influenceoffluidviscousdamperonthedynamicresponseof...

Research ArticleInfluence of Fluid Viscous Damper on the Dynamic Response ofSuspension Bridge under Random Traffic Load

Yue Zhao 1 Pingming Huang1 Guanxu Long1 Yangguang Yuan2 and Yamin Sun34

1School of Highway Changrsquoan University Xirsquoan 710064 China2School of Civil Engineering Xirsquoan University of Architecture and Technology Xirsquoan 710055 China3School of Architecture and Civil Engineering Xirsquoan University of Science and Technology Xirsquoan 710054 China4Postdoctoral Research Station on Civil Engineering Xirsquoan University of Science and Technology Xirsquoan 710054 China

Correspondence should be addressed to Yue Zhao unscmike163com

Received 30 October 2019 Accepted 24 April 2020 Published 26 May 2020

Academic Editor Marco Corradi

Copyright copy 2020 Yue Zhao et al )is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Fluid viscous dampers (FVDs) are widely used in long-span suspension bridges for earthquake resistance To analyze efficientlythe influences of FVDs on the dynamic response of a suspension bridge under high-intensity traffic flow a bridge-vehicle couplingmethod optimized by isoparametric mapping and improved binary search in this work was first developed and validatedAfterwards the traffic flow was simulated on the basis of monitored weigh-in-motion data )e dynamic responses of bridge wereanalyzed by the proposed method under different FVD parameters Results showed that FVDs could positively affect bridgedynamic response under traffic flow )e maximum accumulative longitudinal girder displacement longitudinal girder dis-placement and longitudinal pylon acceleration decreased substantially whereas the midspan girder bending moment pylonbending moment longitudinal pylon displacement and suspender force were less affected )e control efficiency of maximumlongitudinal girder displacement and accumulative girder displacement reached 3367 and 5771 longitudinal pylon ac-celeration and girder bending moment reached 3151 and 714 and the pylon longitudinal displacement pylon bendingmoment and suspender force were less than 3 )e increased damping coefficient and decreased velocity exponent can reducethe bridge dynamic response However when the velocity exponent was 01 an excessive damping coefficient brought littleimprovement and may lead to high-intensity work under traffic flow which will adversely affect component durability )ebenefits of low velocity exponent also reduced when the damping coefficient was high enough so if the velocity exponent has to beincreased the damping coefficient can be enlarged to fit with the velocity exponent )e installation of FVDs influences dynamicresponses of bridge structures in daily operations and this issue warrants investigation )us traffic load should be considered inFVD design because structural responses are perceptibly influenced by FVD parameters

1 Introduction

FVDs were first used in the machinery and military in-dustries On account of their excellent energy dissipatingcapacity they have been applied in structural vibrationcontrol under earthquakesWith structural vibration controlas a key problem in safety operation of long-span bridgesFVDs are installed on many bridges to mitigate vibrationand to serve as an effective seismic control device In 1989FVDs were first used on the Golden Bridge located in USAfor increasing its seismic resistance )e identification testorganized by the Highway Innovative Technology

Evaluation Center had been a valid reference in subsequentapplication and specification preparation of FVDs [1] )etechnology was first brought to China on the ChongqingEgongyan Yangtze River Bridge in 2000 and is now widelyused in Yangtze River bridges [2] Since FVDs are integratedwith the bridge they have various complex effects on bridgeresponses such as girder displacement pylon bendingmoment and expansion joint displacement)e designationof optimum FVD parameters focuses on the response underearthquake excitation and has been investigated compre-hensively )e research methods generally included nu-merical analyses shake table tests and data monitoring

HindawiAdvances in Civil EngineeringVolume 2020 Article ID 1857378 19 pageshttpsdoiorg10115520201857378

[3ndash5] Different techniques have been proposed by scholarsto ascertain the optimal parameters that can achieve idealstructural response [6ndash8]

Although FVDsrsquo effects under earthquake excitationhave been widely studied the dynamic responses of bridgeswhich are also vulnerable to vibrations excited by windtemperature and vehicles have not been thoroughly ex-amined )e most common dynamic response in the bridgelife cycle is the frequent low-speed reciprocating motionwhich is mainly caused by wind and traffic flow [9] )ecorresponding service life of bridge accessories such asexpansion joints is also influenced by this frequent recip-rocating motion FVDs can also absorb and dissipate energyimported from wind and traffic flow because of their dis-sipation capacity )e working conditions of supports ex-pansion joints and suspenders can be improved byconsidering the daily longitudinal girder displacement indetermining FVD parameters the frequent reciprocatingmotion of main girder also can be effectively improved byusing FVDs [10] For a long-span bridge the wind load maybe more remarkable than vehicle load in transverse responseof the bridge Research shows that the dynamic responses oflong-span suspension bridge under high winds and runningtrains are mainly influenced by strong winds when the windis strong but when the wind comes to low intensity trainload plays a decisive role in vertical displacement [11] Forthe bridge dynamic response under wind and traffic flow thewind load plays a major role in bridge lateral displacement)e vertical displacement is mainly controlled by the vehicleload [12] Furthermore scholars have made some studies onwind-induced vibration control and found that FVDs havedifferent influences on various response indicators whichhas an optimum interval for damping parameters [13 14] Inaddition to the wind-induced vibration the vehicle-inducedvibration of long-span bridges is also remarkable and theinfluences of FVDs are still indistinct )e daily movementof bridge structures under social vehicles is an importantfactor related to bridge accessoriesrsquo durability [5] Variedstructural response indexes present different variationtrends with changing parameters of FVDs such as accu-mulative girder displacement girder midspan bendingmoment pylon longitudinal acceleration and pylon bend-ingmoment)e specific influences of FVDs on the dynamicresponse of long-span suspension bridges under traffic loadwhich can be further referenced in the optimization of FVDparameters should hence be analyzed

Because the traffic load of the highway bridge is acomplicated random process with strong randomness ofvehicle type weight and quantity the calculation matrix ofrandom traffic flow-bridge coupling analysis is huge and willtake a lot of computing resources )us a high efficiencyrandom traffic flow-bridge analysis system needs to beestablished )e traditional vehicle-bridge coupling analysissystem which is based on a numerical method mainlyincludes the modal superposition method and full couplingtheory [15])emodal superposition method is more simpleand practical than the latter but the high-order modes ofstructures are hard to obtain thereby negatively affecting thecalculation precision [16] )e full coupling analytical

method has the advantages of clear physical meaning andhigh precision However this method requires huge amountof time and lots of computing resources when applied tolong-span bridges under large and highly random trafficflow )is paper developed a vehicle-bridge interactionsystem with high efficiency to analyze the dynamic responseof a bridge with FVDs under a large and highly randomtraffic flow )is system was established based on theinterhistory iteration method and optimized by iso-parametric mapping and improved binary search )en thetraffic flow was simulated in accordance with the monitoreddata Finally the responses of the large-span bridge undertraffic flow with different parameters of FVDs were studied

2 Establishment of Vehicle-BridgeInteraction System

21 Finite Element Model of Bridge )e bridge model isestablished by the space truss model in ANSYS 150 [17]Spine-beam double-beam and triple-beam models arewidely used in the integral analysis of long-span bridges andthe grillage model and multiscale model are also usedaccording to different requirements [18 19] Main cablesand suspenders are usually simulated by the Link10 elementwhich can simulate the elasticity and plasticity of thestructure )e pylon and girder are simplified into a two-node beam element and modeled by Beam4 element )erailings and deck paving are simulated by the Mass21 ele-ment )e entire structure is distributed into a number ofelements and connected through the node on the boundaryof adjacent elements Given the demand for refined simu-lation of the vehicle-bridge interaction a grillage model isutilized in this article to construct the prototype bridge toobtain detailed component response )e detailed infor-mation and feasibility of the model are proposed and val-idated in succeeding chapters

22 Dynamic Vehicle Models A vehicle is mainly composedof a car body wheels a shock mitigation system and adamped system connecting different components A vehicledynamic model needs to be established before the vehicle-bridge coupling analysis is conducted With these demandssome hypothesis is unavoidable in constructing the dynamicmodels )e mass of damper and spring components areignored compared with the quality of the vehicle bodyVehicle model is generally divided into different rigid bodiesthat are connected by axle mass blocks and by elastic anddamped components According to previous research thecommon Chinese highway vehicles can be divided into 17types of 5 categories in accordance with vehicle axle distanceaxle number axle load and vehicle load based on weigh-in-motion data and Chinese automobile model manual [20 21])e vehicle classification is shown in Table 1

)e dynamic models are also provided based on differentvehicle type Taking three-axle vehicle as an example thevehicle dynamic model of three-axle vehicle (double rearaxle) is shown in Figure 1 [20] )e rigid body is usuallydeemed to have 6 degrees of freedom As the vibration of the

2 Advances in Civil Engineering

vehicle in advancing direction is tiny compared to thedistance travelled the vibration of the vehicle body in ad-vancing direction has few effects on the bridge )e bridge ismainly influenced by the integral movement of the vehicle inadvancing direction In order to reduce the complexity ofvehicle model and the calculation the degree of freedom inadvancing is generally neglected in vehicle-bridge interac-tion analysis [22] )us 5 degrees of freedom (verticalhorizontal head nodding side rolling and head shaking) areconsidered for the integral vehicle Each wheel has 2 degreesof freedom (vertical and horizontal) For the trailer hori-zontal and head nodding degrees of freedom are neglected toderive the equation Each vehicle body has 3 degrees offreedom and each wheel has 1 degree of freedom

23 Load Distribution Based on Quadrilateral IsoparametricMapping For the dynamic analysis under large randomtraffic flow the positioning and loading of the contact pointbetween the vehicle and the bridge should be a crucial issueBridge decks are typically simulated by grillage solid or shellelements in most models )e wheel load is simplified to aconcentrated force acting on the bridge deck and thendistributed to four adjacent nodes by four-node

isoparametric mapping [23] which is the same as that ofdisplacement in a two-dimensional plane as shown inFigures 2 and 3

)e coordinate mapping equations are as follows

x 11139364

i1Ni(ξ η)xi

y 11139364

i1Ni(ξ η)yi

Ni(ξ η) 14

1 + ξξi( 1113857 1 + ηηi( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)

where x and y denote the coordinates of the force point of thewheel load P xi and yi denote the ith quadrilateral elementpointrsquos horizontal and vertical positions ξ and η denote thecorresponding values that x and y map from the quadri-lateral element to the parent element ξi and ηi denote knownquantities namely the ith nodersquos horizontal and verticalpositions of the parent element Ni(ξ η) denotes a functionof isoparametric point (ξ η) function value Ni denotes theith quadrilateral element pointrsquos load distribution coefficient

Table 1 Vehicle classification [20 21]

Type Model number Number of axles DescriptionT1 V1 V2 2 Sedan car and off-road vehicleT2 V3 V4 2 Microbus and midsize truckT3 V5 V6 2 Two-axle bus or truck

T4

V7 3 )ree-axle busV8 3 )ree-axle truck (double front axle)V9 3 )ree-axle truck (double rear axle)V10 4 Four-axle truck

T5

V11 3 )ree-axle trailerV12 4 Four-axle trailerV13 5 Five-axle trailer (double front axle)

V14 V15 5 Five-axle trailer (triple front axle with different axle distance)V16 V17 6 Six-axle trailer (different axle distance)

θvrZvr

K2vuL C2

vuL

K2vlL C2

vlL

Z2vaL

X

Z C3vlLK3

vlL

C3vuLK3

vuL

L2 L1

Z3vaL Z1

vaL

C1vlLK1

vlL

C1vuLK1

vuL

Driver seat

L3

(a)

b1b1

Zvr

Y

Z1vaR

C1vlRK1

vlR

C1vuR

K1vuRK1

vuLC1vuL

K1vlL C1

vlL

Z1vaL

ZY1vaL

K1yuL

C1yuLK1

ylLC1ylL

K1yuR

Y1vaRC1

yuR K1ylRC1ylR

h 1

h v

b2

Yvr

Φvr

(b)

Figure 1 Dynamic analysis vehicle model of three-axle vehicle (double rear axle) [20]

Advances in Civil Engineering 3

)e unknown quantities ξ and η of bilinear equation (1)can be solved by Newton iteration to obtain the distributioncoefficient Ni Afterwards the automatic loading of thewheel load can be realized)e iteration process is as follows

ξk+1

ηk+1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ξk

ηk

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ minus J ξk

ηk1113872 1113873minus 1

11139364

i1Ni ξk

ηk1113872 1113873xi minus x

11139364

i1Ni ξk

ηk1113872 1113873yi minus y

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

J

z 11139364i1 Ni(ξ η)xi1113960 1113961

zξz 1113936

4i1 Ni(ξ η)xi1113960 1113961

zη

z 11139364i1 Ni(ξ η)yi1113960 1113961

zξz 1113936

4i1 Ni(ξ η)yi1113960 1113961

zη

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(2)

where J denotes a Jacobian matrix ξk and ηk denote thecalculated value of the kth iteration )e iteration is stopped

when ξk+1minus ξk

ηk+1 minus ηk

lt ε with the iteration error ε set to 10minus8

)e position information of four adjacent nodes is in-dispensable for realizing the automatic loading of the wheelload P Binary search can effectively halve the scope infinding specific elements in ordered arrays )us it can beextended for searching the coordinate range of the wheelloading For example the process of searching the longi-tudinal wheel loading point x is shown in Figure 4 and thepositioning for the lateral loading point y is the same

)e detailed steps are as follows

① )e node coordinates of the lane on which the ve-hicles drove are imported and sorted in ascendingorder to form a one-dimensional matrix

② Initialization the value of ldquofrontrdquo is set to 1 the valueof ldquolastrdquo is set to the number of columns of the matrixformed in step 1

③ Calculation of ldquomidrdquo ldquomidrdquo ldquoroundrdquo((ldquofrontrdquo+ldquolastrdquo)2) ldquoroundrdquo means rounding off

④ )e search ends when ldquomidrdquo ldquolastrdquo )e upper limitof the wheel x-coordinate in the one-dimensionalmatrix is ldquomidrdquo and the lower limit is ldquomidrdquominus 1Otherwise whether the wheel x-coordinate is be-tween ldquofrontrdquo and ldquomidrdquo is determined If it happensthen ldquolastrdquo ldquomidrdquo otherwise ldquofrontrdquo ldquomidrdquo

⑤ Step 3 is executed until ldquomidrdquo ldquolastrdquo

Equation (1) can be solved by Newton iteration afterfinding four adjacent nodes of the wheel load P )en theinterpolated coefficient at each node can be obtained tocalculate the distribution load In order to reduce thecomputation of bridge-vehicle coupling only the results ofdisplacement and velocity of load acting point are outputAfterwards based on the known interpolated coefficient thevelocity and displacement of load acting point are calculatedto solve the bridge-vehicle interaction force

24 Road Surface Roughness Road surface roughness is animportant factor that could cause a vehiclersquos vertical vi-bration which is directly related to the safety and comfort ofthe vehicle Surface roughness can be described as animplementation of random processes and expressed by apower spectral density function )e recommended powerspectral density function is adopted in this study [24] )efunction Gd (n) is expressed as

Gd(n) Gd n0( 1113857n

n01113888 1113889

minusω

(3)

FF1 F2

F3F4

Figure 2 Contact point load distribution

P (x y)

1 (ndash1 ndash1) 2 (1 ndash1)

3 (1 1)4 (ndash1 1)η

ξ x

y

1 (x1 y1) 2 (x2 y2)

3 (x3 y3)4 (x4 y4)

Figure 3 Isoparametric mapping

Sort onendashdimensional matrix

Element nodecoordinates

Start

mid = round ((front + last)2)

mid = lastx ge t (front)

ampx le t (mid)

ANSYSmodel

Longitudinalcoordinate x of

wheelfront = 1last = length (t)

last = mid

front = mid

Yes

No

Upper limit is ldquomidrdquolower limit is ldquomid ndash 1rdquo

Finish

Yes

No

Figure 4 Flowchart of improved binary search


where Gd (n) denotes the power spectral density function(m3cycle) n denotes spatial frequency per unit length(mminus1) n0 denotes reference spatial frequency n0 01mminus1Gd (n0) denotes power spectral density under referencespatial frequency and ω denotes the frequency exponentand ω 2

Road surface roughness is assumed to be a zero-meanstationary Gaussian random process )us it can be gen-erated through Fourier inversion as follows

r(x) 1113944N

k1

2Gd nk( 1113857Δn1113969

cos 2πnkx + θk( 1113857 (4)

where θk denotes the random phase angle uniformly dis-tributed in [0 2π] nk denotes the kth reference spatialfrequency

)e wheel-bridge relationship is assumed to be the pointcontact in constructing the vehicle-bridge interaction sys-tem When the wheel comes into contact with the roadsurface the vertical displacement of vehicle and bridge is thesame and the bridgersquos vertical displacement is deemed ad-ditional surface roughness to the vehicle )us the addi-tional surface roughness caused by bridge displacement androad surface roughness is superposed to constitute theequivalent roughness which is inputted as the vertical ex-citation source For the ith wheel the formula consideringbridge displacement is expressed as

