Engineeringand Engineering Vibration vol 6 no 4 pp 417ndash424 2007
[103] E Simiu and R H ScanlanWind Effects on Structures 1996[104] Y Cao H Xiang and Y Zhou ldquoSimulation of stochastic wind
velocity field on long-span bridgesrdquo Journal of EngineeringMechanics vol 126 no 1 pp 1ndash6 2000
[105] M Shinozuka and C-M Jan ldquoDigital simulation of randomprocesses and its applicationsrdquo Journal of Sound and Vibrationvol 25 no 1 pp 111ndash128 1972
[106] K M Shum Lateral and torsional vibration control of long spanbridge deck using novel tuned liquid column dampers [PhDthesis] Department of Civil and Structural Engineering TheHong Kong Polytechnic University 2004
[107] Y K Lin and J N Yang ldquoMultimode bridge response to windexcitationsrdquo Journal of EngineeringMechanics vol 109 no 2 pp586ndash603 1983
[108] Y S Lin Self-Excited Bridge Motion in Turbulent Wind 1978[109] D T Lau M S Cheung and S H Cheng ldquo3D flutter analysis
of bridges by spline finite-strip methodrdquo Journal of StructuralEngineering vol 126 no 10 pp 1246ndash1254 2000
[110] L A Balzer ldquoAtmospheric turbulence encountered by high-speed ground transport vehiclesrdquo Journal of Mechanical Engi-neering Science vol 19 pp 227ndash235 1977
[111] R K Cooper ldquoAtmospheric turbulence with respect to movingground vehiclesrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 17 no 2 pp 215ndash238 1984
[112] C J Baker ldquoA simplified analysis of various types of wind-induced road vehicle accidentsrdquo Journal of Wind Engineeringand Industrial Aerodynamics vol 22 no 1 pp 69ndash85 1986
[113] C J Baker ldquoMeasures to control vehicle movement at exposedsites during windy periodsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 25 no 2 pp 151ndash161 1987
[114] C J Baker ldquoHigh sided articulated road vehicles in strong crosswindsrdquo Journal of Wind Engineering and Industrial Aerodynam-ics vol 31 no 1 pp 67ndash85 1988
[115] C J Baker ldquoGround vehicles in high cross winds part I steadyaerodynamic forcesrdquo Journal of Fluids and Structures vol 5 no1 pp 69ndash90 1991
[116] C J Baker ldquoGround vehicles in high cross winds part IIunsteady aerodynamic forcesrdquo Journal of Fluids and Structuresvol 5 no 1 pp 91ndash111 1991
[117] Y L Xu and Q S Ding ldquoInteraction of railway vehicles withtrack in cross-windsrdquo Journal of Fluids and Structures vol 22no 3 pp 295ndash314 2006
[118] S R Chen and J Wu ldquoDynamic performance simulation oflong-span bridge under combined loads of stochastic traffic andwindrdquo Journal of Bridge Engineering vol 15 no 3 pp 219ndash2302010
[119] Y Li S Qiang H Liao and Y L Xu ldquoDynamics of wind-rail vehicle-bridge systemsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 93 no 6 pp 483ndash507 2005
[120] Y L Xu N Zhang and H Xia ldquoVibration of coupled trainand cable-stayed bridge systems in cross windsrdquo EngineeringStructures vol 26 no 10 pp 1389ndash1406 2004
[121] W W Guo Y L Xu H Xia W S Zhang and K M ShumldquoDynamic response of suspension bridge to typhoon and trainsII numerical resultsrdquo Journal of Structural Engineering vol 133no 1 pp 12ndash21 2007
The Scientific World Journal 17
[122] Y L Li H Y Xiang B Wang Y L Xu and S Z QiangldquoDynamic analysis of wind-vehicle-bridge coupling systemduring the meeting of two trainsrdquo Advances in StructuralEngineering vol 16 pp 1663ndash1670 2013
[123] F Dorigatti M Sterling D Rocchi et al ldquoWind tunnelmeasurements of crosswind loads on high sided vehicles overlong span bridgesrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 107 pp 214ndash224 2012
[124] L D Zhu L Li Y L Xu and Q Zhu ldquoWind tunnel investi-gations of aerodynamic coefficients of road vehicles on bridgedeckrdquo Journal of Fluids and Structures vol 30 pp 35ndash50 2012
[125] Y Han J X Hu C S Cai Z Q Chen and C G LildquoExperimental and numerical studies of aerodynamic forces onvehicles and bridgesrdquoWind and Structures vol 17 pp 163ndash1842013
[126] ASCE ldquoCommittee on fatigue and fracture reliability of thecommittee on structural safety and reliability of the structuraldivision fatigue reliability 1ndash4rdquo Journal of Structural Engineer-ing vol 108 pp 3ndash88 1982
[127] BS ldquoBS5400 part 10 code of practice for fatiguerdquo BritishStandard Institute 1980
[128] BS ldquoBS7608 code of practice for fatigue design and assessmentof steel structuresrdquo British Standard Institute 1993
[129] AASHTOGuide Specifications for Fatigue Evaluation of ExistingSteel Bridges 1990
[130] AASHTO Guide Manual for Condition Evaluation and Loadand Resistance Factor Rating (LRFR) of Highway Bridges 2003
[131] S R Chen C S Cai and B Wolshon ldquoFrom normal operationto evacuation Single-vehicle safety under adverse weathertopographic and operational conditionsrdquo Natural HazardsReview vol 10 no 2 pp 68ndash76 2009
[132] S R Chen and C S Cai ldquoEquivalent wheel load approach forslender cable-stayed bridge fatigue assessment under traffic andwind feasibility studyrdquo Journal of Bridge Engineering vol 12 no6 pp 755ndash764 2007
Figure 6 Dynamic model of a high-sided road vehicle [80]
that contain several DOFs have been devised for vehicle-bridge interaction analysis For example Guo and Xu [79]modeled a 17-DOF four-axle heavy tractor-trailer vehicle (seeFigure 5) to investigate the interaction between vehicles anda cable-stayed bridge A total of three DOFs were assignedto rigid bodies representing either the tractor or the trailerto account for vertical rolling and pitching motions Onlyone DOFwas assigned to the rigid body representing the axleset moving in the vertical direction Different vehicle modelsare adopted in wind-vehicle-bridge interaction analyses Xuand Guo [80] modeled a 13-DOF two-axle road vehicle (seeFigure 6) for the dynamic analysis of a coupled road vehicleand bridge system under turbulent wind Five DOFs wereassigned to the vehicle body with respect to its center ofgravity to account for vertical lateral rolling yawing andpitching motions and two DOFs were assigned to the frontand rear axle sets to account for motions in the vertical andlateral directionMoreDOFs are needed to account for lateralcrosswinds
222 Simulation of Road Vehicle Flow On long-span bridgesthere is a high probability of the simultaneous presence ofmultiple road vehicles including heavy trucks This maylead to larger amplitude stress responses and greater fatiguedamage of the local bridge components than would be thecase with only one road vehicle The simulation of roadvehicle flow is thus important in the analysis of the dynamicinteraction between road vehicles and bridges Rather simplepatterns of road vehicle flow have been assumed in most
vehicle-bridge coupled dynamic analyses [79 81 82] in whicheither one or several vehicles are distributed on the bridgein an assumed (usually uniform) pattern Obviously suchassumptions do not represent actual road traffic conditionsRecently Chen and Wu [83] modeled the stochastic trafficload for a long-span bridge based on the cellular automaton(CA) traffic flow simulation technique In this study theysimulated a complicated road vehicle flow on long-spanbridges in terms of vehicle number vehicle type combinationand driver operation characteristics such as lane changingacceleration or deceleration
223 Modeling of Road Surface Roughness Road surfaceroughness is an important factor that greatly affects vehicle-bridge interactions Paultre et al [84] pointed out that roadsurface or pavement roughness can significantly affect theimpact response of a bridge The roughness or surface profiledepends primarily on the workmanship involved in theconstruction of the pavement or roadway and how it is main-tained which although random in nature may contain someinherent frequencies [38] In most cases surface roughnesswhich is three-dimensional in reality is often approximatedby a two-dimensional profile To account for its randomnature the road profile can be modeled as a stationaryGaussian random process and derived using a certain powerspectral density function Other methods similar to this havebeen widely adopted by researchers studying vehicle-inducedbridge vibration [65 70 71 85ndash90] Dodds and Robson [91]developed power spectral density functions that were later
8 The Scientific World Journal
modified and used byWang and Huang [87] and Huang et al[92] This approach was also adopted by literatures [79 81] intheir dynamic analyses of coupled vehicle-bridge and wind-vehicle-bridge systems
23 Dynamic Interaction between Wind and Bridge Whena long-span cable-supported bridge is immersed in a givenflow field the bridge will be subject to mean and fluctuatingwind forces To simulate these forces a linear approximationof the time-averaged static and time-varying buffeting andself-excited force components must be formulated [93 94]As dynamic bridge responses are of concern in this study onlybuffeting and self-excited forces are considered and reviewedin this section
231 Buffeting Forces Buffeting action is a random vibrationcaused by turbulent wind that excites certain modes of vibra-tion across a bridge depending on the spectral distributionof the pressure vectors [95] Although the buffeting responsemay not lead to catastrophic failure it can lead to structuralfatigue and affect the safety of passing vehicles [96] Hencebuffeting analysis has received much attention in recentyears in research into the structural safety of bridges underturbulent wind action [81 95 97ndash102]
By assuming no interaction between buffeting forces andself-excited forces and using quasi-steady aerodynamic forcecoefficients the buffeting forces per unit span Feibf on the 119894thsection of a bridge deck can be expressed as [103]
bf are the buffeting drag lift andmoment respectively 119906
119894and 119908
119894are the horizontal and
vertical components respectively of fluctuating wind at the119894th section 120588 is the air density 119880
119894is the mean wind speed
at the 119894th section of the bridge deck 119861119894and 119871
119894are the width
and length of the 119894th bridge section 119862119863119894 119862119871119894 and 119862
119872119894are
the drag lift and moment coefficients respectively of the119894th bridge segment 1198621015840
119863119894= 119889119862
1198631198941198891205721015840 1198621015840
119871119894= 119889119862
1198711198941198891205721015840 and
1198621015840119872119894
= 1198891198621198721198941198891205721015840 1205721015840 is the angle of attack of a normal
wind incident on the horizontal plane of the deck and 120594119863bu
120594119863bw
120594119871bu
120594119871bw
120594119872bu
and 120594119872bw
are the aerodynamic transferfunctions between the fluctuating wind velocities and thebuffeting forces
It can be found from this equation that a series oftime histories of fluctuating wind velocity 119906
119894 119908119894119879 in the
longitudinal and vertical directions at various points alongthe bridge deck is needed to carry out a detailed buffetinganalysis To simulate the stochastic wind velocity field thefast spectral representation method proposed by Cao et al[104] that is based on the spectral representation methoddeveloped by Shinozuka and Jan [105] is often adopted Thismethod rests on the assumptions that (1) the bridge deck ishorizontal at the same elevation (2) the mean wind speedand wind spectra do not vary along the bridge deck and(3) the distance between any two successive points wherewind speeds are simulated is the same The time historiesof the along-wind component 119906(119905) and the upward windcomponent 119908(119905) at the jth point can be generated using thefollowing equations [104]
119906119895(119905) = radic2 (Δ120596)
119895
sum119898=1
119873119891
sum119896=1
radic119878119906119906(120596119898119896)
times 119866119895119898(120596119898119896) cos (120596
119898119896119905 + 120593119898119896)
(4a)
119908119895(119905) = radic2 (Δ120596)
119895
sum119898=1
119873119891
sum119896=1
radic119878119908119908(120596119898119896)
times 119866119895119898(120596119898119896) cos (120596
119898119896119905 + 120593119898119896)
(4b)
119866119895119898(120596) =
0 when 1 le 119895 lt 119898 le 119899
119862|119895minus119898| when 119897 = 1 119898 le 119895 le 119899
119862|119895minus119898|radic(1 minus 1198622) when 2 le 119898 le 119895 le 119899
(4c)
119862 = exp(minus120582120596119898119896Δ
2120587119880) Δ =
119871
119899119901minus 1
(4d)
120596119898119896= (119896 minus 1) Δ120596 +
119898
119899Δ120596 (119896 = 1 2 119873
119891) (4e)
where Δ120596 = 120596up119873119891 is the frequency interval between thespectral lines119873
119891is the total number of frequency intervals
120596up is the upper cutoff frequency n is the total number ofpoints at which wind speeds are simulated 119878uu and 119878ww arethe along-wind and vertical wind spectrum respectively 120593lmis a random variable that is uniformly distributed between 0and 2120587 L is the span length and 120582 is a parameter that usuallyfalls between 7 and 10
In reality the equivalent buffeting forces in (3) are actuallyassociated with the spatial distribution of the wind pressureson the surface of the bridge deck Ignoring the spatialdistribution or aerodynamic transfer function of the buffetingforces across the cross-section of the bridge deck may havea considerable impact on the accuracy of buffeting response
The Scientific World Journal 9
wi(t) UI + ui(t)
j
j + 1
k
Leibf Lkibf
Meibf
Deibf
Dkibf
ce
Figure 7 Buffeting wind pressures and buffeting forces at nodes [22]
predictions Furthermore the local structural behavior ofthe bridge deck associated with local stresses and strainswhich are prone to causing local damage cannot be predicteddirectly by the current approaches based on equivalentbuffeting forces In this regard Liu et al [22] proposed anapproach to consider the spatial distribution of buffetingforces on a bridge deck structure based on wind pressuredistributions from wind tunnel tests (see Figure 7)
232 Self-Excited Forces In addition to buffeting actionflutter instability caused by self-excited forces induced bywind-structure interactions is an important considerationin the design and construction of long-span suspensionbridges [96] because the additional energy injected into theoscillating structure by the aerodynamic forces increases themagnitude of vibration sometimes to catastrophic levels [95]The self-excited forces on a bridge deck are attributable tothe interactions between wind and the motion of the bridgeWhen the energy of motion extracted from the flow exceedsthe energy dissipated by the system through mechanicaldamping the magnitude of vibration can reach catastrophiclevels [106] Expressing self-excited forces in the form ofindicial functions was first suggested by Scanlan [94] Basedon the assumption that self-excited forces are generated in alinear fashion Lin and Yang [107] simplified the self-excitedforces acting on a bridge deck and expressed them in termsof convolution integrals between the bridge deck motion andthe impulse response functions
119891se119890119863(119905)
=1
21205881198802
int119905
minusinfin
[119868119863ℎ(119905 minus 120591) ℎ
119890(120591) + 119868
119863119902(119905 minus 120591) 119902
119890(120591)
+119861119868119863120579(119905 minus 120591) 120579
119890(120591)
] 119889120591
(5a)
119891se119890119871(119905)
=1
21205881198802
int119905
minusinfin
[119868119871ℎ(119905 minus 120591) ℎ
119890(120591) + 119868
119871119902(119905 minus 120591) 119902
119890(120591)
+119861119868119871120579(119905 minus 120591) 120579
is the impulse function of the self-excited forces in which120595 represents the corresponding force components and heqe and 120579119890 are the equivalent vertical lateral and torsionaldisplacements respectively at the center of elasticity of thebridge deck section The relationship between the aero-dynamic impulse functions and flutter derivatives can beobtained by taking the Fourier transform of (7) [98]
119868119863ℎ(120596) = 119870
2
(119875lowast
6+ 119894119875lowast
5) 119868
119863119902(120596) = 119870
2
(119875lowast
4+ 119894119875lowast
1)
119868119863120579(120596) = 119870
2
(119875lowast
3+ 119894119875lowast
2)
119868119871ℎ(120596) = 119870
2
(119867lowast
4+ 119894119867lowast
1) 119868
119871119902(120596) = 119870
2
(119867lowast
6+ 119894119867lowast
5)
119868119871120579(120596) = 119870
2
(119867lowast
3+ 119894119867lowast
2)
119868119872ℎ(120596) = 119870
2
(119860lowast
4+ 119894119860lowast
1) 119868
119872119902(120596) = 119870
2
(119860lowast
6+ 119894119860lowast
5)
119868119872120579(120596) = 119870
2
(119860lowast
3+ 119894119860lowast
2)
(6)
where the overbars denote the Fourier transform operationthe terms containing 119894 represent imaginary parts119875lowast
120595 119867lowast
120595 and
119860lowast120595(120595 = 1 2 6) are dimensionless flutter derivatives
obtained from wind tunnel tests 119870 = 120596119861119880 is the reducedfrequency and 120596 is the circular frequency of vibration
According to classical airfoil theory the impulse func-tions can reasonably be approximated by a rational function[108]
119868 (120596) = [1198621+ 1198941198622
2120587
]+
119898
sum119897=1
119862119897+2
41205872 + 1198942120587119889119897+2
]1198892119897+2
V2 + 41205872] (7)
where the value of 119898 determines the level of accuracy ofthe approximation 119862
1 1198622 119862119897+2
and 119889119897+2(119897 = 1 2 119898)
are the frequency independent coefficients and ] = 2120587119870
is the reduced mean wind velocity By equating the realand imaginary parts in the comparison of (8) and (9) therelationship between the dimensionless flutter derivativesand the coefficients 119862120595
1 119862120595
2 119862120595
119897+2 and 119889120595
119897+2(where 120595 = Dh
Dq 119863120579 Lh Lq 119871120579 Mh Mq 119872120579 and 119897 = 1 2 119898) canbe establishedThese coefficients are determined by using thenonlinear least-squares method to fit the measured flutter
10 The Scientific World Journal
derivatives at different reduced frequencies The expressionof the aerodynamic impulse functions in the time domaincan be obtained by taking the inverse Fourier transform ofthe impulse functions By substituting the related impulseresponse functions into (5b) the self-excited lift force at the119894th section of bridge deck can then be derived as
119891se119890119871119894
=1
21205881198861198802
119894119861119894119862119871120579
1119894120579119894(119905) + 119862
119871120579
2119894(119861119894
119880119894
) 120579119894(119905)
+ 119862119871120579
3119894(119861119894
119880119894
) 120579119894(119905) +
119898
sum119897=1
119862119871120579
119897+3119894
times int119905
minusinfin
120579119894(119905) exp[minus
119889119871120579119897+3119894119880119894
119861119894
(119905 minus 120591)] 119889120591
+1
21205881198861198802
119894119862119871ℎ
1119894ℎ119894(119905) + 119862
119871ℎ
2119894(119861119894
119880119894
) ℎ119894(119905)
+ 119862119871ℎ
3119894(119861119894
119880119894
) ℎ119894(119905) +
119898
sum119897=1
119862119871ℎ
119897+3119894
times int119905
minusinfin
ℎ119894(119905) exp[minus
119889119871ℎ119897+3119894119880119894
119861119894
(119905 minus 120591)] 119889120591
+1
21205881198861198802
119894
119862119871119902
1119894119902119894(119905) + 119862
119871119902
2119894(119861119894
119880119894
) 119902119894(119905)
+ 119862119871119902
3119894(119861119894
119880119894
) 119902119894(119905) +
119898
sum119897=1
119862119871119902
119897+3119894
times int119905
minusinfin
119902120595(119905) exp[
[
minus119889119871119901
119897+3119894119880119894
119861119894
(119905 minus 120591)]
]
119889120591
(8)
In practice the terms 1198621198711205793119894 119862119871ℎ3119894 and 119862119871119902
3119894 which are related to
additional aerodynamic masses are normally neglected andthe value of 119898 is often taken as 2 [101] Similar formulationsfor self-excited drag and moment can be derived withanalogous definitions The self-excited forces at the 119894th nodeof the bridge deck can thus be expressed as
Feise = EeiXei + GeiXei + Feise (9)
where
Xei =
0
ℎei119902ei120579ei0
0
Eei =1
21205881198802
119894
[[[[[[[[
[
0 0 0 0 0 0
0 119862119871ℎ
1119894119862119871119902
11198941198611198941198621198711205791119894
0 0
0 119862119863ℎ1119894
119862119863119902
11198941198611198941198621198631205791119894
0 0
0 119861119894119862119872ℎ1119894
119861119894119862119872119902
111989411986121198941198621198721205791119894
0 0
0 0 0 0 0 0
0 0 0 0 0 0
]]]]]]]]
]
Gei =1
21205881198802
119894119861119894
[[[[[[[[
[
0 0 0 0 0 0
0 119862119871ℎ
2119894119862119871119902
21198941198611198941198621198711205792119894
0 0
0 119862119863ℎ2119894
119862119863119902
21198941198611198941198621198631205792119894
0 0
0 119861119894119862119872ℎ2119894
119861119894119862119872119902
211989411986121198941198621198721205792119894
0 0
0 0 0 0 0 0
0 0 0 0 0 0
]]]]]]]]
]
Feise =
[[[[[[[
[
0
eise
119863eise
eise0
0
]]]]]]]
]
=
[[[[[[[[[[[[[[[[
[
02
sum119897=1
119862119871119902
l+3119894119881119871119902
l+3119894 +2
sum119897=1
119862119871ℎl+3119894119881119871ℎ
l+3119894 +2
sum119897=1
119862119871120579l+3119894119881119871120579
l+3119894
2
sum119897=1
119862119863119902
l+3119894119881119863119902
l+3119894 +2
sum119897=1
119862119863ℎl+3119894119881119863ℎ
l+3119894 +2
sum119897=1
119862119863120579l+3119894119881119863120579
l+3119894
2
sum119897=1
119862119872119902
l+3119894119881119872119902
l+3119894 +2
sum119897=1
119862119872ℎl+3119894119881119872ℎ
l+3119894 +2
sum119897=1
119862119872120579l+3119894119881119872120579
l+3119894
0
0
]]]]]]]]]]]]]]]]
]
(10)
where 119881120595l+3119894 (120595 = Dh Dq 119863120579 Lh Lq 119871120579 Mh Mq) are theconvolution integrations of the 119894th node and can be calculatedusing a recursive algorithm For example
(11)The self-excited forces expressed by (9) relate to the centerof elasticity of the 119894th deck section Hence the force modelmust be distributed to the nodal points of the section Adistribution based on the rigid body motion relationshipsbetween themotions at the nodal point and those at the centerof elasticity of the deck section [109] was applied by Liu etal [22] In this study by applying the virtual work principlethe self-excited forces at the center of elasticity of the givensection were distributed to all nodes (see Figure 8)
24 Dynamic Interactions in a Wind-Vehicle-Bridge SystemWhen trains and road vehicles are running on long-spanbridges under crosswinds complicated dynamic interac-tions occur among the trains road vehicles cable-supported
The Scientific World Journal 11
Ui h
p
120572
Lsec i
Msec i
Dsec i
Fsekiz
Fsekiy
ce
Figure 8 Self-excited forces at the centre of elasticity and at the nodes in the 119894th deck section [22]
bridge and windThe buffeting response of the bridge due tocrosswind is superimposed on the dynamic response of thebridge due to railway and road vehicles The large vibrationof the bridge will in turn considerably affect the safetyand ride comfort of the drivers of the road vehicles Thusthe dynamic responses of a coupled vehicle-bridge systemunder crosswinds are of great concern to both engineers andresearchers
Detailed reviews of the dynamic interactions betweentrains and bridges between road vehicles and bridges andbetween wind and bridges have been given in the previoussections However the interaction between wind and vehiclesmust also be taken into account in a coupled wind-vehicle-bridge analysis Many studies have investigated wind-vehicleinteractions in the past few decades Balzer [110] developeda theory to estimate the aerodynamic forces on a movingvehicle using Taylorrsquos hypothesis of ldquofrozen turbulencerdquo Forengineering applications Cooper [111] proposed the powerspectral density (PSD) square-root coherence functionphase-lag function and aerodynamic admittance functionto model the unsteady side forces on a moving vehicle andlaid down the foundations for investigating the effects ofwind on a moving vehicle in the frequency domain Bakerdeveloped a theoretical model that describes the dynamics ofvehicles in crosswinds in the time domain [112 113] whichwas later extended to include driver behavior [114] Baker[115 116] further investigated both the steady and unsteadyaerodynamic forces acting on a variety of vehicles and carriedout extensive studies of the interaction between aerodynamicforces and moving vehicles These approaches have all beenapplied in coupled vehicle-bridge analysis For example Xuet al [101] simulated the aerodynamic wind forces actingon running road vehicles using the quasi-steady approachand Xu and Ding [117] derived and simulated the steadyand unsteady aerodynamic forces acting on a moving railwayvehicle in crosswinds in the time domain
Based on these separate studies on the various typesof dynamic interactions among wind vehicles (trains orroad vehicles) and long-span bridges several researchersin the last decade have examined the wind-vehicle-bridgecoupled system as a whole For instance studies have beencarried out on coupled road vehicle and cable-stayed bridgesystems [81 82 118] and on coupled train and cable-supportedbridge systems in crosswinds [101 117 119ndash121] In therecent years several new advances have been made bothin numerical simulation technologies and in wind tunnelmeasurements Chen et al [17] proposed a wind-vehicle-bridge framework which enables considering the dynamiceffects induced by simultaneous actions of railway highway
and wind loading and it was applied to analyze dynamicstress of long suspension bridges Li et al [122] extended thewind-vehicle-bridge couple analysis to the case of two trainsmeeting on a long-span suspension bridge Chen and Wu[118] proposed a semideterministic analytical model whichis able to consider dynamic interactions between the bridgewind and stochastic ldquorealrdquo traffic Based on the wind tunneltests Dorigatti et al [123]measured crosswind loads on high-sided vehicles over long-span bridges taking three differentvehicles (van double deck bus and lorry) and two differentbridge deck configurations into consideration Zhu et al[124] investigated aerodynamic coefficients of road vehiclesby adopting different road vehicles types wind directionsand vehicle positions Li et al [122] studied the effects ofsudden changes of wind loads as the train passing througha bridge tower or two trains passing each other by usingthe wind tunnel test rig with moving train models Hanet al [125] developed an experimental setup for measuringthe aerodynamic characteristics of vehicles and the bridgein wind tunnel and then investigated the influences ofparameters adopted in the tests
3 Applications of Simulation Technology toBridge Assessment
After reviewing the key issues of numerical simulationsfor dynamic response of long-span multiload bridges thissection will review the engineering applications of the newlydeveloped technologies to safety assessment of long-spanbridges such as assessment of fatigue and assessment underextreme events
31 Assessment of FatigueDamage Steel structures are widelyused in long-span bridges Research by the ASCE [126] indi-cates that 80ndash90 of failures in steel structures are related tofatigue and fracture Several disasters resulting from fatigue-induced bridge failure have occurred in history For instance46 people died in the collapse of the Silver Bridge (USA1967) and 32 people lost their lives in the collapse of theSungsoo Grand Bridge (South Korea 1994) These disastersteach us that fatigue is an important aspect of the safety ofsteel bridges and action should be taken to prevent similarfatigue-induced bridge failures In the past few decadesfatigue assessment of steel bridges has attracted increasingattention from both governments and bridge engineers andrelevant provisions have been stipulated in several codes andstandards [127ndash130]
12 The Scientific World Journal
It has great advantages to evaluate fatigue damage of long-span bridges based on numerical simulation especially fora multiload bridge which suffers multiple types of dynamicloading such as railway highway andwind loadingDifferentfrom sudden structural damage fatigue damage accumulateswith load-induced dynamic stress (or stress fluctuation) overthe service life of a bridge lasting for more than 100 yearsThe increase in traffic volume and gross vehicle weight thataccompany economic development is very likely to happenin the long period Numerical simulation technology can bean ideal tool to study influences of traffic growth patternsto fatigue damage of bridge In addition slender long-span bridges built in wind-prone regions also suffer fromconsiderable wind induced vibration which appears withina wide range of wind speeds and lasts for almost the wholedesign life of the bridge Given the simultaneous presenceof multiple vehicles and wind it is necessary to considercombined effects of traffic loading (railway andor highwayloading) and wind loading in the fatigue assessment Sincemultiple loading is concerned in a long time period thereare a large number of loading combinations for multipleloading in different intensities It is almost unavailable forfield measurement to obtain such complete information butnumerical simulation could be a good choice to determinedynamic responses of a long-span bridge under multipleloading
A number of structural health monitoring systems(SHMSs) have been installed on numerous recently builtlong-span bridges and a variety of sensors are used for mon-itoring bridge loadings (eg traffic wind and earthquakes)and conditions (including global and local responses) toensure bridge safety and user comfort under in-service con-ditions Well-known examples include Tsing-Ma Bridge inHong Kong Akashi Kaikyo Bridge in Japan Binzhou YellowRiver Bridge in China and Jindo Bridge in Korea Integrationof numerical simulation technologies and measurement dataof structural health monitoring systems (SHMSs) installedon long-span bridges will make the fatigue assessment morereliable for several reasons (1) it is a perfect validation byusing field measurement data of the different types of loadingas input of numerical simulation and the measured dynamicresponses for comparison with the computed ones (2) a largenumber of measured loading data could be used to establishloading databases or probabilistic models of different loads
In the recent years several researchers [7ndash10] appliedthe newly developed numerical simulation technologies tofatigue assessment of long-span bridges Chen et al [7]proposed a framework for fatigue analysis of a long-spansuspension bridge under railway highway and wind loadingby integrating computer simulation with SHMSs and itwas applied to evaluate fatigue damage of the Tsing MaSuspension Bridge over its design life as a case study Basedon this work Chen et al [8] proposed a framework forfatigue reliability analysis of long suspension bridges undermultiple loading inwhich the probabilisticmodels of railwayhighway and wind loading were established based on themeasurement data acquired from the SHMS of the TsingMa Bridge Wu et al [9] proposed a reliability-based fatigueapproach for slender long-span bridge and the combined
dynamic loading effects from wind and traffic as well asthe associated uncertainties were considered Based on theassumption that dynamic magnification related to vehicledynamics can be neglected in long suspension bridges Chenet al [8] established a framework for fatigue reliability anal-ysis To account for different types of long-span bridges withthe span length ranging from a few hundred to thousands ofmeters Zhang et al [10] proposed a more general frameworkwhich includes multiple random variables for the dynamicloads in a bridgersquos life cycle for the vehicle-bridge-winddynamic system such as road profile vehicle speed andwindvelocity and direction among other effects
32 Assessment under Extreme Events The aforementionedfatigue assessment mainly focuses on damage accumulationinduced by stress fluctuations under normal operationalcondition in a long-term period For long-span bridges inaddition to the normal operational conditions in which windspeeds are small ormoderate and traffic scenarios are normalthere are some extreme event conditions Extreme eventsmayinclude complex traffic congestion on the bridge coupledwith moderate or even strong wind [11] For example severetraffic congestions may be formed on the bridge as a resultof an evacuation or a partial blockage of driving lanes due totraffic accidents construction ormaintenance For hurricaneevacuations there are usually a lot of road vehicles passingthrough the bridge before the landfall of the hurricane whilethe wind speed may become pretty high already [131]
Although the excessive dynamic responses of the bridgesunder extreme events are rare it is also critical since itmay cause critical damage initiation or accumulation onsome local bridgemembers Furthermore the extreme events(eg heavy traffic) may even trigger the collapse of thewhole bridge by breaking the ldquoweakest linkrdquo especially whensome hidden damage or design flaw has not been detectedOne recent example is the Minnesota Bridge failure whichoccurred during rush hours with heavy traffic although trafficloads may not be the direct cause of failure For slender long-span bridges strong wind may also cause threats by workinginteractively with heavy traffic loads Therefore even thoughthe extreme cases associated with congested traffic andorwindy weather may be relatively rare and the durationscould be short it is still important for bridge engineers toappropriately look into these unusual extreme events duringstructural design and life-time management of these criticalinfrastructures [11]
The dynamic performance of long-span bridges undercombined actions of strong winds and running road vehicleshas been studied by many researchers in recent years [17 7981 82 132] Most of them studied bridge dynamic perfor-mance under road traffic inwhich only one or several vehiclesdistributed in an assumed (usually uniform) pattern on long-span bridges were considered Extreme events such as trafficcongestion coupled with strong wind were out of concernin those studies Recently Wu and Chen [11] conducteda research on the assessment of long-span bridges underextreme events which includes complex traffic congestioncoupled with moderate or even strong wind This study
The Scientific World Journal 13
applied the cellular automaton (CA) traffic model to thesimulation of the actual traffic flow through the bridgedefined representative scenarios for the extreme events andnumerically studied the bridge performance under thesepossible extreme events
4 Conclusions and Recommendations
Dynamic responses of long-span bridges are often requiredfor assessing the safety of these bridges and can be determinedby numerical simulation technologies This paper provides adetailed review of key issues involved in dynamic responseanalysis of long-span multiload bridges based on numericalsimulation including dynamic interactions between runningtrains and bridge between running road vehicles and bridgeand between wind and bridge and in the wind-vehicle-bridge coupled systemThen the review work was conductedfor engineering applications of newly developed numericalsimulation technologies to safety assessment of long-spanbridges such as assessment of fatigue damage and assessmentunder extreme event condition Although technologies fornumerical simulation of dynamic responses of long-spanmultiload bridge have achieved great advances in past fewdecades and successfully applied to several important bridgesit is still far from reach its maturity and enable to takeplace of traditional fieldmeasurementThe existing problemsand promising research efforts at least include the followingaspects
(1) After multiple types of dynamic interactions beingconsidered in the complex system computationalefficiency is a bottleneck problem for numericalsimulation of dynamic response of a long-span bridgeTypically when multiple loads are involved a largenumber of loading combinations for multiple load-ings must be considered in the assessment
(2) It is rather complex for the time-depending windloads acting on a long-span bridge and running vehi-cles especially for the case of rapid change of windloads such as a train passing through a bridge toweror two trains passing each other The aerodynamiccharacteristics of vehicles and the bridge under differ-ent loading scenarios can be determined through thewind tunnel testing and used in numerical simulationof dynamic responses of the bridge and vehicles
(3) It is a new trend to integrate numerical simulationtechnologies and measurement data of structuralhealth monitoring systems (SHMSs) installed onlong-span bridges whichmakes the safety assessmentof bridge structures more reliable Measured struc-tural responses could be used to validate numericalsimulation approach and measured loading infor-mation could be used for generating statistical orprobabilistic models of multiple loads
(4) It is important to study dynamic responses of bridgestructures under extreme events such as congestedtraffic coupled with windy weather which happens ina long-span bridge For the assessment under extreme
events using numerical simulation technologies sim-ulation of traffic flow and definition of representativescenarios of the extreme events are key issues
(5) It is necessary to consider the effects of typhoonwinds on the safety assessment of long-span bridgesin a reasonable way Few researches do this mostlybecause a probabilistic distribution of wind speedand direction specifically for typhoons is requiredfor assessment but there are insufficient measuredrecords to establish a reliable probabilistic typhoonwind model
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors wish to acknowledge the financial supportsfrom the National Natural Science Foundation of China(NSFC-51108395 and NSFC-51178366) the FundamentalResearch Funds for theCentral Universities (2012121032) andopen funding from Jiangsu Key Laboratory of EngineeringMechanics Special thanks go to the supervisor of the firstauthor Professor Y L XuHongKong PolytechnicUniversityfor his expert guidance and continuous support at all levelsthroughout his PhD study Sincere thanks should go tothe Highways Department of Hong Kong for providing theauthors with the field measurement data Any opinions andconcluding remarks presented in this paper are entirely thoseof the authors
References
[1] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[2] H Sohn C R Farrar N F Hunter and K Worden ldquoStructuralhealth monitoring using statistical pattern recognition tech-niquesrdquo Journal of Dynamic Systems Measurement and Controlvol 123 no 4 pp 706ndash711 2001
[3] W Fan and P Qiao ldquoVibration-based damage identificationmethods a review and comparative studyrdquo Structural HealthMonitoring vol 10 no 1 pp 83ndash111 2011
[4] X Q Zhu and S S Law ldquoDamage detection in simply supportedconcrete bridge structure under moving vehicular loadsrdquo Jour-nal of Vibration and Acoustics Transactions of the ASME vol129 no 1 pp 58ndash65 2007
[5] J Li and S S Law ldquoDamage identification of a target sub-structure with moving load excitationrdquoMechanical Systems andSignal Processing vol 30 pp 78ndash90 2012
[6] J Li S S Law and H Hao ldquoImproved damage identification inbridge structures subject tomoving loads numerical and exper-imental studiesrdquo International Journal of Mechanical Sciencesvol 74 pp 99ndash111 2013
14 The Scientific World Journal
[7] Z W Chen Y L Xu Y Xia Q Li and K Y Wong ldquoFatigueanalysis of long-span suspension bridges under multiple load-ing case studyrdquo Engineering Structures vol 33 no 12 pp 3246ndash3256 2011
[8] Z W Chen Y L Xu and X M Wang ldquoSHMS-based fatiguereliability analysis of multiloading suspension bridgesrdquo Journalof Structural Engineering-Asce vol 138 pp 299ndash307 2012
[9] J Wu S R Chen and J W van de Lindt ldquoFatigue assessmentof slender long-span bridges reliability approachrdquo Journal ofBridge Engineering vol 17 no 1 pp 47ndash57 2012
[10] W Zhang C S Cai and F Pan ldquoFatigue reliability assessmentfor long-span bridges under combined dynamic loads fromwinds and vehiclesrdquo Journal of Bridge Engineering vol 18 pp735ndash747 2013
[11] JWu and S R Chen ldquoProbabilistic dynamic behavior of a long-span bridge under extreme eventsrdquo Engineering Structures vol33 no 5 pp 1657ndash1665 2011
[12] J M Ko and Y Q Ni ldquoTechnology developments in structuralhealth monitoring of large-scale bridgesrdquo Engineering Struc-tures vol 27 no 12 pp 1715ndash1725 2005
[13] TH YiHN Li andHM Sun ldquoMulti-stage structural damagediagnosis method based on ldquoenergy-damagerdquo theoryrdquo SmartStructures and Systems vol 12 pp 345ndash361 2013
[14] T H Yi H N Li and M Gu ldquoFull-scale measurements ofdynamic response of suspension bridge subjected to environ-mental loads using GPS technologyrdquo Science China Technologi-cal Sciences vol 53 no 2 pp 469ndash479 2010
[15] T H Yi H N Li and M Gu ldquoExperimental assessmentof high-rate GPS receivers for deformation monitoring ofbridgerdquoMeasurement Journal of the InternationalMeasurementConfederation vol 46 pp 420ndash432 2013
[16] X W Ye Y Q Ni K Y Wong and J M Ko ldquoStatistical analysisof stress spectra for fatigue life assessment of steel bridges withstructural health monitoring datardquo Engineering Structures vol45 pp 166ndash176 2012
[17] Z W Chen Y L Xu Q Li and D J Wu ldquoDynamic stressanalysis of long suspension bridges under wind railway andhighway loadingsrdquo Journal of Bridge Engineering vol 16 no 3pp 383ndash391 2011
[18] S G Meisenholder and P Weidlinger ldquoDynamic interactionaspects of cable-stayed guideways for high speed ground trans-portationrdquoAmerican Society ofMechanical Engineers no 74 pp180ndash192 1974
[19] Q H Mao Research on the Highway Bridge Vibration Due toMoving Vehicles Tongji University Shang Hai China 1989
[20] Y L Xu J M Ko and Z Yu ldquoModal analysis of tower-cable system of Tsing Ma long suspension bridgerdquo EngineeringStructures vol 19 pp 857ndash867 1997
[21] W Guo H Xia and Y-L Xu ldquoDynamic response of a long spansuspension bridge and running safety of a train under windactionrdquo Frontiers of Architecture and Civil Engineering in Chinavol 1 no 1 pp 71ndash79 2007
