Dynamic Reliability of Continuous Rigid-Frame Bridges...

Research ArticleDynamic Reliability of Continuous Rigid-Frame Bridges underStochastic Moving Vehicle Loads

Naiwei Lu 12 Kai Wang1 Honghao Wang1 Yang Liu13 Yuan Luo3 and Xinhui Xiao3

1Engineering Research Center of Catastrophic Prophylaxis and Treatment of Road amp Traffic Safety of Ministry of EducationChangsha University of Science amp Technology Changsha 410114 China2Industry Key Laboratory of Traffic Infrastructure Security Risk Management Changsha University of Science and TechnologyChangsha 410114 China3School of Civil Engineering Hunan University of Technology Zhuzhou 412007 China

Correspondence should be addressed to Naiwei Lu lunaiweide163com

Received 17 July 2020 Revised 3 September 2020 Accepted 16 September 2020 Published 25 September 2020

Academic Editor Songye Zhu

Copyright copy 2020 Naiwei Lu 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

)e current volume of freight traffic has increased significantly during the past decades impacted by the fast development of thenational transportation market As a result the phenomena of truck overloading and traffic congestion emerge which haveresulted in numerous bridge collapse events or damage due to truck overloading)us it is an urgent task to evaluate bridge safetyunder actual traffic loads )is study evaluated probabilistic dynamic load effects on rigid-frame bridges under highway trafficmonitoring loads )e site-specific traffic monitoring data of a highway in China were utilized to establish stochastic trafficmodels )e dynamic effect was considered in a vehicle-bridge coupled vibration model and the probability estimation wasconducted based on the first-passage criterion of the girder deflection )e prototype bridge is a continuous rigid-frame bridgewith a midspan length of 200m and a pier height of 182m It is demonstrated that the dynamic traffic load effect follows Gaussiandistribution which can be treated as a stationary random process )e mean value and standard deviation of the deflections are0071m and 0088m respectively )e dynamic reliability index for the first passage of girder deflection is 645 for the currenttraffic condition However the reliability index decreases to 560 in the bridge lifetime accounting for an average traffic volumegrowth ratio of 36

1 Introduction

In the design phase of a bridge the structural safety is usuallyevaluated considering design traffic loads which were for-mulated according to large traffic monitoring data duringthe past decades [1 2] However the current or futurevehicle load might be beyond the design value with the rapiddevelopment of transportation industry As a result nu-merous bridges were collapsed or badly damaged due totruck overloading [3ndash5] In case that growing traffic load isbecoming more and more critical for in-service bridges thebridge safety under actual traffic loads deserves investiga-tion)e traffic load effect on a bridge is stochastic due to therandom nature of traffic loads [6] In addition a movingvehicle leads to dynamic effects motivated by the road

roughness and the bridge deflection In view of the abovereasons the structural dynamic reliability theory is appro-priate for the safety evaluation of existing bridges underactual stochastic traffic loads In addition the design vehicleload model can be calibrated with the target reliability index

In general the dynamic effect induced by a movingvehicle can be evaluated in a vehicle-bridge interactionsystem [7 8] A great number of literatures focus on dy-namic traffic load effects on both short-span and long-spanbridges Chen and Wu [9] investigated dynamic effects of acable-stayed bridge under stochastic vehicle loads and in-vestigated influence of the sparse and dense states of trafficloads on the probabilistic characteristics of bridge load effectLu et al [10] evaluated lifetime deflection of a suspensionbridge considering dynamic and growing traffic loads Zhou

HindawiShock and VibrationVolume 2020 Article ID 8811105 13 pageshttpsdoiorg10115520208811105

and Chen [8] investigated probabilistic characteristics of thedynamic response of a cable-stayed bridge under wind loadand stochastic vehicle load Lu et al [11] investigated thefirst-passage probability of a cable-stayed bridge understochastic vehicle loads Obrien et al [12] investigated thedynamic increment of extreme traffic loading on short-spanbridges Jiang et al [13] investigated the fatigue damage ofcables of cable-stayed bridge under combined effect of trafficand wind Li et al [14] investigated the safety of suspendersof Tsing Ma Bridge under traffic loads However there arerelatively few studies on probabilistic dynamic traffic loadeffects on long-span continuous rigid-frame bridges )eunique feature for the continuous rigid-frame bridge is thehigh-rise piers which are usually space-flexible affecting thedynamic behaviour of the bridge )us the probabilisticdynamic traffic load effects on continuous rigid-framebridges deserve more investigation

Most studies treated the traffic load effect on a bridge as arandom variable In practice the load effect on a bridge is atime-varying random process [15] )e commonly usedtheoretical basis for the probabilistic dynamic analysis ofbridges under vehicle load is the first-passage criterion[16 17] )e first-passage failure probability is usually es-timated based on Ricersquos level-crossing principle [18]However applications of the first-passage theory are mostlyconcentrated in seismic engineering while the application totraffic load analysis is relatively insufficient)e bottleneck isthat the root mean square of traffic load effects is usuallyevaluated in frequency domain while the vehicle-bridgecoupled vibration is mostly analysed in time domain [19])erefore the key point is how to extract the probabilityparameters from the dynamic load effect