Zie Z

ib + ri(x) (5)

where Zie denotes the vertical equivalent roughness con-

sidering bridge displacement ri (x) denotes the surfaceroughness at the wheel-bridge contact point and Zi

b denotesthe displacement at wheel-bridge contact point

25 Motion Equations of Vehicle-Bridge Interaction )einterhistory iteration method is adopted to construct thehighway vehicle-bridge coupling analysis system given thetechniquersquos remarkable efficiency under high-intensitytraffic flow [25] Unlike time step iteration each step of theinterhistory method involves full-time calculation )ebridge dynamic response under external load can be com-puted by any commercial analysis software with severaliteration steps thereby addressing the limitation of theprogramming calculation in the traditional method

)e bridge and vehicle dynamic equations can beexpressed as follows

MveuroZv + Cv

_Zv + KvXv Fv

MbeuroZb + Cb

_Xb + KbXb Fb

⎧⎨

⎩ (6)

where Mv Cv and Kv denote the global mass damping andstiffness matrices of the train subsystem respectivelyMbCband Kb denote the global mass damping and stiffnessmatrices of the bridge subsystem respectively Xv and Xbdenote the displacement vectors of the train subsystem andthe bridge subsystem respectively _Zv and euroZv denote thevelocity vector and acceleration vector of vehicle respec-tively _Zb and euroZb denote the velocity vector and acceleration

vector of bridge respectively and Fv and Fb denote the forcevectors of vehicle and bridge respectively

Wheels are assumed to always be in contact with thebridge deck while the vehicle is moving so the shockmitigation system and suspension system are deformed withbridge vertical deformation and surface roughness )edynamic equations of the vehicle-bridge coupling systemcan be expressed as (7) )e interaction force between thevehicle and the bridge is a function of vehicle motion bridgemotion and road surface roughness stated by Zv Zb and i

Fv Fvi Zv _Zv euroZv Zb _Zb euroZb i1113872 1113873

Fb Fbi Zv _Zv euroZv Zb _Zb euroZb i1113872 1113873

⎧⎪⎨

⎪⎩(7)

)e coupling relationship between the vehicle and bridgesubsystems is established by simultaneously solving (6) and(7) and then solved by the interhistory iteration method)eiteration process is realized by MATLAB R2014a [26] whilethe bridge model constructed in ANSYS 150 is used tooutput the bridge response Each step of interhistory iter-ation for the bridge and the vehicle is separate and involvesfull-time calculation )us the method will occupy reducedcomputer storage generate accurate computations and havethe advantages of a clear aim and ease of operation )econvergence is checked by the interaction force between thevehicle and the bridge and defined as Fi

v minus Fiminus1v lt 01 )e

detailed process is plotted in Figure 5

3 Validation of the Established AnalysisSystem by Field Load Test

31 Prototype Bridge and the FE Model To validate the ve-hicle-bridge interaction analysis system and to analyze theinfluence of FVDs under random traffic flow a three-di-mensional finite element model of a long-span suspensionbridge is constructed in ANSYS 150 as the research object(Figure 6) )e bridge has a span of 896m a steel truss maingirder with the deck slab width of 26m and a rigid centralbuckle )e detailed information of the bridge is as follows

(1) )e sag-span ratio of main cable is 1 10 )e lon-gitudinal distance in each suspender is 128m )esteel girder adopts warren truss stiffening beam withthe height of 65m and width of 26m the minimumsegment length is 64m and the standard segmentlength is 128m )e pylon adopts a concrete portalframe structure One pylon is 1176m high andanother pylon is 1222m high

(2) )e main cable consists of 127 galvanized high-strength prefabricated parallel steel wire strandseach steel wire strand consists of 127 steel wires witha diameter of 51mm Suspender is pin jointed withmain cable and girder and each consists of 91 gal-vanized high-strength prefabricated parallel steelwires with a diameter of 52mm

(3) )e material parameters have been modifiedaccording to the measured features of the bridgeelastic modulus of main cable is 194times1011 Pa


elastic modulus of girder is 213 times1011 Pa elasticmodulus of pylon is 348times1010 Pa the density ofpylon is 2663 kgm3 the density of the truss elementis 8646 kgm3 and the density of main cable is8020 kgm3

)e pylon central buckle and main girder are modeledusing Beam4 element In order to simulate the lane loadfour longitudinal girders were built in the model Each laneload was exerted on the corresponding longitudinal girder)e main cables and suspenders are simulated using Link10element and the railings and deck paving are simulatedusingMass21 element)e boundary conditions of the pylonfoundation are simulated as fixed connection )e linedisplacement of the main cable at the anchorage point isconstrained in three directions )e simulation of the vertexand tangential point of the cable saddle are realized throughthe node coupling )e vertical and horizontal displacementof the end of the girder is limited and the rotational degree offreedom is released

)e damping effect is an output resistive force thatdepends on the relative velocity of the damping deviceEquation (8) is recommended to calculate the produceddamping force [27] FVDs are simulated using Combin37element As shown in Figure 7 the FVDs are longitudinallysettled on both sides of the midspan as a connection of thegirder and the pylon with two dampers on each side

F K|v|αsign(v) (8)

where F denotes the damping force K denotes the dampingcoefficient ] denotes the relative velocity between the endsof the damping device itself sign denotes the sign functionand α denotes the velocity exponent

)e data of the bridge were collected when the prototypebridge just finished the construction Existing researchshows that the conditions of road surface roughness of newbridges in China are most A grade in accordance with ISO8608 [24 28] So the road surface roughness is set to A gradeand generated based on specified parameters in ISO 8608 asshown in Figure 8

32 Accuracy Validation by Field Load Test )e numericalanalysis result is compared with the result of dynamic ex-periment to verify the accuracy of the proposed method)ebasis of the validation is the uniformity of theoretical modeland actual bridge )us the FE model in this article wasconstructed according to measured material properties andcable forces Referring to existing researches the accuracy ofFE model can be considered as acceptable when the errorsbetween the calculated and measured frequencies are lessthan 5 [29 30] )e measured frequencies of the bridge arecompared with the numerical results in Table 2 It can beseen that the numerical results are comparable to the

114m 896m

Main cable

Steel truss girder Midspan

Suspender

Approachbridge

GravityanchorCentral

buckle

214

Tunnelanchorage

208m

Figure 6 Information of bridge model (unit (m))

No

Start

Vehicle motion history

Vehicle-bridgeinteraction force history

Set bridge initialdisplacement as 0

Road surfaceroughness

Bridge motion history

Vehiclemotion state

Bridgemotion state

Yes

Finish

Bridgesubsystem

Vehiclesubsystem

Convergencecheck

Figure 5 Flowchart of interhistory iteration


measured results )e maximum frequency error is 437and all errors are less than 5 which proves the correctnessof the FE model

)en the dynamic strain and vertical acceleration of mid-span are measured to validate the proposed method )emeasuring point of the dynamic strain is set on the top chord ofthe steel trusses in the middle of midspan )e measuring pointof vertical acceleration is set on the bridge deck at the same place

)e driving position of trucks and measuring points areshown in Figure 9 the detailed vehicle dynamic parametersare shown in Table 3 [31] )ree test conditions are settledfor the validation (1) two three-axle trucks passing throughthe bridge at a constant speed of 20 kmh (2) two three-axletrucks passing through the bridge at a constant speed of30 kmh and (3) two three-axle trucks passing through thebridge at a constant speed of 50 kmh

Table 2 Comparison of the bridge frequencies

Vibrating direction Order Measured result Numerical result Error ()

Horizontal 1 0071 007 1412 0183 0175 437

Vertical

1 0131 0134 2292 0185 0182 1623 0292 0282 3424 0394 038 355

Torsional 1 0348 0342 1722 0434 0425 207

Combin 37 element

Link 10

Beam 4

Beam 4Fluid viscous damper

Node JNode I

Spring

Axialforce

Figure 7 Information of bridge FE model

Dec

k ro

ughn

ess (

mm

)

10

5

0

ndash5

ndash100 100 200 300 400 500

Position on bridge (m)600 700 800 900 1000

Figure 8 Road surface roughness of A grade


)e measured results are extracted and compared withthose obtained by the analytical solution on the basis of theproposed method )e comparison of dynamic strain resultsunder different vehicle velocities is shown in Figure 10 It can beseen that the changing trends of two curves are consistent inFigure 10 By comparison the peak value of the dynamic strainof 50kmh is higher than 20kmh and 30kmh)e peak valueof measured dynamic strain of 50 kmh is 3003 while that of20kmh and 30kmh is 266 and 279 It indicates that themaximum dynamic strain increases with the vehicle velocity InFigure 10(a) the error of peak values of two curves is 135 InFigure 10(b) the error of peak values of two curves is 112 InFigure 10(c) the error of peak values of two curves is 03)ecomparison results indicate that the validity and accuracy of theproposed method are acceptable

)e midspan acceleration of measured result and theproposed method is shown in Figure 11 As the noise ofmeasured acceleration is more obvious than that of strain andcannot be eliminated the comparison of acceleration is notgood as that of strain But the variation trends of measuredresult and the proposed method are also consistent )econsistence of changing trends of the vertical acceleration alsoproves the accuracy of the proposed method

4 Traffic Flow Simulation Based onMonitored Data

)e response of long-span suspension bridges under trafficflow is mainly influenced by traffic flow density and totalvehicle weight )e investigation focuses on the FVDsrsquo in-fluence on the bridge dynamic responses under traffic flowthus the key variables are FVD parameters while the trafficflow needs to remain unchanged Monitored traffic data

gathered by weigh-in-motion system are processed to formtypical traffic load to obtain the response of the studiedstructure [32] )e data collection site lies in Zhangjiawanbridge at the G104 highway in Zhejiang province of Chinawhich is a coastal developed area )e vehicles include carsvans trucks and trailers )e WIM data were measured intwo lanes and collected more than 10000 vehicles throughone-month monitoring In order to eliminate the sick datathe vehicles have low weight (less than 05 t) and high speed(more than 120 kmh) filtered in preprocessing the datacoming from weigh-in-motion system

First the monitored data are statically initialized toseparate vehicles in carriageway and passing laneaccording to the statistical property of lane distribution Asthe statistical analysis in hours can reflect the randomnessand the peak hours of traffic flow hourly traffic flow isoften used in traffic load analysis [32 33] )e WIM datawere divided into 24 hours each day and the total vehicleweight of each period was calculated For the bridge thedensity of dynamic response is directly influenced by theexternal load In order to obtain the most obvious re-sponse the vehicles with maximum total vehicle weight inan hour are screened out as the basic data of the traffic flowFigure 12 shows the maximum total vehicle weight of eachhour in a month )e left road of two lanes of the ad-vancing direction in China is passing lane and the right iscarriageway )e maximum vehicle weights in an houralong the passing lane and the carriageway are 111531 tand 75978 t respectively )e corresponding vehicle dataare chosen as the basic data to simulate traffic flow )edetailed information of vehicle type is shown in Figure 13it can be seen that the proportion of small and mediumcars in traffic is larger than trucks

Table 3 Technical parameters of the vehicle [31]

Vehicle model Axle number Axleweight (kg)

Upper stiffnesscoefficient (kNm)

Upper dampingcoefficient (kNmiddotsm)

Lower stiffnesscoefficient (kNm)

Lower dampingcoefficient (kNmiddotsm)

)ree-axle truckFront axle 8000 1577 266 3146 224Middle axle 13500 2362 20 2362 334Rear axle 13500 2362 20 2362 334

114 1122 128 1122 114

714

656 1288 6562600

Suspender

Main cable

20Straingauge

Straingauge

Accelerometer

192

Figure 9 Information of bridge model and dynamic test (unit cm)


Axi

al st

rain

(με)

40

0 20 40 60 80 100Time (s)

120 140 160

20

0

ndash20

ndash40

Measured resultProposed method

(a)

ndash20Axi

al st

rain

(με)

40

20

0

ndash400 20 40 60 80 100

Time (s)


(b)

ndash20Axi

al st

rain

(με)

40

20

0

ndash400 10 20 30

Time (s)40 50


60

(c)

Figure 10 )e comparison of dynamic strain results (a) 20 kmh (b) 30 kmh (c) 50 kmh

00050

00025

00000

Acce

lera

tion

(ms

2 )

ndash00025

ndash000500 20 40 60 80 100

Time (s)120


140 160

(a)

Figure 11 Continued


)e detailed simulation process is shown in Figure 14 Asthe prototype bridge is bidirectional with four lanes and themonitored data only have two lanes the chosen vehicle data

002

001

000

Acce

lera

tion

(ms

2 )ndash001

ndash0020 20 40 60 80

Time (s)


100

(b)

006

003

000

Acce

lera

tion

(ms

2 )

ndash003

ndash0060 10 20 30 5040

Time (s)


60

(c)

Figure 11 )e comparison of midspan acceleration results (a) Under 20 kmh (b) under 30 kmh (c) under 50 kmh

Tota

l veh

icle

wei

ght (t)

1200

Passing laneCarriageway

111531

75968

1000

800

600

400

0 2 4 6 8 10 12 14 16Time (h)

18 20 22 24

Figure 12 Extreme value of vehicle weight in hours

Prop

ortio

n (

)

1008920


2368

235

3158

376

1316

446

1842

024

1316

80

60

40

20

0

Vehicle typeT1 T2 T3 T4 T5

Figure 13 )e vehicle type of selected data


are used to simulate sparse flow and dense flow separatelythat can be loaded on both sides of the bridge simulta-neously )e traffic condition is divided into sparse flow anddense flow that the headway of the former exceeds 3 s andthat of the latter is less than 3 s [34] Statistical analysis on theWIM data indicates that the vehicle headway of the denseand sparse flow obeys gamma distribution )e key pa-rameters are shown in Table 4 )e corresponding vehicleheadways are generated according to statistical propertiesFigure 15 shows the vehicle headways of dense flow inpassing lane )en combined with vehicle data a bidirec-tional traffic flow with a dense upstream flow and a sparsedownstream flow is formed to load on the bridge

5 FVD Effect under Traffic Flow

According to existing research the service life of bridgeaccessories such as expansion joints and FVDs is closelyrelated to the deformation and fatigue properties of thecomponents under external loads [35] An accumulativefriction slide distance that exceeds the design value is animportant precipitating factor and also it will lead tocomponent damage in which traffic flow accounts for mostof the accumulative displacement Maximum girder dis-placement is a critical factor that determines whether theworking condition of a bridge accessory is in the designrange )us the longitudinal accumulative girder dis-placement and maximum longitudinal girder displacement

are selected as the typical indexes of the vibration controleffect of FVDs under traffic flow In JTT 926-2014 thedamping velocity exponent is specified into seven classes(01 02 03 04 05 06 and 1) and no limitation exists forthe damping coefficient whose value commonly ranges from1000 kN(ms)α to 20000 kN(ms)α [13 27] )e dampingcoefficient is set to 1000 kN(ms)α 2500 kN(ms)α5000 kN(ms)α 7000 kN(ms)α 10000 kN(ms)α15000 kN(ms)α and 20000 kN(ms)α to analyze theFVDsrsquo effects with representative combinations ofparameters

250

200

Gamma distributionα = 77043 β = 37311

150

Generated data

Cou

nt

100

50

00 1 2 3

Headway (s)4 5 6 7

Figure 15 Vehicle headways of dense flow in passing lane

Selected trafficdata

Vehicleweight

Dense flow Sparse flow

Bidirectionaltraffic flow

Vehiclelength

Axleload

Axledistance

Physicalcharacteristicsof traffic flow

Vehicletype

Vehiclespeed

Timeheadway

Lanedistribution

Figure 14 Flowchart of traffic flow generating

Table 4 Parameters of vehicle headway

Operation state Distribution type LaneDistribution parametersα β

Sparse flow Gamma distribution Carriageway 02954 00038Passing lane 04005 00149

Dense flow Gamma distribution Carriageway 74072 35829Passing lane 77043 37311


Nonlinear time history analysis under traffic flow isconducted using the bridge-vehicle interaction systemestablished in Section 2 )e main girder of the suspensionbridge is directly influenced by FVDs so the girder responsevaries as shown in Figures 16ndash18 )e installation of FVDsprovides the following benefits longitudinal displacementand accumulative displacement decrease noticeably with anenlargement in the damping coefficient and a decrease in thevelocity exponent)e girder midspan bendingmoment alsodecreases with increased damping coefficient and decreasedvelocity exponent When the velocity exponent exceeds 04

the influence on girder midspan bending moment is notremarkable But when the velocity exponent is below 04 andthe damping coefficient exceeds 7000 kN(ms)α the girdermidspan bending moment noticeably declines with in-creased damping coefficient and decreased velocityexponent

Figures 19 and 20 show that the variation trend oflongitudinal pylon displacement and acceleration increasesaccordingly with the increase in the velocity exponent or thedecrease in the damping coefficient Figure 21 shows that thevariation trend of the pylon bending moment is relatively

5

4Ac

cum

ulat

ive l

ongi

tudi

nal

gird

er d

ispla

cem

ent (

m)

3

2

1200040006000

800010000Damping coefficient (kN(ms) α)

1200014000

1600018000

20000

None1

0605

0403

Velocity exponent

0201

4400

3900

3400

2900

2400

1900

Figure 16 Accumulative longitudinal girder displacement Wgc

Max

imum

long

itudi

nal

gird

er d

ispla

cem

ent (

m)

065

060

055

050

045

040

0352000

4000 60008000

Damping coefficient (kN(ms) α)

1000012000

1400016000

1800020000

None1

0605

04

Velocity exponent

0302

01

0580

0540

0490

0440

0390

Figure 17 Maximum longitudinal girder displacement Wgm


complex it increases with the damping coefficient when thevelocity exponent αlt 03 but decreases and then increaseswith increased damping coefficient when 03le αlt 1