[22] T T Liu Y L Xu W S Zhang K Y Wong H J Zhou and KW Y Chan ldquoBuffeting-induced stresses in a long suspensionbridge structural health monitoring oriented stress analysisrdquoWind and Structures An International Journal vol 12 no 6 pp479ndash504 2009
[23] K Y Wong ldquoStructural identification of Tsing Ma BridgerdquoTransactions Hong Kong Institution of Engineers vol 10 no 1pp 38ndash47 2003
[24] Y L XuQ Li D JWu and ZWChen ldquoStress and accelerationanalysis of coupled vehicle and long-span bridge systems usingthe mode superposition methodrdquo Engineering Structures vol32 no 5 pp 1356ndash1368 2010
[25] Y F Duan Y L Xu Q G Fei et al ldquoAdvanced finite elementmodel of Tsing Ma Bridge for structural health monitoringrdquoInternational Journal of Structural Stability and Dynamics vol11 no 2 pp 313ndash344 2011
[26] Z X Li T Q Zhou T H T Chan and Y Yu ldquoMulti-scalenumerical analysis on dynamic response and local damage inlong-span bridgesrdquo Engineering Structures vol 29 no 7 pp1507ndash1524 2007
[27] W Zhang C S Cai and F Pan ldquoFinite element modeling ofbridges with equivalent orthotropic material method for multi-scale dynamic loadsrdquo Engineering Structures vol 54 pp 82ndash932013
[28] S P Timoshenko ldquoOn the forced vibrations of bridgesrdquo Philo-sophical Magazine vol 6 no 257 pp 1018ndash1019 1922
[29] R S Ayre G Ford and L S Jacobsen ldquoTransverse vibration ofa two-span beam under the action of a moving constant forcerdquoJournal of Applied Mechanics vol 17 pp 1ndash12 1950
[30] R S Ayre andLS Jacobsen ldquoTransverse vibration of a two-spanbeam under the action of a moving alternating forcerdquo Journal ofApplied Mechanics vol 17 pp 283ndash290 1950
[31] L FrybaVibration of Solids and Structures underMoving LoadsASCE Press 1972
[32] J-S Wu and C-W Dai ldquoDynamic response of multispannonuniform beam due to moving loadsrdquo Journal of StructuralEngineering vol 113 no 3 pp 458ndash474 1987
[33] W Weaver S P Timoshenko and D H Young VibrationProblems in Engineering John Wiley amp Sons 1990
[34] N H Galdos D R Schelling and M A Sahin ldquoMethodologyfor impact factor of horizontally curved box bridgesrdquo Journal ofStructural Engineering vol 119 no 6 pp 1917ndash1934 1993
[35] J A Gbadeyan and S T Oni ldquoDynamic behaviour of beamsand rectangular plates under moving loadsrdquo Journal of Soundand Vibration vol 182 no 5 pp 677ndash695 1995
[36] D Y Zheng Y K Cheung F T K Au and Y S ChengldquoVibration of multi-span non-uniform beams under movingloads by using modified beam vibration functionsrdquo Journal ofSound and Vibration vol 212 no 3 pp 455ndash467 1998
[37] G V Rao ldquoLinear dynamics of an elastic beam under movingloadsrdquo Journal of Vibration and Acoustics Transactions of theASME vol 122 no 3 pp 281ndash289 2000
[38] Y B Yang J D Yau and Y S Wu Vehicle-Bridge InteractionDynamic with Applications to High-Speed Railways WorldScientific 2004
[39] E C Ting J Genin and J H Ginsberg ldquoA general algorithm formovingmass problemsrdquo Journal of Sound and Vibration vol 33no 1 pp 49ndash58 1974
[40] S Sadiku and H H E Leipholz ldquoOn the dynamics of elasticsystems with moving concentrated massesrdquo Ingenieur-Archivvol 57 no 3 pp 223ndash242 1987
[41] J E Akin and M Mofid ldquoNumerical solution for response ofbeamswithmovingmassrdquo Journal of Structural Engineering vol115 no 1 pp 120ndash131 1989
[42] M AMahmoud andM A Abou Zaid ldquoDynamic response of abeam with a crack subject to a moving massrdquo Journal of Soundand Vibration vol 256 no 4 pp 591ndash603 2002
[43] V K Garg Dynamics of Railway Vehicle Systems AcademicPress 1994
The Scientific World Journal 15
[44] T-LWang andDHuang ldquoCable-stayed bridge vibration due toroad surface roughnessrdquo Journal of Structural Engineering vol118 no 5 pp 1354ndash1374 1992
[45] Y-B Yang and B-H Lin ldquoVehicle-bridge interaction analysisby dynamic condensation methodrdquo Journal of Structural Engi-neering vol 121 no 11 pp 1636ndash1643 1995
[46] Y B Yang J D Yau and L C Hsu ldquoVibration of simple beamsdue to trainsmoving at high speedsrdquoEngineering Structures vol19 no 11 pp 936ndash943 1997
[47] B Tabarrok and E Esmailzadeh ldquoInduced vibration of bridgestransversed by moving vehiclesrdquo Transactions of the CanadianSociety for Mechanical Engineering B vol 24 no 1 pp 191ndash1982000
[48] C Liu T-L Wang and D Huang ldquoImpact study for multi-girder bridge based on correlated road roughnessrdquo StructuralEngineering and Mechanics vol 11 no 3 pp 259ndash272 2001
[49] K H Chu V K Garg and T L Wang ldquoImpact in railwayprestressed concrete bridgesrdquo Journal of Structural Engineeringvol 112 no 5 pp 1036ndash1051 1986
[50] T-LWang V K Garg and K-H Chu ldquoRailway bridgevehicleinteraction studies with new vehicle modelrdquo Journal of Struc-tural Engineering vol 117 no 7 pp 2099ndash2116 1991
[51] H Xia Y L Xu and T H T Chan ldquoDynamic interaction oflong suspension bridges with running trainsrdquo Journal of Soundand Vibration vol 237 no 2 pp 263ndash280 2000
[52] Q-L Zhang A Vrouwenvelder and J Wardenier ldquoNumericalsimulation of train-bridge interactive dynamicsrdquo Computersand Structures vol 79 no 10 pp 1059ndash1075 2001
[53] H Xia N Zhang and G de Roeck ldquoDynamic analysis of highspeed railway bridge under articulated trainsrdquo Computers andStructures vol 81 no 26-27 pp 2467ndash2478 2003
[54] G Diana F Cheli A Collina R Corradi and S MelzildquoThe development of a numerical model for railway vehiclescomfort assessment through comparison with experimentalmeasurementsrdquoVehicle SystemDynamics vol 38 no 3 pp 165ndash183 2002
[55] Q Li Y L Xu D J Wu and Z W Chen ldquoComputer-aided nonlinear vehicle-bridge interaction analysisrdquo Journal ofVibration and Control vol 16 pp 1791ndash1816 2010
[56] A Wiriyachai K H Chu and V K Garg ldquoBridge impact dueto wheel and track irregularitiesrdquo Journal of the EngineeringMechanics Division vol 108 no 4 pp 648ndash666 1982
[57] L Fryba Dynamics of Railway Bridges Inst of Civil Engineers1996
[58] D Huang and T-L Wang ldquoImpact analysis of cable-stayedbridgesrdquo Computers and Structures vol 43 no 5 pp 897ndash9081992
[59] W M Zhai Vehicle-Track Coupling Dynamics Chinese RailwayPress Beijing China 2007
[60] M Olsson ldquoFinite element modal co-ordinate analysis ofstructures subjected to moving loadsrdquo Journal of Sound andVibration vol 99 no 1 pp 1ndash12 1985
[61] Y-B Yang C-H Chang and J-D Yau ldquoAn element foranalysing vehicle-bridge systems considering vehiclersquos pitchingeffectrdquo International Journal for NumericalMethods in Engineer-ing vol 46 no 7 pp 1031ndash1047 1999
[62] Y B Yang and Y S Wu ldquoA versatile element for analyzingvehicle-bridge interaction responserdquo Engineering Structuresvol 23 no 5 pp 452ndash469 2001
[63] F T K Au J J Wang and Y K Cheung ldquoImpact study ofcable-stayed bridge under railway traffic using various modelsrdquo
Journal of Sound and Vibration vol 240 no 3 pp 447ndash4652001
[64] Y Q Sun andMDhanasekar ldquoA dynamicmodel for the verticalinteraction of the rail track and wagon systemrdquo InternationalJournal of Solids and Structures vol 39 no 5 pp 1337ndash13592002
[65] K Henchi M Fafard M Talbot and G Dhatt ldquoAn efficientalgorithm for dynamic analysis of bridges under moving vehi-cles using a coupledmodal and physical components approachrdquoJournal of Sound and Vibration vol 212 no 4 pp 663ndash6831998
[66] Y L Xu and L YWang ldquoAnalytical study of wind-rain-inducedcable vibration SDOFmodelrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 91 no 1-2 pp 27ndash40 2003
[67] B Biondi G Muscolino and A Sofi ldquoA substructure approachfor the dynamic analysis of train-track-bridge systemrdquoComput-ers and Structures vol 83 no 28ndash30 pp 2271ndash2281 2005
[68] J L Humar and A H Kashif ldquoDynamic response analysis ofslab-type bridgesrdquo Journal of Structural Engineering vol 121 no1 pp 48ndash62 1995
[69] P Lou and Q-Y Zeng ldquoFormulation of equations of verticalmotion for vehicle-track-bridge systemrdquo Journal of the ChinaRailway Society vol 26 no 5 p 71 2004
[70] O Coussy M Said and J-P van Hoove ldquoThe influence ofrandom surface irregularities on the dynamic response ofbridges under suspended moving loadsrdquo Journal of Sound andVibration vol 130 no 2 pp 313ndash320 1989
[71] E S Hwang and A S Nowak ldquoSimulation of dynamic loadfor bridgesrdquo Journal of Structural Engineering vol 117 pp 1413ndash1434 1991
[72] F Yang and G A Fonder ldquoAn iterative solution methodfor dynamic response of bridge-vehicles systemsrdquo EarthquakeEngineering and Structural Dynamics vol 25 pp 195ndash215 1996
[73] W Zhai and Z Cai ldquoDynamic interaction between a lumpedmass vehicle and a discretely supported continuous rail trackrdquoComputers and Structures vol 63 no 5 pp 987ndash997 1997
[74] W M Zhai and C B Cai ldquoTraintrackbridge dynamic inter-actions simulation and applicationsrdquo Vehicle System Dynamicsvol 37 pp 653ndash665 2003
[75] D Bruno F Greco and P Lonetti ldquoDynamic impact analysis oflong span cable-stayed bridges under moving loadsrdquo Engineer-ing Structures vol 30 no 4 pp 1160ndash1177 2008
[76] X D Song D J Wu and Q Li ldquoDynamic impact analysisof double-tower cable-stayed maglev bridges using a simplemodelrdquo Journal of Bridge Engineering vol 19 pp 34ndash43 2014
[77] Y-S Wu and Y-B Yang ldquoSteady-state response and ridingcomfort of trains moving over a series of simply supportedbridgesrdquoEngineering Structures vol 25 no 2 pp 251ndash265 2003
[78] P Antolin N Zhang J M Goicolea H Xia M A Astiz andJ Oliva ldquoConsideration of nonlinear wheel-rail contact forcesfor dynamic vehicle-bridge interaction in high-speed railwaysrdquoJournal of Sound and Vibration vol 332 no 5 pp 1231ndash12512013
[79] W H Guo and Y L Xu ldquoFully computerized approach to studycable-stayed bridge-vehicle interactionrdquo Journal of Sound andVibration vol 248 no 4 pp 745ndash761 2001
[80] Y L Xu and W H Guo ldquoDynamic behaviour of high-sidedroad vehicles subject to a sudden crosswind gustrdquo Wind andStructures vol 6 no 5 pp 325ndash346 2003
[81] Y L Xu and W H Guo ldquoDynamic analysis of coupled roadvehicle and cable-stayed bridge systems under turbulent windrdquoEngineering Structures vol 25 no 4 pp 473ndash486 2003
16 The Scientific World Journal
[82] C S Cai and S R Chen ldquoFramework of vehicle-bridge-winddynamic analysisrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 92 no 7-8 pp 579ndash607 2004
[83] S R Chen and J Wu ldquoModeling stochastic live load for long-span bridge based on microscopic traffic flow simulationrdquoComputers and Structures vol 89 no 9-10 pp 813ndash824 2011
[84] A V Paultre B Yang L A Bergman and C A Tan ldquoBridgedynamics and dynamic amplification factorsmdasha review ofanalytical and experimental findingsrdquoCanadian Journal of CivilEngineering vol 19 no 2 pp 260ndash278 1992
[85] H Honda Y Kajikawa and T Kobori ldquoSpectra of road surfaceroughness of bridgesrdquo Journal of the Structural Division vol 108pp 1956ndash1966 1982
[86] M J Inbanathan and M Wieland ldquoBridge vibrations dueto vehicle moving over rough surfacerdquo Journal of StructuralEngineering vol 113 no 9 pp 1994ndash2008 1987
[87] T-LWang andDHuang ldquoCable-stayed bridge vibration due toroad surface roughnessrdquo Journal of Structural Engineering vol118 no 5 pp 1354ndash1374 1992
[88] P K Chatterjee T K Datta and C S Surana ldquoVibrationsuspension bridges under vehicular movementrdquo Journal ofStructural Engineering vol 120 no 3 pp 681ndash703 1994
[89] D Chang and H Lee ldquoImpact factors for simple-span highwaygirder bridgesrdquo Journal of Structural Engineering vol 120 no 3pp 704ndash715 1994
[90] T-C Pan and J Li ldquoDynamic vehicle element method for tran-sient response of coupled vehicle-structure systemsrdquo Journal ofStructural Engineering vol 128 no 2 pp 214ndash223 2002
[91] C J Dodds and J D Robson ldquoThe description of road surfaceroughnessrdquo Journal of Sound Vibration vol 31 no 2 pp 175ndash183 1973
[92] D Huang T-L Wang and M Shahawy ldquoImpact studies ofmultigirder concrete bridgesrdquo Journal of Structural Engineeringvol 119 no 8 pp 2387ndash2402 1993
[93] A G Davenport ldquoBuffeting of a suspension bridge by stormwindrdquo Journal of Structural Division vol 88 pp 233ndash268 1962
[94] R H Scanlan ldquoThe action of flexible bridges under wind Iflutter theoryrdquo Journal of Sound and Vibration vol 60 no 2pp 187ndash199 1978
[95] Q Ding and P K K Lee ldquoComputer simulation of buffetingactions of suspension bridges under turbulentwindrdquoComputersand Structures vol 76 no 6 pp 787ndash797 2000
[96] V Boonyapinyo T Miyata and H Yamada ldquoAdvanced aerody-namic analysis of suspension bridges by state-space approachrdquoJournal of Structural Engineering vol 125 no 12 pp 1357ndash13661999
[97] Y-H Chen and C-Y Li ldquoDynamic response of elevated high-speed railwayrdquo Journal of Bridge Engineering vol 5 no 2 pp124ndash130 2000
[98] X Chen M Matsumoto and A Kareem ldquoTime domainflutter and buffeting response analysis of bridgesrdquo Journal ofEngineering Mechanics vol 126 no 1 pp 7ndash16 2000
[99] X Chen and A Kareem ldquoEquivalent static wind loads forbuffeting response of bridgesrdquo Journal of Structural Engineeringvol 127 no 12 pp 1467ndash1475 2001
[100] S R Chen and C S Cai ldquoEvolution of long-span bridgeresponse to wind-numerical simulation and discussionrdquo Com-puters and Structures vol 81 no 21 pp 2055ndash2066 2003
[101] Y L XuHXia andQ S Yan ldquoDynamic response of suspensionbridge to high wind and running trainrdquo Journal of BridgeEngineering vol 8 no 1 pp 46ndash55 2003
[102] A Guo Y L Xu and H Li ldquoDynamic performance of cable-stayed bridge tower with multi-stage pendulum mass damperunderwind excitations-II experimentrdquoEarthquake Engineeringand Engineering Vibration vol 6 no 4 pp 417ndash424 2007
[103] E Simiu and R H ScanlanWind Effects on Structures 1996[104] Y Cao H Xiang and Y Zhou ldquoSimulation of stochastic wind
velocity field on long-span bridgesrdquo Journal of EngineeringMechanics vol 126 no 1 pp 1ndash6 2000
[105] M Shinozuka and C-M Jan ldquoDigital simulation of randomprocesses and its applicationsrdquo Journal of Sound and Vibrationvol 25 no 1 pp 111ndash128 1972
[106] K M Shum Lateral and torsional vibration control of long spanbridge deck using novel tuned liquid column dampers [PhDthesis] Department of Civil and Structural Engineering TheHong Kong Polytechnic University 2004
[107] Y K Lin and J N Yang ldquoMultimode bridge response to windexcitationsrdquo Journal of EngineeringMechanics vol 109 no 2 pp586ndash603 1983
[108] Y S Lin Self-Excited Bridge Motion in Turbulent Wind 1978[109] D T Lau M S Cheung and S H Cheng ldquo3D flutter analysis
of bridges by spline finite-strip methodrdquo Journal of StructuralEngineering vol 126 no 10 pp 1246ndash1254 2000
[110] L A Balzer ldquoAtmospheric turbulence encountered by high-speed ground transport vehiclesrdquo Journal of Mechanical Engi-neering Science vol 19 pp 227ndash235 1977
[111] R K Cooper ldquoAtmospheric turbulence with respect to movingground vehiclesrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 17 no 2 pp 215ndash238 1984
[112] C J Baker ldquoA simplified analysis of various types of wind-induced road vehicle accidentsrdquo Journal of Wind Engineeringand Industrial Aerodynamics vol 22 no 1 pp 69ndash85 1986
[113] C J Baker ldquoMeasures to control vehicle movement at exposedsites during windy periodsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 25 no 2 pp 151ndash161 1987
[114] C J Baker ldquoHigh sided articulated road vehicles in strong crosswindsrdquo Journal of Wind Engineering and Industrial Aerodynam-ics vol 31 no 1 pp 67ndash85 1988
[115] C J Baker ldquoGround vehicles in high cross winds part I steadyaerodynamic forcesrdquo Journal of Fluids and Structures vol 5 no1 pp 69ndash90 1991
[116] C J Baker ldquoGround vehicles in high cross winds part IIunsteady aerodynamic forcesrdquo Journal of Fluids and Structuresvol 5 no 1 pp 91ndash111 1991
[117] Y L Xu and Q S Ding ldquoInteraction of railway vehicles withtrack in cross-windsrdquo Journal of Fluids and Structures vol 22no 3 pp 295ndash314 2006
[118] S R Chen and J Wu ldquoDynamic performance simulation oflong-span bridge under combined loads of stochastic traffic andwindrdquo Journal of Bridge Engineering vol 15 no 3 pp 219ndash2302010
[119] Y Li S Qiang H Liao and Y L Xu ldquoDynamics of wind-rail vehicle-bridge systemsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 93 no 6 pp 483ndash507 2005
[120] Y L Xu N Zhang and H Xia ldquoVibration of coupled trainand cable-stayed bridge systems in cross windsrdquo EngineeringStructures vol 26 no 10 pp 1389ndash1406 2004
[121] W W Guo Y L Xu H Xia W S Zhang and K M ShumldquoDynamic response of suspension bridge to typhoon and trainsII numerical resultsrdquo Journal of Structural Engineering vol 133no 1 pp 12ndash21 2007
The Scientific World Journal 17
[122] Y L Li H Y Xiang B Wang Y L Xu and S Z QiangldquoDynamic analysis of wind-vehicle-bridge coupling systemduring the meeting of two trainsrdquo Advances in StructuralEngineering vol 16 pp 1663ndash1670 2013
[123] F Dorigatti M Sterling D Rocchi et al ldquoWind tunnelmeasurements of crosswind loads on high sided vehicles overlong span bridgesrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 107 pp 214ndash224 2012
[124] L D Zhu L Li Y L Xu and Q Zhu ldquoWind tunnel investi-gations of aerodynamic coefficients of road vehicles on bridgedeckrdquo Journal of Fluids and Structures vol 30 pp 35ndash50 2012
[125] Y Han J X Hu C S Cai Z Q Chen and C G LildquoExperimental and numerical studies of aerodynamic forces onvehicles and bridgesrdquoWind and Structures vol 17 pp 163ndash1842013
[126] ASCE ldquoCommittee on fatigue and fracture reliability of thecommittee on structural safety and reliability of the structuraldivision fatigue reliability 1ndash4rdquo Journal of Structural Engineer-ing vol 108 pp 3ndash88 1982
[127] BS ldquoBS5400 part 10 code of practice for fatiguerdquo BritishStandard Institute 1980
[128] BS ldquoBS7608 code of practice for fatigue design and assessmentof steel structuresrdquo British Standard Institute 1993
[129] AASHTOGuide Specifications for Fatigue Evaluation of ExistingSteel Bridges 1990
[130] AASHTO Guide Manual for Condition Evaluation and Loadand Resistance Factor Rating (LRFR) of Highway Bridges 2003
[131] S R Chen C S Cai and B Wolshon ldquoFrom normal operationto evacuation Single-vehicle safety under adverse weathertopographic and operational conditionsrdquo Natural HazardsReview vol 10 no 2 pp 68ndash76 2009
[132] S R Chen and C S Cai ldquoEquivalent wheel load approach forslender cable-stayed bridge fatigue assessment under traffic andwind feasibility studyrdquo Journal of Bridge Engineering vol 12 no6 pp 755ndash764 2007
modified and used byWang and Huang [87] and Huang et al[92] This approach was also adopted by literatures [79 81] intheir dynamic analyses of coupled vehicle-bridge and wind-vehicle-bridge systems
23 Dynamic Interaction between Wind and Bridge Whena long-span cable-supported bridge is immersed in a givenflow field the bridge will be subject to mean and fluctuatingwind forces To simulate these forces a linear approximationof the time-averaged static and time-varying buffeting andself-excited force components must be formulated [93 94]As dynamic bridge responses are of concern in this study onlybuffeting and self-excited forces are considered and reviewedin this section
231 Buffeting Forces Buffeting action is a random vibrationcaused by turbulent wind that excites certain modes of vibra-tion across a bridge depending on the spectral distributionof the pressure vectors [95] Although the buffeting responsemay not lead to catastrophic failure it can lead to structuralfatigue and affect the safety of passing vehicles [96] Hencebuffeting analysis has received much attention in recentyears in research into the structural safety of bridges underturbulent wind action [81 95 97ndash102]
By assuming no interaction between buffeting forces andself-excited forces and using quasi-steady aerodynamic forcecoefficients the buffeting forces per unit span Feibf on the 119894thsection of a bridge deck can be expressed as [103]
bf are the buffeting drag lift andmoment respectively 119906
119894and 119908
119894are the horizontal and
vertical components respectively of fluctuating wind at the119894th section 120588 is the air density 119880
119894is the mean wind speed
at the 119894th section of the bridge deck 119861119894and 119871
119894are the width
and length of the 119894th bridge section 119862119863119894 119862119871119894 and 119862
119872119894are
the drag lift and moment coefficients respectively of the119894th bridge segment 1198621015840
119863119894= 119889119862
1198631198941198891205721015840 1198621015840
119871119894= 119889119862
1198711198941198891205721015840 and
1198621015840119872119894
= 1198891198621198721198941198891205721015840 1205721015840 is the angle of attack of a normal
wind incident on the horizontal plane of the deck and 120594119863bu
120594119863bw
120594119871bu
120594119871bw
120594119872bu
and 120594119872bw
are the aerodynamic transferfunctions between the fluctuating wind velocities and thebuffeting forces
It can be found from this equation that a series oftime histories of fluctuating wind velocity 119906
119894 119908119894119879 in the
longitudinal and vertical directions at various points alongthe bridge deck is needed to carry out a detailed buffetinganalysis To simulate the stochastic wind velocity field thefast spectral representation method proposed by Cao et al[104] that is based on the spectral representation methoddeveloped by Shinozuka and Jan [105] is often adopted Thismethod rests on the assumptions that (1) the bridge deck ishorizontal at the same elevation (2) the mean wind speedand wind spectra do not vary along the bridge deck and(3) the distance between any two successive points wherewind speeds are simulated is the same The time historiesof the along-wind component 119906(119905) and the upward windcomponent 119908(119905) at the jth point can be generated using thefollowing equations [104]
119906119895(119905) = radic2 (Δ120596)
119895
sum119898=1
119873119891
sum119896=1
radic119878119906119906(120596119898119896)
times 119866119895119898(120596119898119896) cos (120596
119898119896119905 + 120593119898119896)
(4a)
119908119895(119905) = radic2 (Δ120596)
119895
sum119898=1
119873119891
sum119896=1
radic119878119908119908(120596119898119896)
times 119866119895119898(120596119898119896) cos (120596
119898119896119905 + 120593119898119896)
(4b)
119866119895119898(120596) =
0 when 1 le 119895 lt 119898 le 119899
119862|119895minus119898| when 119897 = 1 119898 le 119895 le 119899
119862|119895minus119898|radic(1 minus 1198622) when 2 le 119898 le 119895 le 119899
(4c)
119862 = exp(minus120582120596119898119896Δ
2120587119880) Δ =
119871
119899119901minus 1
(4d)
120596119898119896= (119896 minus 1) Δ120596 +
119898
119899Δ120596 (119896 = 1 2 119873
119891) (4e)
where Δ120596 = 120596up119873119891 is the frequency interval between thespectral lines119873
119891is the total number of frequency intervals
120596up is the upper cutoff frequency n is the total number ofpoints at which wind speeds are simulated 119878uu and 119878ww arethe along-wind and vertical wind spectrum respectively 120593lmis a random variable that is uniformly distributed between 0and 2120587 L is the span length and 120582 is a parameter that usuallyfalls between 7 and 10
In reality the equivalent buffeting forces in (3) are actuallyassociated with the spatial distribution of the wind pressureson the surface of the bridge deck Ignoring the spatialdistribution or aerodynamic transfer function of the buffetingforces across the cross-section of the bridge deck may havea considerable impact on the accuracy of buffeting response
The Scientific World Journal 9
wi(t) UI + ui(t)
j
j + 1
k
Leibf Lkibf
Meibf
Deibf
Dkibf
ce
Figure 7 Buffeting wind pressures and buffeting forces at nodes [22]
predictions Furthermore the local structural behavior ofthe bridge deck associated with local stresses and strainswhich are prone to causing local damage cannot be predicteddirectly by the current approaches based on equivalentbuffeting forces In this regard Liu et al [22] proposed anapproach to consider the spatial distribution of buffetingforces on a bridge deck structure based on wind pressuredistributions from wind tunnel tests (see Figure 7)
232 Self-Excited Forces In addition to buffeting actionflutter instability caused by self-excited forces induced bywind-structure interactions is an important considerationin the design and construction of long-span suspensionbridges [96] because the additional energy injected into theoscillating structure by the aerodynamic forces increases themagnitude of vibration sometimes to catastrophic levels [95]The self-excited forces on a bridge deck are attributable tothe interactions between wind and the motion of the bridgeWhen the energy of motion extracted from the flow exceedsthe energy dissipated by the system through mechanicaldamping the magnitude of vibration can reach catastrophiclevels [106] Expressing self-excited forces in the form ofindicial functions was first suggested by Scanlan [94] Basedon the assumption that self-excited forces are generated in alinear fashion Lin and Yang [107] simplified the self-excitedforces acting on a bridge deck and expressed them in termsof convolution integrals between the bridge deck motion andthe impulse response functions
119891se119890119863(119905)
=1
21205881198802
int119905
minusinfin
[119868119863ℎ(119905 minus 120591) ℎ
119890(120591) + 119868
119863119902(119905 minus 120591) 119902
119890(120591)
+119861119868119863120579(119905 minus 120591) 120579
119890(120591)
] 119889120591
(5a)
119891se119890119871(119905)
=1
21205881198802
int119905
minusinfin
[119868119871ℎ(119905 minus 120591) ℎ
119890(120591) + 119868
119871119902(119905 minus 120591) 119902
119890(120591)
+119861119868119871120579(119905 minus 120591) 120579
is the impulse function of the self-excited forces in which120595 represents the corresponding force components and heqe and 120579119890 are the equivalent vertical lateral and torsionaldisplacements respectively at the center of elasticity of thebridge deck section The relationship between the aero-dynamic impulse functions and flutter derivatives can beobtained by taking the Fourier transform of (7) [98]
119868119863ℎ(120596) = 119870
2
(119875lowast
6+ 119894119875lowast
5) 119868
119863119902(120596) = 119870
2
(119875lowast
4+ 119894119875lowast
1)
119868119863120579(120596) = 119870
2
(119875lowast
3+ 119894119875lowast
2)
119868119871ℎ(120596) = 119870
2
(119867lowast
4+ 119894119867lowast
1) 119868
119871119902(120596) = 119870
2
(119867lowast
6+ 119894119867lowast
5)
119868119871120579(120596) = 119870
2
(119867lowast
3+ 119894119867lowast
2)
119868119872ℎ(120596) = 119870
2
(119860lowast
4+ 119894119860lowast
1) 119868
119872119902(120596) = 119870
2
(119860lowast
6+ 119894119860lowast
5)
119868119872120579(120596) = 119870
2
(119860lowast
3+ 119894119860lowast
2)
(6)
where the overbars denote the Fourier transform operationthe terms containing 119894 represent imaginary parts119875lowast
120595 119867lowast
120595 and
119860lowast120595(120595 = 1 2 6) are dimensionless flutter derivatives
obtained from wind tunnel tests 119870 = 120596119861119880 is the reducedfrequency and 120596 is the circular frequency of vibration
According to classical airfoil theory the impulse func-tions can reasonably be approximated by a rational function[108]
119868 (120596) = [1198621+ 1198941198622
2120587
]+
119898
sum119897=1
119862119897+2
41205872 + 1198942120587119889119897+2
]1198892119897+2
V2 + 41205872] (7)
where the value of 119898 determines the level of accuracy ofthe approximation 119862
1 1198622 119862119897+2
and 119889119897+2(119897 = 1 2 119898)
are the frequency independent coefficients and ] = 2120587119870
is the reduced mean wind velocity By equating the realand imaginary parts in the comparison of (8) and (9) therelationship between the dimensionless flutter derivativesand the coefficients 119862120595
1 119862120595
2 119862120595
119897+2 and 119889120595
119897+2(where 120595 = Dh
Dq 119863120579 Lh Lq 119871120579 Mh Mq 119872120579 and 119897 = 1 2 119898) canbe establishedThese coefficients are determined by using thenonlinear least-squares method to fit the measured flutter
10 The Scientific World Journal
derivatives at different reduced frequencies The expressionof the aerodynamic impulse functions in the time domaincan be obtained by taking the inverse Fourier transform ofthe impulse functions By substituting the related impulseresponse functions into (5b) the self-excited lift force at the119894th section of bridge deck can then be derived as
119891se119890119871119894
=1
21205881198861198802
119894119861119894119862119871120579
1119894120579119894(119905) + 119862
119871120579
2119894(119861119894
119880119894
) 120579119894(119905)
+ 119862119871120579
3119894(119861119894
119880119894
) 120579119894(119905) +
119898
sum119897=1
119862119871120579
119897+3119894
times int119905
minusinfin
120579119894(119905) exp[minus
119889119871120579119897+3119894119880119894
119861119894
(119905 minus 120591)] 119889120591
+1
21205881198861198802
119894119862119871ℎ
1119894ℎ119894(119905) + 119862
119871ℎ
2119894(119861119894
119880119894
) ℎ119894(119905)
+ 119862119871ℎ
3119894(119861119894
119880119894
) ℎ119894(119905) +
119898
sum119897=1
119862119871ℎ
119897+3119894
times int119905
minusinfin
ℎ119894(119905) exp[minus
119889119871ℎ119897+3119894119880119894
119861119894
(119905 minus 120591)] 119889120591
+1
21205881198861198802
119894
119862119871119902
1119894119902119894(119905) + 119862
119871119902
2119894(119861119894
119880119894
) 119902119894(119905)
+ 119862119871119902
3119894(119861119894
119880119894
) 119902119894(119905) +
119898
sum119897=1
119862119871119902
119897+3119894
times int119905
minusinfin
119902120595(119905) exp[
[
minus119889119871119901
119897+3119894119880119894
119861119894
(119905 minus 120591)]
]
119889120591
(8)
In practice the terms 1198621198711205793119894 119862119871ℎ3119894 and 119862119871119902
3119894 which are related to
additional aerodynamic masses are normally neglected andthe value of 119898 is often taken as 2 [101] Similar formulationsfor self-excited drag and moment can be derived withanalogous definitions The self-excited forces at the 119894th nodeof the bridge deck can thus be expressed as
Feise = EeiXei + GeiXei + Feise (9)
where
Xei =
0
ℎei119902ei120579ei0
0
Eei =1
21205881198802
119894
[[[[[[[[
[
0 0 0 0 0 0
0 119862119871ℎ
1119894119862119871119902
11198941198611198941198621198711205791119894
0 0
0 119862119863ℎ1119894
119862119863119902
11198941198611198941198621198631205791119894
0 0
0 119861119894119862119872ℎ1119894
119861119894119862119872119902
111989411986121198941198621198721205791119894
0 0
0 0 0 0 0 0
0 0 0 0 0 0
]]]]]]]]
]
Gei =1
21205881198802
119894119861119894
[[[[[[[[
[
0 0 0 0 0 0
0 119862119871ℎ
2119894119862119871119902
21198941198611198941198621198711205792119894
0 0
0 119862119863ℎ2119894
119862119863119902
21198941198611198941198621198631205792119894
0 0
0 119861119894119862119872ℎ2119894
119861119894119862119872119902
211989411986121198941198621198721205792119894
0 0
0 0 0 0 0 0
0 0 0 0 0 0
]]]]]]]]
]
Feise =
[[[[[[[
[
0
eise
119863eise
eise0
0
]]]]]]]
]
=
[[[[[[[[[[[[[[[[
[
02
sum119897=1
119862119871119902
l+3119894119881119871119902
l+3119894 +2
sum119897=1
119862119871ℎl+3119894119881119871ℎ
l+3119894 +2
sum119897=1
119862119871120579l+3119894119881119871120579
l+3119894
2
sum119897=1
119862119863119902
l+3119894119881119863119902
l+3119894 +2
sum119897=1
119862119863ℎl+3119894119881119863ℎ
l+3119894 +2
sum119897=1
119862119863120579l+3119894119881119863120579
l+3119894
2
sum119897=1
119862119872119902
l+3119894119881119872119902
l+3119894 +2
sum119897=1
119862119872ℎl+3119894119881119872ℎ
l+3119894 +2
sum119897=1
119862119872120579l+3119894119881119872120579
l+3119894
0
0
]]]]]]]]]]]]]]]]
]
(10)
where 119881120595l+3119894 (120595 = Dh Dq 119863120579 Lh Lq 119871120579 Mh Mq) are theconvolution integrations of the 119894th node and can be calculatedusing a recursive algorithm For example
(11)The self-excited forces expressed by (9) relate to the centerof elasticity of the 119894th deck section Hence the force modelmust be distributed to the nodal points of the section Adistribution based on the rigid body motion relationshipsbetween themotions at the nodal point and those at the centerof elasticity of the deck section [109] was applied by Liu etal [22] In this study by applying the virtual work principlethe self-excited forces at the center of elasticity of the givensection were distributed to all nodes (see Figure 8)
24 Dynamic Interactions in a Wind-Vehicle-Bridge SystemWhen trains and road vehicles are running on long-spanbridges under crosswinds complicated dynamic interac-tions occur among the trains road vehicles cable-supported
The Scientific World Journal 11
Ui h
p
120572
Lsec i
Msec i
Dsec i
Fsekiz
Fsekiy
ce
Figure 8 Self-excited forces at the centre of elasticity and at the nodes in the 119894th deck section [22]
bridge and windThe buffeting response of the bridge due tocrosswind is superimposed on the dynamic response of thebridge due to railway and road vehicles The large vibrationof the bridge will in turn considerably affect the safetyand ride comfort of the drivers of the road vehicles Thusthe dynamic responses of a coupled vehicle-bridge systemunder crosswinds are of great concern to both engineers andresearchers
Detailed reviews of the dynamic interactions betweentrains and bridges between road vehicles and bridges andbetween wind and bridges have been given in the previoussections However the interaction between wind and vehiclesmust also be taken into account in a coupled wind-vehicle-bridge analysis Many studies have investigated wind-vehicleinteractions in the past few decades Balzer [110] developeda theory to estimate the aerodynamic forces on a movingvehicle using Taylorrsquos hypothesis of ldquofrozen turbulencerdquo Forengineering applications Cooper [111] proposed the powerspectral density (PSD) square-root coherence functionphase-lag function and aerodynamic admittance functionto model the unsteady side forces on a moving vehicle andlaid down the foundations for investigating the effects ofwind on a moving vehicle in the frequency domain Bakerdeveloped a theoretical model that describes the dynamics ofvehicles in crosswinds in the time domain [112 113] whichwas later extended to include driver behavior [114] Baker[115 116] further investigated both the steady and unsteadyaerodynamic forces acting on a variety of vehicles and carriedout extensive studies of the interaction between aerodynamicforces and moving vehicles These approaches have all beenapplied in coupled vehicle-bridge analysis For example Xuet al [101] simulated the aerodynamic wind forces actingon running road vehicles using the quasi-steady approachand Xu and Ding [117] derived and simulated the steadyand unsteady aerodynamic forces acting on a moving railwayvehicle in crosswinds in the time domain
Based on these separate studies on the various typesof dynamic interactions among wind vehicles (trains orroad vehicles) and long-span bridges several researchersin the last decade have examined the wind-vehicle-bridgecoupled system as a whole For instance studies have beencarried out on coupled road vehicle and cable-stayed bridgesystems [81 82 118] and on coupled train and cable-supportedbridge systems in crosswinds [101 117 119ndash121] In therecent years several new advances have been made bothin numerical simulation technologies and in wind tunnelmeasurements Chen et al [17] proposed a wind-vehicle-bridge framework which enables considering the dynamiceffects induced by simultaneous actions of railway highway
and wind loading and it was applied to analyze dynamicstress of long suspension bridges Li et al [122] extended thewind-vehicle-bridge couple analysis to the case of two trainsmeeting on a long-span suspension bridge Chen and Wu[118] proposed a semideterministic analytical model whichis able to consider dynamic interactions between the bridgewind and stochastic ldquorealrdquo traffic Based on the wind tunneltests Dorigatti et al [123]measured crosswind loads on high-sided vehicles over long-span bridges taking three differentvehicles (van double deck bus and lorry) and two differentbridge deck configurations into consideration Zhu et al[124] investigated aerodynamic coefficients of road vehiclesby adopting different road vehicles types wind directionsand vehicle positions Li et al [122] studied the effects ofsudden changes of wind loads as the train passing througha bridge tower or two trains passing each other by usingthe wind tunnel test rig with moving train models Hanet al [125] developed an experimental setup for measuringthe aerodynamic characteristics of vehicles and the bridgein wind tunnel and then investigated the influences ofparameters adopted in the tests
3 Applications of Simulation Technology toBridge Assessment
After reviewing the key issues of numerical simulationsfor dynamic response of long-span multiload bridges thissection will review the engineering applications of the newlydeveloped technologies to safety assessment of long-spanbridges such as assessment of fatigue and assessment underextreme events
31 Assessment of FatigueDamage Steel structures are widelyused in long-span bridges Research by the ASCE [126] indi-cates that 80ndash90 of failures in steel structures are related tofatigue and fracture Several disasters resulting from fatigue-induced bridge failure have occurred in history For instance46 people died in the collapse of the Silver Bridge (USA1967) and 32 people lost their lives in the collapse of theSungsoo Grand Bridge (South Korea 1994) These disastersteach us that fatigue is an important aspect of the safety ofsteel bridges and action should be taken to prevent similarfatigue-induced bridge failures In the past few decadesfatigue assessment of steel bridges has attracted increasingattention from both governments and bridge engineers andrelevant provisions have been stipulated in several codes andstandards [127ndash130]
12 The Scientific World Journal
It has great advantages to evaluate fatigue damage of long-span bridges based on numerical simulation especially fora multiload bridge which suffers multiple types of dynamicloading such as railway highway andwind loadingDifferentfrom sudden structural damage fatigue damage accumulateswith load-induced dynamic stress (or stress fluctuation) overthe service life of a bridge lasting for more than 100 yearsThe increase in traffic volume and gross vehicle weight thataccompany economic development is very likely to happenin the long period Numerical simulation technology can bean ideal tool to study influences of traffic growth patternsto fatigue damage of bridge In addition slender long-span bridges built in wind-prone regions also suffer fromconsiderable wind induced vibration which appears withina wide range of wind speeds and lasts for almost the wholedesign life of the bridge Given the simultaneous presenceof multiple vehicles and wind it is necessary to considercombined effects of traffic loading (railway andor highwayloading) and wind loading in the fatigue assessment Sincemultiple loading is concerned in a long time period thereare a large number of loading combinations for multipleloading in different intensities It is almost unavailable forfield measurement to obtain such complete information butnumerical simulation could be a good choice to determinedynamic responses of a long-span bridge under multipleloading
A number of structural health monitoring systems(SHMSs) have been installed on numerous recently builtlong-span bridges and a variety of sensors are used for mon-itoring bridge loadings (eg traffic wind and earthquakes)and conditions (including global and local responses) toensure bridge safety and user comfort under in-service con-ditions Well-known examples include Tsing-Ma Bridge inHong Kong Akashi Kaikyo Bridge in Japan Binzhou YellowRiver Bridge in China and Jindo Bridge in Korea Integrationof numerical simulation technologies and measurement dataof structural health monitoring systems (SHMSs) installedon long-span bridges will make the fatigue assessment morereliable for several reasons (1) it is a perfect validation byusing field measurement data of the different types of loadingas input of numerical simulation and the measured dynamicresponses for comparison with the computed ones (2) a largenumber of measured loading data could be used to establishloading databases or probabilistic models of different loads
In the recent years several researchers [7ndash10] appliedthe newly developed numerical simulation technologies tofatigue assessment of long-span bridges Chen et al [7]proposed a framework for fatigue analysis of a long-spansuspension bridge under railway highway and wind loadingby integrating computer simulation with SHMSs and itwas applied to evaluate fatigue damage of the Tsing MaSuspension Bridge over its design life as a case study Basedon this work Chen et al [8] proposed a framework forfatigue reliability analysis of long suspension bridges undermultiple loading inwhich the probabilisticmodels of railwayhighway and wind loading were established based on themeasurement data acquired from the SHMS of the TsingMa Bridge Wu et al [9] proposed a reliability-based fatigueapproach for slender long-span bridge and the combined
dynamic loading effects from wind and traffic as well asthe associated uncertainties were considered Based on theassumption that dynamic magnification related to vehicledynamics can be neglected in long suspension bridges Chenet al [8] established a framework for fatigue reliability anal-ysis To account for different types of long-span bridges withthe span length ranging from a few hundred to thousands ofmeters Zhang et al [10] proposed a more general frameworkwhich includes multiple random variables for the dynamicloads in a bridgersquos life cycle for the vehicle-bridge-winddynamic system such as road profile vehicle speed andwindvelocity and direction among other effects