Figure 22 shows that the suspender force is also influ-enced by FVDs which is caused by the reciprocating motionof the main girder )e maximum suspender forces wereinduced by vehicles only it may cause a relative displace-ment between the girder and the main cable )e changinglaw is similar to that of maximum longitudinal pylondisplacement

)e extreme values of each index are shown in Table 5)e FVDs evidently influence the longitudinal girder

displacement and pylon acceleration )e maximum controlefficiency of the girder displacement reaches 3367 )eaccumulated longitudinal girder displacement can be re-duced by 5771 )ese results mean that FVDs can re-markably decrease the daily reciprocating motion of relatedcomponents Although the maximum reduced rate of thepylon longitudinal displacement is only 202 the maxi-mum pylon acceleration can be reduced by 3151 )epylon bending moment and suspender force are less affectedin that the maximum control efficiencies are less than 2

Once the damping coefficient and velocity exponent aredetermined the damping force under normal working

Max

imum

grid

er m

id-s

pan

bend

ing

mom

ent (

kNmiddotm

)

129times103

126

123

120

117

1142000

400060008000

1000012000

1400016000

1800020000

None

1250

1230

1210

1190

1170

Velocity exponent

106

0504

0302

01


Figure 18 Maximum midspan girder bending moment Mg

Max

imum

long

itudi

nal p

ylon

disp

lace

men

t (m

)

0160

0159

0158

0157

0156

0155

01542000

40006000

800010000

1200014000

16000

None

01588

01581

01572

01563

015541

06

Velocity exponent05

0403

0201

1800020000


Figure 19 Maximum longitudinal pylon displacement Wtm


conditions can be calculated by (8) on the basis of thecorresponding damping velocity shown in Figure 23 Asshown in Figure 24 the damping force of a partial dampingcoefficient and velocity exponent is around or exceeds thespecified value for a single FVD in Chinese standard [27])e damping coefficient has an apparent influence on thedamping force of the FVDs only when the velocity exponentis relatively small Preceding results reveal that the longi-tudinal girder displacement is the most significant responseof the structure but its variation tendency is nonlinear Alarge damping coefficient improves the FVD effects but theeffect of improvement is not proportional to the damping

coefficient An excessive damping coefficient brings in-creased damping force so the design force of FVD must behigh enough to burden the external load Otherwise theFVD would be working in limit state under the dense trafficflow which is adverse to its durability

By contrast low velocity exponent results in an evidentimprovement especially for FVD performance and girderdisplacement thereby potentially making FVD a valid vi-bration mitigation device in daily operation Figures 25 and26 show the time history curves of the maximum girderdisplacement and accumulative girder displacement whenthe velocity exponent is 01 )e accumulative girder

Max

imum

long

itudi

nal

pylo

n ac

cele

ratio

n (m

s2 )

0016

0014

0012

0010

00082000

None

0015

0014

0012

0011

00101

0605

Velocity exponent04

0302

01

4000 60008000

1000012000

1400016000

1800020000


Figure 20 Maximum longitudinal pylon acceleration at

Max

imum

pyl

onbe

ndin

g m

omen

t (kN

middotm)

times105

1716

1712

1708

1704

17002000

40006000

800010000

1200014000

16000

Damping coefficient (kN(ms) α) 1800020000

None

1717

1714

1711

1708

17051

0605

04

Velocity exponent03

0201

Figure 21 Maximum pylon bending moment Mt


displacement decreases noticeably when the damping co-efficient Clt 5000 kN(ms)α but when the damping coef-ficient Cgt 5000 kN(ms)α the accumulative girderdisplacement decreases slightly with increased damping

coefficient )e benefits of an excessive damping coefficientare not proportional to the costs of it

But for the low velocity exponent the manufacturingprocess of this type of FVD is complicated and its durability

Susp

ende

r for

ce (k

N)

398times102

396

394

392

3902000 None

3973

3961

3946

3931

3916

Velocity exponent

106

0504

0302

01

40006000 8000


1400016000

1800020000

Figure 22 Maximum suspender force Fs

Table 5 Peak values of bridge dynamic response

Wgm (m) Wgc (m) Mg (kNmiddotm) Wtm (m) at (ms2) Mt (kNmiddotm) Fs (kN)

Minimum value 03931 18845 1157 01554 0010 170548 39165Maximum value 05741 42352 1246 01586 00143 171652 39715Without damper 05926 44561 1246 01586 00146 171481 39725Maximum control efficiency 3367 5771 714 202 3151 054 141

Dam

ping

velo

city

(ms

)

0048

0045

0042

0039

0033

0036

0 2500 5000 7500 10000 12500 15000 17500 20000Damping coefficient (kN(ms)α)

α = 01α = 02α = 03

α = 05α = 06α = 1

α = 04

Figure 23 Maximum value of damping velocity Vc


16000

14000

12000

10000

8000

Dam

ping

forc

e (kN

)

6000

4000

2000

0 2500 5000 7500Damping coefficient (kN(ms)α)

10000 12500 15000 17500 200000

α = 01α = 02α = 03

α = 05α = 06α = 1

α = 04

Figure 24 Maximum value of damping force F

Max

imum

gird

er d

ispla

cem

ent (

m)

045

030

015

000

ndash015

ndash030

ndash045

ndash0600 50

ndash046

C = 1000C = 2500C = 5000

C = 10000C = 15000

C = 20000C = 7000None

Unit kN(ms)α

ndash048

ndash050

ndash006

ndash009

ndash012

ndash01530 40 215 220 225 230

100Time (s)

150 200 250 300

Figure 25 Time history curve of maximum girder displacement (α 01)

Accu

mul

ativ

e gird

er d

ispla

cem

ent (

m)

5

C = 1000C = 2500C = 5000

C = 7000C = 10000

C = 20000None

C = 15000

0 50 100 150 200Time (s)

250 300

Unit kN(ms)α

4

3

2

1

0

Figure 26 Time history curve of accumulative girder displacement (α 01)


may be limited So the influence of the velocity exponent also isinvestigatedWhen the damping coefficient is 5000 kN(ms)αthe time history curves of the maximum girder displacementand accumulative girder displacement are shown in Figures 27and 28 It can be seen that the decreasing of velocity exponenthas little influence on the maximum girder displacement whilethe common vibration is reduced According to Figure 28 theaccumulative girder displacement can be reduced obviouslywith the decreased velocity exponent )at is because trafficload is different from the seismic excitation which has thecharacteristics of short time but strong intensity Traffic load isgenerally a low intensity but long termprocess)e low velocity

exponent brings evident benefits for the bridge response undertraffic flow

According to the results the damping coefficientCgt 5000 kN(ms)α and the velocity exponent α 01 are thebest parameters for FVDs under traffic flow )e dampingcoefficient does not need to be too high However the FVDsapplication should consider a variety of factors such as windand earthquake So if the velocity exponent has to be in-creased combined with Figure 16 the benefit of low velocityexponent reduced when the damping coefficient is highenough and the damping coefficient can be enlarged to fitwith the velocity exponent for the best effect of reducing theresponse under traffic flow

6 Conclusions

In this study bridge dynamic response with FVDs undertraffic flow was analyzed on the basis of monitored data Forrapid analysis under high-intensity traffic flow a bridge-vehicle coupling method optimized by isoparametricmapping and improved binary search was developed andvalidated)en traffic data with extreme total vehicle weightin an hour were screened out from monitoring data tosimulate traffic flow Lastly the simulation analysis of thebridge with FVDs under traffic flow based on monitoreddata was accomplished Various indicators of the bridge areaffected to different degrees by variations in the FVD pa-rameters )e results show that vehicle-bridge couplinganalysis method optimized by isoparametric mapping andimproved binary search based on the interhistory methodwas validated )e methodrsquos findings were consistent withthe experimental results therefore the proposed methodcan be used in vehicle-bridge coupling analysis under high-intensity traffic flow

For the prototype bridge the results show that FVDsrsquosettlement positively impacts bridge dynamic responseunder traffic flow )e traffic load should be considered inFVD design as the dynamic response is remarkably influ-enced by FVD parameters A reasonable FVD design caneffectively reduce the accumulative longitudinal girderdisplacement maximum longitudinal girder displacementand longitudinal pylon acceleration whose maximum con-trol efficiency respectively reaches 5771 3367 and3151 the girder bending moment also can decrease by714 and the maximum control efficiencies of other in-dexes were less than 3

)is study also shows an increase in the damping co-efficient and a decrease in the velocity exponent that canreduce the response of key indexes However an excessivedamping coefficient will bring negligible improvement andmay lead to high-intensity work under traffic flow therebyadversely affecting component durability A low velocityexponent will result in evident improvement especially forFVD performance and girder displacement and plays a rolein maintaining low-velocity movement )e benefits of lowvelocity exponent reduced when the damping coefficient ishigh enough When the velocity exponent has to be in-creased the damping coefficient can be enlarged to fit withthe velocity exponent

Max

imum

grid

er d

ispla

cem

ent (

m)

04

02

00

ndash02

ndash06

ndash04

ndash047

α = 01α = 02α = 03

α = 04α = 05

α = 1None

α = 06

ndash006

ndash009

ndash012

ndash015220 230

ndash048

ndash049

ndash050 30 32 34 36

0 50 100 150Time (s)

200 250 300

Figure 27 Time history curve of maximum girder displacement(C 5000 kN(ms)α)

Accu

mul

ativ

e gird

er d

ispla

cem

ent (

m)

5

4

3

2

1

00 50 100 150

Time (s)200 250 300

α = 01α = 02α = 03

α = 04α = 05

α = 1None

α = 06

Figure 28 Time history curve of accumulative girder displacement(C 5000 kN(ms)α)


Note that the research and conclusions in this paperfocus on the FVDsrsquo influence on the bridge response undertraffic flow In a subsequent research program FVDs effectson bridge dynamic response under the combined action ofwind and traffic loads and the comprehensive optimizationof FVD parameters will be investigated

Appendix

KivuL andKi

vuR denote the vertical spring stiffness of theith axle in upper suspension (Nm)

KivuL K

ivuR (A1)

KiyuL andKi

yuR denote the lateral spring stiffness of theith axle in upper suspension (Nm)

KiyuL K

iyuR (A2)

KivlL andKi

vlR denote the vertical spring stiffness of theith axle in lower suspension (Nm)

KivlL K

ivlR (A3)

KiylL andKi

ylR denote the lateral spring stiffness of theith axle in lower suspension (Nm)

KiylL K

iylR (A4)

CivuL andCi

vuR denote the vertical damping of the ithaxle in upper suspension (Nmiddotsm)

CivuL C

ivuR (A5)

CiyuL andCi

yuR denote the lateral damping of the ith axlein upper suspension (Nmiddotsm)

CiyuL C

iyuR (A6)

CivlL andCi

vlR denote the vertical damping of the ith axlein lower suspension (Nmiddotsm)

CivlL C

ivlR (A7)

CiylL andCi

ylR denote the lateral damping of the ith axlein lower suspension (Nmiddotsm)

CiylL C

iylR (A8)

Li denotes the distance from the center of mass of carbody to the ith axleb1 denotes half of the lateral wheel spacingh1 denotes the vertical distance from the center of massof car body to the upper plane of the springhv denotes the vertical distance from the center of massof car body to road surface

Data Availability

)e rawprocessed data required to reproduce these findingscannot be shared at this time as the data also form part of anongoing study

Conflicts of Interest

)e authors declare that they have no conflicts of interest

Acknowledgments

)e authors appreciate the financial support of the NationalNatural Science Foundation of China (Grant no 51878058)and the Fundamental Research Funds for the CentralUniversities (Grant no 300102219402)

References

[1] I D Aiken and J M Kelly ldquoCyclic dynamic testing of fluidviscous dampersrdquo in Proceedings of the Caltrans fourth seismicresearch conference pp 1ndash9 Sacramento CA USA July 1996

[2] Z H Zhang H W Zhou and X F Wang ldquoApplication anddevelopment of high efficiency viscous Damper on EgongyanYangtze River suspension bridge in Chongqingrdquo in Pro-ceedings of the China Highway Society Paper Collection of2002 National Bridge Academic Conference of Bridge andStructural Engineering Branch of China Highway Society pp470ndash477 Beijing China June 2002 in Chinese

[3] F Ferreira and L Simotildees ldquoOptimum design of a controlledcable-stayed footbridge subject to a running event usingsemiactive and passivemass dampersrdquo Journal of Performanceof Constructed Facilities vol 33 no 3 Article ID 040190252019

[4] J Yi J Li and Z Guan ldquoShake table studies on viscousdampers in seismic control of a single-tower cable-stayedbridge model under near-field ground motionsrdquo Journal ofEarthquake and Tsunami vol 12 no 5 Article ID 18500112018

[5] T Guo J Liu Y F Zhang and S J Pan ldquoDisplacementmonitoring and analysis of expansion joints of long-span steelbridges with viscous dampersrdquo Journal of Bridge Engineeringvol 20 no 9 Article ID 04014099 2015

[6] J Zhong Z Hu W Yuan and L Chen ldquoSystem-basedprobabilistic optimization of fluid viscous dampers equippedin cable-stayed bridgesrdquo Advances in Structural Engineeringvol 21 no 12 pp 1815ndash1825 2018

[7] J S Hwang and Y S Tseng ldquoDesign formulations for sup-plemental viscous dampers to highway bridgesrdquo EarthquakeEngineering amp Structural Dynamics vol 34 no 13pp 1627ndash1642 2010

[8] Y F Huang Y Xu and J Z Li ldquoSimplified parameter designmethod of viscous dampers for cable-stayed bridge undernear-fault ground motionsrdquo China Civil Engineering Journalvol 49 no 9 pp 72ndash77 2016 in Chinese

[9] Z X Liu T Guo L Y Huang and Z H Pan ldquoFatigue lifeevaluation on short suspenders of long-span suspensionbridge with central clampsrdquo Journal of Bridge Engineeringvol 22 no 10 Article ID 04017074 2017

[10] T BWan ldquoPerformance requirements of viscous dampers forimprovement of durability of bridge structuresrdquo BridgeConstruction vol 46 no 4 pp 29ndash34 2016 in Chinese


[11] H Xia Y L Xu and Q S Yan ldquoDynamic response of longspan suspension bridge to high wind and running trainrdquoJournal of the China Railway Society vol 24 no 4 pp 83ndash912002 in Chinese

[12] W S Han H J Liu D H Bao P M Huang and Y G YuanldquoEstablishment and visualization of wind-vehicle-bridgeanalysis system for the large-span steel truss suspensionbridgerdquo China Civil Engineering Journal vol 51 no 3pp 99ndash108 2018 in Chinese

[13] Z Q Wang S D Wu and L C Fan ldquoResearch on viscousdamper parameters of Donghai Bridgerdquo China Journal ofHighway and Transport vol 18 no 3 2005 in Chinese

[14] Y L Ding F F Geng W H Ge J Y Song W H Li andY Q Wang ldquoControl of wind-induced buffeting responses ofa multi-tower cable-stayed bridge using viscous dampersrdquoEngineering Mechanics vol 32 no 4 pp 130ndash137 2015 inChinese

[15] X Z Li L M Zhang and J Zhang ldquoState of the art review andtrend of studies on coupling vibration for vehicle and highwaybridge systemrdquo Engineering Mechanics vol 25 no 3pp 230ndash240 2008 in Chinese

[16] Q L Zou L Deng T D Guo and X F Yin ldquoComparativestudy of different numerical models for vehicle-bridge in-teraction analysisrdquo International Journal of Structural Sta-bility and Dynamics vol 16 no 9 pp 1636ndash1643 2016

[17] ANSYS IncANSYS 150 ANSYS Inc Canonsburg PA USA2013

[18] H Wang and A Q Li Finite Element Analysis and Engi-neering Examples of Long-Span Bridges pp 18ndash94 ChinaArchitechture publishing and Media Co Ltd Beijing China2014 in Chinese

[19] H Wang A Li R Hu and J Li ldquoAccurate stress analysis onsteel box girder of long span suspension bridges based onmulti-scale submodeling methodrdquo Advances in StructuralEngineering vol 13 no 4 pp 727ndash740 2010

[20] W S Han L Ma and B Wang ldquoRefinement analysis anddynamic visualization of traffic-bridge coupling vibrationsystemrdquo China Journal of Highway and Transport vol 26no 4 pp 78ndash87 2013 in Chinese

[21] China Automobile Industry Corporation and China Auto-motive Technology and Research Center (CAIC-CATRC)Manual of Chinese Vehicle Type Shandong Scientific andTechnical Press Shandong China 1993 in Chinese

[22] T L Wang M Shahawy and D Z Huang ldquoDynamic re-sponse of highway trucks due to road surface roughnessrdquoComputers amp Structures vol 49 no 6 pp 1055ndash1067 1993

[23] X S Yi J G Ren and J P Zhou ldquoAn accurately integrated 4-node quadrilateral elementrdquo Journal of National University ofDefense Technology vol 20 no 1 pp 1ndash4 1998 in Chinese

[24] International Organization for Standardization (ISO) Me-chanical Vibration-Road Surface Profiles-Reporting of Mea-sured Data (ISO 8608 1995(E)) International Organizationfor Standardization (ISO) Geneva Switzerland 1995

[25] N Zhang Y Tian and H Xia ldquoA train-bridge dynamicinteraction analysis method and its experimental validationrdquoEngineering vol 2 no 4 pp 528ndash536 2016

[26] Mathworks Inc MTLAB R2014a Mathworks Inc NatickMA USA 2014

[27] Ministry of Transport of the Peoplersquos Republic of China(MTPR) Fluid Viscous Damper for Bridges (JTT 926-2014)Ministry of Transport of the Peoplersquos Republic of China(MTPR) Beijing China 2014 in Chinese

[28] G Y Yang Study on the Interaction of Vehicle Random Loadsand Flexible Pavement School of Civil Engineering CentralSouth University Changsha China 2007 in Chinese

[29] M Huang W Guo H Zhu and L Li ldquoDynamic test andfinite element model updating of bridge structures based onambient vibrationrdquo Frontiers of Architecture and Civil En-gineering in China vol 2 no 2 pp 139ndash144 2008

[30] H Wang A Q Li T Guo and T Y Tao ldquoEstablishment andapplication of the wind and structural health monitoringsystem for the Runyang Yangtze River bridgerdquo Shock ampVibration vol 2014 Article ID 421038 15 pages 2014

[31] W S Han Visualized Simulation and Safety Evaluation ofHeavy Traffic Dynamic Bridge China communication pressBeijing China 2016 in Chinese

[32] W S Han X D Liu G Z Gao and Q Xie ldquoSite-specificextra-heavy truck load characteristics and bridge safety as-sessmentrdquo Journal of Aerospace Engineering vol 31 no 6Article ID 04018098 2018

[33] J-L Martin ldquoRelationship between crash rate and hourlytraffic flow on interurban motorwaysrdquo Accident Analysis ampPrevention vol 34 no 5 pp 619ndash629 2002

[34] Ministry of Transport of the Peoplersquos Republic of China(MTPR) Unified Standard for Reliability Design of HighwayEngineering Structures (GBT 50283-1999) Ministry ofTransport of the Peoplersquos Republic of China (MTPR) BeijingChina 1999 in Chinese

[35] Z Sun and Y F Zhang ldquoFailure mechanism of expansionjoints in a suspension bridgerdquo Journal of Bridge Engineeringvol 21 no 10 Article ID 05016005 2016


[3ndash5] Different techniques have been proposed by scholarsto ascertain the optimal parameters that can achieve idealstructural response [6ndash8]

Although FVDsrsquo effects under earthquake excitationhave been widely studied the dynamic responses of bridgeswhich are also vulnerable to vibrations excited by windtemperature and vehicles have not been thoroughly ex-amined )e most common dynamic response in the bridgelife cycle is the frequent low-speed reciprocating motionwhich is mainly caused by wind and traffic flow [9] )ecorresponding service life of bridge accessories such asexpansion joints is also influenced by this frequent recip-rocating motion FVDs can also absorb and dissipate energyimported from wind and traffic flow because of their dis-sipation capacity )e working conditions of supports ex-pansion joints and suspenders can be improved byconsidering the daily longitudinal girder displacement indetermining FVD parameters the frequent reciprocatingmotion of main girder also can be effectively improved byusing FVDs [10] For a long-span bridge the wind load maybe more remarkable than vehicle load in transverse responseof the bridge Research shows that the dynamic responses oflong-span suspension bridge under high winds and runningtrains are mainly influenced by strong winds when the windis strong but when the wind comes to low intensity trainload plays a decisive role in vertical displacement [11] Forthe bridge dynamic response under wind and traffic flow thewind load plays a major role in bridge lateral displacement)e vertical displacement is mainly controlled by the vehicleload [12] Furthermore scholars have made some studies onwind-induced vibration control and found that FVDs havedifferent influences on various response indicators whichhas an optimum interval for damping parameters [13 14] Inaddition to the wind-induced vibration the vehicle-inducedvibration of long-span bridges is also remarkable and theinfluences of FVDs are still indistinct )e daily movementof bridge structures under social vehicles is an importantfactor related to bridge accessoriesrsquo durability [5] Variedstructural response indexes present different variationtrends with changing parameters of FVDs such as accu-mulative girder displacement girder midspan bendingmoment pylon longitudinal acceleration and pylon bend-ingmoment)e specific influences of FVDs on the dynamicresponse of long-span suspension bridges under traffic loadwhich can be further referenced in the optimization of FVDparameters should hence be analyzed

Because the traffic load of the highway bridge is acomplicated random process with strong randomness ofvehicle type weight and quantity the calculation matrix ofrandom traffic flow-bridge coupling analysis is huge and willtake a lot of computing resources )us a high efficiencyrandom traffic flow-bridge analysis system needs to beestablished )e traditional vehicle-bridge coupling analysissystem which is based on a numerical method mainlyincludes the modal superposition method and full couplingtheory [15])emodal superposition method is more simpleand practical than the latter but the high-order modes ofstructures are hard to obtain thereby negatively affecting thecalculation precision [16] )e full coupling analytical

method has the advantages of clear physical meaning andhigh precision However this method requires huge amountof time and lots of computing resources when applied tolong-span bridges under large and highly random trafficflow )is paper developed a vehicle-bridge interactionsystem with high efficiency to analyze the dynamic responseof a bridge with FVDs under a large and highly randomtraffic flow )is system was established based on theinterhistory iteration method and optimized by iso-parametric mapping and improved binary search )en thetraffic flow was simulated in accordance with the monitoreddata Finally the responses of the large-span bridge undertraffic flow with different parameters of FVDs were studied

2 Establishment of Vehicle-BridgeInteraction System

21 Finite Element Model of Bridge )e bridge model isestablished by the space truss model in ANSYS 150 [17]Spine-beam double-beam and triple-beam models arewidely used in the integral analysis of long-span bridges andthe grillage model and multiscale model are also usedaccording to different requirements [18 19] Main cablesand suspenders are usually simulated by the Link10 elementwhich can simulate the elasticity and plasticity of thestructure )e pylon and girder are simplified into a two-node beam element and modeled by Beam4 element )erailings and deck paving are simulated by the Mass21 ele-ment )e entire structure is distributed into a number ofelements and connected through the node on the boundaryof adjacent elements Given the demand for refined simu-lation of the vehicle-bridge interaction a grillage model isutilized in this article to construct the prototype bridge toobtain detailed component response )e detailed infor-mation and feasibility of the model are proposed and val-idated in succeeding chapters

22 Dynamic Vehicle Models A vehicle is mainly composedof a car body wheels a shock mitigation system and adamped system connecting different components A vehicledynamic model needs to be established before the vehicle-bridge coupling analysis is conducted With these demandssome hypothesis is unavoidable in constructing the dynamicmodels )e mass of damper and spring components areignored compared with the quality of the vehicle bodyVehicle model is generally divided into different rigid bodiesthat are connected by axle mass blocks and by elastic anddamped components According to previous research thecommon Chinese highway vehicles can be divided into 17types of 5 categories in accordance with vehicle axle distanceaxle number axle load and vehicle load based on weigh-in-motion data and Chinese automobile model manual [20 21])e vehicle classification is shown in Table 1

)e dynamic models are also provided based on differentvehicle type Taking three-axle vehicle as an example thevehicle dynamic model of three-axle vehicle (double rearaxle) is shown in Figure 1 [20] )e rigid body is usuallydeemed to have 6 degrees of freedom As the vibration of the






x 11139364

i1Ni(ξ η)xi

y 11139364

i1Ni(ξ η)yi

Ni(ξ η) 14

1 + ξξi( 1113857 1 + ηηi( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)




T4


T5



θvrZvr

K2vuL C2

vuL

K2vlL C2

vlL

Z2vaL

X

Z C3vlLK3

vlL

C3vuLK3

vuL

L2 L1

Z3vaL Z1

vaL

C1vlLK1

vlL

C1vuLK1

vuL

Driver seat

L3

(a)

b1b1

Zvr

Y

Z1vaR

C1vlRK1

vlR

C1vuR

K1vuRK1

vuLC1vuL

K1vlL C1

vlL

Z1vaL

ZY1vaL

K1yuL

C1yuLK1

ylLC1ylL

K1yuR

Y1vaRC1

yuR K1ylRC1ylR

h 1

h v

b2

Yvr

Φvr

(b)




ξk+1

ηk+1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ξk

ηk

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ minus J ξk

ηk1113872 1113873minus 1

11139364

i1Ni ξk

ηk1113872 1113873xi minus x

11139364

i1Ni ξk

ηk1113872 1113873yi minus y

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

J

z 11139364i1 Ni(ξ η)xi1113960 1113961

zξz 1113936

4i1 Ni(ξ η)xi1113960 1113961

zη

z 11139364i1 Ni(ξ η)yi1113960 1113961

zξz 1113936

4i1 Ni(ξ η)yi1113960 1113961

zη

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(2)


when ξk+1minus ξk

ηk+1 minus ηk











Gd(n) Gd n0( 1113857n

n01113888 1113889

minusω

(3)

FF1 F2

F3F4


P (x y)


3 (1 1)4 (ndash1 1)η

ξ x

y

1 (x1 y1) 2 (x2 y2)

3 (x3 y3)4 (x4 y4)




Start



ampx le t (mid)

ANSYSmodel



last = mid

front = mid

Yes

No


Finish

Yes

No





r(x) 1113944N

k1

2Gd nk( 1113857Δn1113969

cos 2πnkx + θk( 1113857 (4)



Zie Z

ib + ri(x) (5)






MveuroZv + Cv

_Zv + KvXv Fv

MbeuroZb + Cb

_Xb + KbXb Fb

⎧⎨

⎩ (6)






⎧⎪⎨

⎪⎩(7)













F K|v|αsign(v) (8)




114m 896m

Main cable


Suspender

Approachbridge


buckle

214

Tunnelanchorage

208m


No

Start






Vehiclemotion state

Bridgemotion state

Yes

Finish

Bridgesubsystem

Vehiclesubsystem

Convergencecheck








Horizontal 1 0071 007 1412 0183 0175 437

Vertical

1 0131 0134 2292 0185 0182 1623 0292 0282 3424 0394 038 355

Torsional 1 0348 0342 1722 0434 0425 207

Combin 37 element

Link 10

Beam 4


Node JNode I

Spring

Axialforce


Dec

k ro

ughn

ess (

mm

)

10

5

0

ndash5

ndash100 100 200 300 400 500

















114 1122 128 1122 114

714

656 1288 6562600

Suspender

Main cable

20Straingauge

Straingauge

Accelerometer

192



Axi

al st

rain

(με)

40

0 20 40 60 80 100Time (s)

120 140 160

20

0

ndash20

ndash40


(a)

ndash20Axi

al st

rain

(με)

40

20

0

ndash400 20 40 60 80 100

Time (s)


(b)

ndash20Axi

al st

rain

(με)

40

20

0

ndash400 10 20 30

Time (s)40 50


60

(c)


00050

00025

00000

Acce

lera

tion

(ms

2 )

ndash00025

ndash000500 20 40 60 80 100

Time (s)120


140 160

(a)

Figure 11 Continued



002

001

000

Acce

lera

tion

(ms

2 )ndash001

ndash0020 20 40 60 80

Time (s)


100

(b)

006

003

000

Acce

lera

tion

(ms

2 )

ndash003

ndash0060 10 20 30 5040

Time (s)


60

(c)


Tota

l veh

icle

wei

ght (t)

1200


111531

75968

1000

800

600

400

0 2 4 6 8 10 12 14 16Time (h)

18 20 22 24


Prop

ortio

n (

)

1008920


2368

235

3158

376

1316

446

1842

024

1316

80

60

40

20

0








250

200


150

Generated data

Cou

nt

100

50

00 1 2 3

Headway (s)4 5 6 7



Vehicleweight



Vehiclelength

Axleload

Axledistance


Vehicletype

Vehiclespeed

Timeheadway

Lanedistribution










5

4Ac

cum

ulat

ive l

ongi

tudi

nal

gird

er d

ispla

cem

ent (

m)

3

2

1200040006000


1200014000

1600018000

20000

None1

0605

0403

Velocity exponent

0201

4400

3900

3400

2900

2400

1900


Max

imum

long

itudi

nal

gird

er d

ispla

cem

ent (

m)

065

060

055

050

045

040

0352000

4000 60008000


1000012000

1400016000

1800020000

None1

0605

04

Velocity exponent

0302

01

0580

0540

0490

0440

0390








Max

imum

grid

er m

id-s

pan

bend

ing

mom

ent (

kNmiddotm

)

129times103

126

123

120

117

1142000

400060008000

1000012000

1400016000

1800020000

None

1250

1230

1210

1190

1170

Velocity exponent

106

0504

0302

01



Max

imum

long

itudi

nal p

ylon

disp

lace

men

t (m

)

0160

0159

0158

0157

0156

0155

01542000

40006000

800010000

1200014000

16000

None

01588

01581

01572

01563

015541

06

Velocity exponent05

0403

0201

1800020000







Max

imum

long

itudi

nal

pylo

n ac

cele

ratio

n (m

s2 )

0016

0014

0012

0010

00082000

None

0015

0014

0012

0011

00101

0605

Velocity exponent04

0302

01

4000 60008000

1000012000

1400016000

1800020000



Max

imum

pyl

onbe

ndin

g m

omen

t (kN

middotm)

times105

1716

1712

1708

1704

17002000

40006000

800010000

1200014000

16000


None

1717

1714

1711

1708

17051

0605

04

Velocity exponent03

0201






Susp

ende

r for

ce (k

N)

398times102

396

394

392

3902000 None

3973

3961

3946

3931

3916

Velocity exponent

106

0504

0302

01

40006000 8000


1400016000

1800020000





Dam

ping

velo

city

(ms

)

0048

0045

0042

0039

0033

0036


α = 01α = 02α = 03

α = 05α = 06α = 1

α = 04



16000

14000

12000

10000

8000

Dam

ping

forc

e (kN

)

6000

4000

2000


10000 12500 15000 17500 200000

α = 01α = 02α = 03

α = 05α = 06α = 1

α = 04


Max

imum

gird

er d

ispla

cem

ent (

m)

045

030

015

000

ndash015

ndash030

ndash045

ndash0600 50

ndash046

C = 1000C = 2500C = 5000

C = 10000C = 15000

C = 20000C = 7000None

Unit kN(ms)α

ndash048

ndash050

ndash006

ndash009

ndash012

ndash01530 40 215 220 225 230

100Time (s)

150 200 250 300


Accu

mul

ativ

e gird

er d

ispla

cem

ent (

m)

5

C = 1000C = 2500C = 5000

C = 7000C = 10000

C = 20000None

C = 15000

0 50 100 150 200Time (s)

250 300

Unit kN(ms)α

4

3

2

1

0






6 Conclusions




Max

imum

grid

er d

ispla

cem

ent (

m)

04

02

00

ndash02

ndash06

ndash04

ndash047

α = 01α = 02α = 03

α = 04α = 05

α = 1None

α = 06

ndash006

ndash009

ndash012

ndash015220 230

ndash048

ndash049

ndash050 30 32 34 36

0 50 100 150Time (s)

200 250 300


Accu

mul

ativ

e gird

er d

ispla

cem

ent (

m)

5

4

3

2

1

00 50 100 150

Time (s)200 250 300

α = 01α = 02α = 03

α = 04α = 05

α = 1None

α = 06




Appendix

KivuL andKi


KivuL K

ivuR (A1)

KiyuL andKi


KiyuL K

iyuR (A2)

KivlL andKi


KivlL K

ivlR (A3)

KiylL andKi


KiylL K

iylR (A4)

CivuL andCi


CivuL C

ivuR (A5)

CiyuL andCi


CiyuL C

iyuR (A6)

CivlL andCi


CivlL C

ivlR (A7)

CiylL andCi


CiylL C

iylR (A8)


Data Availability




Acknowledgments


References










































x 11139364

i1Ni(ξ η)xi

y 11139364

i1Ni(ξ η)yi

Ni(ξ η) 14

1 + ξξi( 1113857 1 + ηηi( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)




T4


T5



θvrZvr

K2vuL C2

vuL

K2vlL C2

vlL

Z2vaL

X

Z C3vlLK3

vlL

C3vuLK3

vuL

L2 L1

Z3vaL Z1

vaL

C1vlLK1

vlL

C1vuLK1

vuL

Driver seat

L3

(a)

b1b1

Zvr

Y

Z1vaR

C1vlRK1

vlR

C1vuR

K1vuRK1

vuLC1vuL

K1vlL C1

vlL

Z1vaL

ZY1vaL

K1yuL

C1yuLK1

ylLC1ylL

K1yuR

Y1vaRC1

yuR K1ylRC1ylR

h 1

h v

b2

Yvr

Φvr

(b)




ξk+1

ηk+1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ξk

ηk

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ minus J ξk

ηk1113872 1113873minus 1

11139364

i1Ni ξk

ηk1113872 1113873xi minus x

11139364

i1Ni ξk

ηk1113872 1113873yi minus y

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

J

z 11139364i1 Ni(ξ η)xi1113960 1113961

zξz 1113936

4i1 Ni(ξ η)xi1113960 1113961

zη

z 11139364i1 Ni(ξ η)yi1113960 1113961

zξz 1113936

4i1 Ni(ξ η)yi1113960 1113961

zη

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(2)


when ξk+1minus ξk

ηk+1 minus ηk











Gd(n) Gd n0( 1113857n

n01113888 1113889

minusω

(3)

FF1 F2

F3F4


P (x y)


3 (1 1)4 (ndash1 1)η

ξ x

y

1 (x1 y1) 2 (x2 y2)