32 Assessment under Extreme Events The aforementionedfatigue assessment mainly focuses on damage accumulationinduced by stress fluctuations under normal operationalcondition in a long-term period For long-span bridges inaddition to the normal operational conditions in which windspeeds are small ormoderate and traffic scenarios are normalthere are some extreme event conditions Extreme eventsmayinclude complex traffic congestion on the bridge coupledwith moderate or even strong wind [11] For example severetraffic congestions may be formed on the bridge as a resultof an evacuation or a partial blockage of driving lanes due totraffic accidents construction ormaintenance For hurricaneevacuations there are usually a lot of road vehicles passingthrough the bridge before the landfall of the hurricane whilethe wind speed may become pretty high already [131]
Although the excessive dynamic responses of the bridgesunder extreme events are rare it is also critical since itmay cause critical damage initiation or accumulation onsome local bridgemembers Furthermore the extreme events(eg heavy traffic) may even trigger the collapse of thewhole bridge by breaking the ldquoweakest linkrdquo especially whensome hidden damage or design flaw has not been detectedOne recent example is the Minnesota Bridge failure whichoccurred during rush hours with heavy traffic although trafficloads may not be the direct cause of failure For slender long-span bridges strong wind may also cause threats by workinginteractively with heavy traffic loads Therefore even thoughthe extreme cases associated with congested traffic andorwindy weather may be relatively rare and the durationscould be short it is still important for bridge engineers toappropriately look into these unusual extreme events duringstructural design and life-time management of these criticalinfrastructures [11]
The dynamic performance of long-span bridges undercombined actions of strong winds and running road vehicleshas been studied by many researchers in recent years [17 7981 82 132] Most of them studied bridge dynamic perfor-mance under road traffic inwhich only one or several vehiclesdistributed in an assumed (usually uniform) pattern on long-span bridges were considered Extreme events such as trafficcongestion coupled with strong wind were out of concernin those studies Recently Wu and Chen [11] conducteda research on the assessment of long-span bridges underextreme events which includes complex traffic congestioncoupled with moderate or even strong wind This study
The Scientific World Journal 13
applied the cellular automaton (CA) traffic model to thesimulation of the actual traffic flow through the bridgedefined representative scenarios for the extreme events andnumerically studied the bridge performance under thesepossible extreme events
4 Conclusions and Recommendations
Dynamic responses of long-span bridges are often requiredfor assessing the safety of these bridges and can be determinedby numerical simulation technologies This paper provides adetailed review of key issues involved in dynamic responseanalysis of long-span multiload bridges based on numericalsimulation including dynamic interactions between runningtrains and bridge between running road vehicles and bridgeand between wind and bridge and in the wind-vehicle-bridge coupled systemThen the review work was conductedfor engineering applications of newly developed numericalsimulation technologies to safety assessment of long-spanbridges such as assessment of fatigue damage and assessmentunder extreme event condition Although technologies fornumerical simulation of dynamic responses of long-spanmultiload bridge have achieved great advances in past fewdecades and successfully applied to several important bridgesit is still far from reach its maturity and enable to takeplace of traditional fieldmeasurementThe existing problemsand promising research efforts at least include the followingaspects
(1) After multiple types of dynamic interactions beingconsidered in the complex system computationalefficiency is a bottleneck problem for numericalsimulation of dynamic response of a long-span bridgeTypically when multiple loads are involved a largenumber of loading combinations for multiple load-ings must be considered in the assessment
(2) It is rather complex for the time-depending windloads acting on a long-span bridge and running vehi-cles especially for the case of rapid change of windloads such as a train passing through a bridge toweror two trains passing each other The aerodynamiccharacteristics of vehicles and the bridge under differ-ent loading scenarios can be determined through thewind tunnel testing and used in numerical simulationof dynamic responses of the bridge and vehicles
(3) It is a new trend to integrate numerical simulationtechnologies and measurement data of structuralhealth monitoring systems (SHMSs) installed onlong-span bridges whichmakes the safety assessmentof bridge structures more reliable Measured struc-tural responses could be used to validate numericalsimulation approach and measured loading infor-mation could be used for generating statistical orprobabilistic models of multiple loads
(4) It is important to study dynamic responses of bridgestructures under extreme events such as congestedtraffic coupled with windy weather which happens ina long-span bridge For the assessment under extreme
events using numerical simulation technologies sim-ulation of traffic flow and definition of representativescenarios of the extreme events are key issues
(5) It is necessary to consider the effects of typhoonwinds on the safety assessment of long-span bridgesin a reasonable way Few researches do this mostlybecause a probabilistic distribution of wind speedand direction specifically for typhoons is requiredfor assessment but there are insufficient measuredrecords to establish a reliable probabilistic typhoonwind model
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors wish to acknowledge the financial supportsfrom the National Natural Science Foundation of China(NSFC-51108395 and NSFC-51178366) the FundamentalResearch Funds for theCentral Universities (2012121032) andopen funding from Jiangsu Key Laboratory of EngineeringMechanics Special thanks go to the supervisor of the firstauthor Professor Y L XuHongKong PolytechnicUniversityfor his expert guidance and continuous support at all levelsthroughout his PhD study Sincere thanks should go tothe Highways Department of Hong Kong for providing theauthors with the field measurement data Any opinions andconcluding remarks presented in this paper are entirely thoseof the authors
References
[1] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[2] H Sohn C R Farrar N F Hunter and K Worden ldquoStructuralhealth monitoring using statistical pattern recognition tech-niquesrdquo Journal of Dynamic Systems Measurement and Controlvol 123 no 4 pp 706ndash711 2001
[3] W Fan and P Qiao ldquoVibration-based damage identificationmethods a review and comparative studyrdquo Structural HealthMonitoring vol 10 no 1 pp 83ndash111 2011
[4] X Q Zhu and S S Law ldquoDamage detection in simply supportedconcrete bridge structure under moving vehicular loadsrdquo Jour-nal of Vibration and Acoustics Transactions of the ASME vol129 no 1 pp 58ndash65 2007
[5] J Li and S S Law ldquoDamage identification of a target sub-structure with moving load excitationrdquoMechanical Systems andSignal Processing vol 30 pp 78ndash90 2012
[6] J Li S S Law and H Hao ldquoImproved damage identification inbridge structures subject tomoving loads numerical and exper-imental studiesrdquo International Journal of Mechanical Sciencesvol 74 pp 99ndash111 2013
14 The Scientific World Journal
[7] Z W Chen Y L Xu Y Xia Q Li and K Y Wong ldquoFatigueanalysis of long-span suspension bridges under multiple load-ing case studyrdquo Engineering Structures vol 33 no 12 pp 3246ndash3256 2011
[8] Z W Chen Y L Xu and X M Wang ldquoSHMS-based fatiguereliability analysis of multiloading suspension bridgesrdquo Journalof Structural Engineering-Asce vol 138 pp 299ndash307 2012
[9] J Wu S R Chen and J W van de Lindt ldquoFatigue assessmentof slender long-span bridges reliability approachrdquo Journal ofBridge Engineering vol 17 no 1 pp 47ndash57 2012
[10] W Zhang C S Cai and F Pan ldquoFatigue reliability assessmentfor long-span bridges under combined dynamic loads fromwinds and vehiclesrdquo Journal of Bridge Engineering vol 18 pp735ndash747 2013
[11] JWu and S R Chen ldquoProbabilistic dynamic behavior of a long-span bridge under extreme eventsrdquo Engineering Structures vol33 no 5 pp 1657ndash1665 2011
[12] J M Ko and Y Q Ni ldquoTechnology developments in structuralhealth monitoring of large-scale bridgesrdquo Engineering Struc-tures vol 27 no 12 pp 1715ndash1725 2005
[13] TH YiHN Li andHM Sun ldquoMulti-stage structural damagediagnosis method based on ldquoenergy-damagerdquo theoryrdquo SmartStructures and Systems vol 12 pp 345ndash361 2013
[14] T H Yi H N Li and M Gu ldquoFull-scale measurements ofdynamic response of suspension bridge subjected to environ-mental loads using GPS technologyrdquo Science China Technologi-cal Sciences vol 53 no 2 pp 469ndash479 2010
[15] T H Yi H N Li and M Gu ldquoExperimental assessmentof high-rate GPS receivers for deformation monitoring ofbridgerdquoMeasurement Journal of the InternationalMeasurementConfederation vol 46 pp 420ndash432 2013
[16] X W Ye Y Q Ni K Y Wong and J M Ko ldquoStatistical analysisof stress spectra for fatigue life assessment of steel bridges withstructural health monitoring datardquo Engineering Structures vol45 pp 166ndash176 2012
[17] Z W Chen Y L Xu Q Li and D J Wu ldquoDynamic stressanalysis of long suspension bridges under wind railway andhighway loadingsrdquo Journal of Bridge Engineering vol 16 no 3pp 383ndash391 2011
[18] S G Meisenholder and P Weidlinger ldquoDynamic interactionaspects of cable-stayed guideways for high speed ground trans-portationrdquoAmerican Society ofMechanical Engineers no 74 pp180ndash192 1974
[19] Q H Mao Research on the Highway Bridge Vibration Due toMoving Vehicles Tongji University Shang Hai China 1989
[20] Y L Xu J M Ko and Z Yu ldquoModal analysis of tower-cable system of Tsing Ma long suspension bridgerdquo EngineeringStructures vol 19 pp 857ndash867 1997
[21] W Guo H Xia and Y-L Xu ldquoDynamic response of a long spansuspension bridge and running safety of a train under windactionrdquo Frontiers of Architecture and Civil Engineering in Chinavol 1 no 1 pp 71ndash79 2007
[22] T T Liu Y L Xu W S Zhang K Y Wong H J Zhou and KW Y Chan ldquoBuffeting-induced stresses in a long suspensionbridge structural health monitoring oriented stress analysisrdquoWind and Structures An International Journal vol 12 no 6 pp479ndash504 2009
[23] K Y Wong ldquoStructural identification of Tsing Ma BridgerdquoTransactions Hong Kong Institution of Engineers vol 10 no 1pp 38ndash47 2003
[24] Y L XuQ Li D JWu and ZWChen ldquoStress and accelerationanalysis of coupled vehicle and long-span bridge systems usingthe mode superposition methodrdquo Engineering Structures vol32 no 5 pp 1356ndash1368 2010
[25] Y F Duan Y L Xu Q G Fei et al ldquoAdvanced finite elementmodel of Tsing Ma Bridge for structural health monitoringrdquoInternational Journal of Structural Stability and Dynamics vol11 no 2 pp 313ndash344 2011
[26] Z X Li T Q Zhou T H T Chan and Y Yu ldquoMulti-scalenumerical analysis on dynamic response and local damage inlong-span bridgesrdquo Engineering Structures vol 29 no 7 pp1507ndash1524 2007
[27] W Zhang C S Cai and F Pan ldquoFinite element modeling ofbridges with equivalent orthotropic material method for multi-scale dynamic loadsrdquo Engineering Structures vol 54 pp 82ndash932013
[28] S P Timoshenko ldquoOn the forced vibrations of bridgesrdquo Philo-sophical Magazine vol 6 no 257 pp 1018ndash1019 1922
[29] R S Ayre G Ford and L S Jacobsen ldquoTransverse vibration ofa two-span beam under the action of a moving constant forcerdquoJournal of Applied Mechanics vol 17 pp 1ndash12 1950
[30] R S Ayre andLS Jacobsen ldquoTransverse vibration of a two-spanbeam under the action of a moving alternating forcerdquo Journal ofApplied Mechanics vol 17 pp 283ndash290 1950
[31] L FrybaVibration of Solids and Structures underMoving LoadsASCE Press 1972
[32] J-S Wu and C-W Dai ldquoDynamic response of multispannonuniform beam due to moving loadsrdquo Journal of StructuralEngineering vol 113 no 3 pp 458ndash474 1987
[33] W Weaver S P Timoshenko and D H Young VibrationProblems in Engineering John Wiley amp Sons 1990
[34] N H Galdos D R Schelling and M A Sahin ldquoMethodologyfor impact factor of horizontally curved box bridgesrdquo Journal ofStructural Engineering vol 119 no 6 pp 1917ndash1934 1993
[35] J A Gbadeyan and S T Oni ldquoDynamic behaviour of beamsand rectangular plates under moving loadsrdquo Journal of Soundand Vibration vol 182 no 5 pp 677ndash695 1995
[36] D Y Zheng Y K Cheung F T K Au and Y S ChengldquoVibration of multi-span non-uniform beams under movingloads by using modified beam vibration functionsrdquo Journal ofSound and Vibration vol 212 no 3 pp 455ndash467 1998
[37] G V Rao ldquoLinear dynamics of an elastic beam under movingloadsrdquo Journal of Vibration and Acoustics Transactions of theASME vol 122 no 3 pp 281ndash289 2000
[38] Y B Yang J D Yau and Y S Wu Vehicle-Bridge InteractionDynamic with Applications to High-Speed Railways WorldScientific 2004
[39] E C Ting J Genin and J H Ginsberg ldquoA general algorithm formovingmass problemsrdquo Journal of Sound and Vibration vol 33no 1 pp 49ndash58 1974
[40] S Sadiku and H H E Leipholz ldquoOn the dynamics of elasticsystems with moving concentrated massesrdquo Ingenieur-Archivvol 57 no 3 pp 223ndash242 1987
[41] J E Akin and M Mofid ldquoNumerical solution for response ofbeamswithmovingmassrdquo Journal of Structural Engineering vol115 no 1 pp 120ndash131 1989
[42] M AMahmoud andM A Abou Zaid ldquoDynamic response of abeam with a crack subject to a moving massrdquo Journal of Soundand Vibration vol 256 no 4 pp 591ndash603 2002
[43] V K Garg Dynamics of Railway Vehicle Systems AcademicPress 1994
The Scientific World Journal 15
[44] T-LWang andDHuang ldquoCable-stayed bridge vibration due toroad surface roughnessrdquo Journal of Structural Engineering vol118 no 5 pp 1354ndash1374 1992
[45] Y-B Yang and B-H Lin ldquoVehicle-bridge interaction analysisby dynamic condensation methodrdquo Journal of Structural Engi-neering vol 121 no 11 pp 1636ndash1643 1995
[46] Y B Yang J D Yau and L C Hsu ldquoVibration of simple beamsdue to trainsmoving at high speedsrdquoEngineering Structures vol19 no 11 pp 936ndash943 1997
[47] B Tabarrok and E Esmailzadeh ldquoInduced vibration of bridgestransversed by moving vehiclesrdquo Transactions of the CanadianSociety for Mechanical Engineering B vol 24 no 1 pp 191ndash1982000
[48] C Liu T-L Wang and D Huang ldquoImpact study for multi-girder bridge based on correlated road roughnessrdquo StructuralEngineering and Mechanics vol 11 no 3 pp 259ndash272 2001
[49] K H Chu V K Garg and T L Wang ldquoImpact in railwayprestressed concrete bridgesrdquo Journal of Structural Engineeringvol 112 no 5 pp 1036ndash1051 1986
[50] T-LWang V K Garg and K-H Chu ldquoRailway bridgevehicleinteraction studies with new vehicle modelrdquo Journal of Struc-tural Engineering vol 117 no 7 pp 2099ndash2116 1991
[51] H Xia Y L Xu and T H T Chan ldquoDynamic interaction oflong suspension bridges with running trainsrdquo Journal of Soundand Vibration vol 237 no 2 pp 263ndash280 2000
[52] Q-L Zhang A Vrouwenvelder and J Wardenier ldquoNumericalsimulation of train-bridge interactive dynamicsrdquo Computersand Structures vol 79 no 10 pp 1059ndash1075 2001
[53] H Xia N Zhang and G de Roeck ldquoDynamic analysis of highspeed railway bridge under articulated trainsrdquo Computers andStructures vol 81 no 26-27 pp 2467ndash2478 2003
[54] G Diana F Cheli A Collina R Corradi and S MelzildquoThe development of a numerical model for railway vehiclescomfort assessment through comparison with experimentalmeasurementsrdquoVehicle SystemDynamics vol 38 no 3 pp 165ndash183 2002
[55] Q Li Y L Xu D J Wu and Z W Chen ldquoComputer-aided nonlinear vehicle-bridge interaction analysisrdquo Journal ofVibration and Control vol 16 pp 1791ndash1816 2010
[56] A Wiriyachai K H Chu and V K Garg ldquoBridge impact dueto wheel and track irregularitiesrdquo Journal of the EngineeringMechanics Division vol 108 no 4 pp 648ndash666 1982
[57] L Fryba Dynamics of Railway Bridges Inst of Civil Engineers1996
[58] D Huang and T-L Wang ldquoImpact analysis of cable-stayedbridgesrdquo Computers and Structures vol 43 no 5 pp 897ndash9081992
[59] W M Zhai Vehicle-Track Coupling Dynamics Chinese RailwayPress Beijing China 2007
[60] M Olsson ldquoFinite element modal co-ordinate analysis ofstructures subjected to moving loadsrdquo Journal of Sound andVibration vol 99 no 1 pp 1ndash12 1985
[61] Y-B Yang C-H Chang and J-D Yau ldquoAn element foranalysing vehicle-bridge systems considering vehiclersquos pitchingeffectrdquo International Journal for NumericalMethods in Engineer-ing vol 46 no 7 pp 1031ndash1047 1999
[62] Y B Yang and Y S Wu ldquoA versatile element for analyzingvehicle-bridge interaction responserdquo Engineering Structuresvol 23 no 5 pp 452ndash469 2001
[63] F T K Au J J Wang and Y K Cheung ldquoImpact study ofcable-stayed bridge under railway traffic using various modelsrdquo
Journal of Sound and Vibration vol 240 no 3 pp 447ndash4652001
[64] Y Q Sun andMDhanasekar ldquoA dynamicmodel for the verticalinteraction of the rail track and wagon systemrdquo InternationalJournal of Solids and Structures vol 39 no 5 pp 1337ndash13592002
[65] K Henchi M Fafard M Talbot and G Dhatt ldquoAn efficientalgorithm for dynamic analysis of bridges under moving vehi-cles using a coupledmodal and physical components approachrdquoJournal of Sound and Vibration vol 212 no 4 pp 663ndash6831998
[66] Y L Xu and L YWang ldquoAnalytical study of wind-rain-inducedcable vibration SDOFmodelrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 91 no 1-2 pp 27ndash40 2003
[67] B Biondi G Muscolino and A Sofi ldquoA substructure approachfor the dynamic analysis of train-track-bridge systemrdquoComput-ers and Structures vol 83 no 28ndash30 pp 2271ndash2281 2005
[68] J L Humar and A H Kashif ldquoDynamic response analysis ofslab-type bridgesrdquo Journal of Structural Engineering vol 121 no1 pp 48ndash62 1995
[69] P Lou and Q-Y Zeng ldquoFormulation of equations of verticalmotion for vehicle-track-bridge systemrdquo Journal of the ChinaRailway Society vol 26 no 5 p 71 2004
[70] O Coussy M Said and J-P van Hoove ldquoThe influence ofrandom surface irregularities on the dynamic response ofbridges under suspended moving loadsrdquo Journal of Sound andVibration vol 130 no 2 pp 313ndash320 1989
[71] E S Hwang and A S Nowak ldquoSimulation of dynamic loadfor bridgesrdquo Journal of Structural Engineering vol 117 pp 1413ndash1434 1991
[72] F Yang and G A Fonder ldquoAn iterative solution methodfor dynamic response of bridge-vehicles systemsrdquo EarthquakeEngineering and Structural Dynamics vol 25 pp 195ndash215 1996
[73] W Zhai and Z Cai ldquoDynamic interaction between a lumpedmass vehicle and a discretely supported continuous rail trackrdquoComputers and Structures vol 63 no 5 pp 987ndash997 1997
[74] W M Zhai and C B Cai ldquoTraintrackbridge dynamic inter-actions simulation and applicationsrdquo Vehicle System Dynamicsvol 37 pp 653ndash665 2003
[75] D Bruno F Greco and P Lonetti ldquoDynamic impact analysis oflong span cable-stayed bridges under moving loadsrdquo Engineer-ing Structures vol 30 no 4 pp 1160ndash1177 2008
[76] X D Song D J Wu and Q Li ldquoDynamic impact analysisof double-tower cable-stayed maglev bridges using a simplemodelrdquo Journal of Bridge Engineering vol 19 pp 34ndash43 2014
[77] Y-S Wu and Y-B Yang ldquoSteady-state response and ridingcomfort of trains moving over a series of simply supportedbridgesrdquoEngineering Structures vol 25 no 2 pp 251ndash265 2003
[78] P Antolin N Zhang J M Goicolea H Xia M A Astiz andJ Oliva ldquoConsideration of nonlinear wheel-rail contact forcesfor dynamic vehicle-bridge interaction in high-speed railwaysrdquoJournal of Sound and Vibration vol 332 no 5 pp 1231ndash12512013
[79] W H Guo and Y L Xu ldquoFully computerized approach to studycable-stayed bridge-vehicle interactionrdquo Journal of Sound andVibration vol 248 no 4 pp 745ndash761 2001
[80] Y L Xu and W H Guo ldquoDynamic behaviour of high-sidedroad vehicles subject to a sudden crosswind gustrdquo Wind andStructures vol 6 no 5 pp 325ndash346 2003
[81] Y L Xu and W H Guo ldquoDynamic analysis of coupled roadvehicle and cable-stayed bridge systems under turbulent windrdquoEngineering Structures vol 25 no 4 pp 473ndash486 2003
16 The Scientific World Journal
[82] C S Cai and S R Chen ldquoFramework of vehicle-bridge-winddynamic analysisrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 92 no 7-8 pp 579ndash607 2004
[83] S R Chen and J Wu ldquoModeling stochastic live load for long-span bridge based on microscopic traffic flow simulationrdquoComputers and Structures vol 89 no 9-10 pp 813ndash824 2011
[84] A V Paultre B Yang L A Bergman and C A Tan ldquoBridgedynamics and dynamic amplification factorsmdasha review ofanalytical and experimental findingsrdquoCanadian Journal of CivilEngineering vol 19 no 2 pp 260ndash278 1992
[85] H Honda Y Kajikawa and T Kobori ldquoSpectra of road surfaceroughness of bridgesrdquo Journal of the Structural Division vol 108pp 1956ndash1966 1982
[86] M J Inbanathan and M Wieland ldquoBridge vibrations dueto vehicle moving over rough surfacerdquo Journal of StructuralEngineering vol 113 no 9 pp 1994ndash2008 1987
[87] T-LWang andDHuang ldquoCable-stayed bridge vibration due toroad surface roughnessrdquo Journal of Structural Engineering vol118 no 5 pp 1354ndash1374 1992
[88] P K Chatterjee T K Datta and C S Surana ldquoVibrationsuspension bridges under vehicular movementrdquo Journal ofStructural Engineering vol 120 no 3 pp 681ndash703 1994
[89] D Chang and H Lee ldquoImpact factors for simple-span highwaygirder bridgesrdquo Journal of Structural Engineering vol 120 no 3pp 704ndash715 1994
[90] T-C Pan and J Li ldquoDynamic vehicle element method for tran-sient response of coupled vehicle-structure systemsrdquo Journal ofStructural Engineering vol 128 no 2 pp 214ndash223 2002
[91] C J Dodds and J D Robson ldquoThe description of road surfaceroughnessrdquo Journal of Sound Vibration vol 31 no 2 pp 175ndash183 1973
[92] D Huang T-L Wang and M Shahawy ldquoImpact studies ofmultigirder concrete bridgesrdquo Journal of Structural Engineeringvol 119 no 8 pp 2387ndash2402 1993
[93] A G Davenport ldquoBuffeting of a suspension bridge by stormwindrdquo Journal of Structural Division vol 88 pp 233ndash268 1962
[94] R H Scanlan ldquoThe action of flexible bridges under wind Iflutter theoryrdquo Journal of Sound and Vibration vol 60 no 2pp 187ndash199 1978
[95] Q Ding and P K K Lee ldquoComputer simulation of buffetingactions of suspension bridges under turbulentwindrdquoComputersand Structures vol 76 no 6 pp 787ndash797 2000
[96] V Boonyapinyo T Miyata and H Yamada ldquoAdvanced aerody-namic analysis of suspension bridges by state-space approachrdquoJournal of Structural Engineering vol 125 no 12 pp 1357ndash13661999
[97] Y-H Chen and C-Y Li ldquoDynamic response of elevated high-speed railwayrdquo Journal of Bridge Engineering vol 5 no 2 pp124ndash130 2000
[98] X Chen M Matsumoto and A Kareem ldquoTime domainflutter and buffeting response analysis of bridgesrdquo Journal ofEngineering Mechanics vol 126 no 1 pp 7ndash16 2000
[99] X Chen and A Kareem ldquoEquivalent static wind loads forbuffeting response of bridgesrdquo Journal of Structural Engineeringvol 127 no 12 pp 1467ndash1475 2001
[100] S R Chen and C S Cai ldquoEvolution of long-span bridgeresponse to wind-numerical simulation and discussionrdquo Com-puters and Structures vol 81 no 21 pp 2055ndash2066 2003
[101] Y L XuHXia andQ S Yan ldquoDynamic response of suspensionbridge to high wind and running trainrdquo Journal of BridgeEngineering vol 8 no 1 pp 46ndash55 2003
[102] A Guo Y L Xu and H Li ldquoDynamic performance of cable-stayed bridge tower with multi-stage pendulum mass damperunderwind excitations-II experimentrdquoEarthquake Engineeringand Engineering Vibration vol 6 no 4 pp 417ndash424 2007
[103] E Simiu and R H ScanlanWind Effects on Structures 1996[104] Y Cao H Xiang and Y Zhou ldquoSimulation of stochastic wind
velocity field on long-span bridgesrdquo Journal of EngineeringMechanics vol 126 no 1 pp 1ndash6 2000
[105] M Shinozuka and C-M Jan ldquoDigital simulation of randomprocesses and its applicationsrdquo Journal of Sound and Vibrationvol 25 no 1 pp 111ndash128 1972
[106] K M Shum Lateral and torsional vibration control of long spanbridge deck using novel tuned liquid column dampers [PhDthesis] Department of Civil and Structural Engineering TheHong Kong Polytechnic University 2004
[107] Y K Lin and J N Yang ldquoMultimode bridge response to windexcitationsrdquo Journal of EngineeringMechanics vol 109 no 2 pp586ndash603 1983
[108] Y S Lin Self-Excited Bridge Motion in Turbulent Wind 1978[109] D T Lau M S Cheung and S H Cheng ldquo3D flutter analysis
of bridges by spline finite-strip methodrdquo Journal of StructuralEngineering vol 126 no 10 pp 1246ndash1254 2000
[110] L A Balzer ldquoAtmospheric turbulence encountered by high-speed ground transport vehiclesrdquo Journal of Mechanical Engi-neering Science vol 19 pp 227ndash235 1977
[111] R K Cooper ldquoAtmospheric turbulence with respect to movingground vehiclesrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 17 no 2 pp 215ndash238 1984
[112] C J Baker ldquoA simplified analysis of various types of wind-induced road vehicle accidentsrdquo Journal of Wind Engineeringand Industrial Aerodynamics vol 22 no 1 pp 69ndash85 1986
[113] C J Baker ldquoMeasures to control vehicle movement at exposedsites during windy periodsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 25 no 2 pp 151ndash161 1987
[114] C J Baker ldquoHigh sided articulated road vehicles in strong crosswindsrdquo Journal of Wind Engineering and Industrial Aerodynam-ics vol 31 no 1 pp 67ndash85 1988
[115] C J Baker ldquoGround vehicles in high cross winds part I steadyaerodynamic forcesrdquo Journal of Fluids and Structures vol 5 no1 pp 69ndash90 1991
[116] C J Baker ldquoGround vehicles in high cross winds part IIunsteady aerodynamic forcesrdquo Journal of Fluids and Structuresvol 5 no 1 pp 91ndash111 1991
[117] Y L Xu and Q S Ding ldquoInteraction of railway vehicles withtrack in cross-windsrdquo Journal of Fluids and Structures vol 22no 3 pp 295ndash314 2006
[118] S R Chen and J Wu ldquoDynamic performance simulation oflong-span bridge under combined loads of stochastic traffic andwindrdquo Journal of Bridge Engineering vol 15 no 3 pp 219ndash2302010
[119] Y Li S Qiang H Liao and Y L Xu ldquoDynamics of wind-rail vehicle-bridge systemsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 93 no 6 pp 483ndash507 2005
[120] Y L Xu N Zhang and H Xia ldquoVibration of coupled trainand cable-stayed bridge systems in cross windsrdquo EngineeringStructures vol 26 no 10 pp 1389ndash1406 2004
[121] W W Guo Y L Xu H Xia W S Zhang and K M ShumldquoDynamic response of suspension bridge to typhoon and trainsII numerical resultsrdquo Journal of Structural Engineering vol 133no 1 pp 12ndash21 2007
The Scientific World Journal 17
[122] Y L Li H Y Xiang B Wang Y L Xu and S Z QiangldquoDynamic analysis of wind-vehicle-bridge coupling systemduring the meeting of two trainsrdquo Advances in StructuralEngineering vol 16 pp 1663ndash1670 2013
[123] F Dorigatti M Sterling D Rocchi et al ldquoWind tunnelmeasurements of crosswind loads on high sided vehicles overlong span bridgesrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 107 pp 214ndash224 2012
[124] L D Zhu L Li Y L Xu and Q Zhu ldquoWind tunnel investi-gations of aerodynamic coefficients of road vehicles on bridgedeckrdquo Journal of Fluids and Structures vol 30 pp 35ndash50 2012
[125] Y Han J X Hu C S Cai Z Q Chen and C G LildquoExperimental and numerical studies of aerodynamic forces onvehicles and bridgesrdquoWind and Structures vol 17 pp 163ndash1842013
[126] ASCE ldquoCommittee on fatigue and fracture reliability of thecommittee on structural safety and reliability of the structuraldivision fatigue reliability 1ndash4rdquo Journal of Structural Engineer-ing vol 108 pp 3ndash88 1982
[127] BS ldquoBS5400 part 10 code of practice for fatiguerdquo BritishStandard Institute 1980
[128] BS ldquoBS7608 code of practice for fatigue design and assessmentof steel structuresrdquo British Standard Institute 1993
[129] AASHTOGuide Specifications for Fatigue Evaluation of ExistingSteel Bridges 1990
[130] AASHTO Guide Manual for Condition Evaluation and Loadand Resistance Factor Rating (LRFR) of Highway Bridges 2003
[131] S R Chen C S Cai and B Wolshon ldquoFrom normal operationto evacuation Single-vehicle safety under adverse weathertopographic and operational conditionsrdquo Natural HazardsReview vol 10 no 2 pp 68ndash76 2009
[132] S R Chen and C S Cai ldquoEquivalent wheel load approach forslender cable-stayed bridge fatigue assessment under traffic andwind feasibility studyrdquo Journal of Bridge Engineering vol 12 no6 pp 755ndash764 2007
Figure 7 Buffeting wind pressures and buffeting forces at nodes [22]
predictions Furthermore the local structural behavior ofthe bridge deck associated with local stresses and strainswhich are prone to causing local damage cannot be predicteddirectly by the current approaches based on equivalentbuffeting forces In this regard Liu et al [22] proposed anapproach to consider the spatial distribution of buffetingforces on a bridge deck structure based on wind pressuredistributions from wind tunnel tests (see Figure 7)
232 Self-Excited Forces In addition to buffeting actionflutter instability caused by self-excited forces induced bywind-structure interactions is an important considerationin the design and construction of long-span suspensionbridges [96] because the additional energy injected into theoscillating structure by the aerodynamic forces increases themagnitude of vibration sometimes to catastrophic levels [95]The self-excited forces on a bridge deck are attributable tothe interactions between wind and the motion of the bridgeWhen the energy of motion extracted from the flow exceedsthe energy dissipated by the system through mechanicaldamping the magnitude of vibration can reach catastrophiclevels [106] Expressing self-excited forces in the form ofindicial functions was first suggested by Scanlan [94] Basedon the assumption that self-excited forces are generated in alinear fashion Lin and Yang [107] simplified the self-excitedforces acting on a bridge deck and expressed them in termsof convolution integrals between the bridge deck motion andthe impulse response functions
119891se119890119863(119905)
=1
21205881198802
int119905
minusinfin
[119868119863ℎ(119905 minus 120591) ℎ
119890(120591) + 119868
119863119902(119905 minus 120591) 119902
119890(120591)
+119861119868119863120579(119905 minus 120591) 120579
119890(120591)
] 119889120591
(5a)
119891se119890119871(119905)
=1
21205881198802
int119905
minusinfin
[119868119871ℎ(119905 minus 120591) ℎ
119890(120591) + 119868
119871119902(119905 minus 120591) 119902
119890(120591)
+119861119868119871120579(119905 minus 120591) 120579
is the impulse function of the self-excited forces in which120595 represents the corresponding force components and heqe and 120579119890 are the equivalent vertical lateral and torsionaldisplacements respectively at the center of elasticity of thebridge deck section The relationship between the aero-dynamic impulse functions and flutter derivatives can beobtained by taking the Fourier transform of (7) [98]
119868119863ℎ(120596) = 119870
2
(119875lowast
6+ 119894119875lowast
5) 119868
119863119902(120596) = 119870
2
(119875lowast
4+ 119894119875lowast
1)
119868119863120579(120596) = 119870
2
(119875lowast
3+ 119894119875lowast
2)
119868119871ℎ(120596) = 119870
2
(119867lowast
4+ 119894119867lowast
1) 119868
119871119902(120596) = 119870
2
(119867lowast
6+ 119894119867lowast
5)
119868119871120579(120596) = 119870
2
(119867lowast
3+ 119894119867lowast
2)
119868119872ℎ(120596) = 119870
2
(119860lowast
4+ 119894119860lowast
1) 119868
119872119902(120596) = 119870
2
(119860lowast
6+ 119894119860lowast
5)
119868119872120579(120596) = 119870
2
(119860lowast
3+ 119894119860lowast
2)
(6)
where the overbars denote the Fourier transform operationthe terms containing 119894 represent imaginary parts119875lowast
120595 119867lowast
120595 and
119860lowast120595(120595 = 1 2 6) are dimensionless flutter derivatives
obtained from wind tunnel tests 119870 = 120596119861119880 is the reducedfrequency and 120596 is the circular frequency of vibration
According to classical airfoil theory the impulse func-tions can reasonably be approximated by a rational function[108]
119868 (120596) = [1198621+ 1198941198622
2120587
]+
119898
sum119897=1
119862119897+2
41205872 + 1198942120587119889119897+2
]1198892119897+2
V2 + 41205872] (7)
where the value of 119898 determines the level of accuracy ofthe approximation 119862
1 1198622 119862119897+2
and 119889119897+2(119897 = 1 2 119898)
are the frequency independent coefficients and ] = 2120587119870
is the reduced mean wind velocity By equating the realand imaginary parts in the comparison of (8) and (9) therelationship between the dimensionless flutter derivativesand the coefficients 119862120595
1 119862120595
2 119862120595
119897+2 and 119889120595
119897+2(where 120595 = Dh
Dq 119863120579 Lh Lq 119871120579 Mh Mq 119872120579 and 119897 = 1 2 119898) canbe establishedThese coefficients are determined by using thenonlinear least-squares method to fit the measured flutter
10 The Scientific World Journal
derivatives at different reduced frequencies The expressionof the aerodynamic impulse functions in the time domaincan be obtained by taking the inverse Fourier transform ofthe impulse functions By substituting the related impulseresponse functions into (5b) the self-excited lift force at the119894th section of bridge deck can then be derived as
119891se119890119871119894
=1
21205881198861198802
119894119861119894119862119871120579
1119894120579119894(119905) + 119862
119871120579
2119894(119861119894
119880119894
) 120579119894(119905)
+ 119862119871120579
3119894(119861119894
119880119894
) 120579119894(119905) +
119898
sum119897=1
119862119871120579
119897+3119894
times int119905
minusinfin
120579119894(119905) exp[minus
119889119871120579119897+3119894119880119894
119861119894
(119905 minus 120591)] 119889120591
+1
21205881198861198802
119894119862119871ℎ
1119894ℎ119894(119905) + 119862
119871ℎ
2119894(119861119894
119880119894
) ℎ119894(119905)
+ 119862119871ℎ
3119894(119861119894
119880119894
) ℎ119894(119905) +
119898
sum119897=1
119862119871ℎ
119897+3119894
times int119905
minusinfin
ℎ119894(119905) exp[minus
119889119871ℎ119897+3119894119880119894
119861119894
(119905 minus 120591)] 119889120591
+1
21205881198861198802
119894
119862119871119902
1119894119902119894(119905) + 119862
119871119902
2119894(119861119894
119880119894
) 119902119894(119905)
+ 119862119871119902
3119894(119861119894
119880119894
) 119902119894(119905) +
119898
sum119897=1
119862119871119902
119897+3119894
times int119905
minusinfin
119902120595(119905) exp[
[
minus119889119871119901
119897+3119894119880119894
119861119894
(119905 minus 120591)]
]
119889120591
(8)
In practice the terms 1198621198711205793119894 119862119871ℎ3119894 and 119862119871119902
3119894 which are related to
additional aerodynamic masses are normally neglected andthe value of 119898 is often taken as 2 [101] Similar formulationsfor self-excited drag and moment can be derived withanalogous definitions The self-excited forces at the 119894th nodeof the bridge deck can thus be expressed as
Feise = EeiXei + GeiXei + Feise (9)
where
Xei =
0
ℎei119902ei120579ei0
0
Eei =1
21205881198802
119894
[[[[[[[[
[
0 0 0 0 0 0
0 119862119871ℎ
1119894119862119871119902
11198941198611198941198621198711205791119894
0 0
0 119862119863ℎ1119894
119862119863119902
11198941198611198941198621198631205791119894
0 0
0 119861119894119862119872ℎ1119894
119861119894119862119872119902
111989411986121198941198621198721205791119894
0 0
0 0 0 0 0 0
0 0 0 0 0 0
]]]]]]]]
]
Gei =1
21205881198802
119894119861119894
[[[[[[[[
[
0 0 0 0 0 0
0 119862119871ℎ
2119894119862119871119902
21198941198611198941198621198711205792119894
0 0
0 119862119863ℎ2119894
119862119863119902
21198941198611198941198621198631205792119894
0 0
0 119861119894119862119872ℎ2119894
119861119894119862119872119902
211989411986121198941198621198721205792119894
0 0
0 0 0 0 0 0
0 0 0 0 0 0
]]]]]]]]
]
Feise =
[[[[[[[
[
0
eise
119863eise
eise0
0
]]]]]]]
]
=
[[[[[[[[[[[[[[[[
[
02
sum119897=1
119862119871119902
l+3119894119881119871119902
l+3119894 +2
sum119897=1
119862119871ℎl+3119894119881119871ℎ
l+3119894 +2
sum119897=1
119862119871120579l+3119894119881119871120579
l+3119894
2
sum119897=1
119862119863119902
l+3119894119881119863119902
l+3119894 +2
sum119897=1
119862119863ℎl+3119894119881119863ℎ
l+3119894 +2
sum119897=1
119862119863120579l+3119894119881119863120579
l+3119894
2
sum119897=1
119862119872119902
l+3119894119881119872119902
l+3119894 +2
sum119897=1
119862119872ℎl+3119894119881119872ℎ
l+3119894 +2
sum119897=1
119862119872120579l+3119894119881119872120579
l+3119894
0
0
]]]]]]]]]]]]]]]]
]
(10)
where 119881120595l+3119894 (120595 = Dh Dq 119863120579 Lh Lq 119871120579 Mh Mq) are theconvolution integrations of the 119894th node and can be calculatedusing a recursive algorithm For example
(11)The self-excited forces expressed by (9) relate to the centerof elasticity of the 119894th deck section Hence the force modelmust be distributed to the nodal points of the section Adistribution based on the rigid body motion relationshipsbetween themotions at the nodal point and those at the centerof elasticity of the deck section [109] was applied by Liu etal [22] In this study by applying the virtual work principlethe self-excited forces at the center of elasticity of the givensection were distributed to all nodes (see Figure 8)
24 Dynamic Interactions in a Wind-Vehicle-Bridge SystemWhen trains and road vehicles are running on long-spanbridges under crosswinds complicated dynamic interac-tions occur among the trains road vehicles cable-supported
The Scientific World Journal 11
Ui h
p
120572
Lsec i
Msec i
Dsec i
Fsekiz
Fsekiy
ce
Figure 8 Self-excited forces at the centre of elasticity and at the nodes in the 119894th deck section [22]
bridge and windThe buffeting response of the bridge due tocrosswind is superimposed on the dynamic response of thebridge due to railway and road vehicles The large vibrationof the bridge will in turn considerably affect the safetyand ride comfort of the drivers of the road vehicles Thusthe dynamic responses of a coupled vehicle-bridge systemunder crosswinds are of great concern to both engineers andresearchers
Detailed reviews of the dynamic interactions betweentrains and bridges between road vehicles and bridges andbetween wind and bridges have been given in the previoussections However the interaction between wind and vehiclesmust also be taken into account in a coupled wind-vehicle-bridge analysis Many studies have investigated wind-vehicleinteractions in the past few decades Balzer [110] developeda theory to estimate the aerodynamic forces on a movingvehicle using Taylorrsquos hypothesis of ldquofrozen turbulencerdquo Forengineering applications Cooper [111] proposed the powerspectral density (PSD) square-root coherence functionphase-lag function and aerodynamic admittance functionto model the unsteady side forces on a moving vehicle andlaid down the foundations for investigating the effects ofwind on a moving vehicle in the frequency domain Bakerdeveloped a theoretical model that describes the dynamics ofvehicles in crosswinds in the time domain [112 113] whichwas later extended to include driver behavior [114] Baker[115 116] further investigated both the steady and unsteadyaerodynamic forces acting on a variety of vehicles and carriedout extensive studies of the interaction between aerodynamicforces and moving vehicles These approaches have all beenapplied in coupled vehicle-bridge analysis For example Xuet al [101] simulated the aerodynamic wind forces actingon running road vehicles using the quasi-steady approachand Xu and Ding [117] derived and simulated the steadyand unsteady aerodynamic forces acting on a moving railwayvehicle in crosswinds in the time domain
Based on these separate studies on the various typesof dynamic interactions among wind vehicles (trains orroad vehicles) and long-span bridges several researchersin the last decade have examined the wind-vehicle-bridgecoupled system as a whole For instance studies have beencarried out on coupled road vehicle and cable-stayed bridgesystems [81 82 118] and on coupled train and cable-supportedbridge systems in crosswinds [101 117 119ndash121] In therecent years several new advances have been made bothin numerical simulation technologies and in wind tunnelmeasurements Chen et al [17] proposed a wind-vehicle-bridge framework which enables considering the dynamiceffects induced by simultaneous actions of railway highway
and wind loading and it was applied to analyze dynamicstress of long suspension bridges Li et al [122] extended thewind-vehicle-bridge couple analysis to the case of two trainsmeeting on a long-span suspension bridge Chen and Wu[118] proposed a semideterministic analytical model whichis able to consider dynamic interactions between the bridgewind and stochastic ldquorealrdquo traffic Based on the wind tunneltests Dorigatti et al [123]measured crosswind loads on high-sided vehicles over long-span bridges taking three differentvehicles (van double deck bus and lorry) and two differentbridge deck configurations into consideration Zhu et al[124] investigated aerodynamic coefficients of road vehiclesby adopting different road vehicles types wind directionsand vehicle positions Li et al [122] studied the effects ofsudden changes of wind loads as the train passing througha bridge tower or two trains passing each other by usingthe wind tunnel test rig with moving train models Hanet al [125] developed an experimental setup for measuringthe aerodynamic characteristics of vehicles and the bridgein wind tunnel and then investigated the influences ofparameters adopted in the tests
3 Applications of Simulation Technology toBridge Assessment
After reviewing the key issues of numerical simulationsfor dynamic response of long-span multiload bridges thissection will review the engineering applications of the newlydeveloped technologies to safety assessment of long-spanbridges such as assessment of fatigue and assessment underextreme events
31 Assessment of FatigueDamage Steel structures are widelyused in long-span bridges Research by the ASCE [126] indi-cates that 80ndash90 of failures in steel structures are related tofatigue and fracture Several disasters resulting from fatigue-induced bridge failure have occurred in history For instance46 people died in the collapse of the Silver Bridge (USA1967) and 32 people lost their lives in the collapse of theSungsoo Grand Bridge (South Korea 1994) These disastersteach us that fatigue is an important aspect of the safety ofsteel bridges and action should be taken to prevent similarfatigue-induced bridge failures In the past few decadesfatigue assessment of steel bridges has attracted increasingattention from both governments and bridge engineers andrelevant provisions have been stipulated in several codes andstandards [127ndash130]
12 The Scientific World Journal
It has great advantages to evaluate fatigue damage of long-span bridges based on numerical simulation especially fora multiload bridge which suffers multiple types of dynamicloading such as railway highway andwind loadingDifferentfrom sudden structural damage fatigue damage accumulateswith load-induced dynamic stress (or stress fluctuation) overthe service life of a bridge lasting for more than 100 yearsThe increase in traffic volume and gross vehicle weight thataccompany economic development is very likely to happenin the long period Numerical simulation technology can bean ideal tool to study influences of traffic growth patternsto fatigue damage of bridge In addition slender long-span bridges built in wind-prone regions also suffer fromconsiderable wind induced vibration which appears withina wide range of wind speeds and lasts for almost the wholedesign life of the bridge Given the simultaneous presenceof multiple vehicles and wind it is necessary to considercombined effects of traffic loading (railway andor highwayloading) and wind loading in the fatigue assessment Sincemultiple loading is concerned in a long time period thereare a large number of loading combinations for multipleloading in different intensities It is almost unavailable forfield measurement to obtain such complete information butnumerical simulation could be a good choice to determinedynamic responses of a long-span bridge under multipleloading
A number of structural health monitoring systems(SHMSs) have been installed on numerous recently builtlong-span bridges and a variety of sensors are used for mon-itoring bridge loadings (eg traffic wind and earthquakes)and conditions (including global and local responses) toensure bridge safety and user comfort under in-service con-ditions Well-known examples include Tsing-Ma Bridge inHong Kong Akashi Kaikyo Bridge in Japan Binzhou YellowRiver Bridge in China and Jindo Bridge in Korea Integrationof numerical simulation technologies and measurement dataof structural health monitoring systems (SHMSs) installedon long-span bridges will make the fatigue assessment morereliable for several reasons (1) it is a perfect validation byusing field measurement data of the different types of loadingas input of numerical simulation and the measured dynamicresponses for comparison with the computed ones (2) a largenumber of measured loading data could be used to establishloading databases or probabilistic models of different loads
In the recent years several researchers [7ndash10] appliedthe newly developed numerical simulation technologies tofatigue assessment of long-span bridges Chen et al [7]proposed a framework for fatigue analysis of a long-spansuspension bridge under railway highway and wind loadingby integrating computer simulation with SHMSs and itwas applied to evaluate fatigue damage of the Tsing MaSuspension Bridge over its design life as a case study Basedon this work Chen et al [8] proposed a framework forfatigue reliability analysis of long suspension bridges undermultiple loading inwhich the probabilisticmodels of railwayhighway and wind loading were established based on themeasurement data acquired from the SHMS of the TsingMa Bridge Wu et al [9] proposed a reliability-based fatigueapproach for slender long-span bridge and the combined
dynamic loading effects from wind and traffic as well asthe associated uncertainties were considered Based on theassumption that dynamic magnification related to vehicledynamics can be neglected in long suspension bridges Chenet al [8] established a framework for fatigue reliability anal-ysis To account for different types of long-span bridges withthe span length ranging from a few hundred to thousands ofmeters Zhang et al [10] proposed a more general frameworkwhich includes multiple random variables for the dynamicloads in a bridgersquos life cycle for the vehicle-bridge-winddynamic system such as road profile vehicle speed andwindvelocity and direction among other effects
32 Assessment under Extreme Events The aforementionedfatigue assessment mainly focuses on damage accumulationinduced by stress fluctuations under normal operationalcondition in a long-term period For long-span bridges inaddition to the normal operational conditions in which windspeeds are small ormoderate and traffic scenarios are normalthere are some extreme event conditions Extreme eventsmayinclude complex traffic congestion on the bridge coupledwith moderate or even strong wind [11] For example severetraffic congestions may be formed on the bridge as a resultof an evacuation or a partial blockage of driving lanes due totraffic accidents construction ormaintenance For hurricaneevacuations there are usually a lot of road vehicles passingthrough the bridge before the landfall of the hurricane whilethe wind speed may become pretty high already [131]
Although the excessive dynamic responses of the bridgesunder extreme events are rare it is also critical since itmay cause critical damage initiation or accumulation onsome local bridgemembers Furthermore the extreme events(eg heavy traffic) may even trigger the collapse of thewhole bridge by breaking the ldquoweakest linkrdquo especially whensome hidden damage or design flaw has not been detectedOne recent example is the Minnesota Bridge failure whichoccurred during rush hours with heavy traffic although trafficloads may not be the direct cause of failure For slender long-span bridges strong wind may also cause threats by workinginteractively with heavy traffic loads Therefore even thoughthe extreme cases associated with congested traffic andorwindy weather may be relatively rare and the durationscould be short it is still important for bridge engineers toappropriately look into these unusual extreme events duringstructural design and life-time management of these criticalinfrastructures [11]
The dynamic performance of long-span bridges undercombined actions of strong winds and running road vehicleshas been studied by many researchers in recent years [17 7981 82 132] Most of them studied bridge dynamic perfor-mance under road traffic inwhich only one or several vehiclesdistributed in an assumed (usually uniform) pattern on long-span bridges were considered Extreme events such as trafficcongestion coupled with strong wind were out of concernin those studies Recently Wu and Chen [11] conducteda research on the assessment of long-span bridges underextreme events which includes complex traffic congestioncoupled with moderate or even strong wind This study
The Scientific World Journal 13
applied the cellular automaton (CA) traffic model to thesimulation of the actual traffic flow through the bridgedefined representative scenarios for the extreme events andnumerically studied the bridge performance under thesepossible extreme events
4 Conclusions and Recommendations
Dynamic responses of long-span bridges are often requiredfor assessing the safety of these bridges and can be determinedby numerical simulation technologies This paper provides adetailed review of key issues involved in dynamic responseanalysis of long-span multiload bridges based on numericalsimulation including dynamic interactions between runningtrains and bridge between running road vehicles and bridgeand between wind and bridge and in the wind-vehicle-bridge coupled systemThen the review work was conductedfor engineering applications of newly developed numericalsimulation technologies to safety assessment of long-spanbridges such as assessment of fatigue damage and assessmentunder extreme event condition Although technologies fornumerical simulation of dynamic responses of long-spanmultiload bridge have achieved great advances in past fewdecades and successfully applied to several important bridgesit is still far from reach its maturity and enable to takeplace of traditional fieldmeasurementThe existing problemsand promising research efforts at least include the followingaspects
(1) After multiple types of dynamic interactions beingconsidered in the complex system computationalefficiency is a bottleneck problem for numericalsimulation of dynamic response of a long-span bridgeTypically when multiple loads are involved a largenumber of loading combinations for multiple load-ings must be considered in the assessment
(2) It is rather complex for the time-depending windloads acting on a long-span bridge and running vehi-cles especially for the case of rapid change of windloads such as a train passing through a bridge toweror two trains passing each other The aerodynamiccharacteristics of vehicles and the bridge under differ-ent loading scenarios can be determined through thewind tunnel testing and used in numerical simulationof dynamic responses of the bridge and vehicles
(3) It is a new trend to integrate numerical simulationtechnologies and measurement data of structuralhealth monitoring systems (SHMSs) installed onlong-span bridges whichmakes the safety assessmentof bridge structures more reliable Measured struc-tural responses could be used to validate numericalsimulation approach and measured loading infor-mation could be used for generating statistical orprobabilistic models of multiple loads
(4) It is important to study dynamic responses of bridgestructures under extreme events such as congestedtraffic coupled with windy weather which happens ina long-span bridge For the assessment under extreme
events using numerical simulation technologies sim-ulation of traffic flow and definition of representativescenarios of the extreme events are key issues
(5) It is necessary to consider the effects of typhoonwinds on the safety assessment of long-span bridgesin a reasonable way Few researches do this mostlybecause a probabilistic distribution of wind speedand direction specifically for typhoons is requiredfor assessment but there are insufficient measuredrecords to establish a reliable probabilistic typhoonwind model
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors wish to acknowledge the financial supportsfrom the National Natural Science Foundation of China(NSFC-51108395 and NSFC-51178366) the FundamentalResearch Funds for theCentral Universities (2012121032) andopen funding from Jiangsu Key Laboratory of EngineeringMechanics Special thanks go to the supervisor of the firstauthor Professor Y L XuHongKong PolytechnicUniversityfor his expert guidance and continuous support at all levelsthroughout his PhD study Sincere thanks should go tothe Highways Department of Hong Kong for providing theauthors with the field measurement data Any opinions andconcluding remarks presented in this paper are entirely thoseof the authors
References
[1] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[2] H Sohn C R Farrar N F Hunter and K Worden ldquoStructuralhealth monitoring using statistical pattern recognition tech-niquesrdquo Journal of Dynamic Systems Measurement and Controlvol 123 no 4 pp 706ndash711 2001
[3] W Fan and P Qiao ldquoVibration-based damage identificationmethods a review and comparative studyrdquo Structural HealthMonitoring vol 10 no 1 pp 83ndash111 2011
[4] X Q Zhu and S S Law ldquoDamage detection in simply supportedconcrete bridge structure under moving vehicular loadsrdquo Jour-nal of Vibration and Acoustics Transactions of the ASME vol129 no 1 pp 58ndash65 2007
[5] J Li and S S Law ldquoDamage identification of a target sub-structure with moving load excitationrdquoMechanical Systems andSignal Processing vol 30 pp 78ndash90 2012
[6] J Li S S Law and H Hao ldquoImproved damage identification inbridge structures subject tomoving loads numerical and exper-imental studiesrdquo International Journal of Mechanical Sciencesvol 74 pp 99ndash111 2013
14 The Scientific World Journal
[7] Z W Chen Y L Xu Y Xia Q Li and K Y Wong ldquoFatigueanalysis of long-span suspension bridges under multiple load-ing case studyrdquo Engineering Structures vol 33 no 12 pp 3246ndash3256 2011
[8] Z W Chen Y L Xu and X M Wang ldquoSHMS-based fatiguereliability analysis of multiloading suspension bridgesrdquo Journalof Structural Engineering-Asce vol 138 pp 299ndash307 2012
[9] J Wu S R Chen and J W van de Lindt ldquoFatigue assessmentof slender long-span bridges reliability approachrdquo Journal ofBridge Engineering vol 17 no 1 pp 47ndash57 2012
[10] W Zhang C S Cai and F Pan ldquoFatigue reliability assessmentfor long-span bridges under combined dynamic loads fromwinds and vehiclesrdquo Journal of Bridge Engineering vol 18 pp735ndash747 2013
[11] JWu and S R Chen ldquoProbabilistic dynamic behavior of a long-span bridge under extreme eventsrdquo Engineering Structures vol33 no 5 pp 1657ndash1665 2011
[12] J M Ko and Y Q Ni ldquoTechnology developments in structuralhealth monitoring of large-scale bridgesrdquo Engineering Struc-tures vol 27 no 12 pp 1715ndash1725 2005
[13] TH YiHN Li andHM Sun ldquoMulti-stage structural damagediagnosis method based on ldquoenergy-damagerdquo theoryrdquo SmartStructures and Systems vol 12 pp 345ndash361 2013
[14] T H Yi H N Li and M Gu ldquoFull-scale measurements ofdynamic response of suspension bridge subjected to environ-mental loads using GPS technologyrdquo Science China Technologi-cal Sciences vol 53 no 2 pp 469ndash479 2010
[15] T H Yi H N Li and M Gu ldquoExperimental assessmentof high-rate GPS receivers for deformation monitoring ofbridgerdquoMeasurement Journal of the InternationalMeasurementConfederation vol 46 pp 420ndash432 2013
[16] X W Ye Y Q Ni K Y Wong and J M Ko ldquoStatistical analysisof stress spectra for fatigue life assessment of steel bridges withstructural health monitoring datardquo Engineering Structures vol45 pp 166ndash176 2012
[17] Z W Chen Y L Xu Q Li and D J Wu ldquoDynamic stressanalysis of long suspension bridges under wind railway andhighway loadingsrdquo Journal of Bridge Engineering vol 16 no 3pp 383ndash391 2011
[18] S G Meisenholder and P Weidlinger ldquoDynamic interactionaspects of cable-stayed guideways for high speed ground trans-portationrdquoAmerican Society ofMechanical Engineers no 74 pp180ndash192 1974
[19] Q H Mao Research on the Highway Bridge Vibration Due toMoving Vehicles Tongji University Shang Hai China 1989
[20] Y L Xu J M Ko and Z Yu ldquoModal analysis of tower-cable system of Tsing Ma long suspension bridgerdquo EngineeringStructures vol 19 pp 857ndash867 1997
[21] W Guo H Xia and Y-L Xu ldquoDynamic response of a long spansuspension bridge and running safety of a train under windactionrdquo Frontiers of Architecture and Civil Engineering in Chinavol 1 no 1 pp 71ndash79 2007
[22] T T Liu Y L Xu W S Zhang K Y Wong H J Zhou and KW Y Chan ldquoBuffeting-induced stresses in a long suspensionbridge structural health monitoring oriented stress analysisrdquoWind and Structures An International Journal vol 12 no 6 pp479ndash504 2009
[23] K Y Wong ldquoStructural identification of Tsing Ma BridgerdquoTransactions Hong Kong Institution of Engineers vol 10 no 1pp 38ndash47 2003
[24] Y L XuQ Li D JWu and ZWChen ldquoStress and accelerationanalysis of coupled vehicle and long-span bridge systems usingthe mode superposition methodrdquo Engineering Structures vol32 no 5 pp 1356ndash1368 2010
[25] Y F Duan Y L Xu Q G Fei et al ldquoAdvanced finite elementmodel of Tsing Ma Bridge for structural health monitoringrdquoInternational Journal of Structural Stability and Dynamics vol11 no 2 pp 313ndash344 2011
[26] Z X Li T Q Zhou T H T Chan and Y Yu ldquoMulti-scalenumerical analysis on dynamic response and local damage inlong-span bridgesrdquo Engineering Structures vol 29 no 7 pp1507ndash1524 2007
[27] W Zhang C S Cai and F Pan ldquoFinite element modeling ofbridges with equivalent orthotropic material method for multi-scale dynamic loadsrdquo Engineering Structures vol 54 pp 82ndash932013
[28] S P Timoshenko ldquoOn the forced vibrations of bridgesrdquo Philo-sophical Magazine vol 6 no 257 pp 1018ndash1019 1922
[29] R S Ayre G Ford and L S Jacobsen ldquoTransverse vibration ofa two-span beam under the action of a moving constant forcerdquoJournal of Applied Mechanics vol 17 pp 1ndash12 1950
[30] R S Ayre andLS Jacobsen ldquoTransverse vibration of a two-spanbeam under the action of a moving alternating forcerdquo Journal ofApplied Mechanics vol 17 pp 283ndash290 1950
[31] L FrybaVibration of Solids and Structures underMoving LoadsASCE Press 1972
[32] J-S Wu and C-W Dai ldquoDynamic response of multispannonuniform beam due to moving loadsrdquo Journal of StructuralEngineering vol 113 no 3 pp 458ndash474 1987
[33] W Weaver S P Timoshenko and D H Young VibrationProblems in Engineering John Wiley amp Sons 1990
[34] N H Galdos D R Schelling and M A Sahin ldquoMethodologyfor impact factor of horizontally curved box bridgesrdquo Journal ofStructural Engineering vol 119 no 6 pp 1917ndash1934 1993
[35] J A Gbadeyan and S T Oni ldquoDynamic behaviour of beamsand rectangular plates under moving loadsrdquo Journal of Soundand Vibration vol 182 no 5 pp 677ndash695 1995
[36] D Y Zheng Y K Cheung F T K Au and Y S ChengldquoVibration of multi-span non-uniform beams under movingloads by using modified beam vibration functionsrdquo Journal ofSound and Vibration vol 212 no 3 pp 455ndash467 1998
[37] G V Rao ldquoLinear dynamics of an elastic beam under movingloadsrdquo Journal of Vibration and Acoustics Transactions of theASME vol 122 no 3 pp 281ndash289 2000
[38] Y B Yang J D Yau and Y S Wu Vehicle-Bridge InteractionDynamic with Applications to High-Speed Railways WorldScientific 2004
[39] E C Ting J Genin and J H Ginsberg ldquoA general algorithm formovingmass problemsrdquo Journal of Sound and Vibration vol 33no 1 pp 49ndash58 1974
[40] S Sadiku and H H E Leipholz ldquoOn the dynamics of elasticsystems with moving concentrated massesrdquo Ingenieur-Archivvol 57 no 3 pp 223ndash242 1987
[41] J E Akin and M Mofid ldquoNumerical solution for response ofbeamswithmovingmassrdquo Journal of Structural Engineering vol115 no 1 pp 120ndash131 1989
[42] M AMahmoud andM A Abou Zaid ldquoDynamic response of abeam with a crack subject to a moving massrdquo Journal of Soundand Vibration vol 256 no 4 pp 591ndash603 2002
[43] V K Garg Dynamics of Railway Vehicle Systems AcademicPress 1994
The Scientific World Journal 15
[44] T-LWang andDHuang ldquoCable-stayed bridge vibration due toroad surface roughnessrdquo Journal of Structural Engineering vol118 no 5 pp 1354ndash1374 1992
[45] Y-B Yang and B-H Lin ldquoVehicle-bridge interaction analysisby dynamic condensation methodrdquo Journal of Structural Engi-neering vol 121 no 11 pp 1636ndash1643 1995
[46] Y B Yang J D Yau and L C Hsu ldquoVibration of simple beamsdue to trainsmoving at high speedsrdquoEngineering Structures vol19 no 11 pp 936ndash943 1997
[47] B Tabarrok and E Esmailzadeh ldquoInduced vibration of bridgestransversed by moving vehiclesrdquo Transactions of the CanadianSociety for Mechanical Engineering B vol 24 no 1 pp 191ndash1982000
[48] C Liu T-L Wang and D Huang ldquoImpact study for multi-girder bridge based on correlated road roughnessrdquo StructuralEngineering and Mechanics vol 11 no 3 pp 259ndash272 2001
[49] K H Chu V K Garg and T L Wang ldquoImpact in railwayprestressed concrete bridgesrdquo Journal of Structural Engineeringvol 112 no 5 pp 1036ndash1051 1986
[50] T-LWang V K Garg and K-H Chu ldquoRailway bridgevehicleinteraction studies with new vehicle modelrdquo Journal of Struc-tural Engineering vol 117 no 7 pp 2099ndash2116 1991
[51] H Xia Y L Xu and T H T Chan ldquoDynamic interaction oflong suspension bridges with running trainsrdquo Journal of Soundand Vibration vol 237 no 2 pp 263ndash280 2000
[52] Q-L Zhang A Vrouwenvelder and J Wardenier ldquoNumericalsimulation of train-bridge interactive dynamicsrdquo Computersand Structures vol 79 no 10 pp 1059ndash1075 2001
[53] H Xia N Zhang and G de Roeck ldquoDynamic analysis of highspeed railway bridge under articulated trainsrdquo Computers andStructures vol 81 no 26-27 pp 2467ndash2478 2003
[54] G Diana F Cheli A Collina R Corradi and S MelzildquoThe development of a numerical model for railway vehiclescomfort assessment through comparison with experimentalmeasurementsrdquoVehicle SystemDynamics vol 38 no 3 pp 165ndash183 2002
[55] Q Li Y L Xu D J Wu and Z W Chen ldquoComputer-aided nonlinear vehicle-bridge interaction analysisrdquo Journal ofVibration and Control vol 16 pp 1791ndash1816 2010
[56] A Wiriyachai K H Chu and V K Garg ldquoBridge impact dueto wheel and track irregularitiesrdquo Journal of the EngineeringMechanics Division vol 108 no 4 pp 648ndash666 1982
[57] L Fryba Dynamics of Railway Bridges Inst of Civil Engineers1996
[58] D Huang and T-L Wang ldquoImpact analysis of cable-stayedbridgesrdquo Computers and Structures vol 43 no 5 pp 897ndash9081992
[59] W M Zhai Vehicle-Track Coupling Dynamics Chinese RailwayPress Beijing China 2007
[60] M Olsson ldquoFinite element modal co-ordinate analysis ofstructures subjected to moving loadsrdquo Journal of Sound andVibration vol 99 no 1 pp 1ndash12 1985
[61] Y-B Yang C-H Chang and J-D Yau ldquoAn element foranalysing vehicle-bridge systems considering vehiclersquos pitchingeffectrdquo International Journal for NumericalMethods in Engineer-ing vol 46 no 7 pp 1031ndash1047 1999
[62] Y B Yang and Y S Wu ldquoA versatile element for analyzingvehicle-bridge interaction responserdquo Engineering Structuresvol 23 no 5 pp 452ndash469 2001
[63] F T K Au J J Wang and Y K Cheung ldquoImpact study ofcable-stayed bridge under railway traffic using various modelsrdquo
Journal of Sound and Vibration vol 240 no 3 pp 447ndash4652001
[64] Y Q Sun andMDhanasekar ldquoA dynamicmodel for the verticalinteraction of the rail track and wagon systemrdquo InternationalJournal of Solids and Structures vol 39 no 5 pp 1337ndash13592002
[65] K Henchi M Fafard M Talbot and G Dhatt ldquoAn efficientalgorithm for dynamic analysis of bridges under moving vehi-cles using a coupledmodal and physical components approachrdquoJournal of Sound and Vibration vol 212 no 4 pp 663ndash6831998
[66] Y L Xu and L YWang ldquoAnalytical study of wind-rain-inducedcable vibration SDOFmodelrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 91 no 1-2 pp 27ndash40 2003
[67] B Biondi G Muscolino and A Sofi ldquoA substructure approachfor the dynamic analysis of train-track-bridge systemrdquoComput-ers and Structures vol 83 no 28ndash30 pp 2271ndash2281 2005
[68] J L Humar and A H Kashif ldquoDynamic response analysis ofslab-type bridgesrdquo Journal of Structural Engineering vol 121 no1 pp 48ndash62 1995
[69] P Lou and Q-Y Zeng ldquoFormulation of equations of verticalmotion for vehicle-track-bridge systemrdquo Journal of the ChinaRailway Society vol 26 no 5 p 71 2004
[70] O Coussy M Said and J-P van Hoove ldquoThe influence ofrandom surface irregularities on the dynamic response ofbridges under suspended moving loadsrdquo Journal of Sound andVibration vol 130 no 2 pp 313ndash320 1989
[71] E S Hwang and A S Nowak ldquoSimulation of dynamic loadfor bridgesrdquo Journal of Structural Engineering vol 117 pp 1413ndash1434 1991
[72] F Yang and G A Fonder ldquoAn iterative solution methodfor dynamic response of bridge-vehicles systemsrdquo EarthquakeEngineering and Structural Dynamics vol 25 pp 195ndash215 1996
[73] W Zhai and Z Cai ldquoDynamic interaction between a lumpedmass vehicle and a discretely supported continuous rail trackrdquoComputers and Structures vol 63 no 5 pp 987ndash997 1997
[74] W M Zhai and C B Cai ldquoTraintrackbridge dynamic inter-actions simulation and applicationsrdquo Vehicle System Dynamicsvol 37 pp 653ndash665 2003
[75] D Bruno F Greco and P Lonetti ldquoDynamic impact analysis oflong span cable-stayed bridges under moving loadsrdquo Engineer-ing Structures vol 30 no 4 pp 1160ndash1177 2008
[76] X D Song D J Wu and Q Li ldquoDynamic impact analysisof double-tower cable-stayed maglev bridges using a simplemodelrdquo Journal of Bridge Engineering vol 19 pp 34ndash43 2014
[77] Y-S Wu and Y-B Yang ldquoSteady-state response and ridingcomfort of trains moving over a series of simply supportedbridgesrdquoEngineering Structures vol 25 no 2 pp 251ndash265 2003
[78] P Antolin N Zhang J M Goicolea H Xia M A Astiz andJ Oliva ldquoConsideration of nonlinear wheel-rail contact forcesfor dynamic vehicle-bridge interaction in high-speed railwaysrdquoJournal of Sound and Vibration vol 332 no 5 pp 1231ndash12512013
[79] W H Guo and Y L Xu ldquoFully computerized approach to studycable-stayed bridge-vehicle interactionrdquo Journal of Sound andVibration vol 248 no 4 pp 745ndash761 2001
[80] Y L Xu and W H Guo ldquoDynamic behaviour of high-sidedroad vehicles subject to a sudden crosswind gustrdquo Wind andStructures vol 6 no 5 pp 325ndash346 2003
[81] Y L Xu and W H Guo ldquoDynamic analysis of coupled roadvehicle and cable-stayed bridge systems under turbulent windrdquoEngineering Structures vol 25 no 4 pp 473ndash486 2003
16 The Scientific World Journal
[82] C S Cai and S R Chen ldquoFramework of vehicle-bridge-winddynamic analysisrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 92 no 7-8 pp 579ndash607 2004
[83] S R Chen and J Wu ldquoModeling stochastic live load for long-span bridge based on microscopic traffic flow simulationrdquoComputers and Structures vol 89 no 9-10 pp 813ndash824 2011
[84] A V Paultre B Yang L A Bergman and C A Tan ldquoBridgedynamics and dynamic amplification factorsmdasha review ofanalytical and experimental findingsrdquoCanadian Journal of CivilEngineering vol 19 no 2 pp 260ndash278 1992
[85] H Honda Y Kajikawa and T Kobori ldquoSpectra of road surfaceroughness of bridgesrdquo Journal of the Structural Division vol 108pp 1956ndash1966 1982
[86] M J Inbanathan and M Wieland ldquoBridge vibrations dueto vehicle moving over rough surfacerdquo Journal of StructuralEngineering vol 113 no 9 pp 1994ndash2008 1987
[87] T-LWang andDHuang ldquoCable-stayed bridge vibration due toroad surface roughnessrdquo Journal of Structural Engineering vol118 no 5 pp 1354ndash1374 1992
[88] P K Chatterjee T K Datta and C S Surana ldquoVibrationsuspension bridges under vehicular movementrdquo Journal ofStructural Engineering vol 120 no 3 pp 681ndash703 1994
[89] D Chang and H Lee ldquoImpact factors for simple-span highwaygirder bridgesrdquo Journal of Structural Engineering vol 120 no 3pp 704ndash715 1994
[90] T-C Pan and J Li ldquoDynamic vehicle element method for tran-sient response of coupled vehicle-structure systemsrdquo Journal ofStructural Engineering vol 128 no 2 pp 214ndash223 2002
[91] C J Dodds and J D Robson ldquoThe description of road surfaceroughnessrdquo Journal of Sound Vibration vol 31 no 2 pp 175ndash183 1973
[92] D Huang T-L Wang and M Shahawy ldquoImpact studies ofmultigirder concrete bridgesrdquo Journal of Structural Engineeringvol 119 no 8 pp 2387ndash2402 1993
[93] A G Davenport ldquoBuffeting of a suspension bridge by stormwindrdquo Journal of Structural Division vol 88 pp 233ndash268 1962
[94] R H Scanlan ldquoThe action of flexible bridges under wind Iflutter theoryrdquo Journal of Sound and Vibration vol 60 no 2pp 187ndash199 1978
[95] Q Ding and P K K Lee ldquoComputer simulation of buffetingactions of suspension bridges under turbulentwindrdquoComputersand Structures vol 76 no 6 pp 787ndash797 2000
[96] V Boonyapinyo T Miyata and H Yamada ldquoAdvanced aerody-namic analysis of suspension bridges by state-space approachrdquoJournal of Structural Engineering vol 125 no 12 pp 1357ndash13661999
[97] Y-H Chen and C-Y Li ldquoDynamic response of elevated high-speed railwayrdquo Journal of Bridge Engineering vol 5 no 2 pp124ndash130 2000
[98] X Chen M Matsumoto and A Kareem ldquoTime domainflutter and buffeting response analysis of bridgesrdquo Journal ofEngineering Mechanics vol 126 no 1 pp 7ndash16 2000
[99] X Chen and A Kareem ldquoEquivalent static wind loads forbuffeting response of bridgesrdquo Journal of Structural Engineeringvol 127 no 12 pp 1467ndash1475 2001
[100] S R Chen and C S Cai ldquoEvolution of long-span bridgeresponse to wind-numerical simulation and discussionrdquo Com-puters and Structures vol 81 no 21 pp 2055ndash2066 2003
[101] Y L XuHXia andQ S Yan ldquoDynamic response of suspensionbridge to high wind and running trainrdquo Journal of BridgeEngineering vol 8 no 1 pp 46ndash55 2003
[102] A Guo Y L Xu and H Li ldquoDynamic performance of cable-stayed bridge tower with multi-stage pendulum mass damperunderwind excitations-II experimentrdquoEarthquake Engineeringand Engineering Vibration vol 6 no 4 pp 417ndash424 2007
[103] E Simiu and R H ScanlanWind Effects on Structures 1996[104] Y Cao H Xiang and Y Zhou ldquoSimulation of stochastic wind
velocity field on long-span bridgesrdquo Journal of EngineeringMechanics vol 126 no 1 pp 1ndash6 2000
[105] M Shinozuka and C-M Jan ldquoDigital simulation of randomprocesses and its applicationsrdquo Journal of Sound and Vibrationvol 25 no 1 pp 111ndash128 1972
[106] K M Shum Lateral and torsional vibration control of long spanbridge deck using novel tuned liquid column dampers [PhDthesis] Department of Civil and Structural Engineering TheHong Kong Polytechnic University 2004
[107] Y K Lin and J N Yang ldquoMultimode bridge response to windexcitationsrdquo Journal of EngineeringMechanics vol 109 no 2 pp586ndash603 1983
[108] Y S Lin Self-Excited Bridge Motion in Turbulent Wind 1978[109] D T Lau M S Cheung and S H Cheng ldquo3D flutter analysis
of bridges by spline finite-strip methodrdquo Journal of StructuralEngineering vol 126 no 10 pp 1246ndash1254 2000
[110] L A Balzer ldquoAtmospheric turbulence encountered by high-speed ground transport vehiclesrdquo Journal of Mechanical Engi-neering Science vol 19 pp 227ndash235 1977
[111] R K Cooper ldquoAtmospheric turbulence with respect to movingground vehiclesrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 17 no 2 pp 215ndash238 1984
[112] C J Baker ldquoA simplified analysis of various types of wind-induced road vehicle accidentsrdquo Journal of Wind Engineeringand Industrial Aerodynamics vol 22 no 1 pp 69ndash85 1986
[113] C J Baker ldquoMeasures to control vehicle movement at exposedsites during windy periodsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 25 no 2 pp 151ndash161 1987
[114] C J Baker ldquoHigh sided articulated road vehicles in strong crosswindsrdquo Journal of Wind Engineering and Industrial Aerodynam-ics vol 31 no 1 pp 67ndash85 1988
[115] C J Baker ldquoGround vehicles in high cross winds part I steadyaerodynamic forcesrdquo Journal of Fluids and Structures vol 5 no1 pp 69ndash90 1991
[116] C J Baker ldquoGround vehicles in high cross winds part IIunsteady aerodynamic forcesrdquo Journal of Fluids and Structuresvol 5 no 1 pp 91ndash111 1991
[117] Y L Xu and Q S Ding ldquoInteraction of railway vehicles withtrack in cross-windsrdquo Journal of Fluids and Structures vol 22no 3 pp 295ndash314 2006
[118] S R Chen and J Wu ldquoDynamic performance simulation oflong-span bridge under combined loads of stochastic traffic andwindrdquo Journal of Bridge Engineering vol 15 no 3 pp 219ndash2302010
[119] Y Li S Qiang H Liao and Y L Xu ldquoDynamics of wind-rail vehicle-bridge systemsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 93 no 6 pp 483ndash507 2005
[120] Y L Xu N Zhang and H Xia ldquoVibration of coupled trainand cable-stayed bridge systems in cross windsrdquo EngineeringStructures vol 26 no 10 pp 1389ndash1406 2004
[121] W W Guo Y L Xu H Xia W S Zhang and K M ShumldquoDynamic response of suspension bridge to typhoon and trainsII numerical resultsrdquo Journal of Structural Engineering vol 133no 1 pp 12ndash21 2007
The Scientific World Journal 17
[122] Y L Li H Y Xiang B Wang Y L Xu and S Z QiangldquoDynamic analysis of wind-vehicle-bridge coupling systemduring the meeting of two trainsrdquo Advances in StructuralEngineering vol 16 pp 1663ndash1670 2013
[123] F Dorigatti M Sterling D Rocchi et al ldquoWind tunnelmeasurements of crosswind loads on high sided vehicles overlong span bridgesrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 107 pp 214ndash224 2012
[124] L D Zhu L Li Y L Xu and Q Zhu ldquoWind tunnel investi-gations of aerodynamic coefficients of road vehicles on bridgedeckrdquo Journal of Fluids and Structures vol 30 pp 35ndash50 2012
[125] Y Han J X Hu C S Cai Z Q Chen and C G LildquoExperimental and numerical studies of aerodynamic forces onvehicles and bridgesrdquoWind and Structures vol 17 pp 163ndash1842013
[126] ASCE ldquoCommittee on fatigue and fracture reliability of thecommittee on structural safety and reliability of the structuraldivision fatigue reliability 1ndash4rdquo Journal of Structural Engineer-ing vol 108 pp 3ndash88 1982
[127] BS ldquoBS5400 part 10 code of practice for fatiguerdquo BritishStandard Institute 1980
[128] BS ldquoBS7608 code of practice for fatigue design and assessmentof steel structuresrdquo British Standard Institute 1993
[129] AASHTOGuide Specifications for Fatigue Evaluation of ExistingSteel Bridges 1990
[130] AASHTO Guide Manual for Condition Evaluation and Loadand Resistance Factor Rating (LRFR) of Highway Bridges 2003
[131] S R Chen C S Cai and B Wolshon ldquoFrom normal operationto evacuation Single-vehicle safety under adverse weathertopographic and operational conditionsrdquo Natural HazardsReview vol 10 no 2 pp 68ndash76 2009
[132] S R Chen and C S Cai ldquoEquivalent wheel load approach forslender cable-stayed bridge fatigue assessment under traffic andwind feasibility studyrdquo Journal of Bridge Engineering vol 12 no6 pp 755ndash764 2007
derivatives at different reduced frequencies The expressionof the aerodynamic impulse functions in the time domaincan be obtained by taking the inverse Fourier transform ofthe impulse functions By substituting the related impulseresponse functions into (5b) the self-excited lift force at the119894th section of bridge deck can then be derived as
119891se119890119871119894
=1
21205881198861198802
119894119861119894119862119871120579
1119894120579119894(119905) + 119862
119871120579
2119894(119861119894
119880119894
) 120579119894(119905)
+ 119862119871120579
3119894(119861119894
119880119894
) 120579119894(119905) +
119898
sum119897=1
119862119871120579
119897+3119894
times int119905
minusinfin
120579119894(119905) exp[minus
119889119871120579119897+3119894119880119894
119861119894
(119905 minus 120591)] 119889120591
+1
21205881198861198802
119894119862119871ℎ
1119894ℎ119894(119905) + 119862
119871ℎ
2119894(119861119894
119880119894
) ℎ119894(119905)
+ 119862119871ℎ
3119894(119861119894
119880119894
) ℎ119894(119905) +
119898
sum119897=1
119862119871ℎ
119897+3119894
times int119905
minusinfin
ℎ119894(119905) exp[minus
119889119871ℎ119897+3119894119880119894
119861119894
(119905 minus 120591)] 119889120591
+1
21205881198861198802
119894
119862119871119902
1119894119902119894(119905) + 119862
119871119902
2119894(119861119894
119880119894
) 119902119894(119905)
+ 119862119871119902
3119894(119861119894
119880119894
) 119902119894(119905) +
119898
sum119897=1
119862119871119902
119897+3119894
times int119905
minusinfin
119902120595(119905) exp[
[
minus119889119871119901
119897+3119894119880119894
119861119894
(119905 minus 120591)]
]
119889120591
(8)
In practice the terms 1198621198711205793119894 119862119871ℎ3119894 and 119862119871119902
3119894 which are related to
additional aerodynamic masses are normally neglected andthe value of 119898 is often taken as 2 [101] Similar formulationsfor self-excited drag and moment can be derived withanalogous definitions The self-excited forces at the 119894th nodeof the bridge deck can thus be expressed as
Feise = EeiXei + GeiXei + Feise (9)
where
Xei =
0
ℎei119902ei120579ei0
0
Eei =1
21205881198802
119894
[[[[[[[[
[
0 0 0 0 0 0
0 119862119871ℎ
1119894119862119871119902
11198941198611198941198621198711205791119894
0 0
0 119862119863ℎ1119894
119862119863119902
11198941198611198941198621198631205791119894
0 0
0 119861119894119862119872ℎ1119894
119861119894119862119872119902
111989411986121198941198621198721205791119894
0 0
0 0 0 0 0 0
0 0 0 0 0 0
]]]]]]]]
]
Gei =1
21205881198802
119894119861119894
[[[[[[[[
[
0 0 0 0 0 0
0 119862119871ℎ
2119894119862119871119902
21198941198611198941198621198711205792119894
0 0
0 119862119863ℎ2119894
119862119863119902
21198941198611198941198621198631205792119894
0 0
0 119861119894119862119872ℎ2119894
119861119894119862119872119902
211989411986121198941198621198721205792119894
0 0
0 0 0 0 0 0
0 0 0 0 0 0
]]]]]]]]
]
Feise =
[[[[[[[
[
0
eise
119863eise
eise0
0
]]]]]]]
]
=
[[[[[[[[[[[[[[[[
[
02
sum119897=1
119862119871119902
l+3119894119881119871119902
l+3119894 +2
sum119897=1
119862119871ℎl+3119894119881119871ℎ
l+3119894 +2
sum119897=1
119862119871120579l+3119894119881119871120579
l+3119894
2
sum119897=1
119862119863119902
l+3119894119881119863119902
l+3119894 +2
sum119897=1
119862119863ℎl+3119894119881119863ℎ
l+3119894 +2
sum119897=1
119862119863120579l+3119894119881119863120579
l+3119894
2
sum119897=1
119862119872119902
l+3119894119881119872119902
l+3119894 +2
sum119897=1
119862119872ℎl+3119894119881119872ℎ
l+3119894 +2
sum119897=1
119862119872120579l+3119894119881119872120579
l+3119894
0
0
]]]]]]]]]]]]]]]]
]
(10)
where 119881120595l+3119894 (120595 = Dh Dq 119863120579 Lh Lq 119871120579 Mh Mq) are theconvolution integrations of the 119894th node and can be calculatedusing a recursive algorithm For example
(11)The self-excited forces expressed by (9) relate to the centerof elasticity of the 119894th deck section Hence the force modelmust be distributed to the nodal points of the section Adistribution based on the rigid body motion relationshipsbetween themotions at the nodal point and those at the centerof elasticity of the deck section [109] was applied by Liu etal [22] In this study by applying the virtual work principlethe self-excited forces at the center of elasticity of the givensection were distributed to all nodes (see Figure 8)
24 Dynamic Interactions in a Wind-Vehicle-Bridge SystemWhen trains and road vehicles are running on long-spanbridges under crosswinds complicated dynamic interac-tions occur among the trains road vehicles cable-supported
The Scientific World Journal 11
Ui h
p
120572
Lsec i
Msec i
Dsec i
Fsekiz
Fsekiy
ce
Figure 8 Self-excited forces at the centre of elasticity and at the nodes in the 119894th deck section [22]
bridge and windThe buffeting response of the bridge due tocrosswind is superimposed on the dynamic response of thebridge due to railway and road vehicles The large vibrationof the bridge will in turn considerably affect the safetyand ride comfort of the drivers of the road vehicles Thusthe dynamic responses of a coupled vehicle-bridge systemunder crosswinds are of great concern to both engineers andresearchers
Detailed reviews of the dynamic interactions betweentrains and bridges between road vehicles and bridges andbetween wind and bridges have been given in the previoussections However the interaction between wind and vehiclesmust also be taken into account in a coupled wind-vehicle-bridge analysis Many studies have investigated wind-vehicleinteractions in the past few decades Balzer [110] developeda theory to estimate the aerodynamic forces on a movingvehicle using Taylorrsquos hypothesis of ldquofrozen turbulencerdquo Forengineering applications Cooper [111] proposed the powerspectral density (PSD) square-root coherence functionphase-lag function and aerodynamic admittance functionto model the unsteady side forces on a moving vehicle andlaid down the foundations for investigating the effects ofwind on a moving vehicle in the frequency domain Bakerdeveloped a theoretical model that describes the dynamics ofvehicles in crosswinds in the time domain [112 113] whichwas later extended to include driver behavior [114] Baker[115 116] further investigated both the steady and unsteadyaerodynamic forces acting on a variety of vehicles and carriedout extensive studies of the interaction between aerodynamicforces and moving vehicles These approaches have all beenapplied in coupled vehicle-bridge analysis For example Xuet al [101] simulated the aerodynamic wind forces actingon running road vehicles using the quasi-steady approachand Xu and Ding [117] derived and simulated the steadyand unsteady aerodynamic forces acting on a moving railwayvehicle in crosswinds in the time domain
Based on these separate studies on the various typesof dynamic interactions among wind vehicles (trains orroad vehicles) and long-span bridges several researchersin the last decade have examined the wind-vehicle-bridgecoupled system as a whole For instance studies have beencarried out on coupled road vehicle and cable-stayed bridgesystems [81 82 118] and on coupled train and cable-supportedbridge systems in crosswinds [101 117 119ndash121] In therecent years several new advances have been made bothin numerical simulation technologies and in wind tunnelmeasurements Chen et al [17] proposed a wind-vehicle-bridge framework which enables considering the dynamiceffects induced by simultaneous actions of railway highway