3 (x3 y3)4 (x4 y4)




Start



ampx le t (mid)

ANSYSmodel



last = mid

front = mid

Yes

No


Finish

Yes

No





r(x) 1113944N

k1

2Gd nk( 1113857Δn1113969

cos 2πnkx + θk( 1113857 (4)



Zie Z

ib + ri(x) (5)






MveuroZv + Cv

_Zv + KvXv Fv

MbeuroZb + Cb

_Xb + KbXb Fb

⎧⎨

⎩ (6)






⎧⎪⎨

⎪⎩(7)













F K|v|αsign(v) (8)




114m 896m

Main cable


Suspender

Approachbridge


buckle

214

Tunnelanchorage

208m


No

Start






Vehiclemotion state

Bridgemotion state

Yes

Finish

Bridgesubsystem

Vehiclesubsystem

Convergencecheck








Horizontal 1 0071 007 1412 0183 0175 437

Vertical

1 0131 0134 2292 0185 0182 1623 0292 0282 3424 0394 038 355

Torsional 1 0348 0342 1722 0434 0425 207

Combin 37 element

Link 10

Beam 4


Node JNode I

Spring

Axialforce


Dec

k ro

ughn

ess (

mm

)

10

5

0

ndash5

ndash100 100 200 300 400 500

















114 1122 128 1122 114

714

656 1288 6562600

Suspender

Main cable

20Straingauge

Straingauge

Accelerometer

192



Axi

al st

rain

(με)

40

0 20 40 60 80 100Time (s)

120 140 160

20

0

ndash20

ndash40


(a)

ndash20Axi

al st

rain

(με)

40

20

0

ndash400 20 40 60 80 100

Time (s)


(b)

ndash20Axi

al st

rain

(με)

40

20

0

ndash400 10 20 30

Time (s)40 50


60

(c)


00050

00025

00000

Acce

lera

tion

(ms

2 )

ndash00025

ndash000500 20 40 60 80 100

Time (s)120


140 160

(a)

Figure 11 Continued



002

001

000

Acce

lera

tion

(ms

2 )ndash001

ndash0020 20 40 60 80

Time (s)


100

(b)

006

003

000

Acce

lera

tion

(ms

2 )

ndash003

ndash0060 10 20 30 5040

Time (s)


60

(c)


Tota

l veh

icle

wei

ght (t)

1200


111531

75968

1000

800

600

400

0 2 4 6 8 10 12 14 16Time (h)

18 20 22 24


Prop

ortio

n (

)

1008920


2368

235

3158

376

1316

446

1842

024

1316

80

60

40

20

0








250

200


150

Generated data

Cou

nt

100

50

00 1 2 3

Headway (s)4 5 6 7



Vehicleweight



Vehiclelength

Axleload

Axledistance


Vehicletype

Vehiclespeed

Timeheadway

Lanedistribution










5

4Ac

cum

ulat

ive l

ongi

tudi

nal

gird

er d

ispla

cem

ent (

m)

3

2

1200040006000


1200014000

1600018000

20000

None1

0605

0403

Velocity exponent

0201

4400

3900

3400

2900

2400

1900


Max

imum

long

itudi

nal

gird

er d

ispla

cem

ent (

m)

065

060

055

050

045

040

0352000

4000 60008000


1000012000

1400016000

1800020000

None1

0605

04

Velocity exponent

0302

01

0580

0540

0490

0440

0390








Max

imum

grid

er m

id-s

pan

bend

ing

mom

ent (

kNmiddotm

)

129times103

126

123

120

117

1142000

400060008000

1000012000

1400016000

1800020000

None

1250

1230

1210

1190

1170

Velocity exponent

106

0504

0302

01



Max

imum

long

itudi

nal p

ylon

disp

lace

men

t (m

)

0160

0159

0158

0157

0156

0155

01542000

40006000

800010000

1200014000

16000

None

01588

01581

01572

01563

015541

06

Velocity exponent05

0403

0201

1800020000







Max

imum

long

itudi

nal

pylo

n ac

cele

ratio

n (m

s2 )

0016

0014

0012

0010

00082000

None

0015

0014

0012

0011

00101

0605

Velocity exponent04

0302

01

4000 60008000

1000012000

1400016000

1800020000



Max

imum

pyl

onbe

ndin

g m

omen

t (kN

middotm)

times105

1716

1712

1708

1704

17002000

40006000

800010000

1200014000

16000


None

1717

1714

1711

1708

17051

0605

04

Velocity exponent03

0201






Susp

ende

r for

ce (k

N)

398times102

396

394

392

3902000 None

3973

3961

3946

3931

3916

Velocity exponent

106

0504

0302

01

40006000 8000


1400016000

1800020000





Dam

ping

velo

city

(ms

)

0048

0045

0042

0039

0033

0036


α = 01α = 02α = 03

α = 05α = 06α = 1

α = 04



16000

14000

12000

10000

8000

Dam

ping

forc

e (kN

)

6000

4000

2000


10000 12500 15000 17500 200000

α = 01α = 02α = 03

α = 05α = 06α = 1

α = 04


Max

imum

gird

er d

ispla

cem

ent (

m)

045

030

015

000

ndash015

ndash030

ndash045

ndash0600 50

ndash046

C = 1000C = 2500C = 5000

C = 10000C = 15000

C = 20000C = 7000None

Unit kN(ms)α

ndash048

ndash050

ndash006

ndash009

ndash012

ndash01530 40 215 220 225 230

100Time (s)

150 200 250 300


Accu

mul

ativ

e gird

er d

ispla

cem

ent (

m)

5

C = 1000C = 2500C = 5000

C = 7000C = 10000

C = 20000None

C = 15000

0 50 100 150 200Time (s)

250 300

Unit kN(ms)α

4

3

2

1

0






6 Conclusions




Max

imum

grid

er d

ispla

cem

ent (

m)

04

02

00

ndash02

ndash06

ndash04

ndash047

α = 01α = 02α = 03

α = 04α = 05

α = 1None

α = 06

ndash006

ndash009

ndash012

ndash015220 230

ndash048

ndash049

ndash050 30 32 34 36

0 50 100 150Time (s)

200 250 300


Accu

mul

ativ

e gird

er d

ispla

cem

ent (

m)

5

4

3

2

1

00 50 100 150

Time (s)200 250 300

α = 01α = 02α = 03

α = 04α = 05

α = 1None

α = 06




Appendix

KivuL andKi


KivuL K

ivuR (A1)

KiyuL andKi


KiyuL K

iyuR (A2)

KivlL andKi


KivlL K

ivlR (A3)

KiylL andKi


KiylL K

iylR (A4)

CivuL andCi


CivuL C

ivuR (A5)

CiyuL andCi


CiyuL C

iyuR (A6)

CivlL andCi


CivlL C

ivlR (A7)

CiylL andCi


CiylL C

iylR (A8)


Data Availability




Acknowledgments


References







































ξk+1

ηk+1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ξk

ηk

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ minus J ξk

ηk1113872 1113873minus 1

11139364

i1Ni ξk

ηk1113872 1113873xi minus x

11139364

i1Ni ξk

ηk1113872 1113873yi minus y

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

J

z 11139364i1 Ni(ξ η)xi1113960 1113961

zξz 1113936

4i1 Ni(ξ η)xi1113960 1113961

zη

z 11139364i1 Ni(ξ η)yi1113960 1113961

zξz 1113936

4i1 Ni(ξ η)yi1113960 1113961

zη

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(2)


when ξk+1minus ξk

ηk+1 minus ηk











Gd(n) Gd n0( 1113857n

n01113888 1113889

minusω

(3)

FF1 F2

F3F4


P (x y)


3 (1 1)4 (ndash1 1)η

ξ x

y

1 (x1 y1) 2 (x2 y2)

3 (x3 y3)4 (x4 y4)




Start



ampx le t (mid)

ANSYSmodel



last = mid

front = mid

Yes

No


Finish

Yes

No





r(x) 1113944N

k1

2Gd nk( 1113857Δn1113969

cos 2πnkx + θk( 1113857 (4)



Zie Z

ib + ri(x) (5)






MveuroZv + Cv

_Zv + KvXv Fv

MbeuroZb + Cb

_Xb + KbXb Fb

⎧⎨

⎩ (6)






⎧⎪⎨

⎪⎩(7)













F K|v|αsign(v) (8)




114m 896m

Main cable


Suspender

Approachbridge


buckle

214

Tunnelanchorage

208m


No

Start






Vehiclemotion state

Bridgemotion state

Yes

Finish

Bridgesubsystem

Vehiclesubsystem

Convergencecheck








Horizontal 1 0071 007 1412 0183 0175 437

Vertical

1 0131 0134 2292 0185 0182 1623 0292 0282 3424 0394 038 355

Torsional 1 0348 0342 1722 0434 0425 207

Combin 37 element

Link 10

Beam 4


Node JNode I

Spring

Axialforce


Dec

k ro

ughn

ess (

mm

)

10

5

0

ndash5

ndash100 100 200 300 400 500

















114 1122 128 1122 114

714

656 1288 6562600

Suspender

Main cable

20Straingauge

Straingauge

Accelerometer

192



Axi

al st

rain

(με)

40

0 20 40 60 80 100Time (s)

120 140 160

20

0

ndash20

ndash40


(a)

ndash20Axi

al st

rain

(με)

40

20

0

ndash400 20 40 60 80 100

Time (s)


(b)

ndash20Axi

al st

rain

(με)

40

20

0

ndash400 10 20 30

Time (s)40 50


60

(c)


00050

00025

00000

Acce

lera

tion

(ms

2 )

ndash00025

ndash000500 20 40 60 80 100

Time (s)120


140 160

(a)

Figure 11 Continued



002

001

000

Acce

lera

tion

(ms

2 )ndash001

ndash0020 20 40 60 80

Time (s)


100

(b)

006

003

000

Acce

lera

tion

(ms

2 )

ndash003

ndash0060 10 20 30 5040

Time (s)


60

(c)


Tota

l veh

icle

wei

ght (t)

1200


111531

75968

1000

800

600

400

0 2 4 6 8 10 12 14 16Time (h)

18 20 22 24


Prop

ortio

n (

)

1008920


2368

235

3158

376

1316

446

1842

024

1316

80

60

40

20

0








250

200


150

Generated data

Cou

nt

100

50

00 1 2 3

Headway (s)4 5 6 7



Vehicleweight



Vehiclelength

Axleload

Axledistance


Vehicletype

Vehiclespeed

Timeheadway

Lanedistribution










5

4Ac

cum

ulat

ive l

ongi

tudi

nal

gird

er d

ispla

cem

ent (

m)

3

2

1200040006000


1200014000

1600018000

20000

None1

0605

0403

Velocity exponent

0201

4400

3900

3400

2900

2400

1900


Max

imum

long

itudi

nal

gird

er d

ispla

cem

ent (

m)

065

060

055

050

045

040

0352000

4000 60008000


1000012000

1400016000

1800020000

None1

0605

04

Velocity exponent

0302

01

0580

0540

0490

0440

0390








Max

imum

grid

er m

id-s

pan

bend

ing

mom

ent (

kNmiddotm

)

129times103

126

123

120

117

1142000

400060008000

1000012000

1400016000

1800020000

None

1250

1230

1210

1190

1170

Velocity exponent

106

0504

0302

01



Max

imum

long

itudi

nal p

ylon

disp

lace

men

t (m

)

0160

0159

0158

0157

0156

0155

01542000

40006000

800010000

1200014000

16000

None

01588

01581

01572

01563

015541

06

Velocity exponent05

0403

0201

1800020000







Max

imum

long

itudi

nal

pylo

n ac

cele

ratio

n (m

s2 )

0016

0014

0012

0010

00082000

None

0015

0014

0012

0011

00101

0605

Velocity exponent04

0302

01

4000 60008000

1000012000

1400016000

1800020000



Max

imum

pyl

onbe

ndin

g m

omen

t (kN

middotm)

times105

1716

1712

1708

1704

17002000

40006000

800010000

1200014000

16000


None

1717

1714

1711

1708

17051

0605

04

Velocity exponent03

0201






Susp

ende

r for

ce (k

N)

398times102

396

394

392

3902000 None

3973

3961

3946

3931

3916

Velocity exponent

106

0504

0302

01

40006000 8000


1400016000

1800020000





Dam

ping

velo

city

(ms

)

0048

0045

0042

0039

0033

0036


α = 01α = 02α = 03

α = 05α = 06α = 1

α = 04



16000

14000

12000

10000

8000

Dam

ping

forc

e (kN

)

6000

4000

2000


10000 12500 15000 17500 200000

α = 01α = 02α = 03

α = 05α = 06α = 1

α = 04


Max

imum

gird

er d

ispla

cem

ent (

m)

045

030

015

000

ndash015

ndash030

ndash045

ndash0600 50

ndash046

C = 1000C = 2500C = 5000

C = 10000C = 15000

C = 20000C = 7000None

Unit kN(ms)α

ndash048

ndash050

ndash006

ndash009

ndash012

ndash01530 40 215 220 225 230

100Time (s)

150 200 250 300


Accu

mul

ativ

e gird

er d

ispla

cem

ent (

m)

5

C = 1000C = 2500C = 5000

C = 7000C = 10000

C = 20000None

C = 15000

0 50 100 150 200Time (s)

250 300

Unit kN(ms)α

4

3

2

1

0






6 Conclusions




Max

imum

grid

er d

ispla

cem

ent (

m)

04

02

00

ndash02

ndash06

ndash04

ndash047

α = 01α = 02α = 03

α = 04α = 05

α = 1None

α = 06

ndash006

ndash009

ndash012

ndash015220 230

ndash048

ndash049

ndash050 30 32 34 36

0 50 100 150Time (s)

200 250 300


Accu

mul

ativ

e gird

er d

ispla

cem

ent (

m)

5

4

3

2

1

00 50 100 150

Time (s)200 250 300

α = 01α = 02α = 03

α = 04α = 05

α = 1None

α = 06




Appendix

KivuL andKi


KivuL K

ivuR (A1)

KiyuL andKi


KiyuL K

iyuR (A2)

KivlL andKi


KivlL K

ivlR (A3)

KiylL andKi


KiylL K

iylR (A4)

CivuL andCi


CivuL C

ivuR (A5)

CiyuL andCi


CiyuL C

iyuR (A6)

CivlL andCi


CivlL C

ivlR (A7)

CiylL andCi


CiylL C

iylR (A8)


Data Availability




Acknowledgments


References








































r(x) 1113944N

k1

2Gd nk( 1113857Δn1113969

cos 2πnkx + θk( 1113857 (4)



Zie Z

ib + ri(x) (5)






MveuroZv + Cv

_Zv + KvXv Fv

MbeuroZb + Cb

_Xb + KbXb Fb

⎧⎨

⎩ (6)






⎧⎪⎨

⎪⎩(7)













F K|v|αsign(v) (8)




114m 896m

Main cable


Suspender

Approachbridge


buckle

214

Tunnelanchorage

208m


No

Start






Vehiclemotion state

Bridgemotion state

Yes

Finish

Bridgesubsystem

Vehiclesubsystem

Convergencecheck








Horizontal 1 0071 007 1412 0183 0175 437

Vertical

1 0131 0134 2292 0185 0182 1623 0292 0282 3424 0394 038 355

Torsional 1 0348 0342 1722 0434 0425 207

Combin 37 element

Link 10

Beam 4


Node JNode I

Spring

Axialforce


Dec

k ro

ughn

ess (

mm

)

10

5

0

ndash5

ndash100 100 200 300 400 500

















114 1122 128 1122 114

714

656 1288 6562600

Suspender

Main cable

20Straingauge

Straingauge

Accelerometer

192



Axi

al st

rain

(με)

40

0 20 40 60 80 100Time (s)

120 140 160

20

0

ndash20

ndash40


(a)

ndash20Axi

al st

rain

(με)

40

20

0

ndash400 20 40 60 80 100

Time (s)


(b)

ndash20Axi

al st

rain

(με)

40

20

0

ndash400 10 20 30

Time (s)40 50


60

(c)


00050

00025

00000

Acce

lera

tion

(ms

2 )

ndash00025

ndash000500 20 40 60 80 100

Time (s)120


140 160

(a)

Figure 11 Continued



002

001

000

Acce

lera

tion

(ms

2 )ndash001

ndash0020 20 40 60 80

Time (s)


100

(b)

006

003

000

Acce

lera

tion

(ms

2 )

ndash003

ndash0060 10 20 30 5040

Time (s)


60

(c)


Tota

l veh

icle

wei

ght (t)

1200


111531

75968

1000

800

600

400

0 2 4 6 8 10 12 14 16Time (h)

18 20 22 24


Prop

ortio

n (

)

1008920


2368

235

3158

376

1316

446

1842

024

1316

80

60

40

20

0








250

200


150

Generated data

Cou

nt

100

50

00 1 2 3

Headway (s)4 5 6 7



Vehicleweight



Vehiclelength

Axleload

Axledistance


Vehicletype

Vehiclespeed

Timeheadway

Lanedistribution










5

4Ac

cum

ulat

ive l

ongi

tudi

nal

gird

er d

ispla

cem

ent (

m)

3

2

1200040006000


1200014000

1600018000

20000

None1

0605

0403

Velocity exponent

0201

4400

3900

3400

2900

2400

1900


Max

imum

long

itudi

nal

gird

er d

ispla

cem

ent (

m)