and wind loading and it was applied to analyze dynamicstress of long suspension bridges Li et al [122] extended thewind-vehicle-bridge couple analysis to the case of two trainsmeeting on a long-span suspension bridge Chen and Wu[118] proposed a semideterministic analytical model whichis able to consider dynamic interactions between the bridgewind and stochastic ldquorealrdquo traffic Based on the wind tunneltests Dorigatti et al [123]measured crosswind loads on high-sided vehicles over long-span bridges taking three differentvehicles (van double deck bus and lorry) and two differentbridge deck configurations into consideration Zhu et al[124] investigated aerodynamic coefficients of road vehiclesby adopting different road vehicles types wind directionsand vehicle positions Li et al [122] studied the effects ofsudden changes of wind loads as the train passing througha bridge tower or two trains passing each other by usingthe wind tunnel test rig with moving train models Hanet al [125] developed an experimental setup for measuringthe aerodynamic characteristics of vehicles and the bridgein wind tunnel and then investigated the influences ofparameters adopted in the tests
3 Applications of Simulation Technology toBridge Assessment
After reviewing the key issues of numerical simulationsfor dynamic response of long-span multiload bridges thissection will review the engineering applications of the newlydeveloped technologies to safety assessment of long-spanbridges such as assessment of fatigue and assessment underextreme events
31 Assessment of FatigueDamage Steel structures are widelyused in long-span bridges Research by the ASCE [126] indi-cates that 80ndash90 of failures in steel structures are related tofatigue and fracture Several disasters resulting from fatigue-induced bridge failure have occurred in history For instance46 people died in the collapse of the Silver Bridge (USA1967) and 32 people lost their lives in the collapse of theSungsoo Grand Bridge (South Korea 1994) These disastersteach us that fatigue is an important aspect of the safety ofsteel bridges and action should be taken to prevent similarfatigue-induced bridge failures In the past few decadesfatigue assessment of steel bridges has attracted increasingattention from both governments and bridge engineers andrelevant provisions have been stipulated in several codes andstandards [127ndash130]
12 The Scientific World Journal
It has great advantages to evaluate fatigue damage of long-span bridges based on numerical simulation especially fora multiload bridge which suffers multiple types of dynamicloading such as railway highway andwind loadingDifferentfrom sudden structural damage fatigue damage accumulateswith load-induced dynamic stress (or stress fluctuation) overthe service life of a bridge lasting for more than 100 yearsThe increase in traffic volume and gross vehicle weight thataccompany economic development is very likely to happenin the long period Numerical simulation technology can bean ideal tool to study influences of traffic growth patternsto fatigue damage of bridge In addition slender long-span bridges built in wind-prone regions also suffer fromconsiderable wind induced vibration which appears withina wide range of wind speeds and lasts for almost the wholedesign life of the bridge Given the simultaneous presenceof multiple vehicles and wind it is necessary to considercombined effects of traffic loading (railway andor highwayloading) and wind loading in the fatigue assessment Sincemultiple loading is concerned in a long time period thereare a large number of loading combinations for multipleloading in different intensities It is almost unavailable forfield measurement to obtain such complete information butnumerical simulation could be a good choice to determinedynamic responses of a long-span bridge under multipleloading
A number of structural health monitoring systems(SHMSs) have been installed on numerous recently builtlong-span bridges and a variety of sensors are used for mon-itoring bridge loadings (eg traffic wind and earthquakes)and conditions (including global and local responses) toensure bridge safety and user comfort under in-service con-ditions Well-known examples include Tsing-Ma Bridge inHong Kong Akashi Kaikyo Bridge in Japan Binzhou YellowRiver Bridge in China and Jindo Bridge in Korea Integrationof numerical simulation technologies and measurement dataof structural health monitoring systems (SHMSs) installedon long-span bridges will make the fatigue assessment morereliable for several reasons (1) it is a perfect validation byusing field measurement data of the different types of loadingas input of numerical simulation and the measured dynamicresponses for comparison with the computed ones (2) a largenumber of measured loading data could be used to establishloading databases or probabilistic models of different loads
In the recent years several researchers [7ndash10] appliedthe newly developed numerical simulation technologies tofatigue assessment of long-span bridges Chen et al [7]proposed a framework for fatigue analysis of a long-spansuspension bridge under railway highway and wind loadingby integrating computer simulation with SHMSs and itwas applied to evaluate fatigue damage of the Tsing MaSuspension Bridge over its design life as a case study Basedon this work Chen et al [8] proposed a framework forfatigue reliability analysis of long suspension bridges undermultiple loading inwhich the probabilisticmodels of railwayhighway and wind loading were established based on themeasurement data acquired from the SHMS of the TsingMa Bridge Wu et al [9] proposed a reliability-based fatigueapproach for slender long-span bridge and the combined
dynamic loading effects from wind and traffic as well asthe associated uncertainties were considered Based on theassumption that dynamic magnification related to vehicledynamics can be neglected in long suspension bridges Chenet al [8] established a framework for fatigue reliability anal-ysis To account for different types of long-span bridges withthe span length ranging from a few hundred to thousands ofmeters Zhang et al [10] proposed a more general frameworkwhich includes multiple random variables for the dynamicloads in a bridgersquos life cycle for the vehicle-bridge-winddynamic system such as road profile vehicle speed andwindvelocity and direction among other effects
32 Assessment under Extreme Events The aforementionedfatigue assessment mainly focuses on damage accumulationinduced by stress fluctuations under normal operationalcondition in a long-term period For long-span bridges inaddition to the normal operational conditions in which windspeeds are small ormoderate and traffic scenarios are normalthere are some extreme event conditions Extreme eventsmayinclude complex traffic congestion on the bridge coupledwith moderate or even strong wind [11] For example severetraffic congestions may be formed on the bridge as a resultof an evacuation or a partial blockage of driving lanes due totraffic accidents construction ormaintenance For hurricaneevacuations there are usually a lot of road vehicles passingthrough the bridge before the landfall of the hurricane whilethe wind speed may become pretty high already [131]
Although the excessive dynamic responses of the bridgesunder extreme events are rare it is also critical since itmay cause critical damage initiation or accumulation onsome local bridgemembers Furthermore the extreme events(eg heavy traffic) may even trigger the collapse of thewhole bridge by breaking the ldquoweakest linkrdquo especially whensome hidden damage or design flaw has not been detectedOne recent example is the Minnesota Bridge failure whichoccurred during rush hours with heavy traffic although trafficloads may not be the direct cause of failure For slender long-span bridges strong wind may also cause threats by workinginteractively with heavy traffic loads Therefore even thoughthe extreme cases associated with congested traffic andorwindy weather may be relatively rare and the durationscould be short it is still important for bridge engineers toappropriately look into these unusual extreme events duringstructural design and life-time management of these criticalinfrastructures [11]
The dynamic performance of long-span bridges undercombined actions of strong winds and running road vehicleshas been studied by many researchers in recent years [17 7981 82 132] Most of them studied bridge dynamic perfor-mance under road traffic inwhich only one or several vehiclesdistributed in an assumed (usually uniform) pattern on long-span bridges were considered Extreme events such as trafficcongestion coupled with strong wind were out of concernin those studies Recently Wu and Chen [11] conducteda research on the assessment of long-span bridges underextreme events which includes complex traffic congestioncoupled with moderate or even strong wind This study
The Scientific World Journal 13
applied the cellular automaton (CA) traffic model to thesimulation of the actual traffic flow through the bridgedefined representative scenarios for the extreme events andnumerically studied the bridge performance under thesepossible extreme events
4 Conclusions and Recommendations
Dynamic responses of long-span bridges are often requiredfor assessing the safety of these bridges and can be determinedby numerical simulation technologies This paper provides adetailed review of key issues involved in dynamic responseanalysis of long-span multiload bridges based on numericalsimulation including dynamic interactions between runningtrains and bridge between running road vehicles and bridgeand between wind and bridge and in the wind-vehicle-bridge coupled systemThen the review work was conductedfor engineering applications of newly developed numericalsimulation technologies to safety assessment of long-spanbridges such as assessment of fatigue damage and assessmentunder extreme event condition Although technologies fornumerical simulation of dynamic responses of long-spanmultiload bridge have achieved great advances in past fewdecades and successfully applied to several important bridgesit is still far from reach its maturity and enable to takeplace of traditional fieldmeasurementThe existing problemsand promising research efforts at least include the followingaspects
(1) After multiple types of dynamic interactions beingconsidered in the complex system computationalefficiency is a bottleneck problem for numericalsimulation of dynamic response of a long-span bridgeTypically when multiple loads are involved a largenumber of loading combinations for multiple load-ings must be considered in the assessment
(2) It is rather complex for the time-depending windloads acting on a long-span bridge and running vehi-cles especially for the case of rapid change of windloads such as a train passing through a bridge toweror two trains passing each other The aerodynamiccharacteristics of vehicles and the bridge under differ-ent loading scenarios can be determined through thewind tunnel testing and used in numerical simulationof dynamic responses of the bridge and vehicles
(3) It is a new trend to integrate numerical simulationtechnologies and measurement data of structuralhealth monitoring systems (SHMSs) installed onlong-span bridges whichmakes the safety assessmentof bridge structures more reliable Measured struc-tural responses could be used to validate numericalsimulation approach and measured loading infor-mation could be used for generating statistical orprobabilistic models of multiple loads
(4) It is important to study dynamic responses of bridgestructures under extreme events such as congestedtraffic coupled with windy weather which happens ina long-span bridge For the assessment under extreme
events using numerical simulation technologies sim-ulation of traffic flow and definition of representativescenarios of the extreme events are key issues
(5) It is necessary to consider the effects of typhoonwinds on the safety assessment of long-span bridgesin a reasonable way Few researches do this mostlybecause a probabilistic distribution of wind speedand direction specifically for typhoons is requiredfor assessment but there are insufficient measuredrecords to establish a reliable probabilistic typhoonwind model
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors wish to acknowledge the financial supportsfrom the National Natural Science Foundation of China(NSFC-51108395 and NSFC-51178366) the FundamentalResearch Funds for theCentral Universities (2012121032) andopen funding from Jiangsu Key Laboratory of EngineeringMechanics Special thanks go to the supervisor of the firstauthor Professor Y L XuHongKong PolytechnicUniversityfor his expert guidance and continuous support at all levelsthroughout his PhD study Sincere thanks should go tothe Highways Department of Hong Kong for providing theauthors with the field measurement data Any opinions andconcluding remarks presented in this paper are entirely thoseof the authors
References
[1] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[2] H Sohn C R Farrar N F Hunter and K Worden ldquoStructuralhealth monitoring using statistical pattern recognition tech-niquesrdquo Journal of Dynamic Systems Measurement and Controlvol 123 no 4 pp 706ndash711 2001
[3] W Fan and P Qiao ldquoVibration-based damage identificationmethods a review and comparative studyrdquo Structural HealthMonitoring vol 10 no 1 pp 83ndash111 2011
[4] X Q Zhu and S S Law ldquoDamage detection in simply supportedconcrete bridge structure under moving vehicular loadsrdquo Jour-nal of Vibration and Acoustics Transactions of the ASME vol129 no 1 pp 58ndash65 2007
[5] J Li and S S Law ldquoDamage identification of a target sub-structure with moving load excitationrdquoMechanical Systems andSignal Processing vol 30 pp 78ndash90 2012
[6] J Li S S Law and H Hao ldquoImproved damage identification inbridge structures subject tomoving loads numerical and exper-imental studiesrdquo International Journal of Mechanical Sciencesvol 74 pp 99ndash111 2013
14 The Scientific World Journal
[7] Z W Chen Y L Xu Y Xia Q Li and K Y Wong ldquoFatigueanalysis of long-span suspension bridges under multiple load-ing case studyrdquo Engineering Structures vol 33 no 12 pp 3246ndash3256 2011
[8] Z W Chen Y L Xu and X M Wang ldquoSHMS-based fatiguereliability analysis of multiloading suspension bridgesrdquo Journalof Structural Engineering-Asce vol 138 pp 299ndash307 2012
[9] J Wu S R Chen and J W van de Lindt ldquoFatigue assessmentof slender long-span bridges reliability approachrdquo Journal ofBridge Engineering vol 17 no 1 pp 47ndash57 2012
[10] W Zhang C S Cai and F Pan ldquoFatigue reliability assessmentfor long-span bridges under combined dynamic loads fromwinds and vehiclesrdquo Journal of Bridge Engineering vol 18 pp735ndash747 2013
[11] JWu and S R Chen ldquoProbabilistic dynamic behavior of a long-span bridge under extreme eventsrdquo Engineering Structures vol33 no 5 pp 1657ndash1665 2011
[12] J M Ko and Y Q Ni ldquoTechnology developments in structuralhealth monitoring of large-scale bridgesrdquo Engineering Struc-tures vol 27 no 12 pp 1715ndash1725 2005
[13] TH YiHN Li andHM Sun ldquoMulti-stage structural damagediagnosis method based on ldquoenergy-damagerdquo theoryrdquo SmartStructures and Systems vol 12 pp 345ndash361 2013
[14] T H Yi H N Li and M Gu ldquoFull-scale measurements ofdynamic response of suspension bridge subjected to environ-mental loads using GPS technologyrdquo Science China Technologi-cal Sciences vol 53 no 2 pp 469ndash479 2010
[15] T H Yi H N Li and M Gu ldquoExperimental assessmentof high-rate GPS receivers for deformation monitoring ofbridgerdquoMeasurement Journal of the InternationalMeasurementConfederation vol 46 pp 420ndash432 2013
[16] X W Ye Y Q Ni K Y Wong and J M Ko ldquoStatistical analysisof stress spectra for fatigue life assessment of steel bridges withstructural health monitoring datardquo Engineering Structures vol45 pp 166ndash176 2012
[17] Z W Chen Y L Xu Q Li and D J Wu ldquoDynamic stressanalysis of long suspension bridges under wind railway andhighway loadingsrdquo Journal of Bridge Engineering vol 16 no 3pp 383ndash391 2011
[18] S G Meisenholder and P Weidlinger ldquoDynamic interactionaspects of cable-stayed guideways for high speed ground trans-portationrdquoAmerican Society ofMechanical Engineers no 74 pp180ndash192 1974
[19] Q H Mao Research on the Highway Bridge Vibration Due toMoving Vehicles Tongji University Shang Hai China 1989
[20] Y L Xu J M Ko and Z Yu ldquoModal analysis of tower-cable system of Tsing Ma long suspension bridgerdquo EngineeringStructures vol 19 pp 857ndash867 1997
[21] W Guo H Xia and Y-L Xu ldquoDynamic response of a long spansuspension bridge and running safety of a train under windactionrdquo Frontiers of Architecture and Civil Engineering in Chinavol 1 no 1 pp 71ndash79 2007
[22] T T Liu Y L Xu W S Zhang K Y Wong H J Zhou and KW Y Chan ldquoBuffeting-induced stresses in a long suspensionbridge structural health monitoring oriented stress analysisrdquoWind and Structures An International Journal vol 12 no 6 pp479ndash504 2009
[23] K Y Wong ldquoStructural identification of Tsing Ma BridgerdquoTransactions Hong Kong Institution of Engineers vol 10 no 1pp 38ndash47 2003
[24] Y L XuQ Li D JWu and ZWChen ldquoStress and accelerationanalysis of coupled vehicle and long-span bridge systems usingthe mode superposition methodrdquo Engineering Structures vol32 no 5 pp 1356ndash1368 2010
[25] Y F Duan Y L Xu Q G Fei et al ldquoAdvanced finite elementmodel of Tsing Ma Bridge for structural health monitoringrdquoInternational Journal of Structural Stability and Dynamics vol11 no 2 pp 313ndash344 2011
[26] Z X Li T Q Zhou T H T Chan and Y Yu ldquoMulti-scalenumerical analysis on dynamic response and local damage inlong-span bridgesrdquo Engineering Structures vol 29 no 7 pp1507ndash1524 2007
[27] W Zhang C S Cai and F Pan ldquoFinite element modeling ofbridges with equivalent orthotropic material method for multi-scale dynamic loadsrdquo Engineering Structures vol 54 pp 82ndash932013
[28] S P Timoshenko ldquoOn the forced vibrations of bridgesrdquo Philo-sophical Magazine vol 6 no 257 pp 1018ndash1019 1922
[29] R S Ayre G Ford and L S Jacobsen ldquoTransverse vibration ofa two-span beam under the action of a moving constant forcerdquoJournal of Applied Mechanics vol 17 pp 1ndash12 1950
[30] R S Ayre andLS Jacobsen ldquoTransverse vibration of a two-spanbeam under the action of a moving alternating forcerdquo Journal ofApplied Mechanics vol 17 pp 283ndash290 1950
[31] L FrybaVibration of Solids and Structures underMoving LoadsASCE Press 1972
[32] J-S Wu and C-W Dai ldquoDynamic response of multispannonuniform beam due to moving loadsrdquo Journal of StructuralEngineering vol 113 no 3 pp 458ndash474 1987
[33] W Weaver S P Timoshenko and D H Young VibrationProblems in Engineering John Wiley amp Sons 1990
[34] N H Galdos D R Schelling and M A Sahin ldquoMethodologyfor impact factor of horizontally curved box bridgesrdquo Journal ofStructural Engineering vol 119 no 6 pp 1917ndash1934 1993
[35] J A Gbadeyan and S T Oni ldquoDynamic behaviour of beamsand rectangular plates under moving loadsrdquo Journal of Soundand Vibration vol 182 no 5 pp 677ndash695 1995
[36] D Y Zheng Y K Cheung F T K Au and Y S ChengldquoVibration of multi-span non-uniform beams under movingloads by using modified beam vibration functionsrdquo Journal ofSound and Vibration vol 212 no 3 pp 455ndash467 1998
[37] G V Rao ldquoLinear dynamics of an elastic beam under movingloadsrdquo Journal of Vibration and Acoustics Transactions of theASME vol 122 no 3 pp 281ndash289 2000
[38] Y B Yang J D Yau and Y S Wu Vehicle-Bridge InteractionDynamic with Applications to High-Speed Railways WorldScientific 2004
[39] E C Ting J Genin and J H Ginsberg ldquoA general algorithm formovingmass problemsrdquo Journal of Sound and Vibration vol 33no 1 pp 49ndash58 1974
[40] S Sadiku and H H E Leipholz ldquoOn the dynamics of elasticsystems with moving concentrated massesrdquo Ingenieur-Archivvol 57 no 3 pp 223ndash242 1987
[41] J E Akin and M Mofid ldquoNumerical solution for response ofbeamswithmovingmassrdquo Journal of Structural Engineering vol115 no 1 pp 120ndash131 1989
[42] M AMahmoud andM A Abou Zaid ldquoDynamic response of abeam with a crack subject to a moving massrdquo Journal of Soundand Vibration vol 256 no 4 pp 591ndash603 2002
[43] V K Garg Dynamics of Railway Vehicle Systems AcademicPress 1994
The Scientific World Journal 15
[44] T-LWang andDHuang ldquoCable-stayed bridge vibration due toroad surface roughnessrdquo Journal of Structural Engineering vol118 no 5 pp 1354ndash1374 1992
[45] Y-B Yang and B-H Lin ldquoVehicle-bridge interaction analysisby dynamic condensation methodrdquo Journal of Structural Engi-neering vol 121 no 11 pp 1636ndash1643 1995
[46] Y B Yang J D Yau and L C Hsu ldquoVibration of simple beamsdue to trainsmoving at high speedsrdquoEngineering Structures vol19 no 11 pp 936ndash943 1997
[47] B Tabarrok and E Esmailzadeh ldquoInduced vibration of bridgestransversed by moving vehiclesrdquo Transactions of the CanadianSociety for Mechanical Engineering B vol 24 no 1 pp 191ndash1982000
[48] C Liu T-L Wang and D Huang ldquoImpact study for multi-girder bridge based on correlated road roughnessrdquo StructuralEngineering and Mechanics vol 11 no 3 pp 259ndash272 2001
[49] K H Chu V K Garg and T L Wang ldquoImpact in railwayprestressed concrete bridgesrdquo Journal of Structural Engineeringvol 112 no 5 pp 1036ndash1051 1986
[50] T-LWang V K Garg and K-H Chu ldquoRailway bridgevehicleinteraction studies with new vehicle modelrdquo Journal of Struc-tural Engineering vol 117 no 7 pp 2099ndash2116 1991
[51] H Xia Y L Xu and T H T Chan ldquoDynamic interaction oflong suspension bridges with running trainsrdquo Journal of Soundand Vibration vol 237 no 2 pp 263ndash280 2000
[52] Q-L Zhang A Vrouwenvelder and J Wardenier ldquoNumericalsimulation of train-bridge interactive dynamicsrdquo Computersand Structures vol 79 no 10 pp 1059ndash1075 2001
[53] H Xia N Zhang and G de Roeck ldquoDynamic analysis of highspeed railway bridge under articulated trainsrdquo Computers andStructures vol 81 no 26-27 pp 2467ndash2478 2003
[54] G Diana F Cheli A Collina R Corradi and S MelzildquoThe development of a numerical model for railway vehiclescomfort assessment through comparison with experimentalmeasurementsrdquoVehicle SystemDynamics vol 38 no 3 pp 165ndash183 2002
[55] Q Li Y L Xu D J Wu and Z W Chen ldquoComputer-aided nonlinear vehicle-bridge interaction analysisrdquo Journal ofVibration and Control vol 16 pp 1791ndash1816 2010
[56] A Wiriyachai K H Chu and V K Garg ldquoBridge impact dueto wheel and track irregularitiesrdquo Journal of the EngineeringMechanics Division vol 108 no 4 pp 648ndash666 1982
[57] L Fryba Dynamics of Railway Bridges Inst of Civil Engineers1996
[58] D Huang and T-L Wang ldquoImpact analysis of cable-stayedbridgesrdquo Computers and Structures vol 43 no 5 pp 897ndash9081992
[59] W M Zhai Vehicle-Track Coupling Dynamics Chinese RailwayPress Beijing China 2007
[60] M Olsson ldquoFinite element modal co-ordinate analysis ofstructures subjected to moving loadsrdquo Journal of Sound andVibration vol 99 no 1 pp 1ndash12 1985
[61] Y-B Yang C-H Chang and J-D Yau ldquoAn element foranalysing vehicle-bridge systems considering vehiclersquos pitchingeffectrdquo International Journal for NumericalMethods in Engineer-ing vol 46 no 7 pp 1031ndash1047 1999
[62] Y B Yang and Y S Wu ldquoA versatile element for analyzingvehicle-bridge interaction responserdquo Engineering Structuresvol 23 no 5 pp 452ndash469 2001
[63] F T K Au J J Wang and Y K Cheung ldquoImpact study ofcable-stayed bridge under railway traffic using various modelsrdquo
Journal of Sound and Vibration vol 240 no 3 pp 447ndash4652001
[64] Y Q Sun andMDhanasekar ldquoA dynamicmodel for the verticalinteraction of the rail track and wagon systemrdquo InternationalJournal of Solids and Structures vol 39 no 5 pp 1337ndash13592002
[65] K Henchi M Fafard M Talbot and G Dhatt ldquoAn efficientalgorithm for dynamic analysis of bridges under moving vehi-cles using a coupledmodal and physical components approachrdquoJournal of Sound and Vibration vol 212 no 4 pp 663ndash6831998
[66] Y L Xu and L YWang ldquoAnalytical study of wind-rain-inducedcable vibration SDOFmodelrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 91 no 1-2 pp 27ndash40 2003
[67] B Biondi G Muscolino and A Sofi ldquoA substructure approachfor the dynamic analysis of train-track-bridge systemrdquoComput-ers and Structures vol 83 no 28ndash30 pp 2271ndash2281 2005
[68] J L Humar and A H Kashif ldquoDynamic response analysis ofslab-type bridgesrdquo Journal of Structural Engineering vol 121 no1 pp 48ndash62 1995
[69] P Lou and Q-Y Zeng ldquoFormulation of equations of verticalmotion for vehicle-track-bridge systemrdquo Journal of the ChinaRailway Society vol 26 no 5 p 71 2004
[70] O Coussy M Said and J-P van Hoove ldquoThe influence ofrandom surface irregularities on the dynamic response ofbridges under suspended moving loadsrdquo Journal of Sound andVibration vol 130 no 2 pp 313ndash320 1989
[71] E S Hwang and A S Nowak ldquoSimulation of dynamic loadfor bridgesrdquo Journal of Structural Engineering vol 117 pp 1413ndash1434 1991
[72] F Yang and G A Fonder ldquoAn iterative solution methodfor dynamic response of bridge-vehicles systemsrdquo EarthquakeEngineering and Structural Dynamics vol 25 pp 195ndash215 1996
[73] W Zhai and Z Cai ldquoDynamic interaction between a lumpedmass vehicle and a discretely supported continuous rail trackrdquoComputers and Structures vol 63 no 5 pp 987ndash997 1997
[74] W M Zhai and C B Cai ldquoTraintrackbridge dynamic inter-actions simulation and applicationsrdquo Vehicle System Dynamicsvol 37 pp 653ndash665 2003
[75] D Bruno F Greco and P Lonetti ldquoDynamic impact analysis oflong span cable-stayed bridges under moving loadsrdquo Engineer-ing Structures vol 30 no 4 pp 1160ndash1177 2008
[76] X D Song D J Wu and Q Li ldquoDynamic impact analysisof double-tower cable-stayed maglev bridges using a simplemodelrdquo Journal of Bridge Engineering vol 19 pp 34ndash43 2014
[77] Y-S Wu and Y-B Yang ldquoSteady-state response and ridingcomfort of trains moving over a series of simply supportedbridgesrdquoEngineering Structures vol 25 no 2 pp 251ndash265 2003
[78] P Antolin N Zhang J M Goicolea H Xia M A Astiz andJ Oliva ldquoConsideration of nonlinear wheel-rail contact forcesfor dynamic vehicle-bridge interaction in high-speed railwaysrdquoJournal of Sound and Vibration vol 332 no 5 pp 1231ndash12512013
[79] W H Guo and Y L Xu ldquoFully computerized approach to studycable-stayed bridge-vehicle interactionrdquo Journal of Sound andVibration vol 248 no 4 pp 745ndash761 2001
[80] Y L Xu and W H Guo ldquoDynamic behaviour of high-sidedroad vehicles subject to a sudden crosswind gustrdquo Wind andStructures vol 6 no 5 pp 325ndash346 2003
[81] Y L Xu and W H Guo ldquoDynamic analysis of coupled roadvehicle and cable-stayed bridge systems under turbulent windrdquoEngineering Structures vol 25 no 4 pp 473ndash486 2003
16 The Scientific World Journal
[82] C S Cai and S R Chen ldquoFramework of vehicle-bridge-winddynamic analysisrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 92 no 7-8 pp 579ndash607 2004
[83] S R Chen and J Wu ldquoModeling stochastic live load for long-span bridge based on microscopic traffic flow simulationrdquoComputers and Structures vol 89 no 9-10 pp 813ndash824 2011
[84] A V Paultre B Yang L A Bergman and C A Tan ldquoBridgedynamics and dynamic amplification factorsmdasha review ofanalytical and experimental findingsrdquoCanadian Journal of CivilEngineering vol 19 no 2 pp 260ndash278 1992
[85] H Honda Y Kajikawa and T Kobori ldquoSpectra of road surfaceroughness of bridgesrdquo Journal of the Structural Division vol 108pp 1956ndash1966 1982
[86] M J Inbanathan and M Wieland ldquoBridge vibrations dueto vehicle moving over rough surfacerdquo Journal of StructuralEngineering vol 113 no 9 pp 1994ndash2008 1987
[87] T-LWang andDHuang ldquoCable-stayed bridge vibration due toroad surface roughnessrdquo Journal of Structural Engineering vol118 no 5 pp 1354ndash1374 1992
[88] P K Chatterjee T K Datta and C S Surana ldquoVibrationsuspension bridges under vehicular movementrdquo Journal ofStructural Engineering vol 120 no 3 pp 681ndash703 1994
[89] D Chang and H Lee ldquoImpact factors for simple-span highwaygirder bridgesrdquo Journal of Structural Engineering vol 120 no 3pp 704ndash715 1994
[90] T-C Pan and J Li ldquoDynamic vehicle element method for tran-sient response of coupled vehicle-structure systemsrdquo Journal ofStructural Engineering vol 128 no 2 pp 214ndash223 2002
[91] C J Dodds and J D Robson ldquoThe description of road surfaceroughnessrdquo Journal of Sound Vibration vol 31 no 2 pp 175ndash183 1973
[92] D Huang T-L Wang and M Shahawy ldquoImpact studies ofmultigirder concrete bridgesrdquo Journal of Structural Engineeringvol 119 no 8 pp 2387ndash2402 1993
[93] A G Davenport ldquoBuffeting of a suspension bridge by stormwindrdquo Journal of Structural Division vol 88 pp 233ndash268 1962
[94] R H Scanlan ldquoThe action of flexible bridges under wind Iflutter theoryrdquo Journal of Sound and Vibration vol 60 no 2pp 187ndash199 1978
[95] Q Ding and P K K Lee ldquoComputer simulation of buffetingactions of suspension bridges under turbulentwindrdquoComputersand Structures vol 76 no 6 pp 787ndash797 2000
[96] V Boonyapinyo T Miyata and H Yamada ldquoAdvanced aerody-namic analysis of suspension bridges by state-space approachrdquoJournal of Structural Engineering vol 125 no 12 pp 1357ndash13661999
[97] Y-H Chen and C-Y Li ldquoDynamic response of elevated high-speed railwayrdquo Journal of Bridge Engineering vol 5 no 2 pp124ndash130 2000
[98] X Chen M Matsumoto and A Kareem ldquoTime domainflutter and buffeting response analysis of bridgesrdquo Journal ofEngineering Mechanics vol 126 no 1 pp 7ndash16 2000
[99] X Chen and A Kareem ldquoEquivalent static wind loads forbuffeting response of bridgesrdquo Journal of Structural Engineeringvol 127 no 12 pp 1467ndash1475 2001
[100] S R Chen and C S Cai ldquoEvolution of long-span bridgeresponse to wind-numerical simulation and discussionrdquo Com-puters and Structures vol 81 no 21 pp 2055ndash2066 2003
[101] Y L XuHXia andQ S Yan ldquoDynamic response of suspensionbridge to high wind and running trainrdquo Journal of BridgeEngineering vol 8 no 1 pp 46ndash55 2003
[102] A Guo Y L Xu and H Li ldquoDynamic performance of cable-stayed bridge tower with multi-stage pendulum mass damperunderwind excitations-II experimentrdquoEarthquake Engineeringand Engineering Vibration vol 6 no 4 pp 417ndash424 2007
[103] E Simiu and R H ScanlanWind Effects on Structures 1996[104] Y Cao H Xiang and Y Zhou ldquoSimulation of stochastic wind
velocity field on long-span bridgesrdquo Journal of EngineeringMechanics vol 126 no 1 pp 1ndash6 2000
[105] M Shinozuka and C-M Jan ldquoDigital simulation of randomprocesses and its applicationsrdquo Journal of Sound and Vibrationvol 25 no 1 pp 111ndash128 1972
[106] K M Shum Lateral and torsional vibration control of long spanbridge deck using novel tuned liquid column dampers [PhDthesis] Department of Civil and Structural Engineering TheHong Kong Polytechnic University 2004
[107] Y K Lin and J N Yang ldquoMultimode bridge response to windexcitationsrdquo Journal of EngineeringMechanics vol 109 no 2 pp586ndash603 1983
[108] Y S Lin Self-Excited Bridge Motion in Turbulent Wind 1978[109] D T Lau M S Cheung and S H Cheng ldquo3D flutter analysis
of bridges by spline finite-strip methodrdquo Journal of StructuralEngineering vol 126 no 10 pp 1246ndash1254 2000
[110] L A Balzer ldquoAtmospheric turbulence encountered by high-speed ground transport vehiclesrdquo Journal of Mechanical Engi-neering Science vol 19 pp 227ndash235 1977
[111] R K Cooper ldquoAtmospheric turbulence with respect to movingground vehiclesrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 17 no 2 pp 215ndash238 1984
[112] C J Baker ldquoA simplified analysis of various types of wind-induced road vehicle accidentsrdquo Journal of Wind Engineeringand Industrial Aerodynamics vol 22 no 1 pp 69ndash85 1986
[113] C J Baker ldquoMeasures to control vehicle movement at exposedsites during windy periodsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 25 no 2 pp 151ndash161 1987
[114] C J Baker ldquoHigh sided articulated road vehicles in strong crosswindsrdquo Journal of Wind Engineering and Industrial Aerodynam-ics vol 31 no 1 pp 67ndash85 1988
[115] C J Baker ldquoGround vehicles in high cross winds part I steadyaerodynamic forcesrdquo Journal of Fluids and Structures vol 5 no1 pp 69ndash90 1991
[116] C J Baker ldquoGround vehicles in high cross winds part IIunsteady aerodynamic forcesrdquo Journal of Fluids and Structuresvol 5 no 1 pp 91ndash111 1991
[117] Y L Xu and Q S Ding ldquoInteraction of railway vehicles withtrack in cross-windsrdquo Journal of Fluids and Structures vol 22no 3 pp 295ndash314 2006
[118] S R Chen and J Wu ldquoDynamic performance simulation oflong-span bridge under combined loads of stochastic traffic andwindrdquo Journal of Bridge Engineering vol 15 no 3 pp 219ndash2302010
[119] Y Li S Qiang H Liao and Y L Xu ldquoDynamics of wind-rail vehicle-bridge systemsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 93 no 6 pp 483ndash507 2005
[120] Y L Xu N Zhang and H Xia ldquoVibration of coupled trainand cable-stayed bridge systems in cross windsrdquo EngineeringStructures vol 26 no 10 pp 1389ndash1406 2004
[121] W W Guo Y L Xu H Xia W S Zhang and K M ShumldquoDynamic response of suspension bridge to typhoon and trainsII numerical resultsrdquo Journal of Structural Engineering vol 133no 1 pp 12ndash21 2007
The Scientific World Journal 17
[122] Y L Li H Y Xiang B Wang Y L Xu and S Z QiangldquoDynamic analysis of wind-vehicle-bridge coupling systemduring the meeting of two trainsrdquo Advances in StructuralEngineering vol 16 pp 1663ndash1670 2013
[123] F Dorigatti M Sterling D Rocchi et al ldquoWind tunnelmeasurements of crosswind loads on high sided vehicles overlong span bridgesrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 107 pp 214ndash224 2012
[124] L D Zhu L Li Y L Xu and Q Zhu ldquoWind tunnel investi-gations of aerodynamic coefficients of road vehicles on bridgedeckrdquo Journal of Fluids and Structures vol 30 pp 35ndash50 2012
[125] Y Han J X Hu C S Cai Z Q Chen and C G LildquoExperimental and numerical studies of aerodynamic forces onvehicles and bridgesrdquoWind and Structures vol 17 pp 163ndash1842013
[126] ASCE ldquoCommittee on fatigue and fracture reliability of thecommittee on structural safety and reliability of the structuraldivision fatigue reliability 1ndash4rdquo Journal of Structural Engineer-ing vol 108 pp 3ndash88 1982
[127] BS ldquoBS5400 part 10 code of practice for fatiguerdquo BritishStandard Institute 1980
[128] BS ldquoBS7608 code of practice for fatigue design and assessmentof steel structuresrdquo British Standard Institute 1993
[129] AASHTOGuide Specifications for Fatigue Evaluation of ExistingSteel Bridges 1990
[130] AASHTO Guide Manual for Condition Evaluation and Loadand Resistance Factor Rating (LRFR) of Highway Bridges 2003
[131] S R Chen C S Cai and B Wolshon ldquoFrom normal operationto evacuation Single-vehicle safety under adverse weathertopographic and operational conditionsrdquo Natural HazardsReview vol 10 no 2 pp 68ndash76 2009
[132] S R Chen and C S Cai ldquoEquivalent wheel load approach forslender cable-stayed bridge fatigue assessment under traffic andwind feasibility studyrdquo Journal of Bridge Engineering vol 12 no6 pp 755ndash764 2007
Figure 8 Self-excited forces at the centre of elasticity and at the nodes in the 119894th deck section [22]
bridge and windThe buffeting response of the bridge due tocrosswind is superimposed on the dynamic response of thebridge due to railway and road vehicles The large vibrationof the bridge will in turn considerably affect the safetyand ride comfort of the drivers of the road vehicles Thusthe dynamic responses of a coupled vehicle-bridge systemunder crosswinds are of great concern to both engineers andresearchers
Detailed reviews of the dynamic interactions betweentrains and bridges between road vehicles and bridges andbetween wind and bridges have been given in the previoussections However the interaction between wind and vehiclesmust also be taken into account in a coupled wind-vehicle-bridge analysis Many studies have investigated wind-vehicleinteractions in the past few decades Balzer [110] developeda theory to estimate the aerodynamic forces on a movingvehicle using Taylorrsquos hypothesis of ldquofrozen turbulencerdquo Forengineering applications Cooper [111] proposed the powerspectral density (PSD) square-root coherence functionphase-lag function and aerodynamic admittance functionto model the unsteady side forces on a moving vehicle andlaid down the foundations for investigating the effects ofwind on a moving vehicle in the frequency domain Bakerdeveloped a theoretical model that describes the dynamics ofvehicles in crosswinds in the time domain [112 113] whichwas later extended to include driver behavior [114] Baker[115 116] further investigated both the steady and unsteadyaerodynamic forces acting on a variety of vehicles and carriedout extensive studies of the interaction between aerodynamicforces and moving vehicles These approaches have all beenapplied in coupled vehicle-bridge analysis For example Xuet al [101] simulated the aerodynamic wind forces actingon running road vehicles using the quasi-steady approachand Xu and Ding [117] derived and simulated the steadyand unsteady aerodynamic forces acting on a moving railwayvehicle in crosswinds in the time domain
Based on these separate studies on the various typesof dynamic interactions among wind vehicles (trains orroad vehicles) and long-span bridges several researchersin the last decade have examined the wind-vehicle-bridgecoupled system as a whole For instance studies have beencarried out on coupled road vehicle and cable-stayed bridgesystems [81 82 118] and on coupled train and cable-supportedbridge systems in crosswinds [101 117 119ndash121] In therecent years several new advances have been made bothin numerical simulation technologies and in wind tunnelmeasurements Chen et al [17] proposed a wind-vehicle-bridge framework which enables considering the dynamiceffects induced by simultaneous actions of railway highway
and wind loading and it was applied to analyze dynamicstress of long suspension bridges Li et al [122] extended thewind-vehicle-bridge couple analysis to the case of two trainsmeeting on a long-span suspension bridge Chen and Wu[118] proposed a semideterministic analytical model whichis able to consider dynamic interactions between the bridgewind and stochastic ldquorealrdquo traffic Based on the wind tunneltests Dorigatti et al [123]measured crosswind loads on high-sided vehicles over long-span bridges taking three differentvehicles (van double deck bus and lorry) and two differentbridge deck configurations into consideration Zhu et al[124] investigated aerodynamic coefficients of road vehiclesby adopting different road vehicles types wind directionsand vehicle positions Li et al [122] studied the effects ofsudden changes of wind loads as the train passing througha bridge tower or two trains passing each other by usingthe wind tunnel test rig with moving train models Hanet al [125] developed an experimental setup for measuringthe aerodynamic characteristics of vehicles and the bridgein wind tunnel and then investigated the influences ofparameters adopted in the tests
3 Applications of Simulation Technology toBridge Assessment
After reviewing the key issues of numerical simulationsfor dynamic response of long-span multiload bridges thissection will review the engineering applications of the newlydeveloped technologies to safety assessment of long-spanbridges such as assessment of fatigue and assessment underextreme events
31 Assessment of FatigueDamage Steel structures are widelyused in long-span bridges Research by the ASCE [126] indi-cates that 80ndash90 of failures in steel structures are related tofatigue and fracture Several disasters resulting from fatigue-induced bridge failure have occurred in history For instance46 people died in the collapse of the Silver Bridge (USA1967) and 32 people lost their lives in the collapse of theSungsoo Grand Bridge (South Korea 1994) These disastersteach us that fatigue is an important aspect of the safety ofsteel bridges and action should be taken to prevent similarfatigue-induced bridge failures In the past few decadesfatigue assessment of steel bridges has attracted increasingattention from both governments and bridge engineers andrelevant provisions have been stipulated in several codes andstandards [127ndash130]
12 The Scientific World Journal
It has great advantages to evaluate fatigue damage of long-span bridges based on numerical simulation especially fora multiload bridge which suffers multiple types of dynamicloading such as railway highway andwind loadingDifferentfrom sudden structural damage fatigue damage accumulateswith load-induced dynamic stress (or stress fluctuation) overthe service life of a bridge lasting for more than 100 yearsThe increase in traffic volume and gross vehicle weight thataccompany economic development is very likely to happenin the long period Numerical simulation technology can bean ideal tool to study influences of traffic growth patternsto fatigue damage of bridge In addition slender long-span bridges built in wind-prone regions also suffer fromconsiderable wind induced vibration which appears withina wide range of wind speeds and lasts for almost the wholedesign life of the bridge Given the simultaneous presenceof multiple vehicles and wind it is necessary to considercombined effects of traffic loading (railway andor highwayloading) and wind loading in the fatigue assessment Sincemultiple loading is concerned in a long time period thereare a large number of loading combinations for multipleloading in different intensities It is almost unavailable forfield measurement to obtain such complete information butnumerical simulation could be a good choice to determinedynamic responses of a long-span bridge under multipleloading
A number of structural health monitoring systems(SHMSs) have been installed on numerous recently builtlong-span bridges and a variety of sensors are used for mon-itoring bridge loadings (eg traffic wind and earthquakes)and conditions (including global and local responses) toensure bridge safety and user comfort under in-service con-ditions Well-known examples include Tsing-Ma Bridge inHong Kong Akashi Kaikyo Bridge in Japan Binzhou YellowRiver Bridge in China and Jindo Bridge in Korea Integrationof numerical simulation technologies and measurement dataof structural health monitoring systems (SHMSs) installedon long-span bridges will make the fatigue assessment morereliable for several reasons (1) it is a perfect validation byusing field measurement data of the different types of loadingas input of numerical simulation and the measured dynamicresponses for comparison with the computed ones (2) a largenumber of measured loading data could be used to establishloading databases or probabilistic models of different loads
In the recent years several researchers [7ndash10] appliedthe newly developed numerical simulation technologies tofatigue assessment of long-span bridges Chen et al [7]proposed a framework for fatigue analysis of a long-spansuspension bridge under railway highway and wind loadingby integrating computer simulation with SHMSs and itwas applied to evaluate fatigue damage of the Tsing MaSuspension Bridge over its design life as a case study Basedon this work Chen et al [8] proposed a framework forfatigue reliability analysis of long suspension bridges undermultiple loading inwhich the probabilisticmodels of railwayhighway and wind loading were established based on themeasurement data acquired from the SHMS of the TsingMa Bridge Wu et al [9] proposed a reliability-based fatigueapproach for slender long-span bridge and the combined
dynamic loading effects from wind and traffic as well asthe associated uncertainties were considered Based on theassumption that dynamic magnification related to vehicledynamics can be neglected in long suspension bridges Chenet al [8] established a framework for fatigue reliability anal-ysis To account for different types of long-span bridges withthe span length ranging from a few hundred to thousands ofmeters Zhang et al [10] proposed a more general frameworkwhich includes multiple random variables for the dynamicloads in a bridgersquos life cycle for the vehicle-bridge-winddynamic system such as road profile vehicle speed andwindvelocity and direction among other effects
32 Assessment under Extreme Events The aforementionedfatigue assessment mainly focuses on damage accumulationinduced by stress fluctuations under normal operationalcondition in a long-term period For long-span bridges inaddition to the normal operational conditions in which windspeeds are small ormoderate and traffic scenarios are normalthere are some extreme event conditions Extreme eventsmayinclude complex traffic congestion on the bridge coupledwith moderate or even strong wind [11] For example severetraffic congestions may be formed on the bridge as a resultof an evacuation or a partial blockage of driving lanes due totraffic accidents construction ormaintenance For hurricaneevacuations there are usually a lot of road vehicles passingthrough the bridge before the landfall of the hurricane whilethe wind speed may become pretty high already [131]
Although the excessive dynamic responses of the bridgesunder extreme events are rare it is also critical since itmay cause critical damage initiation or accumulation onsome local bridgemembers Furthermore the extreme events(eg heavy traffic) may even trigger the collapse of thewhole bridge by breaking the ldquoweakest linkrdquo especially whensome hidden damage or design flaw has not been detectedOne recent example is the Minnesota Bridge failure whichoccurred during rush hours with heavy traffic although trafficloads may not be the direct cause of failure For slender long-span bridges strong wind may also cause threats by workinginteractively with heavy traffic loads Therefore even thoughthe extreme cases associated with congested traffic andorwindy weather may be relatively rare and the durationscould be short it is still important for bridge engineers toappropriately look into these unusual extreme events duringstructural design and life-time management of these criticalinfrastructures [11]
The dynamic performance of long-span bridges undercombined actions of strong winds and running road vehicleshas been studied by many researchers in recent years [17 7981 82 132] Most of them studied bridge dynamic perfor-mance under road traffic inwhich only one or several vehiclesdistributed in an assumed (usually uniform) pattern on long-span bridges were considered Extreme events such as trafficcongestion coupled with strong wind were out of concernin those studies Recently Wu and Chen [11] conducteda research on the assessment of long-span bridges underextreme events which includes complex traffic congestioncoupled with moderate or even strong wind This study
The Scientific World Journal 13
applied the cellular automaton (CA) traffic model to thesimulation of the actual traffic flow through the bridgedefined representative scenarios for the extreme events andnumerically studied the bridge performance under thesepossible extreme events
4 Conclusions and Recommendations
Dynamic responses of long-span bridges are often requiredfor assessing the safety of these bridges and can be determinedby numerical simulation technologies This paper provides adetailed review of key issues involved in dynamic responseanalysis of long-span multiload bridges based on numericalsimulation including dynamic interactions between runningtrains and bridge between running road vehicles and bridgeand between wind and bridge and in the wind-vehicle-bridge coupled systemThen the review work was conductedfor engineering applications of newly developed numericalsimulation technologies to safety assessment of long-spanbridges such as assessment of fatigue damage and assessmentunder extreme event condition Although technologies fornumerical simulation of dynamic responses of long-spanmultiload bridge have achieved great advances in past fewdecades and successfully applied to several important bridgesit is still far from reach its maturity and enable to takeplace of traditional fieldmeasurementThe existing problemsand promising research efforts at least include the followingaspects
(1) After multiple types of dynamic interactions beingconsidered in the complex system computationalefficiency is a bottleneck problem for numericalsimulation of dynamic response of a long-span bridgeTypically when multiple loads are involved a largenumber of loading combinations for multiple load-ings must be considered in the assessment
(2) It is rather complex for the time-depending windloads acting on a long-span bridge and running vehi-cles especially for the case of rapid change of windloads such as a train passing through a bridge toweror two trains passing each other The aerodynamiccharacteristics of vehicles and the bridge under differ-ent loading scenarios can be determined through thewind tunnel testing and used in numerical simulationof dynamic responses of the bridge and vehicles
(3) It is a new trend to integrate numerical simulationtechnologies and measurement data of structuralhealth monitoring systems (SHMSs) installed onlong-span bridges whichmakes the safety assessmentof bridge structures more reliable Measured struc-tural responses could be used to validate numericalsimulation approach and measured loading infor-mation could be used for generating statistical orprobabilistic models of multiple loads
(4) It is important to study dynamic responses of bridgestructures under extreme events such as congestedtraffic coupled with windy weather which happens ina long-span bridge For the assessment under extreme
events using numerical simulation technologies sim-ulation of traffic flow and definition of representativescenarios of the extreme events are key issues
(5) It is necessary to consider the effects of typhoonwinds on the safety assessment of long-span bridgesin a reasonable way Few researches do this mostlybecause a probabilistic distribution of wind speedand direction specifically for typhoons is requiredfor assessment but there are insufficient measuredrecords to establish a reliable probabilistic typhoonwind model
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors wish to acknowledge the financial supportsfrom the National Natural Science Foundation of China(NSFC-51108395 and NSFC-51178366) the FundamentalResearch Funds for theCentral Universities (2012121032) andopen funding from Jiangsu Key Laboratory of EngineeringMechanics Special thanks go to the supervisor of the firstauthor Professor Y L XuHongKong PolytechnicUniversityfor his expert guidance and continuous support at all levelsthroughout his PhD study Sincere thanks should go tothe Highways Department of Hong Kong for providing theauthors with the field measurement data Any opinions andconcluding remarks presented in this paper are entirely thoseof the authors
References
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[2] H Sohn C R Farrar N F Hunter and K Worden ldquoStructuralhealth monitoring using statistical pattern recognition tech-niquesrdquo Journal of Dynamic Systems Measurement and Controlvol 123 no 4 pp 706ndash711 2001
[3] W Fan and P Qiao ldquoVibration-based damage identificationmethods a review and comparative studyrdquo Structural HealthMonitoring vol 10 no 1 pp 83ndash111 2011
[4] X Q Zhu and S S Law ldquoDamage detection in simply supportedconcrete bridge structure under moving vehicular loadsrdquo Jour-nal of Vibration and Acoustics Transactions of the ASME vol129 no 1 pp 58ndash65 2007
[5] J Li and S S Law ldquoDamage identification of a target sub-structure with moving load excitationrdquoMechanical Systems andSignal Processing vol 30 pp 78ndash90 2012
[6] J Li S S Law and H Hao ldquoImproved damage identification inbridge structures subject tomoving loads numerical and exper-imental studiesrdquo International Journal of Mechanical Sciencesvol 74 pp 99ndash111 2013
14 The Scientific World Journal
[7] Z W Chen Y L Xu Y Xia Q Li and K Y Wong ldquoFatigueanalysis of long-span suspension bridges under multiple load-ing case studyrdquo Engineering Structures vol 33 no 12 pp 3246ndash3256 2011
[8] Z W Chen Y L Xu and X M Wang ldquoSHMS-based fatiguereliability analysis of multiloading suspension bridgesrdquo Journalof Structural Engineering-Asce vol 138 pp 299ndash307 2012
[9] J Wu S R Chen and J W van de Lindt ldquoFatigue assessmentof slender long-span bridges reliability approachrdquo Journal ofBridge Engineering vol 17 no 1 pp 47ndash57 2012
[10] W Zhang C S Cai and F Pan ldquoFatigue reliability assessmentfor long-span bridges under combined dynamic loads fromwinds and vehiclesrdquo Journal of Bridge Engineering vol 18 pp735ndash747 2013
[11] JWu and S R Chen ldquoProbabilistic dynamic behavior of a long-span bridge under extreme eventsrdquo Engineering Structures vol33 no 5 pp 1657ndash1665 2011
[12] J M Ko and Y Q Ni ldquoTechnology developments in structuralhealth monitoring of large-scale bridgesrdquo Engineering Struc-tures vol 27 no 12 pp 1715ndash1725 2005
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[15] T H Yi H N Li and M Gu ldquoExperimental assessmentof high-rate GPS receivers for deformation monitoring ofbridgerdquoMeasurement Journal of the InternationalMeasurementConfederation vol 46 pp 420ndash432 2013
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[17] Z W Chen Y L Xu Q Li and D J Wu ldquoDynamic stressanalysis of long suspension bridges under wind railway andhighway loadingsrdquo Journal of Bridge Engineering vol 16 no 3pp 383ndash391 2011
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[22] T T Liu Y L Xu W S Zhang K Y Wong H J Zhou and KW Y Chan ldquoBuffeting-induced stresses in a long suspensionbridge structural health monitoring oriented stress analysisrdquoWind and Structures An International Journal vol 12 no 6 pp479ndash504 2009
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[26] Z X Li T Q Zhou T H T Chan and Y Yu ldquoMulti-scalenumerical analysis on dynamic response and local damage inlong-span bridgesrdquo Engineering Structures vol 29 no 7 pp1507ndash1524 2007
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[38] Y B Yang J D Yau and Y S Wu Vehicle-Bridge InteractionDynamic with Applications to High-Speed Railways WorldScientific 2004
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[40] S Sadiku and H H E Leipholz ldquoOn the dynamics of elasticsystems with moving concentrated massesrdquo Ingenieur-Archivvol 57 no 3 pp 223ndash242 1987
[41] J E Akin and M Mofid ldquoNumerical solution for response ofbeamswithmovingmassrdquo Journal of Structural Engineering vol115 no 1 pp 120ndash131 1989
[42] M AMahmoud andM A Abou Zaid ldquoDynamic response of abeam with a crack subject to a moving massrdquo Journal of Soundand Vibration vol 256 no 4 pp 591ndash603 2002
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[47] B Tabarrok and E Esmailzadeh ldquoInduced vibration of bridgestransversed by moving vehiclesrdquo Transactions of the CanadianSociety for Mechanical Engineering B vol 24 no 1 pp 191ndash1982000
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[51] H Xia Y L Xu and T H T Chan ldquoDynamic interaction oflong suspension bridges with running trainsrdquo Journal of Soundand Vibration vol 237 no 2 pp 263ndash280 2000
[52] Q-L Zhang A Vrouwenvelder and J Wardenier ldquoNumericalsimulation of train-bridge interactive dynamicsrdquo Computersand Structures vol 79 no 10 pp 1059ndash1075 2001
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[55] Q Li Y L Xu D J Wu and Z W Chen ldquoComputer-aided nonlinear vehicle-bridge interaction analysisrdquo Journal ofVibration and Control vol 16 pp 1791ndash1816 2010
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Journal of Sound and Vibration vol 240 no 3 pp 447ndash4652001
[64] Y Q Sun andMDhanasekar ldquoA dynamicmodel for the verticalinteraction of the rail track and wagon systemrdquo InternationalJournal of Solids and Structures vol 39 no 5 pp 1337ndash13592002
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[70] O Coussy M Said and J-P van Hoove ldquoThe influence ofrandom surface irregularities on the dynamic response ofbridges under suspended moving loadsrdquo Journal of Sound andVibration vol 130 no 2 pp 313ndash320 1989
[71] E S Hwang and A S Nowak ldquoSimulation of dynamic loadfor bridgesrdquo Journal of Structural Engineering vol 117 pp 1413ndash1434 1991
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[73] W Zhai and Z Cai ldquoDynamic interaction between a lumpedmass vehicle and a discretely supported continuous rail trackrdquoComputers and Structures vol 63 no 5 pp 987ndash997 1997
[74] W M Zhai and C B Cai ldquoTraintrackbridge dynamic inter-actions simulation and applicationsrdquo Vehicle System Dynamicsvol 37 pp 653ndash665 2003
[75] D Bruno F Greco and P Lonetti ldquoDynamic impact analysis oflong span cable-stayed bridges under moving loadsrdquo Engineer-ing Structures vol 30 no 4 pp 1160ndash1177 2008
[76] X D Song D J Wu and Q Li ldquoDynamic impact analysisof double-tower cable-stayed maglev bridges using a simplemodelrdquo Journal of Bridge Engineering vol 19 pp 34ndash43 2014
[77] Y-S Wu and Y-B Yang ldquoSteady-state response and ridingcomfort of trains moving over a series of simply supportedbridgesrdquoEngineering Structures vol 25 no 2 pp 251ndash265 2003
[78] P Antolin N Zhang J M Goicolea H Xia M A Astiz andJ Oliva ldquoConsideration of nonlinear wheel-rail contact forcesfor dynamic vehicle-bridge interaction in high-speed railwaysrdquoJournal of Sound and Vibration vol 332 no 5 pp 1231ndash12512013
[79] W H Guo and Y L Xu ldquoFully computerized approach to studycable-stayed bridge-vehicle interactionrdquo Journal of Sound andVibration vol 248 no 4 pp 745ndash761 2001
[80] Y L Xu and W H Guo ldquoDynamic behaviour of high-sidedroad vehicles subject to a sudden crosswind gustrdquo Wind andStructures vol 6 no 5 pp 325ndash346 2003
[81] Y L Xu and W H Guo ldquoDynamic analysis of coupled roadvehicle and cable-stayed bridge systems under turbulent windrdquoEngineering Structures vol 25 no 4 pp 473ndash486 2003
16 The Scientific World Journal
[82] C S Cai and S R Chen ldquoFramework of vehicle-bridge-winddynamic analysisrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 92 no 7-8 pp 579ndash607 2004
[83] S R Chen and J Wu ldquoModeling stochastic live load for long-span bridge based on microscopic traffic flow simulationrdquoComputers and Structures vol 89 no 9-10 pp 813ndash824 2011
[84] A V Paultre B Yang L A Bergman and C A Tan ldquoBridgedynamics and dynamic amplification factorsmdasha review ofanalytical and experimental findingsrdquoCanadian Journal of CivilEngineering vol 19 no 2 pp 260ndash278 1992
[85] H Honda Y Kajikawa and T Kobori ldquoSpectra of road surfaceroughness of bridgesrdquo Journal of the Structural Division vol 108pp 1956ndash1966 1982
[86] M J Inbanathan and M Wieland ldquoBridge vibrations dueto vehicle moving over rough surfacerdquo Journal of StructuralEngineering vol 113 no 9 pp 1994ndash2008 1987
[87] T-LWang andDHuang ldquoCable-stayed bridge vibration due toroad surface roughnessrdquo Journal of Structural Engineering vol118 no 5 pp 1354ndash1374 1992
[88] P K Chatterjee T K Datta and C S Surana ldquoVibrationsuspension bridges under vehicular movementrdquo Journal ofStructural Engineering vol 120 no 3 pp 681ndash703 1994
[89] D Chang and H Lee ldquoImpact factors for simple-span highwaygirder bridgesrdquo Journal of Structural Engineering vol 120 no 3pp 704ndash715 1994
[90] T-C Pan and J Li ldquoDynamic vehicle element method for tran-sient response of coupled vehicle-structure systemsrdquo Journal ofStructural Engineering vol 128 no 2 pp 214ndash223 2002
[91] C J Dodds and J D Robson ldquoThe description of road surfaceroughnessrdquo Journal of Sound Vibration vol 31 no 2 pp 175ndash183 1973
[92] D Huang T-L Wang and M Shahawy ldquoImpact studies ofmultigirder concrete bridgesrdquo Journal of Structural Engineeringvol 119 no 8 pp 2387ndash2402 1993
[93] A G Davenport ldquoBuffeting of a suspension bridge by stormwindrdquo Journal of Structural Division vol 88 pp 233ndash268 1962
[94] R H Scanlan ldquoThe action of flexible bridges under wind Iflutter theoryrdquo Journal of Sound and Vibration vol 60 no 2pp 187ndash199 1978
[95] Q Ding and P K K Lee ldquoComputer simulation of buffetingactions of suspension bridges under turbulentwindrdquoComputersand Structures vol 76 no 6 pp 787ndash797 2000
[96] V Boonyapinyo T Miyata and H Yamada ldquoAdvanced aerody-namic analysis of suspension bridges by state-space approachrdquoJournal of Structural Engineering vol 125 no 12 pp 1357ndash13661999
[97] Y-H Chen and C-Y Li ldquoDynamic response of elevated high-speed railwayrdquo Journal of Bridge Engineering vol 5 no 2 pp124ndash130 2000
[98] X Chen M Matsumoto and A Kareem ldquoTime domainflutter and buffeting response analysis of bridgesrdquo Journal ofEngineering Mechanics vol 126 no 1 pp 7ndash16 2000
[99] X Chen and A Kareem ldquoEquivalent static wind loads forbuffeting response of bridgesrdquo Journal of Structural Engineeringvol 127 no 12 pp 1467ndash1475 2001
[100] S R Chen and C S Cai ldquoEvolution of long-span bridgeresponse to wind-numerical simulation and discussionrdquo Com-puters and Structures vol 81 no 21 pp 2055ndash2066 2003
[101] Y L XuHXia andQ S Yan ldquoDynamic response of suspensionbridge to high wind and running trainrdquo Journal of BridgeEngineering vol 8 no 1 pp 46ndash55 2003
[102] A Guo Y L Xu and H Li ldquoDynamic performance of cable-stayed bridge tower with multi-stage pendulum mass damperunderwind excitations-II experimentrdquoEarthquake Engineeringand Engineering Vibration vol 6 no 4 pp 417ndash424 2007
[103] E Simiu and R H ScanlanWind Effects on Structures 1996[104] Y Cao H Xiang and Y Zhou ldquoSimulation of stochastic wind
velocity field on long-span bridgesrdquo Journal of EngineeringMechanics vol 126 no 1 pp 1ndash6 2000
[105] M Shinozuka and C-M Jan ldquoDigital simulation of randomprocesses and its applicationsrdquo Journal of Sound and Vibrationvol 25 no 1 pp 111ndash128 1972
[106] K M Shum Lateral and torsional vibration control of long spanbridge deck using novel tuned liquid column dampers [PhDthesis] Department of Civil and Structural Engineering TheHong Kong Polytechnic University 2004
[107] Y K Lin and J N Yang ldquoMultimode bridge response to windexcitationsrdquo Journal of EngineeringMechanics vol 109 no 2 pp586ndash603 1983
[108] Y S Lin Self-Excited Bridge Motion in Turbulent Wind 1978[109] D T Lau M S Cheung and S H Cheng ldquo3D flutter analysis
of bridges by spline finite-strip methodrdquo Journal of StructuralEngineering vol 126 no 10 pp 1246ndash1254 2000
[110] L A Balzer ldquoAtmospheric turbulence encountered by high-speed ground transport vehiclesrdquo Journal of Mechanical Engi-neering Science vol 19 pp 227ndash235 1977
[111] R K Cooper ldquoAtmospheric turbulence with respect to movingground vehiclesrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 17 no 2 pp 215ndash238 1984
[112] C J Baker ldquoA simplified analysis of various types of wind-induced road vehicle accidentsrdquo Journal of Wind Engineeringand Industrial Aerodynamics vol 22 no 1 pp 69ndash85 1986
[113] C J Baker ldquoMeasures to control vehicle movement at exposedsites during windy periodsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 25 no 2 pp 151ndash161 1987
[114] C J Baker ldquoHigh sided articulated road vehicles in strong crosswindsrdquo Journal of Wind Engineering and Industrial Aerodynam-ics vol 31 no 1 pp 67ndash85 1988
[115] C J Baker ldquoGround vehicles in high cross winds part I steadyaerodynamic forcesrdquo Journal of Fluids and Structures vol 5 no1 pp 69ndash90 1991
[116] C J Baker ldquoGround vehicles in high cross winds part IIunsteady aerodynamic forcesrdquo Journal of Fluids and Structuresvol 5 no 1 pp 91ndash111 1991
[117] Y L Xu and Q S Ding ldquoInteraction of railway vehicles withtrack in cross-windsrdquo Journal of Fluids and Structures vol 22no 3 pp 295ndash314 2006
[118] S R Chen and J Wu ldquoDynamic performance simulation oflong-span bridge under combined loads of stochastic traffic andwindrdquo Journal of Bridge Engineering vol 15 no 3 pp 219ndash2302010
[119] Y Li S Qiang H Liao and Y L Xu ldquoDynamics of wind-rail vehicle-bridge systemsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 93 no 6 pp 483ndash507 2005
[120] Y L Xu N Zhang and H Xia ldquoVibration of coupled trainand cable-stayed bridge systems in cross windsrdquo EngineeringStructures vol 26 no 10 pp 1389ndash1406 2004
[121] W W Guo Y L Xu H Xia W S Zhang and K M ShumldquoDynamic response of suspension bridge to typhoon and trainsII numerical resultsrdquo Journal of Structural Engineering vol 133no 1 pp 12ndash21 2007
The Scientific World Journal 17
[122] Y L Li H Y Xiang B Wang Y L Xu and S Z QiangldquoDynamic analysis of wind-vehicle-bridge coupling systemduring the meeting of two trainsrdquo Advances in StructuralEngineering vol 16 pp 1663ndash1670 2013
[123] F Dorigatti M Sterling D Rocchi et al ldquoWind tunnelmeasurements of crosswind loads on high sided vehicles overlong span bridgesrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 107 pp 214ndash224 2012
[124] L D Zhu L Li Y L Xu and Q Zhu ldquoWind tunnel investi-gations of aerodynamic coefficients of road vehicles on bridgedeckrdquo Journal of Fluids and Structures vol 30 pp 35ndash50 2012
[125] Y Han J X Hu C S Cai Z Q Chen and C G LildquoExperimental and numerical studies of aerodynamic forces onvehicles and bridgesrdquoWind and Structures vol 17 pp 163ndash1842013
[126] ASCE ldquoCommittee on fatigue and fracture reliability of thecommittee on structural safety and reliability of the structuraldivision fatigue reliability 1ndash4rdquo Journal of Structural Engineer-ing vol 108 pp 3ndash88 1982
[127] BS ldquoBS5400 part 10 code of practice for fatiguerdquo BritishStandard Institute 1980
[128] BS ldquoBS7608 code of practice for fatigue design and assessmentof steel structuresrdquo British Standard Institute 1993
[129] AASHTOGuide Specifications for Fatigue Evaluation of ExistingSteel Bridges 1990
[130] AASHTO Guide Manual for Condition Evaluation and Loadand Resistance Factor Rating (LRFR) of Highway Bridges 2003
[131] S R Chen C S Cai and B Wolshon ldquoFrom normal operationto evacuation Single-vehicle safety under adverse weathertopographic and operational conditionsrdquo Natural HazardsReview vol 10 no 2 pp 68ndash76 2009
[132] S R Chen and C S Cai ldquoEquivalent wheel load approach forslender cable-stayed bridge fatigue assessment under traffic andwind feasibility studyrdquo Journal of Bridge Engineering vol 12 no6 pp 755ndash764 2007
It has great advantages to evaluate fatigue damage of long-span bridges based on numerical simulation especially fora multiload bridge which suffers multiple types of dynamicloading such as railway highway andwind loadingDifferentfrom sudden structural damage fatigue damage accumulateswith load-induced dynamic stress (or stress fluctuation) overthe service life of a bridge lasting for more than 100 yearsThe increase in traffic volume and gross vehicle weight thataccompany economic development is very likely to happenin the long period Numerical simulation technology can bean ideal tool to study influences of traffic growth patternsto fatigue damage of bridge In addition slender long-span bridges built in wind-prone regions also suffer fromconsiderable wind induced vibration which appears withina wide range of wind speeds and lasts for almost the wholedesign life of the bridge Given the simultaneous presenceof multiple vehicles and wind it is necessary to considercombined effects of traffic loading (railway andor highwayloading) and wind loading in the fatigue assessment Sincemultiple loading is concerned in a long time period thereare a large number of loading combinations for multipleloading in different intensities It is almost unavailable forfield measurement to obtain such complete information butnumerical simulation could be a good choice to determinedynamic responses of a long-span bridge under multipleloading
A number of structural health monitoring systems(SHMSs) have been installed on numerous recently builtlong-span bridges and a variety of sensors are used for mon-itoring bridge loadings (eg traffic wind and earthquakes)and conditions (including global and local responses) toensure bridge safety and user comfort under in-service con-ditions Well-known examples include Tsing-Ma Bridge inHong Kong Akashi Kaikyo Bridge in Japan Binzhou YellowRiver Bridge in China and Jindo Bridge in Korea Integrationof numerical simulation technologies and measurement dataof structural health monitoring systems (SHMSs) installedon long-span bridges will make the fatigue assessment morereliable for several reasons (1) it is a perfect validation byusing field measurement data of the different types of loadingas input of numerical simulation and the measured dynamicresponses for comparison with the computed ones (2) a largenumber of measured loading data could be used to establishloading databases or probabilistic models of different loads
In the recent years several researchers [7ndash10] appliedthe newly developed numerical simulation technologies tofatigue assessment of long-span bridges Chen et al [7]proposed a framework for fatigue analysis of a long-spansuspension bridge under railway highway and wind loadingby integrating computer simulation with SHMSs and itwas applied to evaluate fatigue damage of the Tsing MaSuspension Bridge over its design life as a case study Basedon this work Chen et al [8] proposed a framework forfatigue reliability analysis of long suspension bridges undermultiple loading inwhich the probabilisticmodels of railwayhighway and wind loading were established based on themeasurement data acquired from the SHMS of the TsingMa Bridge Wu et al [9] proposed a reliability-based fatigueapproach for slender long-span bridge and the combined
dynamic loading effects from wind and traffic as well asthe associated uncertainties were considered Based on theassumption that dynamic magnification related to vehicledynamics can be neglected in long suspension bridges Chenet al [8] established a framework for fatigue reliability anal-ysis To account for different types of long-span bridges withthe span length ranging from a few hundred to thousands ofmeters Zhang et al [10] proposed a more general frameworkwhich includes multiple random variables for the dynamicloads in a bridgersquos life cycle for the vehicle-bridge-winddynamic system such as road profile vehicle speed andwindvelocity and direction among other effects
32 Assessment under Extreme Events The aforementionedfatigue assessment mainly focuses on damage accumulationinduced by stress fluctuations under normal operationalcondition in a long-term period For long-span bridges inaddition to the normal operational conditions in which windspeeds are small ormoderate and traffic scenarios are normalthere are some extreme event conditions Extreme eventsmayinclude complex traffic congestion on the bridge coupledwith moderate or even strong wind [11] For example severetraffic congestions may be formed on the bridge as a resultof an evacuation or a partial blockage of driving lanes due totraffic accidents construction ormaintenance For hurricaneevacuations there are usually a lot of road vehicles passingthrough the bridge before the landfall of the hurricane whilethe wind speed may become pretty high already [131]
Although the excessive dynamic responses of the bridgesunder extreme events are rare it is also critical since itmay cause critical damage initiation or accumulation onsome local bridgemembers Furthermore the extreme events(eg heavy traffic) may even trigger the collapse of thewhole bridge by breaking the ldquoweakest linkrdquo especially whensome hidden damage or design flaw has not been detectedOne recent example is the Minnesota Bridge failure whichoccurred during rush hours with heavy traffic although trafficloads may not be the direct cause of failure For slender long-span bridges strong wind may also cause threats by workinginteractively with heavy traffic loads Therefore even thoughthe extreme cases associated with congested traffic andorwindy weather may be relatively rare and the durationscould be short it is still important for bridge engineers toappropriately look into these unusual extreme events duringstructural design and life-time management of these criticalinfrastructures [11]
The dynamic performance of long-span bridges undercombined actions of strong winds and running road vehicleshas been studied by many researchers in recent years [17 7981 82 132] Most of them studied bridge dynamic perfor-mance under road traffic inwhich only one or several vehiclesdistributed in an assumed (usually uniform) pattern on long-span bridges were considered Extreme events such as trafficcongestion coupled with strong wind were out of concernin those studies Recently Wu and Chen [11] conducteda research on the assessment of long-span bridges underextreme events which includes complex traffic congestioncoupled with moderate or even strong wind This study
The Scientific World Journal 13
applied the cellular automaton (CA) traffic model to thesimulation of the actual traffic flow through the bridgedefined representative scenarios for the extreme events andnumerically studied the bridge performance under thesepossible extreme events
4 Conclusions and Recommendations
Dynamic responses of long-span bridges are often requiredfor assessing the safety of these bridges and can be determinedby numerical simulation technologies This paper provides adetailed review of key issues involved in dynamic responseanalysis of long-span multiload bridges based on numericalsimulation including dynamic interactions between runningtrains and bridge between running road vehicles and bridgeand between wind and bridge and in the wind-vehicle-bridge coupled systemThen the review work was conductedfor engineering applications of newly developed numericalsimulation technologies to safety assessment of long-spanbridges such as assessment of fatigue damage and assessmentunder extreme event condition Although technologies fornumerical simulation of dynamic responses of long-spanmultiload bridge have achieved great advances in past fewdecades and successfully applied to several important bridgesit is still far from reach its maturity and enable to takeplace of traditional fieldmeasurementThe existing problemsand promising research efforts at least include the followingaspects
(1) After multiple types of dynamic interactions beingconsidered in the complex system computationalefficiency is a bottleneck problem for numericalsimulation of dynamic response of a long-span bridgeTypically when multiple loads are involved a largenumber of loading combinations for multiple load-ings must be considered in the assessment
(2) It is rather complex for the time-depending windloads acting on a long-span bridge and running vehi-cles especially for the case of rapid change of windloads such as a train passing through a bridge toweror two trains passing each other The aerodynamiccharacteristics of vehicles and the bridge under differ-ent loading scenarios can be determined through thewind tunnel testing and used in numerical simulationof dynamic responses of the bridge and vehicles
(3) It is a new trend to integrate numerical simulationtechnologies and measurement data of structuralhealth monitoring systems (SHMSs) installed onlong-span bridges whichmakes the safety assessmentof bridge structures more reliable Measured struc-tural responses could be used to validate numericalsimulation approach and measured loading infor-mation could be used for generating statistical orprobabilistic models of multiple loads
(4) It is important to study dynamic responses of bridgestructures under extreme events such as congestedtraffic coupled with windy weather which happens ina long-span bridge For the assessment under extreme
events using numerical simulation technologies sim-ulation of traffic flow and definition of representativescenarios of the extreme events are key issues
(5) It is necessary to consider the effects of typhoonwinds on the safety assessment of long-span bridgesin a reasonable way Few researches do this mostlybecause a probabilistic distribution of wind speedand direction specifically for typhoons is requiredfor assessment but there are insufficient measuredrecords to establish a reliable probabilistic typhoonwind model
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors wish to acknowledge the financial supportsfrom the National Natural Science Foundation of China(NSFC-51108395 and NSFC-51178366) the FundamentalResearch Funds for theCentral Universities (2012121032) andopen funding from Jiangsu Key Laboratory of EngineeringMechanics Special thanks go to the supervisor of the firstauthor Professor Y L XuHongKong PolytechnicUniversityfor his expert guidance and continuous support at all levelsthroughout his PhD study Sincere thanks should go tothe Highways Department of Hong Kong for providing theauthors with the field measurement data Any opinions andconcluding remarks presented in this paper are entirely thoseof the authors
References
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[2] H Sohn C R Farrar N F Hunter and K Worden ldquoStructuralhealth monitoring using statistical pattern recognition tech-niquesrdquo Journal of Dynamic Systems Measurement and Controlvol 123 no 4 pp 706ndash711 2001
[3] W Fan and P Qiao ldquoVibration-based damage identificationmethods a review and comparative studyrdquo Structural HealthMonitoring vol 10 no 1 pp 83ndash111 2011
[4] X Q Zhu and S S Law ldquoDamage detection in simply supportedconcrete bridge structure under moving vehicular loadsrdquo Jour-nal of Vibration and Acoustics Transactions of the ASME vol129 no 1 pp 58ndash65 2007
[5] J Li and S S Law ldquoDamage identification of a target sub-structure with moving load excitationrdquoMechanical Systems andSignal Processing vol 30 pp 78ndash90 2012
[6] J Li S S Law and H Hao ldquoImproved damage identification inbridge structures subject tomoving loads numerical and exper-imental studiesrdquo International Journal of Mechanical Sciencesvol 74 pp 99ndash111 2013
14 The Scientific World Journal
[7] Z W Chen Y L Xu Y Xia Q Li and K Y Wong ldquoFatigueanalysis of long-span suspension bridges under multiple load-ing case studyrdquo Engineering Structures vol 33 no 12 pp 3246ndash3256 2011
[8] Z W Chen Y L Xu and X M Wang ldquoSHMS-based fatiguereliability analysis of multiloading suspension bridgesrdquo Journalof Structural Engineering-Asce vol 138 pp 299ndash307 2012
[9] J Wu S R Chen and J W van de Lindt ldquoFatigue assessmentof slender long-span bridges reliability approachrdquo Journal ofBridge Engineering vol 17 no 1 pp 47ndash57 2012
[10] W Zhang C S Cai and F Pan ldquoFatigue reliability assessmentfor long-span bridges under combined dynamic loads fromwinds and vehiclesrdquo Journal of Bridge Engineering vol 18 pp735ndash747 2013
[11] JWu and S R Chen ldquoProbabilistic dynamic behavior of a long-span bridge under extreme eventsrdquo Engineering Structures vol33 no 5 pp 1657ndash1665 2011
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[15] T H Yi H N Li and M Gu ldquoExperimental assessmentof high-rate GPS receivers for deformation monitoring ofbridgerdquoMeasurement Journal of the InternationalMeasurementConfederation vol 46 pp 420ndash432 2013
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[22] T T Liu Y L Xu W S Zhang K Y Wong H J Zhou and KW Y Chan ldquoBuffeting-induced stresses in a long suspensionbridge structural health monitoring oriented stress analysisrdquoWind and Structures An International Journal vol 12 no 6 pp479ndash504 2009
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[26] Z X Li T Q Zhou T H T Chan and Y Yu ldquoMulti-scalenumerical analysis on dynamic response and local damage inlong-span bridgesrdquo Engineering Structures vol 29 no 7 pp1507ndash1524 2007
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[35] J A Gbadeyan and S T Oni ldquoDynamic behaviour of beamsand rectangular plates under moving loadsrdquo Journal of Soundand Vibration vol 182 no 5 pp 677ndash695 1995
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[37] G V Rao ldquoLinear dynamics of an elastic beam under movingloadsrdquo Journal of Vibration and Acoustics Transactions of theASME vol 122 no 3 pp 281ndash289 2000
[38] Y B Yang J D Yau and Y S Wu Vehicle-Bridge InteractionDynamic with Applications to High-Speed Railways WorldScientific 2004
[39] E C Ting J Genin and J H Ginsberg ldquoA general algorithm formovingmass problemsrdquo Journal of Sound and Vibration vol 33no 1 pp 49ndash58 1974
[40] S Sadiku and H H E Leipholz ldquoOn the dynamics of elasticsystems with moving concentrated massesrdquo Ingenieur-Archivvol 57 no 3 pp 223ndash242 1987
[41] J E Akin and M Mofid ldquoNumerical solution for response ofbeamswithmovingmassrdquo Journal of Structural Engineering vol115 no 1 pp 120ndash131 1989
[42] M AMahmoud andM A Abou Zaid ldquoDynamic response of abeam with a crack subject to a moving massrdquo Journal of Soundand Vibration vol 256 no 4 pp 591ndash603 2002
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[44] T-LWang andDHuang ldquoCable-stayed bridge vibration due toroad surface roughnessrdquo Journal of Structural Engineering vol118 no 5 pp 1354ndash1374 1992
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[47] B Tabarrok and E Esmailzadeh ldquoInduced vibration of bridgestransversed by moving vehiclesrdquo Transactions of the CanadianSociety for Mechanical Engineering B vol 24 no 1 pp 191ndash1982000
[48] C Liu T-L Wang and D Huang ldquoImpact study for multi-girder bridge based on correlated road roughnessrdquo StructuralEngineering and Mechanics vol 11 no 3 pp 259ndash272 2001
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[50] T-LWang V K Garg and K-H Chu ldquoRailway bridgevehicleinteraction studies with new vehicle modelrdquo Journal of Struc-tural Engineering vol 117 no 7 pp 2099ndash2116 1991
[51] H Xia Y L Xu and T H T Chan ldquoDynamic interaction oflong suspension bridges with running trainsrdquo Journal of Soundand Vibration vol 237 no 2 pp 263ndash280 2000
[52] Q-L Zhang A Vrouwenvelder and J Wardenier ldquoNumericalsimulation of train-bridge interactive dynamicsrdquo Computersand Structures vol 79 no 10 pp 1059ndash1075 2001
[53] H Xia N Zhang and G de Roeck ldquoDynamic analysis of highspeed railway bridge under articulated trainsrdquo Computers andStructures vol 81 no 26-27 pp 2467ndash2478 2003
[54] G Diana F Cheli A Collina R Corradi and S MelzildquoThe development of a numerical model for railway vehiclescomfort assessment through comparison with experimentalmeasurementsrdquoVehicle SystemDynamics vol 38 no 3 pp 165ndash183 2002
[55] Q Li Y L Xu D J Wu and Z W Chen ldquoComputer-aided nonlinear vehicle-bridge interaction analysisrdquo Journal ofVibration and Control vol 16 pp 1791ndash1816 2010
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[62] Y B Yang and Y S Wu ldquoA versatile element for analyzingvehicle-bridge interaction responserdquo Engineering Structuresvol 23 no 5 pp 452ndash469 2001
[63] F T K Au J J Wang and Y K Cheung ldquoImpact study ofcable-stayed bridge under railway traffic using various modelsrdquo
Journal of Sound and Vibration vol 240 no 3 pp 447ndash4652001
[64] Y Q Sun andMDhanasekar ldquoA dynamicmodel for the verticalinteraction of the rail track and wagon systemrdquo InternationalJournal of Solids and Structures vol 39 no 5 pp 1337ndash13592002
[65] K Henchi M Fafard M Talbot and G Dhatt ldquoAn efficientalgorithm for dynamic analysis of bridges under moving vehi-cles using a coupledmodal and physical components approachrdquoJournal of Sound and Vibration vol 212 no 4 pp 663ndash6831998
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[67] B Biondi G Muscolino and A Sofi ldquoA substructure approachfor the dynamic analysis of train-track-bridge systemrdquoComput-ers and Structures vol 83 no 28ndash30 pp 2271ndash2281 2005
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[70] O Coussy M Said and J-P van Hoove ldquoThe influence ofrandom surface irregularities on the dynamic response ofbridges under suspended moving loadsrdquo Journal of Sound andVibration vol 130 no 2 pp 313ndash320 1989