065

060

055

050

045

040

0352000

4000 60008000


1000012000

1400016000

1800020000

None1

0605

04

Velocity exponent

0302

01

0580

0540

0490

0440

0390








Max

imum

grid

er m

id-s

pan

bend

ing

mom

ent (

kNmiddotm

)

129times103

126

123

120

117

1142000

400060008000

1000012000

1400016000

1800020000

None

1250

1230

1210

1190

1170

Velocity exponent

106

0504

0302

01



Max

imum

long

itudi

nal p

ylon

disp

lace

men

t (m

)

0160

0159

0158

0157

0156

0155

01542000

40006000

800010000

1200014000

16000

None

01588

01581

01572

01563

015541

06

Velocity exponent05

0403

0201

1800020000







Max

imum

long

itudi

nal

pylo

n ac

cele

ratio

n (m

s2 )

0016

0014

0012

0010

00082000

None

0015

0014

0012

0011

00101

0605

Velocity exponent04

0302

01

4000 60008000

1000012000

1400016000

1800020000



Max

imum

pyl

onbe

ndin

g m

omen

t (kN

middotm)

times105

1716

1712

1708

1704

17002000

40006000

800010000

1200014000

16000


None

1717

1714

1711

1708

17051

0605

04

Velocity exponent03

0201






Susp

ende

r for

ce (k

N)

398times102

396

394

392

3902000 None

3973

3961

3946

3931

3916

Velocity exponent

106

0504

0302

01

40006000 8000


1400016000

1800020000





Dam

ping

velo

city

(ms

)

0048

0045

0042

0039

0033

0036


α = 01α = 02α = 03

α = 05α = 06α = 1

α = 04



16000

14000

12000

10000

8000

Dam

ping

forc

e (kN

)

6000

4000

2000


10000 12500 15000 17500 200000

α = 01α = 02α = 03

α = 05α = 06α = 1

α = 04


Max

imum

gird

er d

ispla

cem

ent (

m)

045

030

015

000

ndash015

ndash030

ndash045

ndash0600 50

ndash046

C = 1000C = 2500C = 5000

C = 10000C = 15000

C = 20000C = 7000None

Unit kN(ms)α

ndash048

ndash050

ndash006

ndash009

ndash012

ndash01530 40 215 220 225 230

100Time (s)

150 200 250 300


Accu

mul

ativ

e gird

er d

ispla

cem

ent (

m)

5

C = 1000C = 2500C = 5000

C = 7000C = 10000

C = 20000None

C = 15000

0 50 100 150 200Time (s)

250 300

Unit kN(ms)α

4

3

2

1

0






6 Conclusions




Max

imum

grid

er d

ispla

cem

ent (

m)

04

02

00

ndash02

ndash06

ndash04

ndash047

α = 01α = 02α = 03

α = 04α = 05

α = 1None

α = 06

ndash006

ndash009

ndash012

ndash015220 230

ndash048

ndash049

ndash050 30 32 34 36

0 50 100 150Time (s)

200 250 300


Accu

mul

ativ

e gird

er d

ispla

cem

ent (

m)

5

4

3

2

1

00 50 100 150

Time (s)200 250 300

α = 01α = 02α = 03

α = 04α = 05

α = 1None

α = 06




Appendix

KivuL andKi


KivuL K

ivuR (A1)

KiyuL andKi


KiyuL K

iyuR (A2)

KivlL andKi


KivlL K

ivlR (A3)

KiylL andKi


KiylL K

iylR (A4)

CivuL andCi


CivuL C

ivuR (A5)

CiyuL andCi


CiyuL C

iyuR (A6)

CivlL andCi


CivlL C

ivlR (A7)

CiylL andCi


CiylL C

iylR (A8)


Data Availability




Acknowledgments


References









































F K|v|αsign(v) (8)




114m 896m

Main cable


Suspender

Approachbridge


buckle

214

Tunnelanchorage

208m


No

Start






Vehiclemotion state

Bridgemotion state

Yes

Finish

Bridgesubsystem

Vehiclesubsystem

Convergencecheck








Horizontal 1 0071 007 1412 0183 0175 437

Vertical

1 0131 0134 2292 0185 0182 1623 0292 0282 3424 0394 038 355

Torsional 1 0348 0342 1722 0434 0425 207

Combin 37 element

Link 10

Beam 4


Node JNode I

Spring

Axialforce


Dec

k ro

ughn

ess (

mm

)

10

5

0

ndash5

ndash100 100 200 300 400 500

















114 1122 128 1122 114

714

656 1288 6562600

Suspender

Main cable

20Straingauge

Straingauge

Accelerometer

192



Axi

al st

rain

(με)

40

0 20 40 60 80 100Time (s)

120 140 160

20

0

ndash20

ndash40


(a)

ndash20Axi

al st

rain

(με)

40

20

0

ndash400 20 40 60 80 100

Time (s)


(b)

ndash20Axi

al st

rain

(με)

40

20

0

ndash400 10 20 30

Time (s)40 50


60

(c)


00050

00025

00000

Acce

lera

tion

(ms

2 )

ndash00025

ndash000500 20 40 60 80 100

Time (s)120


140 160

(a)

Figure 11 Continued



002

001

000

Acce

lera

tion

(ms

2 )ndash001

ndash0020 20 40 60 80

Time (s)


100

(b)

006

003

000

Acce

lera

tion

(ms

2 )

ndash003

ndash0060 10 20 30 5040

Time (s)


60

(c)


Tota

l veh

icle

wei

ght (t)

1200


111531

75968

1000

800

600

400

0 2 4 6 8 10 12 14 16Time (h)

18 20 22 24


Prop

ortio

n (

)

1008920


2368

235

3158

376

1316

446

1842

024

1316

80

60

40

20

0








250

200


150

Generated data

Cou

nt

100

50

00 1 2 3

Headway (s)4 5 6 7



Vehicleweight



Vehiclelength

Axleload

Axledistance


Vehicletype

Vehiclespeed

Timeheadway

Lanedistribution










5

4Ac

cum

ulat

ive l

ongi

tudi

nal

gird

er d

ispla

cem

ent (

m)

3

2

1200040006000


1200014000

1600018000

20000

None1

0605

0403

Velocity exponent

0201

4400

3900

3400

2900

2400

1900


Max

imum

long

itudi

nal

gird

er d

ispla

cem

ent (

m)

065

060

055

050

045

040

0352000

4000 60008000


1000012000

1400016000

1800020000

None1

0605

04

Velocity exponent

0302

01

0580

0540

0490

0440

0390








Max

imum

grid

er m

id-s

pan

bend

ing

mom

ent (

kNmiddotm

)

129times103

126

123

120

117

1142000

400060008000

1000012000

1400016000

1800020000

None

1250

1230

1210

1190

1170

Velocity exponent

106

0504

0302

01



Max

imum

long

itudi

nal p

ylon

disp

lace

men

t (m

)

0160

0159

0158

0157

0156

0155

01542000

40006000

800010000

1200014000

16000

None

01588

01581

01572

01563

015541

06

Velocity exponent05

0403

0201

1800020000







Max

imum

long

itudi

nal

pylo

n ac

cele

ratio

n (m

s2 )

0016

0014

0012

0010

00082000

None

0015

0014

0012

0011

00101

0605

Velocity exponent04

0302

01

4000 60008000

1000012000

1400016000

1800020000



Max

imum

pyl

onbe

ndin

g m

omen

t (kN

middotm)

times105

1716

1712

1708

1704

17002000

40006000

800010000

1200014000

16000


None

1717

1714

1711

1708

17051

0605

04

Velocity exponent03

0201






Susp

ende

r for

ce (k

N)

398times102

396

394

392

3902000 None

3973

3961

3946

3931

3916

Velocity exponent

106

0504

0302

01

40006000 8000


1400016000

1800020000





Dam

ping

velo

city

(ms

)

0048

0045

0042

0039

0033

0036


α = 01α = 02α = 03

α = 05α = 06α = 1

α = 04



16000

14000

12000

10000

8000

Dam

ping

forc

e (kN

)

6000

4000

2000


10000 12500 15000 17500 200000

α = 01α = 02α = 03

α = 05α = 06α = 1

α = 04


Max

imum

gird

er d

ispla

cem

ent (

m)

045

030

015

000

ndash015

ndash030

ndash045

ndash0600 50

ndash046

C = 1000C = 2500C = 5000

C = 10000C = 15000

C = 20000C = 7000None

Unit kN(ms)α

ndash048

ndash050

ndash006

ndash009

ndash012

ndash01530 40 215 220 225 230

100Time (s)

150 200 250 300


Accu

mul

ativ

e gird

er d

ispla

cem

ent (

m)

5

C = 1000C = 2500C = 5000

C = 7000C = 10000

C = 20000None

C = 15000

0 50 100 150 200Time (s)

250 300

Unit kN(ms)α

4

3

2

1

0






6 Conclusions




Max

imum

grid

er d

ispla

cem

ent (

m)

04

02

00

ndash02

ndash06

ndash04

ndash047

α = 01α = 02α = 03

α = 04α = 05

α = 1None

α = 06

ndash006

ndash009

ndash012

ndash015220 230

ndash048

ndash049

ndash050 30 32 34 36

0 50 100 150Time (s)

200 250 300


Accu

mul

ativ

e gird

er d

ispla

cem

ent (

m)

5

4

3

2

1

00 50 100 150

Time (s)200 250 300

α = 01α = 02α = 03

α = 04α = 05

α = 1None

α = 06




Appendix

KivuL andKi


KivuL K

ivuR (A1)

KiyuL andKi


KiyuL K

iyuR (A2)

KivlL andKi


KivlL K

ivlR (A3)

KiylL andKi


KiylL K

iylR (A4)

CivuL andCi


CivuL C

ivuR (A5)

CiyuL andCi


CiyuL C

iyuR (A6)

CivlL andCi


CivlL C

ivlR (A7)

CiylL andCi


CiylL C

iylR (A8)


Data Availability




Acknowledgments


References











































Horizontal 1 0071 007 1412 0183 0175 437

Vertical

1 0131 0134 2292 0185 0182 1623 0292 0282 3424 0394 038 355

Torsional 1 0348 0342 1722 0434 0425 207

Combin 37 element

Link 10

Beam 4


Node JNode I

Spring

Axialforce


Dec

k ro

ughn

ess (

mm

)

10

5

0

ndash5

ndash100 100 200 300 400 500

















114 1122 128 1122 114

714

656 1288 6562600

Suspender

Main cable

20Straingauge

Straingauge

Accelerometer

192



Axi

al st

rain

(με)

40

0 20 40 60 80 100Time (s)

120 140 160

20

0

ndash20

ndash40


(a)

ndash20Axi

al st

rain

(με)

40

20

0

ndash400 20 40 60 80 100

Time (s)


(b)

ndash20Axi

al st

rain

(με)

40

20

0

ndash400 10 20 30

Time (s)40 50


60

(c)


00050

00025

00000

Acce

lera

tion

(ms

2 )

ndash00025

ndash000500 20 40 60 80 100

Time (s)120


140 160

(a)

Figure 11 Continued



002

001

000

Acce

lera

tion

(ms

2 )ndash001

ndash0020 20 40 60 80

Time (s)


100

(b)

006

003

000

Acce

lera

tion

(ms

2 )

ndash003

ndash0060 10 20 30 5040

Time (s)


60

(c)


Tota

l veh

icle

wei

ght (t)

1200


111531

75968

1000

800

600

400

0 2 4 6 8 10 12 14 16Time (h)

18 20 22 24


Prop

ortio

n (

)

1008920


2368

235

3158

376

1316

446

1842

024

1316

80

60

40

20

0








250

200


150

Generated data

Cou

nt

100

50

00 1 2 3

Headway (s)4 5 6 7



Vehicleweight



Vehiclelength

Axleload

Axledistance


Vehicletype

Vehiclespeed

Timeheadway

Lanedistribution










5

4Ac

cum

ulat

ive l

ongi

tudi

nal

gird

er d

ispla

cem

ent (

m)

3

2

1200040006000


1200014000

1600018000

20000

None1

0605

0403

Velocity exponent

0201

4400

3900

3400

2900

2400

1900


Max

imum

long

itudi

nal

gird

er d

ispla

cem

ent (

m)

065

060

055

050

045

040

0352000

4000 60008000


1000012000

1400016000

1800020000

None1

0605

04

Velocity exponent

0302

01

0580

0540

0490

0440

0390








Max

imum

grid

er m

id-s

pan

bend

ing

mom

ent (

kNmiddotm

)

129times103

126

123

120

117

1142000

400060008000

1000012000

1400016000

1800020000

None

1250

1230

1210

1190

1170

Velocity exponent

106

0504

0302

01



Max

imum

long

itudi

nal p

ylon

disp

lace

men

t (m

)

0160

0159

0158

0157

0156

0155

01542000

40006000

800010000

1200014000

16000

None

01588

01581

01572

01563

015541

06

Velocity exponent05

0403

0201

1800020000







Max

imum

long

itudi

nal

pylo

n ac

cele

ratio

n (m

s2 )

0016

0014

0012

0010

00082000

None

0015

0014

0012

0011

00101

0605

Velocity exponent04

0302

01

4000 60008000

1000012000

1400016000

1800020000



Max

imum

pyl

onbe

ndin

g m

omen

t (kN

middotm)

times105

1716

1712

1708

1704

17002000

40006000

800010000

1200014000

16000


None

1717

1714

1711

1708

17051

0605

04

Velocity exponent03

0201






Susp

ende

r for

ce (k

N)

398times102

396

394

392

3902000 None

3973

3961

3946

3931

3916

Velocity exponent

106

0504

0302

01

40006000 8000


1400016000

1800020000





Dam

ping

velo

city

(ms

)

0048

0045

0042

0039

0033

0036


α = 01α = 02α = 03

α = 05α = 06α = 1

α = 04



16000

14000

12000

10000

8000

Dam

ping

forc

e (kN

)

6000

4000

2000


10000 12500 15000 17500 200000

α = 01α = 02α = 03

α = 05α = 06α = 1

α = 04


Max

imum

gird

er d

ispla

cem

ent (

m)

045

030

015

000

ndash015

ndash030

ndash045

ndash0600 50

ndash046

C = 1000C = 2500C = 5000

C = 10000C = 15000

C = 20000C = 7000None

Unit kN(ms)α

ndash048

ndash050

ndash006

ndash009

ndash012

ndash01530 40 215 220 225 230

100Time (s)

150 200 250 300


Accu

mul

ativ

e gird

er d

ispla

cem

ent (

m)

5

C = 1000C = 2500C = 5000

C = 7000C = 10000

C = 20000None

C = 15000

0 50 100 150 200Time (s)

250 300

Unit kN(ms)α

4

3

2

1

0






6 Conclusions




Max

imum

grid

er d

ispla

cem

ent (

m)

04

02

00

ndash02

ndash06

ndash04

ndash047

α = 01α = 02α = 03

α = 04α = 05

α = 1None

α = 06

ndash006

ndash009

ndash012

ndash015220 230

ndash048

ndash049

ndash050 30 32 34 36

0 50 100 150Time (s)

200 250 300


Accu

mul

ativ

e gird

er d

ispla

cem

ent (

m)

5

4

3

2

1

00 50 100 150

Time (s)200 250 300

α = 01α = 02α = 03

α = 04α = 05

α = 1None

α = 06




Appendix

KivuL andKi


KivuL K

ivuR (A1)

KiyuL andKi


KiyuL K

iyuR (A2)

KivlL andKi


KivlL K

ivlR (A3)

KiylL andKi


KiylL K

iylR (A4)

CivuL andCi


CivuL C

ivuR (A5)

CiyuL andCi


CiyuL C

iyuR (A6)

CivlL andCi


CivlL C

ivlR (A7)

CiylL andCi


CiylL C

iylR (A8)


Data Availability




Acknowledgments


References



















































114 1122 128 1122 114

714

656 1288 6562600

Suspender

Main cable

20Straingauge

Straingauge

Accelerometer

192



Axi

al st

rain

(με)

40

0 20 40 60 80 100Time (s)

120 140 160

20

0

ndash20

ndash40


(a)

ndash20Axi

al st

rain

(με)

40

20

0

ndash400 20 40 60 80 100

Time (s)


(b)

ndash20Axi

al st

rain

(με)

40

20

0

ndash400 10 20 30

Time (s)40 50


60

(c)


00050

00025

00000

Acce

lera

tion

(ms

2 )

ndash00025

ndash000500 20 40 60 80 100

Time (s)120


140 160

(a)

Figure 11 Continued



002

001

000

Acce

lera

tion

(ms

2 )ndash001

ndash0020 20 40 60 80

Time (s)


100

(b)

006

003

000

Acce

lera

tion

(ms

2 )

ndash003

ndash0060 10 20 30 5040

Time (s)


60

(c)


Tota

l veh

icle

wei

ght (t)

1200


111531

75968

1000

800

600

400

0 2 4 6 8 10 12 14 16Time (h)

18 20 22 24


Prop

ortio

n (

)

1008920


2368

235

3158

376

1316

446

1842

024

1316

80

60

40

20

0








250

200


150

Generated data

Cou

nt

100

50

00 1 2 3

Headway (s)4 5 6 7



Vehicleweight



Vehiclelength

Axleload

Axledistance


Vehicletype

Vehiclespeed

Timeheadway

Lanedistribution










5

4Ac

cum

ulat

ive l

ongi

tudi

nal

gird

er d

ispla

cem

ent (

m)