[71] E S Hwang and A S Nowak ldquoSimulation of dynamic loadfor bridgesrdquo Journal of Structural Engineering vol 117 pp 1413ndash1434 1991
[72] F Yang and G A Fonder ldquoAn iterative solution methodfor dynamic response of bridge-vehicles systemsrdquo EarthquakeEngineering and Structural Dynamics vol 25 pp 195ndash215 1996
[73] W Zhai and Z Cai ldquoDynamic interaction between a lumpedmass vehicle and a discretely supported continuous rail trackrdquoComputers and Structures vol 63 no 5 pp 987ndash997 1997
[74] W M Zhai and C B Cai ldquoTraintrackbridge dynamic inter-actions simulation and applicationsrdquo Vehicle System Dynamicsvol 37 pp 653ndash665 2003
[75] D Bruno F Greco and P Lonetti ldquoDynamic impact analysis oflong span cable-stayed bridges under moving loadsrdquo Engineer-ing Structures vol 30 no 4 pp 1160ndash1177 2008
[76] X D Song D J Wu and Q Li ldquoDynamic impact analysisof double-tower cable-stayed maglev bridges using a simplemodelrdquo Journal of Bridge Engineering vol 19 pp 34ndash43 2014
[77] Y-S Wu and Y-B Yang ldquoSteady-state response and ridingcomfort of trains moving over a series of simply supportedbridgesrdquoEngineering Structures vol 25 no 2 pp 251ndash265 2003
[78] P Antolin N Zhang J M Goicolea H Xia M A Astiz andJ Oliva ldquoConsideration of nonlinear wheel-rail contact forcesfor dynamic vehicle-bridge interaction in high-speed railwaysrdquoJournal of Sound and Vibration vol 332 no 5 pp 1231ndash12512013
[79] W H Guo and Y L Xu ldquoFully computerized approach to studycable-stayed bridge-vehicle interactionrdquo Journal of Sound andVibration vol 248 no 4 pp 745ndash761 2001
[80] Y L Xu and W H Guo ldquoDynamic behaviour of high-sidedroad vehicles subject to a sudden crosswind gustrdquo Wind andStructures vol 6 no 5 pp 325ndash346 2003
[81] Y L Xu and W H Guo ldquoDynamic analysis of coupled roadvehicle and cable-stayed bridge systems under turbulent windrdquoEngineering Structures vol 25 no 4 pp 473ndash486 2003
16 The Scientific World Journal
[82] C S Cai and S R Chen ldquoFramework of vehicle-bridge-winddynamic analysisrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 92 no 7-8 pp 579ndash607 2004
[83] S R Chen and J Wu ldquoModeling stochastic live load for long-span bridge based on microscopic traffic flow simulationrdquoComputers and Structures vol 89 no 9-10 pp 813ndash824 2011
[84] A V Paultre B Yang L A Bergman and C A Tan ldquoBridgedynamics and dynamic amplification factorsmdasha review ofanalytical and experimental findingsrdquoCanadian Journal of CivilEngineering vol 19 no 2 pp 260ndash278 1992
[85] H Honda Y Kajikawa and T Kobori ldquoSpectra of road surfaceroughness of bridgesrdquo Journal of the Structural Division vol 108pp 1956ndash1966 1982
[86] M J Inbanathan and M Wieland ldquoBridge vibrations dueto vehicle moving over rough surfacerdquo Journal of StructuralEngineering vol 113 no 9 pp 1994ndash2008 1987
[87] T-LWang andDHuang ldquoCable-stayed bridge vibration due toroad surface roughnessrdquo Journal of Structural Engineering vol118 no 5 pp 1354ndash1374 1992
[88] P K Chatterjee T K Datta and C S Surana ldquoVibrationsuspension bridges under vehicular movementrdquo Journal ofStructural Engineering vol 120 no 3 pp 681ndash703 1994
[89] D Chang and H Lee ldquoImpact factors for simple-span highwaygirder bridgesrdquo Journal of Structural Engineering vol 120 no 3pp 704ndash715 1994
[90] T-C Pan and J Li ldquoDynamic vehicle element method for tran-sient response of coupled vehicle-structure systemsrdquo Journal ofStructural Engineering vol 128 no 2 pp 214ndash223 2002
[91] C J Dodds and J D Robson ldquoThe description of road surfaceroughnessrdquo Journal of Sound Vibration vol 31 no 2 pp 175ndash183 1973
[92] D Huang T-L Wang and M Shahawy ldquoImpact studies ofmultigirder concrete bridgesrdquo Journal of Structural Engineeringvol 119 no 8 pp 2387ndash2402 1993
[93] A G Davenport ldquoBuffeting of a suspension bridge by stormwindrdquo Journal of Structural Division vol 88 pp 233ndash268 1962
[94] R H Scanlan ldquoThe action of flexible bridges under wind Iflutter theoryrdquo Journal of Sound and Vibration vol 60 no 2pp 187ndash199 1978
[95] Q Ding and P K K Lee ldquoComputer simulation of buffetingactions of suspension bridges under turbulentwindrdquoComputersand Structures vol 76 no 6 pp 787ndash797 2000
[96] V Boonyapinyo T Miyata and H Yamada ldquoAdvanced aerody-namic analysis of suspension bridges by state-space approachrdquoJournal of Structural Engineering vol 125 no 12 pp 1357ndash13661999
[97] Y-H Chen and C-Y Li ldquoDynamic response of elevated high-speed railwayrdquo Journal of Bridge Engineering vol 5 no 2 pp124ndash130 2000
[98] X Chen M Matsumoto and A Kareem ldquoTime domainflutter and buffeting response analysis of bridgesrdquo Journal ofEngineering Mechanics vol 126 no 1 pp 7ndash16 2000
[99] X Chen and A Kareem ldquoEquivalent static wind loads forbuffeting response of bridgesrdquo Journal of Structural Engineeringvol 127 no 12 pp 1467ndash1475 2001
[100] S R Chen and C S Cai ldquoEvolution of long-span bridgeresponse to wind-numerical simulation and discussionrdquo Com-puters and Structures vol 81 no 21 pp 2055ndash2066 2003
[101] Y L XuHXia andQ S Yan ldquoDynamic response of suspensionbridge to high wind and running trainrdquo Journal of BridgeEngineering vol 8 no 1 pp 46ndash55 2003
[102] A Guo Y L Xu and H Li ldquoDynamic performance of cable-stayed bridge tower with multi-stage pendulum mass damperunderwind excitations-II experimentrdquoEarthquake Engineeringand Engineering Vibration vol 6 no 4 pp 417ndash424 2007
[103] E Simiu and R H ScanlanWind Effects on Structures 1996[104] Y Cao H Xiang and Y Zhou ldquoSimulation of stochastic wind
velocity field on long-span bridgesrdquo Journal of EngineeringMechanics vol 126 no 1 pp 1ndash6 2000
[105] M Shinozuka and C-M Jan ldquoDigital simulation of randomprocesses and its applicationsrdquo Journal of Sound and Vibrationvol 25 no 1 pp 111ndash128 1972
[106] K M Shum Lateral and torsional vibration control of long spanbridge deck using novel tuned liquid column dampers [PhDthesis] Department of Civil and Structural Engineering TheHong Kong Polytechnic University 2004
[107] Y K Lin and J N Yang ldquoMultimode bridge response to windexcitationsrdquo Journal of EngineeringMechanics vol 109 no 2 pp586ndash603 1983
[108] Y S Lin Self-Excited Bridge Motion in Turbulent Wind 1978[109] D T Lau M S Cheung and S H Cheng ldquo3D flutter analysis
of bridges by spline finite-strip methodrdquo Journal of StructuralEngineering vol 126 no 10 pp 1246ndash1254 2000
[110] L A Balzer ldquoAtmospheric turbulence encountered by high-speed ground transport vehiclesrdquo Journal of Mechanical Engi-neering Science vol 19 pp 227ndash235 1977
[111] R K Cooper ldquoAtmospheric turbulence with respect to movingground vehiclesrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 17 no 2 pp 215ndash238 1984
[112] C J Baker ldquoA simplified analysis of various types of wind-induced road vehicle accidentsrdquo Journal of Wind Engineeringand Industrial Aerodynamics vol 22 no 1 pp 69ndash85 1986
[113] C J Baker ldquoMeasures to control vehicle movement at exposedsites during windy periodsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 25 no 2 pp 151ndash161 1987
[114] C J Baker ldquoHigh sided articulated road vehicles in strong crosswindsrdquo Journal of Wind Engineering and Industrial Aerodynam-ics vol 31 no 1 pp 67ndash85 1988
[115] C J Baker ldquoGround vehicles in high cross winds part I steadyaerodynamic forcesrdquo Journal of Fluids and Structures vol 5 no1 pp 69ndash90 1991
[116] C J Baker ldquoGround vehicles in high cross winds part IIunsteady aerodynamic forcesrdquo Journal of Fluids and Structuresvol 5 no 1 pp 91ndash111 1991
[117] Y L Xu and Q S Ding ldquoInteraction of railway vehicles withtrack in cross-windsrdquo Journal of Fluids and Structures vol 22no 3 pp 295ndash314 2006
[118] S R Chen and J Wu ldquoDynamic performance simulation oflong-span bridge under combined loads of stochastic traffic andwindrdquo Journal of Bridge Engineering vol 15 no 3 pp 219ndash2302010
[119] Y Li S Qiang H Liao and Y L Xu ldquoDynamics of wind-rail vehicle-bridge systemsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 93 no 6 pp 483ndash507 2005
[120] Y L Xu N Zhang and H Xia ldquoVibration of coupled trainand cable-stayed bridge systems in cross windsrdquo EngineeringStructures vol 26 no 10 pp 1389ndash1406 2004
[121] W W Guo Y L Xu H Xia W S Zhang and K M ShumldquoDynamic response of suspension bridge to typhoon and trainsII numerical resultsrdquo Journal of Structural Engineering vol 133no 1 pp 12ndash21 2007
The Scientific World Journal 17
[122] Y L Li H Y Xiang B Wang Y L Xu and S Z QiangldquoDynamic analysis of wind-vehicle-bridge coupling systemduring the meeting of two trainsrdquo Advances in StructuralEngineering vol 16 pp 1663ndash1670 2013
[123] F Dorigatti M Sterling D Rocchi et al ldquoWind tunnelmeasurements of crosswind loads on high sided vehicles overlong span bridgesrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 107 pp 214ndash224 2012
[124] L D Zhu L Li Y L Xu and Q Zhu ldquoWind tunnel investi-gations of aerodynamic coefficients of road vehicles on bridgedeckrdquo Journal of Fluids and Structures vol 30 pp 35ndash50 2012
[125] Y Han J X Hu C S Cai Z Q Chen and C G LildquoExperimental and numerical studies of aerodynamic forces onvehicles and bridgesrdquoWind and Structures vol 17 pp 163ndash1842013
[126] ASCE ldquoCommittee on fatigue and fracture reliability of thecommittee on structural safety and reliability of the structuraldivision fatigue reliability 1ndash4rdquo Journal of Structural Engineer-ing vol 108 pp 3ndash88 1982
[127] BS ldquoBS5400 part 10 code of practice for fatiguerdquo BritishStandard Institute 1980
[128] BS ldquoBS7608 code of practice for fatigue design and assessmentof steel structuresrdquo British Standard Institute 1993
[129] AASHTOGuide Specifications for Fatigue Evaluation of ExistingSteel Bridges 1990
[130] AASHTO Guide Manual for Condition Evaluation and Loadand Resistance Factor Rating (LRFR) of Highway Bridges 2003
[131] S R Chen C S Cai and B Wolshon ldquoFrom normal operationto evacuation Single-vehicle safety under adverse weathertopographic and operational conditionsrdquo Natural HazardsReview vol 10 no 2 pp 68ndash76 2009
[132] S R Chen and C S Cai ldquoEquivalent wheel load approach forslender cable-stayed bridge fatigue assessment under traffic andwind feasibility studyrdquo Journal of Bridge Engineering vol 12 no6 pp 755ndash764 2007
applied the cellular automaton (CA) traffic model to thesimulation of the actual traffic flow through the bridgedefined representative scenarios for the extreme events andnumerically studied the bridge performance under thesepossible extreme events
4 Conclusions and Recommendations
Dynamic responses of long-span bridges are often requiredfor assessing the safety of these bridges and can be determinedby numerical simulation technologies This paper provides adetailed review of key issues involved in dynamic responseanalysis of long-span multiload bridges based on numericalsimulation including dynamic interactions between runningtrains and bridge between running road vehicles and bridgeand between wind and bridge and in the wind-vehicle-bridge coupled systemThen the review work was conductedfor engineering applications of newly developed numericalsimulation technologies to safety assessment of long-spanbridges such as assessment of fatigue damage and assessmentunder extreme event condition Although technologies fornumerical simulation of dynamic responses of long-spanmultiload bridge have achieved great advances in past fewdecades and successfully applied to several important bridgesit is still far from reach its maturity and enable to takeplace of traditional fieldmeasurementThe existing problemsand promising research efforts at least include the followingaspects
(1) After multiple types of dynamic interactions beingconsidered in the complex system computationalefficiency is a bottleneck problem for numericalsimulation of dynamic response of a long-span bridgeTypically when multiple loads are involved a largenumber of loading combinations for multiple load-ings must be considered in the assessment
(2) It is rather complex for the time-depending windloads acting on a long-span bridge and running vehi-cles especially for the case of rapid change of windloads such as a train passing through a bridge toweror two trains passing each other The aerodynamiccharacteristics of vehicles and the bridge under differ-ent loading scenarios can be determined through thewind tunnel testing and used in numerical simulationof dynamic responses of the bridge and vehicles
(3) It is a new trend to integrate numerical simulationtechnologies and measurement data of structuralhealth monitoring systems (SHMSs) installed onlong-span bridges whichmakes the safety assessmentof bridge structures more reliable Measured struc-tural responses could be used to validate numericalsimulation approach and measured loading infor-mation could be used for generating statistical orprobabilistic models of multiple loads
(4) It is important to study dynamic responses of bridgestructures under extreme events such as congestedtraffic coupled with windy weather which happens ina long-span bridge For the assessment under extreme
events using numerical simulation technologies sim-ulation of traffic flow and definition of representativescenarios of the extreme events are key issues
(5) It is necessary to consider the effects of typhoonwinds on the safety assessment of long-span bridgesin a reasonable way Few researches do this mostlybecause a probabilistic distribution of wind speedand direction specifically for typhoons is requiredfor assessment but there are insufficient measuredrecords to establish a reliable probabilistic typhoonwind model
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors wish to acknowledge the financial supportsfrom the National Natural Science Foundation of China(NSFC-51108395 and NSFC-51178366) the FundamentalResearch Funds for theCentral Universities (2012121032) andopen funding from Jiangsu Key Laboratory of EngineeringMechanics Special thanks go to the supervisor of the firstauthor Professor Y L XuHongKong PolytechnicUniversityfor his expert guidance and continuous support at all levelsthroughout his PhD study Sincere thanks should go tothe Highways Department of Hong Kong for providing theauthors with the field measurement data Any opinions andconcluding remarks presented in this paper are entirely thoseof the authors
References
[1] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoShock and Vibration Digest vol 30 no 2 pp 91ndash105 1998
[2] H Sohn C R Farrar N F Hunter and K Worden ldquoStructuralhealth monitoring using statistical pattern recognition tech-niquesrdquo Journal of Dynamic Systems Measurement and Controlvol 123 no 4 pp 706ndash711 2001
[3] W Fan and P Qiao ldquoVibration-based damage identificationmethods a review and comparative studyrdquo Structural HealthMonitoring vol 10 no 1 pp 83ndash111 2011
[4] X Q Zhu and S S Law ldquoDamage detection in simply supportedconcrete bridge structure under moving vehicular loadsrdquo Jour-nal of Vibration and Acoustics Transactions of the ASME vol129 no 1 pp 58ndash65 2007
[5] J Li and S S Law ldquoDamage identification of a target sub-structure with moving load excitationrdquoMechanical Systems andSignal Processing vol 30 pp 78ndash90 2012
[6] J Li S S Law and H Hao ldquoImproved damage identification inbridge structures subject tomoving loads numerical and exper-imental studiesrdquo International Journal of Mechanical Sciencesvol 74 pp 99ndash111 2013
14 The Scientific World Journal
[7] Z W Chen Y L Xu Y Xia Q Li and K Y Wong ldquoFatigueanalysis of long-span suspension bridges under multiple load-ing case studyrdquo Engineering Structures vol 33 no 12 pp 3246ndash3256 2011
[8] Z W Chen Y L Xu and X M Wang ldquoSHMS-based fatiguereliability analysis of multiloading suspension bridgesrdquo Journalof Structural Engineering-Asce vol 138 pp 299ndash307 2012
[9] J Wu S R Chen and J W van de Lindt ldquoFatigue assessmentof slender long-span bridges reliability approachrdquo Journal ofBridge Engineering vol 17 no 1 pp 47ndash57 2012
[10] W Zhang C S Cai and F Pan ldquoFatigue reliability assessmentfor long-span bridges under combined dynamic loads fromwinds and vehiclesrdquo Journal of Bridge Engineering vol 18 pp735ndash747 2013
[11] JWu and S R Chen ldquoProbabilistic dynamic behavior of a long-span bridge under extreme eventsrdquo Engineering Structures vol33 no 5 pp 1657ndash1665 2011
[12] J M Ko and Y Q Ni ldquoTechnology developments in structuralhealth monitoring of large-scale bridgesrdquo Engineering Struc-tures vol 27 no 12 pp 1715ndash1725 2005
[13] TH YiHN Li andHM Sun ldquoMulti-stage structural damagediagnosis method based on ldquoenergy-damagerdquo theoryrdquo SmartStructures and Systems vol 12 pp 345ndash361 2013
[14] T H Yi H N Li and M Gu ldquoFull-scale measurements ofdynamic response of suspension bridge subjected to environ-mental loads using GPS technologyrdquo Science China Technologi-cal Sciences vol 53 no 2 pp 469ndash479 2010
[15] T H Yi H N Li and M Gu ldquoExperimental assessmentof high-rate GPS receivers for deformation monitoring ofbridgerdquoMeasurement Journal of the InternationalMeasurementConfederation vol 46 pp 420ndash432 2013
[16] X W Ye Y Q Ni K Y Wong and J M Ko ldquoStatistical analysisof stress spectra for fatigue life assessment of steel bridges withstructural health monitoring datardquo Engineering Structures vol45 pp 166ndash176 2012
[17] Z W Chen Y L Xu Q Li and D J Wu ldquoDynamic stressanalysis of long suspension bridges under wind railway andhighway loadingsrdquo Journal of Bridge Engineering vol 16 no 3pp 383ndash391 2011
[18] S G Meisenholder and P Weidlinger ldquoDynamic interactionaspects of cable-stayed guideways for high speed ground trans-portationrdquoAmerican Society ofMechanical Engineers no 74 pp180ndash192 1974
[19] Q H Mao Research on the Highway Bridge Vibration Due toMoving Vehicles Tongji University Shang Hai China 1989
[20] Y L Xu J M Ko and Z Yu ldquoModal analysis of tower-cable system of Tsing Ma long suspension bridgerdquo EngineeringStructures vol 19 pp 857ndash867 1997
[21] W Guo H Xia and Y-L Xu ldquoDynamic response of a long spansuspension bridge and running safety of a train under windactionrdquo Frontiers of Architecture and Civil Engineering in Chinavol 1 no 1 pp 71ndash79 2007
[22] T T Liu Y L Xu W S Zhang K Y Wong H J Zhou and KW Y Chan ldquoBuffeting-induced stresses in a long suspensionbridge structural health monitoring oriented stress analysisrdquoWind and Structures An International Journal vol 12 no 6 pp479ndash504 2009
[23] K Y Wong ldquoStructural identification of Tsing Ma BridgerdquoTransactions Hong Kong Institution of Engineers vol 10 no 1pp 38ndash47 2003
[24] Y L XuQ Li D JWu and ZWChen ldquoStress and accelerationanalysis of coupled vehicle and long-span bridge systems usingthe mode superposition methodrdquo Engineering Structures vol32 no 5 pp 1356ndash1368 2010
[25] Y F Duan Y L Xu Q G Fei et al ldquoAdvanced finite elementmodel of Tsing Ma Bridge for structural health monitoringrdquoInternational Journal of Structural Stability and Dynamics vol11 no 2 pp 313ndash344 2011
[26] Z X Li T Q Zhou T H T Chan and Y Yu ldquoMulti-scalenumerical analysis on dynamic response and local damage inlong-span bridgesrdquo Engineering Structures vol 29 no 7 pp1507ndash1524 2007
[27] W Zhang C S Cai and F Pan ldquoFinite element modeling ofbridges with equivalent orthotropic material method for multi-scale dynamic loadsrdquo Engineering Structures vol 54 pp 82ndash932013
[28] S P Timoshenko ldquoOn the forced vibrations of bridgesrdquo Philo-sophical Magazine vol 6 no 257 pp 1018ndash1019 1922
[29] R S Ayre G Ford and L S Jacobsen ldquoTransverse vibration ofa two-span beam under the action of a moving constant forcerdquoJournal of Applied Mechanics vol 17 pp 1ndash12 1950
[30] R S Ayre andLS Jacobsen ldquoTransverse vibration of a two-spanbeam under the action of a moving alternating forcerdquo Journal ofApplied Mechanics vol 17 pp 283ndash290 1950
[31] L FrybaVibration of Solids and Structures underMoving LoadsASCE Press 1972
[32] J-S Wu and C-W Dai ldquoDynamic response of multispannonuniform beam due to moving loadsrdquo Journal of StructuralEngineering vol 113 no 3 pp 458ndash474 1987
[33] W Weaver S P Timoshenko and D H Young VibrationProblems in Engineering John Wiley amp Sons 1990
[34] N H Galdos D R Schelling and M A Sahin ldquoMethodologyfor impact factor of horizontally curved box bridgesrdquo Journal ofStructural Engineering vol 119 no 6 pp 1917ndash1934 1993
[35] J A Gbadeyan and S T Oni ldquoDynamic behaviour of beamsand rectangular plates under moving loadsrdquo Journal of Soundand Vibration vol 182 no 5 pp 677ndash695 1995
[36] D Y Zheng Y K Cheung F T K Au and Y S ChengldquoVibration of multi-span non-uniform beams under movingloads by using modified beam vibration functionsrdquo Journal ofSound and Vibration vol 212 no 3 pp 455ndash467 1998
[37] G V Rao ldquoLinear dynamics of an elastic beam under movingloadsrdquo Journal of Vibration and Acoustics Transactions of theASME vol 122 no 3 pp 281ndash289 2000
[38] Y B Yang J D Yau and Y S Wu Vehicle-Bridge InteractionDynamic with Applications to High-Speed Railways WorldScientific 2004
[39] E C Ting J Genin and J H Ginsberg ldquoA general algorithm formovingmass problemsrdquo Journal of Sound and Vibration vol 33no 1 pp 49ndash58 1974
[40] S Sadiku and H H E Leipholz ldquoOn the dynamics of elasticsystems with moving concentrated massesrdquo Ingenieur-Archivvol 57 no 3 pp 223ndash242 1987
[41] J E Akin and M Mofid ldquoNumerical solution for response ofbeamswithmovingmassrdquo Journal of Structural Engineering vol115 no 1 pp 120ndash131 1989
[42] M AMahmoud andM A Abou Zaid ldquoDynamic response of abeam with a crack subject to a moving massrdquo Journal of Soundand Vibration vol 256 no 4 pp 591ndash603 2002
[43] V K Garg Dynamics of Railway Vehicle Systems AcademicPress 1994
The Scientific World Journal 15
[44] T-LWang andDHuang ldquoCable-stayed bridge vibration due toroad surface roughnessrdquo Journal of Structural Engineering vol118 no 5 pp 1354ndash1374 1992
[45] Y-B Yang and B-H Lin ldquoVehicle-bridge interaction analysisby dynamic condensation methodrdquo Journal of Structural Engi-neering vol 121 no 11 pp 1636ndash1643 1995
[46] Y B Yang J D Yau and L C Hsu ldquoVibration of simple beamsdue to trainsmoving at high speedsrdquoEngineering Structures vol19 no 11 pp 936ndash943 1997
[47] B Tabarrok and E Esmailzadeh ldquoInduced vibration of bridgestransversed by moving vehiclesrdquo Transactions of the CanadianSociety for Mechanical Engineering B vol 24 no 1 pp 191ndash1982000
[48] C Liu T-L Wang and D Huang ldquoImpact study for multi-girder bridge based on correlated road roughnessrdquo StructuralEngineering and Mechanics vol 11 no 3 pp 259ndash272 2001
[49] K H Chu V K Garg and T L Wang ldquoImpact in railwayprestressed concrete bridgesrdquo Journal of Structural Engineeringvol 112 no 5 pp 1036ndash1051 1986
[50] T-LWang V K Garg and K-H Chu ldquoRailway bridgevehicleinteraction studies with new vehicle modelrdquo Journal of Struc-tural Engineering vol 117 no 7 pp 2099ndash2116 1991
[51] H Xia Y L Xu and T H T Chan ldquoDynamic interaction oflong suspension bridges with running trainsrdquo Journal of Soundand Vibration vol 237 no 2 pp 263ndash280 2000
[52] Q-L Zhang A Vrouwenvelder and J Wardenier ldquoNumericalsimulation of train-bridge interactive dynamicsrdquo Computersand Structures vol 79 no 10 pp 1059ndash1075 2001
[53] H Xia N Zhang and G de Roeck ldquoDynamic analysis of highspeed railway bridge under articulated trainsrdquo Computers andStructures vol 81 no 26-27 pp 2467ndash2478 2003
[54] G Diana F Cheli A Collina R Corradi and S MelzildquoThe development of a numerical model for railway vehiclescomfort assessment through comparison with experimentalmeasurementsrdquoVehicle SystemDynamics vol 38 no 3 pp 165ndash183 2002
[55] Q Li Y L Xu D J Wu and Z W Chen ldquoComputer-aided nonlinear vehicle-bridge interaction analysisrdquo Journal ofVibration and Control vol 16 pp 1791ndash1816 2010
[56] A Wiriyachai K H Chu and V K Garg ldquoBridge impact dueto wheel and track irregularitiesrdquo Journal of the EngineeringMechanics Division vol 108 no 4 pp 648ndash666 1982
[57] L Fryba Dynamics of Railway Bridges Inst of Civil Engineers1996
[58] D Huang and T-L Wang ldquoImpact analysis of cable-stayedbridgesrdquo Computers and Structures vol 43 no 5 pp 897ndash9081992
[59] W M Zhai Vehicle-Track Coupling Dynamics Chinese RailwayPress Beijing China 2007
[60] M Olsson ldquoFinite element modal co-ordinate analysis ofstructures subjected to moving loadsrdquo Journal of Sound andVibration vol 99 no 1 pp 1ndash12 1985
[61] Y-B Yang C-H Chang and J-D Yau ldquoAn element foranalysing vehicle-bridge systems considering vehiclersquos pitchingeffectrdquo International Journal for NumericalMethods in Engineer-ing vol 46 no 7 pp 1031ndash1047 1999
[62] Y B Yang and Y S Wu ldquoA versatile element for analyzingvehicle-bridge interaction responserdquo Engineering Structuresvol 23 no 5 pp 452ndash469 2001
[63] F T K Au J J Wang and Y K Cheung ldquoImpact study ofcable-stayed bridge under railway traffic using various modelsrdquo
Journal of Sound and Vibration vol 240 no 3 pp 447ndash4652001
[64] Y Q Sun andMDhanasekar ldquoA dynamicmodel for the verticalinteraction of the rail track and wagon systemrdquo InternationalJournal of Solids and Structures vol 39 no 5 pp 1337ndash13592002
[65] K Henchi M Fafard M Talbot and G Dhatt ldquoAn efficientalgorithm for dynamic analysis of bridges under moving vehi-cles using a coupledmodal and physical components approachrdquoJournal of Sound and Vibration vol 212 no 4 pp 663ndash6831998
[66] Y L Xu and L YWang ldquoAnalytical study of wind-rain-inducedcable vibration SDOFmodelrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 91 no 1-2 pp 27ndash40 2003
[67] B Biondi G Muscolino and A Sofi ldquoA substructure approachfor the dynamic analysis of train-track-bridge systemrdquoComput-ers and Structures vol 83 no 28ndash30 pp 2271ndash2281 2005
[68] J L Humar and A H Kashif ldquoDynamic response analysis ofslab-type bridgesrdquo Journal of Structural Engineering vol 121 no1 pp 48ndash62 1995
[69] P Lou and Q-Y Zeng ldquoFormulation of equations of verticalmotion for vehicle-track-bridge systemrdquo Journal of the ChinaRailway Society vol 26 no 5 p 71 2004
[70] O Coussy M Said and J-P van Hoove ldquoThe influence ofrandom surface irregularities on the dynamic response ofbridges under suspended moving loadsrdquo Journal of Sound andVibration vol 130 no 2 pp 313ndash320 1989
[71] E S Hwang and A S Nowak ldquoSimulation of dynamic loadfor bridgesrdquo Journal of Structural Engineering vol 117 pp 1413ndash1434 1991
[72] F Yang and G A Fonder ldquoAn iterative solution methodfor dynamic response of bridge-vehicles systemsrdquo EarthquakeEngineering and Structural Dynamics vol 25 pp 195ndash215 1996
[73] W Zhai and Z Cai ldquoDynamic interaction between a lumpedmass vehicle and a discretely supported continuous rail trackrdquoComputers and Structures vol 63 no 5 pp 987ndash997 1997
[74] W M Zhai and C B Cai ldquoTraintrackbridge dynamic inter-actions simulation and applicationsrdquo Vehicle System Dynamicsvol 37 pp 653ndash665 2003
[75] D Bruno F Greco and P Lonetti ldquoDynamic impact analysis oflong span cable-stayed bridges under moving loadsrdquo Engineer-ing Structures vol 30 no 4 pp 1160ndash1177 2008
[76] X D Song D J Wu and Q Li ldquoDynamic impact analysisof double-tower cable-stayed maglev bridges using a simplemodelrdquo Journal of Bridge Engineering vol 19 pp 34ndash43 2014
[77] Y-S Wu and Y-B Yang ldquoSteady-state response and ridingcomfort of trains moving over a series of simply supportedbridgesrdquoEngineering Structures vol 25 no 2 pp 251ndash265 2003
[78] P Antolin N Zhang J M Goicolea H Xia M A Astiz andJ Oliva ldquoConsideration of nonlinear wheel-rail contact forcesfor dynamic vehicle-bridge interaction in high-speed railwaysrdquoJournal of Sound and Vibration vol 332 no 5 pp 1231ndash12512013
[79] W H Guo and Y L Xu ldquoFully computerized approach to studycable-stayed bridge-vehicle interactionrdquo Journal of Sound andVibration vol 248 no 4 pp 745ndash761 2001
[80] Y L Xu and W H Guo ldquoDynamic behaviour of high-sidedroad vehicles subject to a sudden crosswind gustrdquo Wind andStructures vol 6 no 5 pp 325ndash346 2003
[81] Y L Xu and W H Guo ldquoDynamic analysis of coupled roadvehicle and cable-stayed bridge systems under turbulent windrdquoEngineering Structures vol 25 no 4 pp 473ndash486 2003
16 The Scientific World Journal
[82] C S Cai and S R Chen ldquoFramework of vehicle-bridge-winddynamic analysisrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 92 no 7-8 pp 579ndash607 2004
[83] S R Chen and J Wu ldquoModeling stochastic live load for long-span bridge based on microscopic traffic flow simulationrdquoComputers and Structures vol 89 no 9-10 pp 813ndash824 2011
[84] A V Paultre B Yang L A Bergman and C A Tan ldquoBridgedynamics and dynamic amplification factorsmdasha review ofanalytical and experimental findingsrdquoCanadian Journal of CivilEngineering vol 19 no 2 pp 260ndash278 1992
[85] H Honda Y Kajikawa and T Kobori ldquoSpectra of road surfaceroughness of bridgesrdquo Journal of the Structural Division vol 108pp 1956ndash1966 1982
[86] M J Inbanathan and M Wieland ldquoBridge vibrations dueto vehicle moving over rough surfacerdquo Journal of StructuralEngineering vol 113 no 9 pp 1994ndash2008 1987
[87] T-LWang andDHuang ldquoCable-stayed bridge vibration due toroad surface roughnessrdquo Journal of Structural Engineering vol118 no 5 pp 1354ndash1374 1992
[88] P K Chatterjee T K Datta and C S Surana ldquoVibrationsuspension bridges under vehicular movementrdquo Journal ofStructural Engineering vol 120 no 3 pp 681ndash703 1994
[89] D Chang and H Lee ldquoImpact factors for simple-span highwaygirder bridgesrdquo Journal of Structural Engineering vol 120 no 3pp 704ndash715 1994
[90] T-C Pan and J Li ldquoDynamic vehicle element method for tran-sient response of coupled vehicle-structure systemsrdquo Journal ofStructural Engineering vol 128 no 2 pp 214ndash223 2002
[91] C J Dodds and J D Robson ldquoThe description of road surfaceroughnessrdquo Journal of Sound Vibration vol 31 no 2 pp 175ndash183 1973
[92] D Huang T-L Wang and M Shahawy ldquoImpact studies ofmultigirder concrete bridgesrdquo Journal of Structural Engineeringvol 119 no 8 pp 2387ndash2402 1993
[93] A G Davenport ldquoBuffeting of a suspension bridge by stormwindrdquo Journal of Structural Division vol 88 pp 233ndash268 1962
[94] R H Scanlan ldquoThe action of flexible bridges under wind Iflutter theoryrdquo Journal of Sound and Vibration vol 60 no 2pp 187ndash199 1978
[95] Q Ding and P K K Lee ldquoComputer simulation of buffetingactions of suspension bridges under turbulentwindrdquoComputersand Structures vol 76 no 6 pp 787ndash797 2000
[96] V Boonyapinyo T Miyata and H Yamada ldquoAdvanced aerody-namic analysis of suspension bridges by state-space approachrdquoJournal of Structural Engineering vol 125 no 12 pp 1357ndash13661999
[97] Y-H Chen and C-Y Li ldquoDynamic response of elevated high-speed railwayrdquo Journal of Bridge Engineering vol 5 no 2 pp124ndash130 2000
[98] X Chen M Matsumoto and A Kareem ldquoTime domainflutter and buffeting response analysis of bridgesrdquo Journal ofEngineering Mechanics vol 126 no 1 pp 7ndash16 2000
[99] X Chen and A Kareem ldquoEquivalent static wind loads forbuffeting response of bridgesrdquo Journal of Structural Engineeringvol 127 no 12 pp 1467ndash1475 2001
[100] S R Chen and C S Cai ldquoEvolution of long-span bridgeresponse to wind-numerical simulation and discussionrdquo Com-puters and Structures vol 81 no 21 pp 2055ndash2066 2003
[101] Y L XuHXia andQ S Yan ldquoDynamic response of suspensionbridge to high wind and running trainrdquo Journal of BridgeEngineering vol 8 no 1 pp 46ndash55 2003
[102] A Guo Y L Xu and H Li ldquoDynamic performance of cable-stayed bridge tower with multi-stage pendulum mass damperunderwind excitations-II experimentrdquoEarthquake Engineeringand Engineering Vibration vol 6 no 4 pp 417ndash424 2007
[103] E Simiu and R H ScanlanWind Effects on Structures 1996[104] Y Cao H Xiang and Y Zhou ldquoSimulation of stochastic wind
velocity field on long-span bridgesrdquo Journal of EngineeringMechanics vol 126 no 1 pp 1ndash6 2000
[105] M Shinozuka and C-M Jan ldquoDigital simulation of randomprocesses and its applicationsrdquo Journal of Sound and Vibrationvol 25 no 1 pp 111ndash128 1972
[106] K M Shum Lateral and torsional vibration control of long spanbridge deck using novel tuned liquid column dampers [PhDthesis] Department of Civil and Structural Engineering TheHong Kong Polytechnic University 2004
[107] Y K Lin and J N Yang ldquoMultimode bridge response to windexcitationsrdquo Journal of EngineeringMechanics vol 109 no 2 pp586ndash603 1983
[108] Y S Lin Self-Excited Bridge Motion in Turbulent Wind 1978[109] D T Lau M S Cheung and S H Cheng ldquo3D flutter analysis
of bridges by spline finite-strip methodrdquo Journal of StructuralEngineering vol 126 no 10 pp 1246ndash1254 2000
[110] L A Balzer ldquoAtmospheric turbulence encountered by high-speed ground transport vehiclesrdquo Journal of Mechanical Engi-neering Science vol 19 pp 227ndash235 1977
[111] R K Cooper ldquoAtmospheric turbulence with respect to movingground vehiclesrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 17 no 2 pp 215ndash238 1984
[112] C J Baker ldquoA simplified analysis of various types of wind-induced road vehicle accidentsrdquo Journal of Wind Engineeringand Industrial Aerodynamics vol 22 no 1 pp 69ndash85 1986
[113] C J Baker ldquoMeasures to control vehicle movement at exposedsites during windy periodsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 25 no 2 pp 151ndash161 1987
[114] C J Baker ldquoHigh sided articulated road vehicles in strong crosswindsrdquo Journal of Wind Engineering and Industrial Aerodynam-ics vol 31 no 1 pp 67ndash85 1988
[115] C J Baker ldquoGround vehicles in high cross winds part I steadyaerodynamic forcesrdquo Journal of Fluids and Structures vol 5 no1 pp 69ndash90 1991
[116] C J Baker ldquoGround vehicles in high cross winds part IIunsteady aerodynamic forcesrdquo Journal of Fluids and Structuresvol 5 no 1 pp 91ndash111 1991
[117] Y L Xu and Q S Ding ldquoInteraction of railway vehicles withtrack in cross-windsrdquo Journal of Fluids and Structures vol 22no 3 pp 295ndash314 2006
[118] S R Chen and J Wu ldquoDynamic performance simulation oflong-span bridge under combined loads of stochastic traffic andwindrdquo Journal of Bridge Engineering vol 15 no 3 pp 219ndash2302010
[119] Y Li S Qiang H Liao and Y L Xu ldquoDynamics of wind-rail vehicle-bridge systemsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 93 no 6 pp 483ndash507 2005
[120] Y L Xu N Zhang and H Xia ldquoVibration of coupled trainand cable-stayed bridge systems in cross windsrdquo EngineeringStructures vol 26 no 10 pp 1389ndash1406 2004
[121] W W Guo Y L Xu H Xia W S Zhang and K M ShumldquoDynamic response of suspension bridge to typhoon and trainsII numerical resultsrdquo Journal of Structural Engineering vol 133no 1 pp 12ndash21 2007
The Scientific World Journal 17
[122] Y L Li H Y Xiang B Wang Y L Xu and S Z QiangldquoDynamic analysis of wind-vehicle-bridge coupling systemduring the meeting of two trainsrdquo Advances in StructuralEngineering vol 16 pp 1663ndash1670 2013
[123] F Dorigatti M Sterling D Rocchi et al ldquoWind tunnelmeasurements of crosswind loads on high sided vehicles overlong span bridgesrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 107 pp 214ndash224 2012
[124] L D Zhu L Li Y L Xu and Q Zhu ldquoWind tunnel investi-gations of aerodynamic coefficients of road vehicles on bridgedeckrdquo Journal of Fluids and Structures vol 30 pp 35ndash50 2012
[125] Y Han J X Hu C S Cai Z Q Chen and C G LildquoExperimental and numerical studies of aerodynamic forces onvehicles and bridgesrdquoWind and Structures vol 17 pp 163ndash1842013
[126] ASCE ldquoCommittee on fatigue and fracture reliability of thecommittee on structural safety and reliability of the structuraldivision fatigue reliability 1ndash4rdquo Journal of Structural Engineer-ing vol 108 pp 3ndash88 1982
[127] BS ldquoBS5400 part 10 code of practice for fatiguerdquo BritishStandard Institute 1980
[128] BS ldquoBS7608 code of practice for fatigue design and assessmentof steel structuresrdquo British Standard Institute 1993
[129] AASHTOGuide Specifications for Fatigue Evaluation of ExistingSteel Bridges 1990
[130] AASHTO Guide Manual for Condition Evaluation and Loadand Resistance Factor Rating (LRFR) of Highway Bridges 2003
[131] S R Chen C S Cai and B Wolshon ldquoFrom normal operationto evacuation Single-vehicle safety under adverse weathertopographic and operational conditionsrdquo Natural HazardsReview vol 10 no 2 pp 68ndash76 2009
[132] S R Chen and C S Cai ldquoEquivalent wheel load approach forslender cable-stayed bridge fatigue assessment under traffic andwind feasibility studyrdquo Journal of Bridge Engineering vol 12 no6 pp 755ndash764 2007
[7] Z W Chen Y L Xu Y Xia Q Li and K Y Wong ldquoFatigueanalysis of long-span suspension bridges under multiple load-ing case studyrdquo Engineering Structures vol 33 no 12 pp 3246ndash3256 2011
[8] Z W Chen Y L Xu and X M Wang ldquoSHMS-based fatiguereliability analysis of multiloading suspension bridgesrdquo Journalof Structural Engineering-Asce vol 138 pp 299ndash307 2012
[9] J Wu S R Chen and J W van de Lindt ldquoFatigue assessmentof slender long-span bridges reliability approachrdquo Journal ofBridge Engineering vol 17 no 1 pp 47ndash57 2012
[10] W Zhang C S Cai and F Pan ldquoFatigue reliability assessmentfor long-span bridges under combined dynamic loads fromwinds and vehiclesrdquo Journal of Bridge Engineering vol 18 pp735ndash747 2013
[11] JWu and S R Chen ldquoProbabilistic dynamic behavior of a long-span bridge under extreme eventsrdquo Engineering Structures vol33 no 5 pp 1657ndash1665 2011
[12] J M Ko and Y Q Ni ldquoTechnology developments in structuralhealth monitoring of large-scale bridgesrdquo Engineering Struc-tures vol 27 no 12 pp 1715ndash1725 2005
[13] TH YiHN Li andHM Sun ldquoMulti-stage structural damagediagnosis method based on ldquoenergy-damagerdquo theoryrdquo SmartStructures and Systems vol 12 pp 345ndash361 2013
[14] T H Yi H N Li and M Gu ldquoFull-scale measurements ofdynamic response of suspension bridge subjected to environ-mental loads using GPS technologyrdquo Science China Technologi-cal Sciences vol 53 no 2 pp 469ndash479 2010
[15] T H Yi H N Li and M Gu ldquoExperimental assessmentof high-rate GPS receivers for deformation monitoring ofbridgerdquoMeasurement Journal of the InternationalMeasurementConfederation vol 46 pp 420ndash432 2013
[16] X W Ye Y Q Ni K Y Wong and J M Ko ldquoStatistical analysisof stress spectra for fatigue life assessment of steel bridges withstructural health monitoring datardquo Engineering Structures vol45 pp 166ndash176 2012
[17] Z W Chen Y L Xu Q Li and D J Wu ldquoDynamic stressanalysis of long suspension bridges under wind railway andhighway loadingsrdquo Journal of Bridge Engineering vol 16 no 3pp 383ndash391 2011
[18] S G Meisenholder and P Weidlinger ldquoDynamic interactionaspects of cable-stayed guideways for high speed ground trans-portationrdquoAmerican Society ofMechanical Engineers no 74 pp180ndash192 1974
[19] Q H Mao Research on the Highway Bridge Vibration Due toMoving Vehicles Tongji University Shang Hai China 1989
[20] Y L Xu J M Ko and Z Yu ldquoModal analysis of tower-cable system of Tsing Ma long suspension bridgerdquo EngineeringStructures vol 19 pp 857ndash867 1997
[21] W Guo H Xia and Y-L Xu ldquoDynamic response of a long spansuspension bridge and running safety of a train under windactionrdquo Frontiers of Architecture and Civil Engineering in Chinavol 1 no 1 pp 71ndash79 2007
[22] T T Liu Y L Xu W S Zhang K Y Wong H J Zhou and KW Y Chan ldquoBuffeting-induced stresses in a long suspensionbridge structural health monitoring oriented stress analysisrdquoWind and Structures An International Journal vol 12 no 6 pp479ndash504 2009
[23] K Y Wong ldquoStructural identification of Tsing Ma BridgerdquoTransactions Hong Kong Institution of Engineers vol 10 no 1pp 38ndash47 2003
[24] Y L XuQ Li D JWu and ZWChen ldquoStress and accelerationanalysis of coupled vehicle and long-span bridge systems usingthe mode superposition methodrdquo Engineering Structures vol32 no 5 pp 1356ndash1368 2010
[25] Y F Duan Y L Xu Q G Fei et al ldquoAdvanced finite elementmodel of Tsing Ma Bridge for structural health monitoringrdquoInternational Journal of Structural Stability and Dynamics vol11 no 2 pp 313ndash344 2011
[26] Z X Li T Q Zhou T H T Chan and Y Yu ldquoMulti-scalenumerical analysis on dynamic response and local damage inlong-span bridgesrdquo Engineering Structures vol 29 no 7 pp1507ndash1524 2007
[27] W Zhang C S Cai and F Pan ldquoFinite element modeling ofbridges with equivalent orthotropic material method for multi-scale dynamic loadsrdquo Engineering Structures vol 54 pp 82ndash932013
[28] S P Timoshenko ldquoOn the forced vibrations of bridgesrdquo Philo-sophical Magazine vol 6 no 257 pp 1018ndash1019 1922
[29] R S Ayre G Ford and L S Jacobsen ldquoTransverse vibration ofa two-span beam under the action of a moving constant forcerdquoJournal of Applied Mechanics vol 17 pp 1ndash12 1950
[30] R S Ayre andLS Jacobsen ldquoTransverse vibration of a two-spanbeam under the action of a moving alternating forcerdquo Journal ofApplied Mechanics vol 17 pp 283ndash290 1950
[31] L FrybaVibration of Solids and Structures underMoving LoadsASCE Press 1972
[32] J-S Wu and C-W Dai ldquoDynamic response of multispannonuniform beam due to moving loadsrdquo Journal of StructuralEngineering vol 113 no 3 pp 458ndash474 1987
[33] W Weaver S P Timoshenko and D H Young VibrationProblems in Engineering John Wiley amp Sons 1990
[34] N H Galdos D R Schelling and M A Sahin ldquoMethodologyfor impact factor of horizontally curved box bridgesrdquo Journal ofStructural Engineering vol 119 no 6 pp 1917ndash1934 1993
[35] J A Gbadeyan and S T Oni ldquoDynamic behaviour of beamsand rectangular plates under moving loadsrdquo Journal of Soundand Vibration vol 182 no 5 pp 677ndash695 1995
[36] D Y Zheng Y K Cheung F T K Au and Y S ChengldquoVibration of multi-span non-uniform beams under movingloads by using modified beam vibration functionsrdquo Journal ofSound and Vibration vol 212 no 3 pp 455ndash467 1998
[37] G V Rao ldquoLinear dynamics of an elastic beam under movingloadsrdquo Journal of Vibration and Acoustics Transactions of theASME vol 122 no 3 pp 281ndash289 2000
[38] Y B Yang J D Yau and Y S Wu Vehicle-Bridge InteractionDynamic with Applications to High-Speed Railways WorldScientific 2004
[39] E C Ting J Genin and J H Ginsberg ldquoA general algorithm formovingmass problemsrdquo Journal of Sound and Vibration vol 33no 1 pp 49ndash58 1974
[40] S Sadiku and H H E Leipholz ldquoOn the dynamics of elasticsystems with moving concentrated massesrdquo Ingenieur-Archivvol 57 no 3 pp 223ndash242 1987
[41] J E Akin and M Mofid ldquoNumerical solution for response ofbeamswithmovingmassrdquo Journal of Structural Engineering vol115 no 1 pp 120ndash131 1989
[42] M AMahmoud andM A Abou Zaid ldquoDynamic response of abeam with a crack subject to a moving massrdquo Journal of Soundand Vibration vol 256 no 4 pp 591ndash603 2002
[43] V K Garg Dynamics of Railway Vehicle Systems AcademicPress 1994
The Scientific World Journal 15
[44] T-LWang andDHuang ldquoCable-stayed bridge vibration due toroad surface roughnessrdquo Journal of Structural Engineering vol118 no 5 pp 1354ndash1374 1992
[45] Y-B Yang and B-H Lin ldquoVehicle-bridge interaction analysisby dynamic condensation methodrdquo Journal of Structural Engi-neering vol 121 no 11 pp 1636ndash1643 1995
[46] Y B Yang J D Yau and L C Hsu ldquoVibration of simple beamsdue to trainsmoving at high speedsrdquoEngineering Structures vol19 no 11 pp 936ndash943 1997
[47] B Tabarrok and E Esmailzadeh ldquoInduced vibration of bridgestransversed by moving vehiclesrdquo Transactions of the CanadianSociety for Mechanical Engineering B vol 24 no 1 pp 191ndash1982000
[48] C Liu T-L Wang and D Huang ldquoImpact study for multi-girder bridge based on correlated road roughnessrdquo StructuralEngineering and Mechanics vol 11 no 3 pp 259ndash272 2001
[49] K H Chu V K Garg and T L Wang ldquoImpact in railwayprestressed concrete bridgesrdquo Journal of Structural Engineeringvol 112 no 5 pp 1036ndash1051 1986
[50] T-LWang V K Garg and K-H Chu ldquoRailway bridgevehicleinteraction studies with new vehicle modelrdquo Journal of Struc-tural Engineering vol 117 no 7 pp 2099ndash2116 1991
[51] H Xia Y L Xu and T H T Chan ldquoDynamic interaction oflong suspension bridges with running trainsrdquo Journal of Soundand Vibration vol 237 no 2 pp 263ndash280 2000
[52] Q-L Zhang A Vrouwenvelder and J Wardenier ldquoNumericalsimulation of train-bridge interactive dynamicsrdquo Computersand Structures vol 79 no 10 pp 1059ndash1075 2001
[53] H Xia N Zhang and G de Roeck ldquoDynamic analysis of highspeed railway bridge under articulated trainsrdquo Computers andStructures vol 81 no 26-27 pp 2467ndash2478 2003
[54] G Diana F Cheli A Collina R Corradi and S MelzildquoThe development of a numerical model for railway vehiclescomfort assessment through comparison with experimentalmeasurementsrdquoVehicle SystemDynamics vol 38 no 3 pp 165ndash183 2002
[55] Q Li Y L Xu D J Wu and Z W Chen ldquoComputer-aided nonlinear vehicle-bridge interaction analysisrdquo Journal ofVibration and Control vol 16 pp 1791ndash1816 2010
[56] A Wiriyachai K H Chu and V K Garg ldquoBridge impact dueto wheel and track irregularitiesrdquo Journal of the EngineeringMechanics Division vol 108 no 4 pp 648ndash666 1982
[57] L Fryba Dynamics of Railway Bridges Inst of Civil Engineers1996
[58] D Huang and T-L Wang ldquoImpact analysis of cable-stayedbridgesrdquo Computers and Structures vol 43 no 5 pp 897ndash9081992
[59] W M Zhai Vehicle-Track Coupling Dynamics Chinese RailwayPress Beijing China 2007
[60] M Olsson ldquoFinite element modal co-ordinate analysis ofstructures subjected to moving loadsrdquo Journal of Sound andVibration vol 99 no 1 pp 1ndash12 1985
[61] Y-B Yang C-H Chang and J-D Yau ldquoAn element foranalysing vehicle-bridge systems considering vehiclersquos pitchingeffectrdquo International Journal for NumericalMethods in Engineer-ing vol 46 no 7 pp 1031ndash1047 1999
[62] Y B Yang and Y S Wu ldquoA versatile element for analyzingvehicle-bridge interaction responserdquo Engineering Structuresvol 23 no 5 pp 452ndash469 2001
[63] F T K Au J J Wang and Y K Cheung ldquoImpact study ofcable-stayed bridge under railway traffic using various modelsrdquo
Journal of Sound and Vibration vol 240 no 3 pp 447ndash4652001
[64] Y Q Sun andMDhanasekar ldquoA dynamicmodel for the verticalinteraction of the rail track and wagon systemrdquo InternationalJournal of Solids and Structures vol 39 no 5 pp 1337ndash13592002
[65] K Henchi M Fafard M Talbot and G Dhatt ldquoAn efficientalgorithm for dynamic analysis of bridges under moving vehi-cles using a coupledmodal and physical components approachrdquoJournal of Sound and Vibration vol 212 no 4 pp 663ndash6831998
[66] Y L Xu and L YWang ldquoAnalytical study of wind-rain-inducedcable vibration SDOFmodelrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 91 no 1-2 pp 27ndash40 2003
[67] B Biondi G Muscolino and A Sofi ldquoA substructure approachfor the dynamic analysis of train-track-bridge systemrdquoComput-ers and Structures vol 83 no 28ndash30 pp 2271ndash2281 2005
[68] J L Humar and A H Kashif ldquoDynamic response analysis ofslab-type bridgesrdquo Journal of Structural Engineering vol 121 no1 pp 48ndash62 1995
[69] P Lou and Q-Y Zeng ldquoFormulation of equations of verticalmotion for vehicle-track-bridge systemrdquo Journal of the ChinaRailway Society vol 26 no 5 p 71 2004
[70] O Coussy M Said and J-P van Hoove ldquoThe influence ofrandom surface irregularities on the dynamic response ofbridges under suspended moving loadsrdquo Journal of Sound andVibration vol 130 no 2 pp 313ndash320 1989
[71] E S Hwang and A S Nowak ldquoSimulation of dynamic loadfor bridgesrdquo Journal of Structural Engineering vol 117 pp 1413ndash1434 1991
[72] F Yang and G A Fonder ldquoAn iterative solution methodfor dynamic response of bridge-vehicles systemsrdquo EarthquakeEngineering and Structural Dynamics vol 25 pp 195ndash215 1996
[73] W Zhai and Z Cai ldquoDynamic interaction between a lumpedmass vehicle and a discretely supported continuous rail trackrdquoComputers and Structures vol 63 no 5 pp 987ndash997 1997
[74] W M Zhai and C B Cai ldquoTraintrackbridge dynamic inter-actions simulation and applicationsrdquo Vehicle System Dynamicsvol 37 pp 653ndash665 2003
[75] D Bruno F Greco and P Lonetti ldquoDynamic impact analysis oflong span cable-stayed bridges under moving loadsrdquo Engineer-ing Structures vol 30 no 4 pp 1160ndash1177 2008
[76] X D Song D J Wu and Q Li ldquoDynamic impact analysisof double-tower cable-stayed maglev bridges using a simplemodelrdquo Journal of Bridge Engineering vol 19 pp 34ndash43 2014
[77] Y-S Wu and Y-B Yang ldquoSteady-state response and ridingcomfort of trains moving over a series of simply supportedbridgesrdquoEngineering Structures vol 25 no 2 pp 251ndash265 2003
[78] P Antolin N Zhang J M Goicolea H Xia M A Astiz andJ Oliva ldquoConsideration of nonlinear wheel-rail contact forcesfor dynamic vehicle-bridge interaction in high-speed railwaysrdquoJournal of Sound and Vibration vol 332 no 5 pp 1231ndash12512013
[79] W H Guo and Y L Xu ldquoFully computerized approach to studycable-stayed bridge-vehicle interactionrdquo Journal of Sound andVibration vol 248 no 4 pp 745ndash761 2001
[80] Y L Xu and W H Guo ldquoDynamic behaviour of high-sidedroad vehicles subject to a sudden crosswind gustrdquo Wind andStructures vol 6 no 5 pp 325ndash346 2003
[81] Y L Xu and W H Guo ldquoDynamic analysis of coupled roadvehicle and cable-stayed bridge systems under turbulent windrdquoEngineering Structures vol 25 no 4 pp 473ndash486 2003
16 The Scientific World Journal
[82] C S Cai and S R Chen ldquoFramework of vehicle-bridge-winddynamic analysisrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 92 no 7-8 pp 579ndash607 2004
[83] S R Chen and J Wu ldquoModeling stochastic live load for long-span bridge based on microscopic traffic flow simulationrdquoComputers and Structures vol 89 no 9-10 pp 813ndash824 2011
[84] A V Paultre B Yang L A Bergman and C A Tan ldquoBridgedynamics and dynamic amplification factorsmdasha review ofanalytical and experimental findingsrdquoCanadian Journal of CivilEngineering vol 19 no 2 pp 260ndash278 1992
[85] H Honda Y Kajikawa and T Kobori ldquoSpectra of road surfaceroughness of bridgesrdquo Journal of the Structural Division vol 108pp 1956ndash1966 1982
[86] M J Inbanathan and M Wieland ldquoBridge vibrations dueto vehicle moving over rough surfacerdquo Journal of StructuralEngineering vol 113 no 9 pp 1994ndash2008 1987
[87] T-LWang andDHuang ldquoCable-stayed bridge vibration due toroad surface roughnessrdquo Journal of Structural Engineering vol118 no 5 pp 1354ndash1374 1992
[88] P K Chatterjee T K Datta and C S Surana ldquoVibrationsuspension bridges under vehicular movementrdquo Journal ofStructural Engineering vol 120 no 3 pp 681ndash703 1994
[89] D Chang and H Lee ldquoImpact factors for simple-span highwaygirder bridgesrdquo Journal of Structural Engineering vol 120 no 3pp 704ndash715 1994
[90] T-C Pan and J Li ldquoDynamic vehicle element method for tran-sient response of coupled vehicle-structure systemsrdquo Journal ofStructural Engineering vol 128 no 2 pp 214ndash223 2002
[91] C J Dodds and J D Robson ldquoThe description of road surfaceroughnessrdquo Journal of Sound Vibration vol 31 no 2 pp 175ndash183 1973
[92] D Huang T-L Wang and M Shahawy ldquoImpact studies ofmultigirder concrete bridgesrdquo Journal of Structural Engineeringvol 119 no 8 pp 2387ndash2402 1993
[93] A G Davenport ldquoBuffeting of a suspension bridge by stormwindrdquo Journal of Structural Division vol 88 pp 233ndash268 1962
[94] R H Scanlan ldquoThe action of flexible bridges under wind Iflutter theoryrdquo Journal of Sound and Vibration vol 60 no 2pp 187ndash199 1978
[95] Q Ding and P K K Lee ldquoComputer simulation of buffetingactions of suspension bridges under turbulentwindrdquoComputersand Structures vol 76 no 6 pp 787ndash797 2000
[96] V Boonyapinyo T Miyata and H Yamada ldquoAdvanced aerody-namic analysis of suspension bridges by state-space approachrdquoJournal of Structural Engineering vol 125 no 12 pp 1357ndash13661999
[97] Y-H Chen and C-Y Li ldquoDynamic response of elevated high-speed railwayrdquo Journal of Bridge Engineering vol 5 no 2 pp124ndash130 2000
[98] X Chen M Matsumoto and A Kareem ldquoTime domainflutter and buffeting response analysis of bridgesrdquo Journal ofEngineering Mechanics vol 126 no 1 pp 7ndash16 2000
[99] X Chen and A Kareem ldquoEquivalent static wind loads forbuffeting response of bridgesrdquo Journal of Structural Engineeringvol 127 no 12 pp 1467ndash1475 2001
[100] S R Chen and C S Cai ldquoEvolution of long-span bridgeresponse to wind-numerical simulation and discussionrdquo Com-puters and Structures vol 81 no 21 pp 2055ndash2066 2003
[101] Y L XuHXia andQ S Yan ldquoDynamic response of suspensionbridge to high wind and running trainrdquo Journal of BridgeEngineering vol 8 no 1 pp 46ndash55 2003
[102] A Guo Y L Xu and H Li ldquoDynamic performance of cable-stayed bridge tower with multi-stage pendulum mass damperunderwind excitations-II experimentrdquoEarthquake Engineeringand Engineering Vibration vol 6 no 4 pp 417ndash424 2007
[103] E Simiu and R H ScanlanWind Effects on Structures 1996[104] Y Cao H Xiang and Y Zhou ldquoSimulation of stochastic wind
velocity field on long-span bridgesrdquo Journal of EngineeringMechanics vol 126 no 1 pp 1ndash6 2000
[105] M Shinozuka and C-M Jan ldquoDigital simulation of randomprocesses and its applicationsrdquo Journal of Sound and Vibrationvol 25 no 1 pp 111ndash128 1972
[106] K M Shum Lateral and torsional vibration control of long spanbridge deck using novel tuned liquid column dampers [PhDthesis] Department of Civil and Structural Engineering TheHong Kong Polytechnic University 2004
[107] Y K Lin and J N Yang ldquoMultimode bridge response to windexcitationsrdquo Journal of EngineeringMechanics vol 109 no 2 pp586ndash603 1983
[108] Y S Lin Self-Excited Bridge Motion in Turbulent Wind 1978[109] D T Lau M S Cheung and S H Cheng ldquo3D flutter analysis
of bridges by spline finite-strip methodrdquo Journal of StructuralEngineering vol 126 no 10 pp 1246ndash1254 2000
[110] L A Balzer ldquoAtmospheric turbulence encountered by high-speed ground transport vehiclesrdquo Journal of Mechanical Engi-neering Science vol 19 pp 227ndash235 1977
[111] R K Cooper ldquoAtmospheric turbulence with respect to movingground vehiclesrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 17 no 2 pp 215ndash238 1984
[112] C J Baker ldquoA simplified analysis of various types of wind-induced road vehicle accidentsrdquo Journal of Wind Engineeringand Industrial Aerodynamics vol 22 no 1 pp 69ndash85 1986
[113] C J Baker ldquoMeasures to control vehicle movement at exposedsites during windy periodsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 25 no 2 pp 151ndash161 1987
[114] C J Baker ldquoHigh sided articulated road vehicles in strong crosswindsrdquo Journal of Wind Engineering and Industrial Aerodynam-ics vol 31 no 1 pp 67ndash85 1988
[115] C J Baker ldquoGround vehicles in high cross winds part I steadyaerodynamic forcesrdquo Journal of Fluids and Structures vol 5 no1 pp 69ndash90 1991
[116] C J Baker ldquoGround vehicles in high cross winds part IIunsteady aerodynamic forcesrdquo Journal of Fluids and Structuresvol 5 no 1 pp 91ndash111 1991
[117] Y L Xu and Q S Ding ldquoInteraction of railway vehicles withtrack in cross-windsrdquo Journal of Fluids and Structures vol 22no 3 pp 295ndash314 2006
[118] S R Chen and J Wu ldquoDynamic performance simulation oflong-span bridge under combined loads of stochastic traffic andwindrdquo Journal of Bridge Engineering vol 15 no 3 pp 219ndash2302010
[119] Y Li S Qiang H Liao and Y L Xu ldquoDynamics of wind-rail vehicle-bridge systemsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 93 no 6 pp 483ndash507 2005
[120] Y L Xu N Zhang and H Xia ldquoVibration of coupled trainand cable-stayed bridge systems in cross windsrdquo EngineeringStructures vol 26 no 10 pp 1389ndash1406 2004
[121] W W Guo Y L Xu H Xia W S Zhang and K M ShumldquoDynamic response of suspension bridge to typhoon and trainsII numerical resultsrdquo Journal of Structural Engineering vol 133no 1 pp 12ndash21 2007
The Scientific World Journal 17
[122] Y L Li H Y Xiang B Wang Y L Xu and S Z QiangldquoDynamic analysis of wind-vehicle-bridge coupling systemduring the meeting of two trainsrdquo Advances in StructuralEngineering vol 16 pp 1663ndash1670 2013
[123] F Dorigatti M Sterling D Rocchi et al ldquoWind tunnelmeasurements of crosswind loads on high sided vehicles overlong span bridgesrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 107 pp 214ndash224 2012
[124] L D Zhu L Li Y L Xu and Q Zhu ldquoWind tunnel investi-gations of aerodynamic coefficients of road vehicles on bridgedeckrdquo Journal of Fluids and Structures vol 30 pp 35ndash50 2012
[125] Y Han J X Hu C S Cai Z Q Chen and C G LildquoExperimental and numerical studies of aerodynamic forces onvehicles and bridgesrdquoWind and Structures vol 17 pp 163ndash1842013
[126] ASCE ldquoCommittee on fatigue and fracture reliability of thecommittee on structural safety and reliability of the structuraldivision fatigue reliability 1ndash4rdquo Journal of Structural Engineer-ing vol 108 pp 3ndash88 1982
[127] BS ldquoBS5400 part 10 code of practice for fatiguerdquo BritishStandard Institute 1980
[128] BS ldquoBS7608 code of practice for fatigue design and assessmentof steel structuresrdquo British Standard Institute 1993
[129] AASHTOGuide Specifications for Fatigue Evaluation of ExistingSteel Bridges 1990
[130] AASHTO Guide Manual for Condition Evaluation and Loadand Resistance Factor Rating (LRFR) of Highway Bridges 2003
[131] S R Chen C S Cai and B Wolshon ldquoFrom normal operationto evacuation Single-vehicle safety under adverse weathertopographic and operational conditionsrdquo Natural HazardsReview vol 10 no 2 pp 68ndash76 2009
[132] S R Chen and C S Cai ldquoEquivalent wheel load approach forslender cable-stayed bridge fatigue assessment under traffic andwind feasibility studyrdquo Journal of Bridge Engineering vol 12 no6 pp 755ndash764 2007
[44] T-LWang andDHuang ldquoCable-stayed bridge vibration due toroad surface roughnessrdquo Journal of Structural Engineering vol118 no 5 pp 1354ndash1374 1992
[45] Y-B Yang and B-H Lin ldquoVehicle-bridge interaction analysisby dynamic condensation methodrdquo Journal of Structural Engi-neering vol 121 no 11 pp 1636ndash1643 1995
[46] Y B Yang J D Yau and L C Hsu ldquoVibration of simple beamsdue to trainsmoving at high speedsrdquoEngineering Structures vol19 no 11 pp 936ndash943 1997
[47] B Tabarrok and E Esmailzadeh ldquoInduced vibration of bridgestransversed by moving vehiclesrdquo Transactions of the CanadianSociety for Mechanical Engineering B vol 24 no 1 pp 191ndash1982000
[48] C Liu T-L Wang and D Huang ldquoImpact study for multi-girder bridge based on correlated road roughnessrdquo StructuralEngineering and Mechanics vol 11 no 3 pp 259ndash272 2001
[49] K H Chu V K Garg and T L Wang ldquoImpact in railwayprestressed concrete bridgesrdquo Journal of Structural Engineeringvol 112 no 5 pp 1036ndash1051 1986
[50] T-LWang V K Garg and K-H Chu ldquoRailway bridgevehicleinteraction studies with new vehicle modelrdquo Journal of Struc-tural Engineering vol 117 no 7 pp 2099ndash2116 1991
[51] H Xia Y L Xu and T H T Chan ldquoDynamic interaction oflong suspension bridges with running trainsrdquo Journal of Soundand Vibration vol 237 no 2 pp 263ndash280 2000
[52] Q-L Zhang A Vrouwenvelder and J Wardenier ldquoNumericalsimulation of train-bridge interactive dynamicsrdquo Computersand Structures vol 79 no 10 pp 1059ndash1075 2001
[53] H Xia N Zhang and G de Roeck ldquoDynamic analysis of highspeed railway bridge under articulated trainsrdquo Computers andStructures vol 81 no 26-27 pp 2467ndash2478 2003
[54] G Diana F Cheli A Collina R Corradi and S MelzildquoThe development of a numerical model for railway vehiclescomfort assessment through comparison with experimentalmeasurementsrdquoVehicle SystemDynamics vol 38 no 3 pp 165ndash183 2002
[55] Q Li Y L Xu D J Wu and Z W Chen ldquoComputer-aided nonlinear vehicle-bridge interaction analysisrdquo Journal ofVibration and Control vol 16 pp 1791ndash1816 2010
[56] A Wiriyachai K H Chu and V K Garg ldquoBridge impact dueto wheel and track irregularitiesrdquo Journal of the EngineeringMechanics Division vol 108 no 4 pp 648ndash666 1982
[57] L Fryba Dynamics of Railway Bridges Inst of Civil Engineers1996
[58] D Huang and T-L Wang ldquoImpact analysis of cable-stayedbridgesrdquo Computers and Structures vol 43 no 5 pp 897ndash9081992
[59] W M Zhai Vehicle-Track Coupling Dynamics Chinese RailwayPress Beijing China 2007
[60] M Olsson ldquoFinite element modal co-ordinate analysis ofstructures subjected to moving loadsrdquo Journal of Sound andVibration vol 99 no 1 pp 1ndash12 1985
[61] Y-B Yang C-H Chang and J-D Yau ldquoAn element foranalysing vehicle-bridge systems considering vehiclersquos pitchingeffectrdquo International Journal for NumericalMethods in Engineer-ing vol 46 no 7 pp 1031ndash1047 1999
[62] Y B Yang and Y S Wu ldquoA versatile element for analyzingvehicle-bridge interaction responserdquo Engineering Structuresvol 23 no 5 pp 452ndash469 2001
[63] F T K Au J J Wang and Y K Cheung ldquoImpact study ofcable-stayed bridge under railway traffic using various modelsrdquo
Journal of Sound and Vibration vol 240 no 3 pp 447ndash4652001
[64] Y Q Sun andMDhanasekar ldquoA dynamicmodel for the verticalinteraction of the rail track and wagon systemrdquo InternationalJournal of Solids and Structures vol 39 no 5 pp 1337ndash13592002
[65] K Henchi M Fafard M Talbot and G Dhatt ldquoAn efficientalgorithm for dynamic analysis of bridges under moving vehi-cles using a coupledmodal and physical components approachrdquoJournal of Sound and Vibration vol 212 no 4 pp 663ndash6831998
[66] Y L Xu and L YWang ldquoAnalytical study of wind-rain-inducedcable vibration SDOFmodelrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 91 no 1-2 pp 27ndash40 2003
[67] B Biondi G Muscolino and A Sofi ldquoA substructure approachfor the dynamic analysis of train-track-bridge systemrdquoComput-ers and Structures vol 83 no 28ndash30 pp 2271ndash2281 2005
[68] J L Humar and A H Kashif ldquoDynamic response analysis ofslab-type bridgesrdquo Journal of Structural Engineering vol 121 no1 pp 48ndash62 1995
[69] P Lou and Q-Y Zeng ldquoFormulation of equations of verticalmotion for vehicle-track-bridge systemrdquo Journal of the ChinaRailway Society vol 26 no 5 p 71 2004
[70] O Coussy M Said and J-P van Hoove ldquoThe influence ofrandom surface irregularities on the dynamic response ofbridges under suspended moving loadsrdquo Journal of Sound andVibration vol 130 no 2 pp 313ndash320 1989
[71] E S Hwang and A S Nowak ldquoSimulation of dynamic loadfor bridgesrdquo Journal of Structural Engineering vol 117 pp 1413ndash1434 1991
[72] F Yang and G A Fonder ldquoAn iterative solution methodfor dynamic response of bridge-vehicles systemsrdquo EarthquakeEngineering and Structural Dynamics vol 25 pp 195ndash215 1996
[73] W Zhai and Z Cai ldquoDynamic interaction between a lumpedmass vehicle and a discretely supported continuous rail trackrdquoComputers and Structures vol 63 no 5 pp 987ndash997 1997
[74] W M Zhai and C B Cai ldquoTraintrackbridge dynamic inter-actions simulation and applicationsrdquo Vehicle System Dynamicsvol 37 pp 653ndash665 2003
[75] D Bruno F Greco and P Lonetti ldquoDynamic impact analysis oflong span cable-stayed bridges under moving loadsrdquo Engineer-ing Structures vol 30 no 4 pp 1160ndash1177 2008
[76] X D Song D J Wu and Q Li ldquoDynamic impact analysisof double-tower cable-stayed maglev bridges using a simplemodelrdquo Journal of Bridge Engineering vol 19 pp 34ndash43 2014
[77] Y-S Wu and Y-B Yang ldquoSteady-state response and ridingcomfort of trains moving over a series of simply supportedbridgesrdquoEngineering Structures vol 25 no 2 pp 251ndash265 2003
[78] P Antolin N Zhang J M Goicolea H Xia M A Astiz andJ Oliva ldquoConsideration of nonlinear wheel-rail contact forcesfor dynamic vehicle-bridge interaction in high-speed railwaysrdquoJournal of Sound and Vibration vol 332 no 5 pp 1231ndash12512013
[79] W H Guo and Y L Xu ldquoFully computerized approach to studycable-stayed bridge-vehicle interactionrdquo Journal of Sound andVibration vol 248 no 4 pp 745ndash761 2001
[80] Y L Xu and W H Guo ldquoDynamic behaviour of high-sidedroad vehicles subject to a sudden crosswind gustrdquo Wind andStructures vol 6 no 5 pp 325ndash346 2003
[81] Y L Xu and W H Guo ldquoDynamic analysis of coupled roadvehicle and cable-stayed bridge systems under turbulent windrdquoEngineering Structures vol 25 no 4 pp 473ndash486 2003
16 The Scientific World Journal
[82] C S Cai and S R Chen ldquoFramework of vehicle-bridge-winddynamic analysisrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 92 no 7-8 pp 579ndash607 2004
[83] S R Chen and J Wu ldquoModeling stochastic live load for long-span bridge based on microscopic traffic flow simulationrdquoComputers and Structures vol 89 no 9-10 pp 813ndash824 2011
[84] A V Paultre B Yang L A Bergman and C A Tan ldquoBridgedynamics and dynamic amplification factorsmdasha review ofanalytical and experimental findingsrdquoCanadian Journal of CivilEngineering vol 19 no 2 pp 260ndash278 1992
[85] H Honda Y Kajikawa and T Kobori ldquoSpectra of road surfaceroughness of bridgesrdquo Journal of the Structural Division vol 108pp 1956ndash1966 1982
[86] M J Inbanathan and M Wieland ldquoBridge vibrations dueto vehicle moving over rough surfacerdquo Journal of StructuralEngineering vol 113 no 9 pp 1994ndash2008 1987
[87] T-LWang andDHuang ldquoCable-stayed bridge vibration due toroad surface roughnessrdquo Journal of Structural Engineering vol118 no 5 pp 1354ndash1374 1992
[88] P K Chatterjee T K Datta and C S Surana ldquoVibrationsuspension bridges under vehicular movementrdquo Journal ofStructural Engineering vol 120 no 3 pp 681ndash703 1994
[89] D Chang and H Lee ldquoImpact factors for simple-span highwaygirder bridgesrdquo Journal of Structural Engineering vol 120 no 3pp 704ndash715 1994
[90] T-C Pan and J Li ldquoDynamic vehicle element method for tran-sient response of coupled vehicle-structure systemsrdquo Journal ofStructural Engineering vol 128 no 2 pp 214ndash223 2002
[91] C J Dodds and J D Robson ldquoThe description of road surfaceroughnessrdquo Journal of Sound Vibration vol 31 no 2 pp 175ndash183 1973
[92] D Huang T-L Wang and M Shahawy ldquoImpact studies ofmultigirder concrete bridgesrdquo Journal of Structural Engineeringvol 119 no 8 pp 2387ndash2402 1993
[93] A G Davenport ldquoBuffeting of a suspension bridge by stormwindrdquo Journal of Structural Division vol 88 pp 233ndash268 1962
[94] R H Scanlan ldquoThe action of flexible bridges under wind Iflutter theoryrdquo Journal of Sound and Vibration vol 60 no 2pp 187ndash199 1978
[95] Q Ding and P K K Lee ldquoComputer simulation of buffetingactions of suspension bridges under turbulentwindrdquoComputersand Structures vol 76 no 6 pp 787ndash797 2000
[96] V Boonyapinyo T Miyata and H Yamada ldquoAdvanced aerody-namic analysis of suspension bridges by state-space approachrdquoJournal of Structural Engineering vol 125 no 12 pp 1357ndash13661999
[97] Y-H Chen and C-Y Li ldquoDynamic response of elevated high-speed railwayrdquo Journal of Bridge Engineering vol 5 no 2 pp124ndash130 2000
[98] X Chen M Matsumoto and A Kareem ldquoTime domainflutter and buffeting response analysis of bridgesrdquo Journal ofEngineering Mechanics vol 126 no 1 pp 7ndash16 2000
[99] X Chen and A Kareem ldquoEquivalent static wind loads forbuffeting response of bridgesrdquo Journal of Structural Engineeringvol 127 no 12 pp 1467ndash1475 2001
[100] S R Chen and C S Cai ldquoEvolution of long-span bridgeresponse to wind-numerical simulation and discussionrdquo Com-puters and Structures vol 81 no 21 pp 2055ndash2066 2003
[101] Y L XuHXia andQ S Yan ldquoDynamic response of suspensionbridge to high wind and running trainrdquo Journal of BridgeEngineering vol 8 no 1 pp 46ndash55 2003
[102] A Guo Y L Xu and H Li ldquoDynamic performance of cable-stayed bridge tower with multi-stage pendulum mass damperunderwind excitations-II experimentrdquoEarthquake Engineeringand Engineering Vibration vol 6 no 4 pp 417ndash424 2007
[103] E Simiu and R H ScanlanWind Effects on Structures 1996[104] Y Cao H Xiang and Y Zhou ldquoSimulation of stochastic wind
velocity field on long-span bridgesrdquo Journal of EngineeringMechanics vol 126 no 1 pp 1ndash6 2000
[105] M Shinozuka and C-M Jan ldquoDigital simulation of randomprocesses and its applicationsrdquo Journal of Sound and Vibrationvol 25 no 1 pp 111ndash128 1972
[106] K M Shum Lateral and torsional vibration control of long spanbridge deck using novel tuned liquid column dampers [PhDthesis] Department of Civil and Structural Engineering TheHong Kong Polytechnic University 2004
[107] Y K Lin and J N Yang ldquoMultimode bridge response to windexcitationsrdquo Journal of EngineeringMechanics vol 109 no 2 pp586ndash603 1983
[108] Y S Lin Self-Excited Bridge Motion in Turbulent Wind 1978[109] D T Lau M S Cheung and S H Cheng ldquo3D flutter analysis
of bridges by spline finite-strip methodrdquo Journal of StructuralEngineering vol 126 no 10 pp 1246ndash1254 2000
[110] L A Balzer ldquoAtmospheric turbulence encountered by high-speed ground transport vehiclesrdquo Journal of Mechanical Engi-neering Science vol 19 pp 227ndash235 1977
[111] R K Cooper ldquoAtmospheric turbulence with respect to movingground vehiclesrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 17 no 2 pp 215ndash238 1984
[112] C J Baker ldquoA simplified analysis of various types of wind-induced road vehicle accidentsrdquo Journal of Wind Engineeringand Industrial Aerodynamics vol 22 no 1 pp 69ndash85 1986
[113] C J Baker ldquoMeasures to control vehicle movement at exposedsites during windy periodsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 25 no 2 pp 151ndash161 1987
[114] C J Baker ldquoHigh sided articulated road vehicles in strong crosswindsrdquo Journal of Wind Engineering and Industrial Aerodynam-ics vol 31 no 1 pp 67ndash85 1988
[115] C J Baker ldquoGround vehicles in high cross winds part I steadyaerodynamic forcesrdquo Journal of Fluids and Structures vol 5 no1 pp 69ndash90 1991
[116] C J Baker ldquoGround vehicles in high cross winds part IIunsteady aerodynamic forcesrdquo Journal of Fluids and Structuresvol 5 no 1 pp 91ndash111 1991
[117] Y L Xu and Q S Ding ldquoInteraction of railway vehicles withtrack in cross-windsrdquo Journal of Fluids and Structures vol 22no 3 pp 295ndash314 2006
[118] S R Chen and J Wu ldquoDynamic performance simulation oflong-span bridge under combined loads of stochastic traffic andwindrdquo Journal of Bridge Engineering vol 15 no 3 pp 219ndash2302010
[119] Y Li S Qiang H Liao and Y L Xu ldquoDynamics of wind-rail vehicle-bridge systemsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 93 no 6 pp 483ndash507 2005
[120] Y L Xu N Zhang and H Xia ldquoVibration of coupled trainand cable-stayed bridge systems in cross windsrdquo EngineeringStructures vol 26 no 10 pp 1389ndash1406 2004
[121] W W Guo Y L Xu H Xia W S Zhang and K M ShumldquoDynamic response of suspension bridge to typhoon and trainsII numerical resultsrdquo Journal of Structural Engineering vol 133no 1 pp 12ndash21 2007
The Scientific World Journal 17
[122] Y L Li H Y Xiang B Wang Y L Xu and S Z QiangldquoDynamic analysis of wind-vehicle-bridge coupling systemduring the meeting of two trainsrdquo Advances in StructuralEngineering vol 16 pp 1663ndash1670 2013
[123] F Dorigatti M Sterling D Rocchi et al ldquoWind tunnelmeasurements of crosswind loads on high sided vehicles overlong span bridgesrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 107 pp 214ndash224 2012
[124] L D Zhu L Li Y L Xu and Q Zhu ldquoWind tunnel investi-gations of aerodynamic coefficients of road vehicles on bridgedeckrdquo Journal of Fluids and Structures vol 30 pp 35ndash50 2012
[125] Y Han J X Hu C S Cai Z Q Chen and C G LildquoExperimental and numerical studies of aerodynamic forces onvehicles and bridgesrdquoWind and Structures vol 17 pp 163ndash1842013
[126] ASCE ldquoCommittee on fatigue and fracture reliability of thecommittee on structural safety and reliability of the structuraldivision fatigue reliability 1ndash4rdquo Journal of Structural Engineer-ing vol 108 pp 3ndash88 1982
[127] BS ldquoBS5400 part 10 code of practice for fatiguerdquo BritishStandard Institute 1980
[128] BS ldquoBS7608 code of practice for fatigue design and assessmentof steel structuresrdquo British Standard Institute 1993
[129] AASHTOGuide Specifications for Fatigue Evaluation of ExistingSteel Bridges 1990
[130] AASHTO Guide Manual for Condition Evaluation and Loadand Resistance Factor Rating (LRFR) of Highway Bridges 2003
[131] S R Chen C S Cai and B Wolshon ldquoFrom normal operationto evacuation Single-vehicle safety under adverse weathertopographic and operational conditionsrdquo Natural HazardsReview vol 10 no 2 pp 68ndash76 2009
[132] S R Chen and C S Cai ldquoEquivalent wheel load approach forslender cable-stayed bridge fatigue assessment under traffic andwind feasibility studyrdquo Journal of Bridge Engineering vol 12 no6 pp 755ndash764 2007
[82] C S Cai and S R Chen ldquoFramework of vehicle-bridge-winddynamic analysisrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 92 no 7-8 pp 579ndash607 2004
[83] S R Chen and J Wu ldquoModeling stochastic live load for long-span bridge based on microscopic traffic flow simulationrdquoComputers and Structures vol 89 no 9-10 pp 813ndash824 2011
[84] A V Paultre B Yang L A Bergman and C A Tan ldquoBridgedynamics and dynamic amplification factorsmdasha review ofanalytical and experimental findingsrdquoCanadian Journal of CivilEngineering vol 19 no 2 pp 260ndash278 1992
[85] H Honda Y Kajikawa and T Kobori ldquoSpectra of road surfaceroughness of bridgesrdquo Journal of the Structural Division vol 108pp 1956ndash1966 1982
[86] M J Inbanathan and M Wieland ldquoBridge vibrations dueto vehicle moving over rough surfacerdquo Journal of StructuralEngineering vol 113 no 9 pp 1994ndash2008 1987
[87] T-LWang andDHuang ldquoCable-stayed bridge vibration due toroad surface roughnessrdquo Journal of Structural Engineering vol118 no 5 pp 1354ndash1374 1992
[88] P K Chatterjee T K Datta and C S Surana ldquoVibrationsuspension bridges under vehicular movementrdquo Journal ofStructural Engineering vol 120 no 3 pp 681ndash703 1994
[89] D Chang and H Lee ldquoImpact factors for simple-span highwaygirder bridgesrdquo Journal of Structural Engineering vol 120 no 3pp 704ndash715 1994
[90] T-C Pan and J Li ldquoDynamic vehicle element method for tran-sient response of coupled vehicle-structure systemsrdquo Journal ofStructural Engineering vol 128 no 2 pp 214ndash223 2002
[91] C J Dodds and J D Robson ldquoThe description of road surfaceroughnessrdquo Journal of Sound Vibration vol 31 no 2 pp 175ndash183 1973
[92] D Huang T-L Wang and M Shahawy ldquoImpact studies ofmultigirder concrete bridgesrdquo Journal of Structural Engineeringvol 119 no 8 pp 2387ndash2402 1993
[93] A G Davenport ldquoBuffeting of a suspension bridge by stormwindrdquo Journal of Structural Division vol 88 pp 233ndash268 1962
[94] R H Scanlan ldquoThe action of flexible bridges under wind Iflutter theoryrdquo Journal of Sound and Vibration vol 60 no 2pp 187ndash199 1978
[95] Q Ding and P K K Lee ldquoComputer simulation of buffetingactions of suspension bridges under turbulentwindrdquoComputersand Structures vol 76 no 6 pp 787ndash797 2000
[96] V Boonyapinyo T Miyata and H Yamada ldquoAdvanced aerody-namic analysis of suspension bridges by state-space approachrdquoJournal of Structural Engineering vol 125 no 12 pp 1357ndash13661999
[97] Y-H Chen and C-Y Li ldquoDynamic response of elevated high-speed railwayrdquo Journal of Bridge Engineering vol 5 no 2 pp124ndash130 2000
[98] X Chen M Matsumoto and A Kareem ldquoTime domainflutter and buffeting response analysis of bridgesrdquo Journal ofEngineering Mechanics vol 126 no 1 pp 7ndash16 2000
[99] X Chen and A Kareem ldquoEquivalent static wind loads forbuffeting response of bridgesrdquo Journal of Structural Engineeringvol 127 no 12 pp 1467ndash1475 2001
[100] S R Chen and C S Cai ldquoEvolution of long-span bridgeresponse to wind-numerical simulation and discussionrdquo Com-puters and Structures vol 81 no 21 pp 2055ndash2066 2003
[101] Y L XuHXia andQ S Yan ldquoDynamic response of suspensionbridge to high wind and running trainrdquo Journal of BridgeEngineering vol 8 no 1 pp 46ndash55 2003
[102] A Guo Y L Xu and H Li ldquoDynamic performance of cable-stayed bridge tower with multi-stage pendulum mass damperunderwind excitations-II experimentrdquoEarthquake Engineeringand Engineering Vibration vol 6 no 4 pp 417ndash424 2007
[103] E Simiu and R H ScanlanWind Effects on Structures 1996[104] Y Cao H Xiang and Y Zhou ldquoSimulation of stochastic wind
velocity field on long-span bridgesrdquo Journal of EngineeringMechanics vol 126 no 1 pp 1ndash6 2000
[105] M Shinozuka and C-M Jan ldquoDigital simulation of randomprocesses and its applicationsrdquo Journal of Sound and Vibrationvol 25 no 1 pp 111ndash128 1972
[106] K M Shum Lateral and torsional vibration control of long spanbridge deck using novel tuned liquid column dampers [PhDthesis] Department of Civil and Structural Engineering TheHong Kong Polytechnic University 2004
[107] Y K Lin and J N Yang ldquoMultimode bridge response to windexcitationsrdquo Journal of EngineeringMechanics vol 109 no 2 pp586ndash603 1983
[108] Y S Lin Self-Excited Bridge Motion in Turbulent Wind 1978[109] D T Lau M S Cheung and S H Cheng ldquo3D flutter analysis
of bridges by spline finite-strip methodrdquo Journal of StructuralEngineering vol 126 no 10 pp 1246ndash1254 2000
[110] L A Balzer ldquoAtmospheric turbulence encountered by high-speed ground transport vehiclesrdquo Journal of Mechanical Engi-neering Science vol 19 pp 227ndash235 1977
[111] R K Cooper ldquoAtmospheric turbulence with respect to movingground vehiclesrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 17 no 2 pp 215ndash238 1984
[112] C J Baker ldquoA simplified analysis of various types of wind-induced road vehicle accidentsrdquo Journal of Wind Engineeringand Industrial Aerodynamics vol 22 no 1 pp 69ndash85 1986
[113] C J Baker ldquoMeasures to control vehicle movement at exposedsites during windy periodsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 25 no 2 pp 151ndash161 1987
[114] C J Baker ldquoHigh sided articulated road vehicles in strong crosswindsrdquo Journal of Wind Engineering and Industrial Aerodynam-ics vol 31 no 1 pp 67ndash85 1988
[115] C J Baker ldquoGround vehicles in high cross winds part I steadyaerodynamic forcesrdquo Journal of Fluids and Structures vol 5 no1 pp 69ndash90 1991
[116] C J Baker ldquoGround vehicles in high cross winds part IIunsteady aerodynamic forcesrdquo Journal of Fluids and Structuresvol 5 no 1 pp 91ndash111 1991
[117] Y L Xu and Q S Ding ldquoInteraction of railway vehicles withtrack in cross-windsrdquo Journal of Fluids and Structures vol 22no 3 pp 295ndash314 2006
[118] S R Chen and J Wu ldquoDynamic performance simulation oflong-span bridge under combined loads of stochastic traffic andwindrdquo Journal of Bridge Engineering vol 15 no 3 pp 219ndash2302010
[119] Y Li S Qiang H Liao and Y L Xu ldquoDynamics of wind-rail vehicle-bridge systemsrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 93 no 6 pp 483ndash507 2005
[120] Y L Xu N Zhang and H Xia ldquoVibration of coupled trainand cable-stayed bridge systems in cross windsrdquo EngineeringStructures vol 26 no 10 pp 1389ndash1406 2004
[121] W W Guo Y L Xu H Xia W S Zhang and K M ShumldquoDynamic response of suspension bridge to typhoon and trainsII numerical resultsrdquo Journal of Structural Engineering vol 133no 1 pp 12ndash21 2007
The Scientific World Journal 17
[122] Y L Li H Y Xiang B Wang Y L Xu and S Z QiangldquoDynamic analysis of wind-vehicle-bridge coupling systemduring the meeting of two trainsrdquo Advances in StructuralEngineering vol 16 pp 1663ndash1670 2013
[123] F Dorigatti M Sterling D Rocchi et al ldquoWind tunnelmeasurements of crosswind loads on high sided vehicles overlong span bridgesrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 107 pp 214ndash224 2012
[124] L D Zhu L Li Y L Xu and Q Zhu ldquoWind tunnel investi-gations of aerodynamic coefficients of road vehicles on bridgedeckrdquo Journal of Fluids and Structures vol 30 pp 35ndash50 2012
[125] Y Han J X Hu C S Cai Z Q Chen and C G LildquoExperimental and numerical studies of aerodynamic forces onvehicles and bridgesrdquoWind and Structures vol 17 pp 163ndash1842013
[126] ASCE ldquoCommittee on fatigue and fracture reliability of thecommittee on structural safety and reliability of the structuraldivision fatigue reliability 1ndash4rdquo Journal of Structural Engineer-ing vol 108 pp 3ndash88 1982
[127] BS ldquoBS5400 part 10 code of practice for fatiguerdquo BritishStandard Institute 1980
[128] BS ldquoBS7608 code of practice for fatigue design and assessmentof steel structuresrdquo British Standard Institute 1993
[129] AASHTOGuide Specifications for Fatigue Evaluation of ExistingSteel Bridges 1990
[130] AASHTO Guide Manual for Condition Evaluation and Loadand Resistance Factor Rating (LRFR) of Highway Bridges 2003
[131] S R Chen C S Cai and B Wolshon ldquoFrom normal operationto evacuation Single-vehicle safety under adverse weathertopographic and operational conditionsrdquo Natural HazardsReview vol 10 no 2 pp 68ndash76 2009
[132] S R Chen and C S Cai ldquoEquivalent wheel load approach forslender cable-stayed bridge fatigue assessment under traffic andwind feasibility studyrdquo Journal of Bridge Engineering vol 12 no6 pp 755ndash764 2007
[122] Y L Li H Y Xiang B Wang Y L Xu and S Z QiangldquoDynamic analysis of wind-vehicle-bridge coupling systemduring the meeting of two trainsrdquo Advances in StructuralEngineering vol 16 pp 1663ndash1670 2013
[123] F Dorigatti M Sterling D Rocchi et al ldquoWind tunnelmeasurements of crosswind loads on high sided vehicles overlong span bridgesrdquo Journal of Wind Engineering and IndustrialAerodynamics vol 107 pp 214ndash224 2012
[124] L D Zhu L Li Y L Xu and Q Zhu ldquoWind tunnel investi-gations of aerodynamic coefficients of road vehicles on bridgedeckrdquo Journal of Fluids and Structures vol 30 pp 35ndash50 2012
[125] Y Han J X Hu C S Cai Z Q Chen and C G LildquoExperimental and numerical studies of aerodynamic forces onvehicles and bridgesrdquoWind and Structures vol 17 pp 163ndash1842013
[126] ASCE ldquoCommittee on fatigue and fracture reliability of thecommittee on structural safety and reliability of the structuraldivision fatigue reliability 1ndash4rdquo Journal of Structural Engineer-ing vol 108 pp 3ndash88 1982
[127] BS ldquoBS5400 part 10 code of practice for fatiguerdquo BritishStandard Institute 1980
[128] BS ldquoBS7608 code of practice for fatigue design and assessmentof steel structuresrdquo British Standard Institute 1993
[129] AASHTOGuide Specifications for Fatigue Evaluation of ExistingSteel Bridges 1990
[130] AASHTO Guide Manual for Condition Evaluation and Loadand Resistance Factor Rating (LRFR) of Highway Bridges 2003
[131] S R Chen C S Cai and B Wolshon ldquoFrom normal operationto evacuation Single-vehicle safety under adverse weathertopographic and operational conditionsrdquo Natural HazardsReview vol 10 no 2 pp 68ndash76 2009
[132] S R Chen and C S Cai ldquoEquivalent wheel load approach forslender cable-stayed bridge fatigue assessment under traffic andwind feasibility studyrdquo Journal of Bridge Engineering vol 12 no6 pp 755ndash764 2007