3

2

1200040006000


1200014000

1600018000

20000

None1

0605

0403

Velocity exponent

0201

4400

3900

3400

2900

2400

1900


Max

imum

long

itudi

nal

gird

er d

ispla

cem

ent (

m)

065

060

055

050

045

040

0352000

4000 60008000


1000012000

1400016000

1800020000

None1

0605

04

Velocity exponent

0302

01

0580

0540

0490

0440

0390








Max

imum

grid

er m

id-s

pan

bend

ing

mom

ent (

kNmiddotm

)

129times103

126

123

120

117

1142000

400060008000

1000012000

1400016000

1800020000

None

1250

1230

1210

1190

1170

Velocity exponent

106

0504

0302

01



Max

imum

long

itudi

nal p

ylon

disp

lace

men

t (m

)

0160

0159

0158

0157

0156

0155

01542000

40006000

800010000

1200014000

16000

None

01588

01581

01572

01563

015541

06

Velocity exponent05

0403

0201

1800020000







Max

imum

long

itudi

nal

pylo

n ac

cele

ratio

n (m

s2 )

0016

0014

0012

0010

00082000

None

0015

0014

0012

0011

00101

0605

Velocity exponent04

0302

01

4000 60008000

1000012000

1400016000

1800020000



Max

imum

pyl

onbe

ndin

g m

omen

t (kN

middotm)

times105

1716

1712

1708

1704

17002000

40006000

800010000

1200014000

16000


None

1717

1714

1711

1708

17051

0605

04

Velocity exponent03

0201






Susp

ende

r for

ce (k

N)

398times102

396

394

392

3902000 None

3973

3961

3946

3931

3916

Velocity exponent

106

0504

0302

01

40006000 8000


1400016000

1800020000





Dam

ping

velo

city

(ms

)

0048

0045

0042

0039

0033

0036


α = 01α = 02α = 03

α = 05α = 06α = 1

α = 04



16000

14000

12000

10000

8000

Dam

ping

forc

e (kN

)

6000

4000

2000


10000 12500 15000 17500 200000

α = 01α = 02α = 03

α = 05α = 06α = 1

α = 04


Max

imum

gird

er d

ispla

cem

ent (

m)

045

030

015

000

ndash015

ndash030

ndash045

ndash0600 50

ndash046

C = 1000C = 2500C = 5000

C = 10000C = 15000

C = 20000C = 7000None

Unit kN(ms)α

ndash048

ndash050

ndash006

ndash009

ndash012

ndash01530 40 215 220 225 230

100Time (s)

150 200 250 300


Accu

mul

ativ

e gird

er d

ispla

cem

ent (

m)

5

C = 1000C = 2500C = 5000

C = 7000C = 10000

C = 20000None

C = 15000

0 50 100 150 200Time (s)

250 300

Unit kN(ms)α

4

3

2

1

0






6 Conclusions




Max

imum

grid

er d

ispla

cem

ent (

m)

04

02

00

ndash02

ndash06

ndash04

ndash047

α = 01α = 02α = 03

α = 04α = 05

α = 1None

α = 06

ndash006

ndash009

ndash012

ndash015220 230

ndash048

ndash049

ndash050 30 32 34 36

0 50 100 150Time (s)

200 250 300


Accu

mul

ativ

e gird

er d

ispla

cem

ent (

m)

5

4

3

2

1

00 50 100 150

Time (s)200 250 300

α = 01α = 02α = 03

α = 04α = 05

α = 1None

α = 06




Appendix

KivuL andKi


KivuL K

ivuR (A1)

KiyuL andKi


KiyuL K

iyuR (A2)

KivlL andKi


KivlL K

ivlR (A3)

KiylL andKi


KiylL K

iylR (A4)

CivuL andCi


CivuL C

ivuR (A5)

CiyuL andCi


CiyuL C

iyuR (A6)

CivlL andCi


CivlL C

ivlR (A7)

CiylL andCi


CiylL C

iylR (A8)


Data Availability




Acknowledgments


References






































Axi

al st

rain

(με)

40

0 20 40 60 80 100Time (s)

120 140 160

20

0

ndash20

ndash40


(a)

ndash20Axi

al st

rain

(με)

40

20

0

ndash400 20 40 60 80 100

Time (s)


(b)

ndash20Axi

al st

rain

(με)

40

20

0

ndash400 10 20 30

Time (s)40 50


60

(c)


00050

00025

00000

Acce

lera

tion

(ms

2 )

ndash00025

ndash000500 20 40 60 80 100

Time (s)120


140 160

(a)

Figure 11 Continued



002

001

000

Acce

lera

tion

(ms

2 )ndash001

ndash0020 20 40 60 80

Time (s)


100

(b)

006

003

000

Acce

lera

tion

(ms

2 )

ndash003

ndash0060 10 20 30 5040

Time (s)


60

(c)


Tota

l veh

icle

wei

ght (t)

1200


111531

75968

1000

800

600

400

0 2 4 6 8 10 12 14 16Time (h)

18 20 22 24


Prop

ortio

n (

)

1008920


2368

235

3158

376

1316

446

1842

024

1316

80

60

40

20

0








250

200


150

Generated data

Cou

nt

100

50

00 1 2 3

Headway (s)4 5 6 7



Vehicleweight



Vehiclelength

Axleload

Axledistance


Vehicletype

Vehiclespeed

Timeheadway

Lanedistribution










5

4Ac

cum

ulat

ive l

ongi

tudi

nal

gird

er d

ispla

cem

ent (

m)

3

2

1200040006000


1200014000

1600018000

20000

None1

0605

0403

Velocity exponent

0201

4400

3900

3400

2900

2400

1900


Max

imum

long

itudi

nal

gird

er d

ispla

cem

ent (

m)

065

060

055

050

045

040

0352000

4000 60008000


1000012000

1400016000

1800020000

None1

0605

04

Velocity exponent

0302

01

0580

0540

0490

0440

0390








Max

imum

grid

er m

id-s

pan

bend

ing

mom

ent (

kNmiddotm

)

129times103

126

123

120

117

1142000

400060008000

1000012000

1400016000

1800020000

None

1250

1230

1210

1190

1170

Velocity exponent

106

0504

0302

01



Max

imum

long

itudi

nal p

ylon

disp

lace

men

t (m

)

0160

0159

0158

0157

0156

0155

01542000

40006000

800010000

1200014000

16000

None

01588

01581

01572

01563

015541

06

Velocity exponent05

0403

0201

1800020000







Max

imum

long

itudi

nal

pylo

n ac

cele

ratio

n (m

s2 )

0016

0014

0012

0010

00082000

None

0015

0014

0012

0011

00101

0605

Velocity exponent04

0302

01

4000 60008000

1000012000

1400016000

1800020000



Max

imum

pyl

onbe

ndin

g m

omen

t (kN

middotm)

times105

1716

1712

1708

1704

17002000

40006000

800010000

1200014000

16000


None

1717

1714

1711

1708

17051

0605

04

Velocity exponent03

0201






Susp

ende

r for

ce (k

N)

398times102

396

394

392

3902000 None

3973

3961

3946

3931

3916

Velocity exponent

106

0504

0302

01

40006000 8000


1400016000

1800020000





Dam

ping

velo

city

(ms

)

0048

0045

0042

0039

0033

0036


α = 01α = 02α = 03

α = 05α = 06α = 1

α = 04



16000

14000

12000

10000

8000

Dam

ping

forc

e (kN

)

6000

4000

2000


10000 12500 15000 17500 200000

α = 01α = 02α = 03

α = 05α = 06α = 1

α = 04


Max

imum

gird

er d

ispla

cem

ent (

m)

045

030

015

000

ndash015

ndash030

ndash045

ndash0600 50

ndash046

C = 1000C = 2500C = 5000

C = 10000C = 15000

C = 20000C = 7000None

Unit kN(ms)α

ndash048

ndash050

ndash006

ndash009

ndash012

ndash01530 40 215 220 225 230

100Time (s)

150 200 250 300


Accu

mul

ativ

e gird

er d

ispla

cem

ent (

m)

5

C = 1000C = 2500C = 5000

C = 7000C = 10000

C = 20000None

C = 15000

0 50 100 150 200Time (s)

250 300

Unit kN(ms)α

4

3

2

1

0






6 Conclusions




Max

imum

grid

er d

ispla

cem

ent (

m)

04

02

00

ndash02

ndash06

ndash04

ndash047

α = 01α = 02α = 03

α = 04α = 05

α = 1None

α = 06

ndash006

ndash009

ndash012

ndash015220 230

ndash048

ndash049

ndash050 30 32 34 36

0 50 100 150Time (s)

200 250 300


Accu

mul

ativ

e gird

er d

ispla

cem

ent (

m)

5

4

3

2

1

00 50 100 150

Time (s)200 250 300

α = 01α = 02α = 03

α = 04α = 05

α = 1None

α = 06




Appendix

KivuL andKi


KivuL K

ivuR (A1)

KiyuL andKi


KiyuL K

iyuR (A2)

KivlL andKi


KivlL K

ivlR (A3)

KiylL andKi


KiylL K

iylR (A4)

CivuL andCi


CivuL C

ivuR (A5)

CiyuL andCi


CiyuL C

iyuR (A6)

CivlL andCi


CivlL C

ivlR (A7)

CiylL andCi


CiylL C

iylR (A8)


Data Availability




Acknowledgments


References







































002

001

000

Acce

lera

tion

(ms

2 )ndash001

ndash0020 20 40 60 80

Time (s)


100

(b)

006

003

000

Acce

lera

tion

(ms

2 )

ndash003

ndash0060 10 20 30 5040

Time (s)


60

(c)


Tota

l veh

icle

wei

ght (t)

1200


111531

75968

1000

800

600

400

0 2 4 6 8 10 12 14 16Time (h)

18 20 22 24


Prop

ortio

n (

)

1008920


2368

235

3158

376

1316

446

1842

024

1316

80

60

40

20

0








250

200


150

Generated data

Cou

nt

100

50

00 1 2 3

Headway (s)4 5 6 7



Vehicleweight



Vehiclelength

Axleload

Axledistance


Vehicletype

Vehiclespeed

Timeheadway

Lanedistribution










5

4Ac

cum

ulat

ive l

ongi

tudi

nal

gird

er d

ispla

cem

ent (

m)

3

2

1200040006000


1200014000

1600018000

20000

None1

0605

0403

Velocity exponent

0201

4400

3900

3400

2900

2400

1900


Max

imum

long

itudi

nal

gird

er d

ispla

cem

ent (

m)

065

060

055

050

045

040

0352000

4000 60008000


1000012000

1400016000

1800020000

None1

0605

04

Velocity exponent

0302

01

0580

0540

0490

0440

0390








Max

imum

grid

er m

id-s

pan

bend

ing

mom

ent (

kNmiddotm

)

129times103

126

123

120

117

1142000

400060008000

1000012000

1400016000

1800020000

None

1250

1230

1210

1190

1170

Velocity exponent

106

0504

0302

01



Max

imum

long

itudi

nal p

ylon

disp

lace

men

t (m

)

0160

0159

0158

0157

0156

0155

01542000

40006000

800010000

1200014000

16000

None

01588

01581

01572

01563

015541

06

Velocity exponent05

0403

0201

1800020000







Max

imum

long

itudi

nal

pylo

n ac

cele

ratio

n (m

s2 )

0016

0014

0012

0010

00082000

None

0015

0014

0012

0011

00101

0605

Velocity exponent04

0302

01

4000 60008000

1000012000

1400016000

1800020000



Max

imum

pyl

onbe

ndin

g m

omen

t (kN

middotm)

times105

1716

1712

1708

1704

17002000

40006000

800010000

1200014000

16000


None

1717

1714

1711

1708

17051

0605

04

Velocity exponent03

0201






Susp

ende

r for

ce (k

N)

398times102

396

394

392

3902000 None

3973

3961

3946

3931

3916

Velocity exponent

106

0504

0302

01

40006000 8000


1400016000

1800020000





Dam

ping

velo

city

(ms

)

0048

0045

0042

0039

0033

0036


α = 01α = 02α = 03

α = 05α = 06α = 1

α = 04



16000

14000

12000

10000

8000

Dam

ping

forc

e (kN

)

6000

4000

2000


10000 12500 15000 17500 200000

α = 01α = 02α = 03

α = 05α = 06α = 1

α = 04


Max

imum

gird

er d

ispla

cem

ent (

m)

045

030

015

000

ndash015

ndash030

ndash045

ndash0600 50

ndash046

C = 1000C = 2500C = 5000

C = 10000C = 15000

C = 20000C = 7000None

Unit kN(ms)α

ndash048

ndash050

ndash006

ndash009

ndash012

ndash01530 40 215 220 225 230

100Time (s)

150 200 250 300


Accu

mul

ativ

e gird

er d

ispla

cem

ent (

m)

5

C = 1000C = 2500C = 5000

C = 7000C = 10000

C = 20000None

C = 15000

0 50 100 150 200Time (s)

250 300

Unit kN(ms)α

4

3

2

1

0






6 Conclusions




Max

imum

grid

er d

ispla

cem

ent (

m)

04

02

00

ndash02

ndash06

ndash04

ndash047

α = 01α = 02α = 03

α = 04α = 05

α = 1None

α = 06

ndash006

ndash009

ndash012

ndash015220 230

ndash048

ndash049

ndash050 30 32 34 36

0 50 100 150Time (s)

200 250 300


Accu

mul

ativ

e gird

er d

ispla

cem

ent (

m)

5

4

3

2

1

00 50 100 150

Time (s)200 250 300

α = 01α = 02α = 03

α = 04α = 05

α = 1None

α = 06




Appendix

KivuL andKi


KivuL K

ivuR (A1)

KiyuL andKi


KiyuL K

iyuR (A2)

KivlL andKi


KivlL K

ivlR (A3)

KiylL andKi


KiylL K

iylR (A4)

CivuL andCi


CivuL C

ivuR (A5)

CiyuL andCi


CiyuL C

iyuR (A6)

CivlL andCi


CivlL C

ivlR (A7)

CiylL andCi


CiylL C

iylR (A8)


Data Availability




Acknowledgments


References










































250

200


150

Generated data

Cou

nt

100

50

00 1 2 3

Headway (s)4 5 6 7



Vehicleweight



Vehiclelength

Axleload

Axledistance


Vehicletype

Vehiclespeed

Timeheadway

Lanedistribution










5

4Ac

cum

ulat

ive l

ongi

tudi

nal

gird

er d

ispla

cem

ent (

m)

3

2

1200040006000


1200014000

1600018000

20000

None1

0605

0403

Velocity exponent

0201

4400

3900

3400

2900

2400

1900


Max

imum

long

itudi

nal

gird

er d

ispla

cem

ent (

m)

065

060

055

050

045

040

0352000

4000 60008000


1000012000

1400016000

1800020000

None1

0605

04

Velocity exponent

0302

01

0580

0540

0490

0440

0390








Max

imum

grid

er m

id-s

pan

bend

ing

mom

ent (

kNmiddotm

)

129times103

126

123

120

117

1142000

400060008000

1000012000

1400016000

1800020000

None

1250

1230

1210

1190

1170

Velocity exponent

106

0504

0302

01



Max

imum

long

itudi

nal p

ylon

disp

lace

men

t (m

)

0160

0159

0158

0157

0156

0155

01542000

40006000

800010000

1200014000

16000

None

01588

01581

01572

01563

015541

06

Velocity exponent05

0403

0201

1800020000







Max

imum

long

itudi

nal

pylo

n ac

cele

ratio

n (m

s2 )

0016

0014

0012

0010

00082000

None

0015

0014

0012

0011

00101

0605

Velocity exponent04

0302

01

4000 60008000

1000012000

1400016000

1800020000



Max

imum

pyl

onbe

ndin

g m

omen

t (kN

middotm)

times105

1716

1712

1708

1704

17002000

40006000

800010000

1200014000

16000


None

1717

1714

1711

1708

17051

0605

04

Velocity exponent03

0201






Susp

ende

r for

ce (k

N)

398times102

396

394

392

3902000 None

3973

3961

3946

3931

3916

Velocity exponent

106

0504

0302

01

40006000 8000


1400016000

1800020000





Dam

ping

velo

city

(ms

)

0048

0045

0042

0039

0033

0036


α = 01α = 02α = 03

α = 05α = 06α = 1

α = 04



16000

14000

12000

10000

8000

Dam

ping

forc

e (kN

)

6000

4000

2000


10000 12500 15000 17500 200000

α = 01α = 02α = 03

α = 05α = 06α = 1

α = 04


Max

imum

gird

er d

ispla

cem

ent (

m)

045

030

015

000

ndash015

ndash030

ndash045

ndash0600 50

ndash046

C = 1000C = 2500C = 5000

C = 10000C = 15000

C = 20000C = 7000None

Unit kN(ms)α

ndash048

ndash050

ndash006

ndash009

ndash012

ndash01530 40 215 220 225 230

100Time (s)

150 200 250 300


Accu

mul

ativ

e gird

er d

ispla

cem

ent (

m)

5

C = 1000C = 2500C = 5000

C = 7000C = 10000

C = 20000None

C = 15000

0 50 100 150 200Time (s)

250 300

Unit kN(ms)α

4

3

2

1

0






6 Conclusions




Max

imum

grid

er d

ispla

cem

ent (

m)

04

02

00

ndash02

ndash06

ndash04

ndash047

α = 01α = 02α = 03

α = 04α = 05

α = 1None

α = 06

ndash006

ndash009

ndash012

ndash015220 230

ndash048

ndash049

ndash050 30 32 34 36

0 50 100 150Time (s)

200 250 300


Accu

mul

ativ

e gird

er d

ispla

cem

ent (

m)

5

4

3

2

1

00 50 100 150

Time (s)200 250 300

α = 01α = 02α = 03

α = 04α = 05

α = 1None

α = 06




Appendix

KivuL andKi


KivuL K

ivuR (A1)

KiyuL andKi


KiyuL K

iyuR (A2)

KivlL andKi


KivlL K

ivlR (A3)

KiylL andKi


KiylL K

iylR (A4)

CivuL andCi


CivuL C

ivuR (A5)

CiyuL andCi


CiyuL C

iyuR (A6)

CivlL andCi


CivlL C

ivlR (A7)

CiylL andCi


CiylL C

iylR (A8)


Data Availability




Acknowledgments


References









































5

4Ac

cum

ulat

ive l

ongi

tudi

nal

gird

er d

ispla

cem

ent (

m)

3

2

1200040006000


1200014000

1600018000

20000

None1

0605

0403

Velocity exponent

0201

4400

3900

3400

2900

2400

1900


Max

imum

long

itudi

nal

gird

er d

ispla

cem

ent (

m)

065

060

055

050

045

040

0352000

4000 60008000


1000012000

1400016000

1800020000

None1

0605

04

Velocity exponent

0302

01

0580

0540

0490

0440

0390








Max

imum

grid

er m

id-s

pan

bend

ing

mom

ent (

kNmiddotm

)

129times103

126

123

120

117

1142000

400060008000

1000012000

1400016000

1800020000

None

1250

1230

1210

1190

1170

Velocity exponent

106

0504

0302

01



Max

imum

long

itudi

nal p

ylon

disp

lace

men

t (m

)

0160

0159

0158

0157

0156

0155

01542000

40006000

800010000

1200014000

16000

None

01588

01581

01572

01563

015541

06

Velocity exponent05

0403

0201

1800020000







Max

imum

long

itudi

nal

pylo

n ac

cele

ratio

n (m

s2 )

0016

0014

0012

0010

00082000

None

0015

0014

0012

0011

00101

0605

Velocity exponent04

0302

01

4000 60008000

1000012000

1400016000

1800020000



Max

imum

pyl

onbe

ndin

g m

omen

t (kN

middotm)

times105

1716

1712

1708

1704

17002000

40006000

800010000

1200014000

16000


None

1717

1714

1711

1708

17051

0605

04

Velocity exponent03

0201






Susp

ende

r for

ce (k

N)

398times102

396

394

392

3902000 None

3973

3961

3946

3931

3916

Velocity exponent

106

0504

0302

01

40006000 8000


1400016000

1800020000





Dam

ping

velo

city

(ms

)

0048

0045

0042

0039

0033

0036


α = 01α = 02α = 03

α = 05α = 06α = 1

α = 04



16000

14000

12000

10000

8000

Dam

ping

forc

e (kN

)

6000

4000

2000


10000 12500 15000 17500 200000

α = 01α = 02α = 03

α = 05α = 06α = 1

α = 04


Max

imum

gird

er d

ispla

cem

ent (

m)

045

030

015

000

ndash015

ndash030

ndash045

ndash0600 50

ndash046

C = 1000C = 2500C = 5000

C = 10000C = 15000

C = 20000C = 7000None

Unit kN(ms)α

ndash048

ndash050

ndash006

ndash009

ndash012

ndash01530 40 215 220 225 230

100Time (s)

150 200 250 300


Accu

mul

ativ

e gird

er d

ispla

cem

ent (

m)

5

C = 1000C = 2500C = 5000

C = 7000C = 10000

C = 20000None

C = 15000

0 50 100 150 200Time (s)

250 300

Unit kN(ms)α

4

3

2

1

0






6 Conclusions




Max

imum

grid

er d

ispla

cem

ent (

m)

04

02

00

ndash02

ndash06

ndash04

ndash047

α = 01α = 02α = 03

α = 04α = 05

α = 1None

α = 06

ndash006

ndash009

ndash012

ndash015220 230

ndash048

ndash049

ndash050 30 32 34 36

0 50 100 150Time (s)

200 250 300


Accu

mul

ativ

e gird

er d

ispla

cem

ent (

m)

5

4

3

2

1

00 50 100 150

Time (s)200 250 300

α = 01α = 02α = 03

α = 04α = 05

α = 1None

α = 06




Appendix

KivuL andKi


KivuL K

ivuR (A1)

KiyuL andKi


KiyuL K

iyuR (A2)

KivlL andKi


KivlL K

ivlR (A3)

KiylL andKi


KiylL K

iylR (A4)

CivuL andCi


CivuL C

ivuR (A5)

CiyuL andCi


CiyuL C

iyuR (A6)

CivlL andCi


CivlL C

ivlR (A7)

CiylL andCi


CiylL C

iylR (A8)


Data Availability




Acknowledgments


References











































Max

imum

grid

er m

id-s

pan

bend

ing

mom

ent (

kNmiddotm

)

129times103

126

123

120

117

1142000

400060008000

1000012000

1400016000

1800020000

None

1250

1230

1210

1190

1170

Velocity exponent

106

0504

0302

01



Max

imum

long

itudi

nal p

ylon

disp

lace

men

t (m

)

0160

0159

0158

0157

0156

0155

01542000

40006000

800010000

1200014000

16000

None

01588

01581

01572

01563

015541

06

Velocity exponent05

0403

0201

1800020000







Max

imum

long

itudi

nal

pylo

n ac

cele

ratio

n (m

s2 )

0016

0014

0012

0010

00082000

None

0015

0014

0012

0011

00101

0605

Velocity exponent04

0302

01

4000 60008000

1000012000

1400016000

1800020000



Max

imum

pyl

onbe

ndin

g m

omen

t (kN

middotm)

times105

1716

1712

1708

1704

17002000

40006000

800010000

1200014000

16000


None

1717

1714

1711

1708

17051

0605

04

Velocity exponent03

0201






Susp

ende

r for

ce (k

N)

398times102

396

394

392

3902000 None

3973

3961

3946

3931

3916

Velocity exponent

106

0504

0302

01

40006000 8000


1400016000

1800020000





Dam

ping

velo

city

(ms

)

0048

0045

0042

0039

0033

0036


α = 01α = 02α = 03

α = 05α = 06α = 1

α = 04



16000

14000

12000

10000

8000

Dam

ping

forc

e (kN

)

6000

4000

2000


10000 12500 15000 17500 200000

α = 01α = 02α = 03

α = 05α = 06α = 1

α = 04


Max

imum

gird

er d

ispla

cem

ent (

m)

045

030

015

000

ndash015

ndash030

ndash045

ndash0600 50

ndash046

C = 1000C = 2500C = 5000

C = 10000C = 15000

C = 20000C = 7000None

Unit kN(ms)α

ndash048

ndash050

ndash006

ndash009

ndash012

ndash01530 40 215 220 225 230

100Time (s)

150 200 250 300


Accu

mul

ativ

e gird

er d

ispla

cem

ent (

m)

5

C = 1000C = 2500C = 5000

C = 7000C = 10000

C = 20000None

C = 15000

0 50 100 150 200Time (s)

250 300

Unit kN(ms)α

4

3

2

1

0






6 Conclusions




Max

imum

grid

er d

ispla

cem

ent (

m)

04

02

00

ndash02

ndash06

ndash04

ndash047

α = 01α = 02α = 03

α = 04α = 05

α = 1None

α = 06

ndash006

ndash009

ndash012

ndash015220 230

ndash048

ndash049

ndash050 30 32 34 36

0 50 100 150Time (s)

200 250 300


Accu

mul

ativ

e gird

er d

ispla

cem

ent (

m)

5

4

3

2

1

00 50 100 150

Time (s)200 250 300

α = 01α = 02α = 03

α = 04α = 05

α = 1None

α = 06




Appendix

KivuL andKi


KivuL K

ivuR (A1)

KiyuL andKi


KiyuL K

iyuR (A2)

KivlL andKi


KivlL K

ivlR (A3)

KiylL andKi


KiylL K

iylR (A4)

CivuL andCi


CivuL C

ivuR (A5)

CiyuL andCi


CiyuL C

iyuR (A6)

CivlL andCi


CivlL C

ivlR (A7)

CiylL andCi


CiylL C

iylR (A8)


Data Availability




Acknowledgments


References









































Max

imum

long

itudi

nal

pylo

n ac

cele

ratio

n (m

s2 )

0016

0014

0012

0010

00082000

None

0015

0014

0012

0011

00101

0605

Velocity exponent04

0302

01

4000 60008000

1000012000

1400016000

1800020000



Max

imum

pyl

onbe

ndin

g m

omen

t (kN

middotm)

times105

1716

1712

1708

1704

17002000

40006000

800010000

1200014000

16000


None

1717

1714

1711

1708

17051

0605

04

Velocity exponent03

0201






Susp

ende

r for

ce (k

N)

398times102

396

394

392

3902000 None

3973

3961

3946

3931

3916

Velocity exponent

106

0504

0302

01

40006000 8000


1400016000

1800020000





Dam

ping

velo

city

(ms

)

0048

0045

0042

0039

0033

0036


α = 01α = 02α = 03

α = 05α = 06α = 1

α = 04



16000

14000

12000

10000

8000

Dam

ping

forc

e (kN

)

6000

4000

2000


10000 12500 15000 17500 200000

α = 01α = 02α = 03

α = 05α = 06α = 1

α = 04


Max

imum

gird

er d

ispla

cem

ent (

m)

045

030

015

000

ndash015

ndash030

ndash045

ndash0600 50

ndash046

C = 1000C = 2500C = 5000

C = 10000C = 15000

C = 20000C = 7000None

Unit kN(ms)α

ndash048

ndash050

ndash006

ndash009

ndash012

ndash01530 40 215 220 225 230

100Time (s)

150 200 250 300


Accu

mul

ativ

e gird

er d

ispla

cem

ent (

m)

5

C = 1000C = 2500C = 5000

C = 7000C = 10000

C = 20000None

C = 15000

0 50 100 150 200Time (s)

250 300

Unit kN(ms)α

4

3

2

1

0






6 Conclusions




Max

imum

grid

er d

ispla

cem

ent (

m)

04

02

00

ndash02

ndash06

ndash04

ndash047

α = 01α = 02α = 03

α = 04α = 05

α = 1None

α = 06

ndash006

ndash009

ndash012

ndash015220 230

ndash048

ndash049

ndash050 30 32 34 36

0 50 100 150Time (s)

200 250 300


Accu

mul

ativ

e gird

er d

ispla

cem

ent (

m)

5

4

3

2

1

00 50 100 150

Time (s)200 250 300

α = 01α = 02α = 03

α = 04α = 05

α = 1None

α = 06




Appendix

KivuL andKi


KivuL K

ivuR (A1)

KiyuL andKi


KiyuL K

iyuR (A2)

KivlL andKi


KivlL K

ivlR (A3)

KiylL andKi


KiylL K

iylR (A4)

CivuL andCi


CivuL C

ivuR (A5)

CiyuL andCi


CiyuL C

iyuR (A6)

CivlL andCi


CivlL C

ivlR (A7)

CiylL andCi


CiylL C

iylR (A8)


Data Availability




Acknowledgments


References









































Susp

ende

r for

ce (k

N)

398times102

396

394

392

3902000 None

3973

3961

3946

3931

3916

Velocity exponent

106

0504

0302

01

40006000 8000


1400016000

1800020000





Dam

ping

velo

city

(ms

)

0048

0045

0042

0039

0033

0036


α = 01α = 02α = 03

α = 05α = 06α = 1

α = 04



16000

14000

12000

10000

8000

Dam

ping

forc

e (kN

)

6000

4000

2000


10000 12500 15000 17500 200000

α = 01α = 02α = 03

α = 05α = 06α = 1

α = 04


Max

imum

gird

er d

ispla

cem

ent (

m)

045

030

015

000

ndash015

ndash030

ndash045

ndash0600 50

ndash046

C = 1000C = 2500C = 5000

C = 10000C = 15000

C = 20000C = 7000None

Unit kN(ms)α

ndash048

ndash050

ndash006

ndash009

ndash012

ndash01530 40 215 220 225 230

100Time (s)

150 200 250 300


Accu

mul

ativ

e gird

er d

ispla

cem

ent (

m)

5

C = 1000C = 2500C = 5000

C = 7000C = 10000

C = 20000None

C = 15000

0 50 100 150 200Time (s)

250 300

Unit kN(ms)α

4

3

2

1

0






6 Conclusions




Max

imum

grid

er d

ispla

cem

ent (

m)

04

02

00

ndash02

ndash06

ndash04

ndash047

α = 01α = 02α = 03

α = 04α = 05

α = 1None

α = 06

ndash006

ndash009

ndash012

ndash015220 230

ndash048

ndash049

ndash050 30 32 34 36

0 50 100 150Time (s)

200 250 300


Accu

mul

ativ

e gird

er d

ispla

cem

ent (

m)

5

4

3

2

1

00 50 100 150

Time (s)200 250 300

α = 01α = 02α = 03

α = 04α = 05

α = 1None

α = 06




Appendix

KivuL andKi


KivuL K

ivuR (A1)

KiyuL andKi


KiyuL K

iyuR (A2)

KivlL andKi


KivlL K

ivlR (A3)

KiylL andKi


KiylL K

iylR (A4)

CivuL andCi


CivuL C

ivuR (A5)

CiyuL andCi


CiyuL C

iyuR (A6)

CivlL andCi


CivlL C

ivlR (A7)

CiylL andCi


CiylL C

iylR (A8)


Data Availability




Acknowledgments


References






































16000

14000

12000

10000

8000

Dam

ping

forc

e (kN

)

6000

4000

2000


10000 12500 15000 17500 200000

α = 01α = 02α = 03

α = 05α = 06α = 1

α = 04


Max

imum

gird

er d

ispla

cem

ent (

m)

045

030

015

000

ndash015

ndash030

ndash045

ndash0600 50

ndash046

C = 1000C = 2500C = 5000

C = 10000C = 15000

C = 20000C = 7000None

Unit kN(ms)α

ndash048

ndash050

ndash006

ndash009

ndash012

ndash01530 40 215 220 225 230

100Time (s)

150 200 250 300


Accu

mul

ativ

e gird

er d

ispla

cem

ent (

m)

5

C = 1000C = 2500C = 5000

C = 7000C = 10000

C = 20000None

C = 15000

0 50 100 150 200Time (s)

250 300

Unit kN(ms)α

4

3

2

1

0






6 Conclusions




Max

imum

grid

er d

ispla

cem

ent (

m)

04

02

00

ndash02

ndash06

ndash04

ndash047

α = 01α = 02α = 03

α = 04α = 05

α = 1None

α = 06

ndash006

ndash009

ndash012

ndash015220 230

ndash048

ndash049

ndash050 30 32 34 36

0 50 100 150Time (s)

200 250 300


Accu

mul

ativ

e gird

er d

ispla

cem

ent (

m)

5

4

3

2

1

00 50 100 150

Time (s)200 250 300

α = 01α = 02α = 03

α = 04α = 05

α = 1None

α = 06




Appendix

KivuL andKi


KivuL K

ivuR (A1)

KiyuL andKi


KiyuL K

iyuR (A2)

KivlL andKi


KivlL K

ivlR (A3)

KiylL andKi


KiylL K

iylR (A4)

CivuL andCi


CivuL C

ivuR (A5)

CiyuL andCi


CiyuL C

iyuR (A6)

CivlL andCi


CivlL C

ivlR (A7)

CiylL andCi


CiylL C

iylR (A8)


Data Availability




Acknowledgments


References









































6 Conclusions




Max

imum

grid

er d

ispla

cem

ent (

m)

04

02

00

ndash02

ndash06

ndash04

ndash047

α = 01α = 02α = 03

α = 04α = 05

α = 1None

α = 06

ndash006

ndash009

ndash012

ndash015220 230

ndash048

ndash049

ndash050 30 32 34 36

0 50 100 150Time (s)

200 250 300


Accu

mul

ativ

e gird

er d

ispla

cem

ent (

m)

5

4

3

2

1

00 50 100 150

Time (s)200 250 300

α = 01α = 02α = 03

α = 04α = 05

α = 1None

α = 06




Appendix

KivuL andKi


KivuL K

ivuR (A1)

KiyuL andKi


KiyuL K

iyuR (A2)

KivlL andKi


KivlL K

ivlR (A3)

KiylL andKi


KiylL K

iylR (A4)

CivuL andCi


CivuL C

ivuR (A5)

CiyuL andCi


CiyuL C

iyuR (A6)

CivlL andCi


CivlL C

ivlR (A7)

CiylL andCi


CiylL C

iylR (A8)


Data Availability




Acknowledgments


References







































Appendix

KivuL andKi


KivuL K

ivuR (A1)

KiyuL andKi


KiyuL K

iyuR (A2)

KivlL andKi


KivlL K

ivlR (A3)

KiylL andKi


KiylL K

iylR (A4)

CivuL andCi


CivuL C

ivuR (A5)

CiyuL andCi


CiyuL C

iyuR (A6)

CivlL andCi


CivlL C

ivlR (A7)

CiylL andCi


CiylL C

iylR (A8)


Data Availability




Acknowledgments


References






































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