)is study evaluated probabilistic dynamic traffic loadeffects on rigid-frame bridges under highway traffic moni-toring loads )e site-specific traffic monitoring data wereutilized to establish stochastic traffic models )e dynamiceffect was considered in a vehicle-bridge coupled vibrationmodel and the probability estimation was conducted basedon the first-passage criterion of the girder deflection )eprototype bridge is a continuous rigid-frame bridge withmid-span length of 200m and pier height of 182m It isdemonstrated that the dynamic traffic load effect followsGaussian distribution and thus can be treated as a stationaryrandom process Parametric studies were conducted ac-counting for the road roughness condition the bridge spanlength and the traffic growth ratio

2 Theoretical Basis of Probabilistic Traffic-Bridge Interaction Analysis

21 Traffic-Bridge InteractionModel A vehicle passing on abridge leads to the vibration of the bridge due to the roadsurface roughness [20] Meanwhile the vibration of thebridge impacts the vibration of the vehicle )us thevehicle-bridge interaction is a coupled vibration systemFigure 1 shows the plane motion model of two-axle ve-hicle where x represents the road roughness and C and Krepresent the damping and stiffness parameters re-spectively )e vehicle model has four degrees of freedom

including vertical motion of two axles and the motion androtation of the vehicle body

Based on the simplified model as shown in Figure 1 thedifferential equation of motion of the vehicle-bridge coupledsystem can be written as [21]

Mv eurouv + Cv _uv + Kvuv Fvg + Fvb (1a)

Mb euroub + Cb _ub + Kbub Fbg + Fbv (1b)

where Mv and Mb represent the mass matrix of the vehicleand that of the bridge respectively Cv and Cb represent thedamping matrix of the vehicle and that of the bridge re-spectively Kv and Kb represent the stiffness matrix of thevehicle and that of the bridge respectively uv and ub rep-resent the displacement vector of the vehicle and that of thebridge respectively Fvb and Fbv represent the interactionforces between the vehicle and the bridge Fvg and Fbgrepresent the weight of the vehicle and that of the bridgerespectively

In order to consider the multiple-vehicle effect on thebridge response this study utilized an equivalent dynamicwheel load (EDWL) approach proposed by Chen and Cai[22] Based on the EDWL approach the force of the jthvehicle on a bridge can be simplified as a time-varying forcewhich can be written as

F(t) wheelEq 1113944

nv

j11 minus Rj(t)1113960 1113961Gj middot 1113944

n

k1hk xj(t) + αk xj(t)dj(t)1113960 11139611113960 11139611113966 1113967

⎧⎨

⎩

⎫⎬

⎭

(2a)

Rj(t) EDWLj(t)

Gj

(2b)

EDWLj(t) 1113944

na

i1K

ivlY

ivl + C

ivl

_Yi

vl1113874 1113875 (2c)

where Rj represents wheel weight ratio of the jth vehicle Gjrepresents the GVW of the jth vehicle xj and dj representlongitudinal and lateral positions on the bridge respectivelyhk and ak represent the kth-order vertical and torsionalbending modes of the bridge n and nv respectively rep-resent modal orders of bridges and the number of vehicleson the bridges Ki

vl and Civl represent stiffness and damping

matrixes of the vehicle respectively Yivl and _Y

i

vl respectivelyrepresent the vertical displacement and the velocity of the ithaxle EDWLj(t) represents the equivalent dynamic axis loadof the jth vehicle at time t

)e effectiveness of EDWL approach has been verified byWu and Chen [23] and Lu et al [24] for long-span cable-stayed bridges and suspension bridges respectively )isstudy develops the EDWL approach in probability analysisand reliability evaluation for continuous rigid-frame bridges

22 First-Passage Criterion )e first-passage criterion isusually utilized for Ricersquos level-crossing probability analysisof a random process )e principle of Ricersquos level-crossing isshown in Figure 2 where t is time x is a random process

2 Shock and Vibration

that is the traffic load effect in the present study and v(x) isthe level-crossing rate fitted to the histograms of the numberof crossings )us the critical content of the level-crossingtheory is to count the number of crossings for the randomprocess

Suggest that the traffic load effect can be assumed as aGaussian random process which will be demonstrated in thecase study For a random process x(t) Rice [25] provided thenumber of crossings for a level b during time T which iswritten as

Nb(T) 1113946T

01113946

+infin

minusinfin| _x(t)|px _x(b _x t)d _xdt (3)

where px _x(b _x t) is a joint probabilistic density function forx(t) and _x(t) )is function is difficult to estimate for anonstationary process Fortunately the traffic load effect on

a bridge can be treated as a stationary random process asmentioned by researchers [26 27] On this basis the level-crossing rate vb(t) for a threshold b can be written as

vb(t) 1113946infin

minusinfin| _x|px _x(b _x t)d _x (4)

For a Gaussian stationary random process with the meanvalue of zero the level-crossing probability can be simplifiedfurther as

vb 12π

σ _x

σx

exp minusb2

2σ2x1113890 1113891 (5)

where vb is a constant for level b and σx is the root meansquare value In addition the number of level-crossings ofstochastic traffic load effects can be assumed as Poisson

brvbarEgraver1

x

lv

b2lv

Ks2 Cs2

Kt2 Ct2

y2 y1

Ks1 Cs1

Kt1 Ct1

b1lv

Zv

Figure 1 Vehicle-bridge coupled vibration system

t

Upcrossing

Downcrossing

Level 1

Level 2

x

(a)

x

v(x)

Ricersquos fitting

Interval length

Optimal starting point

(b)

Figure 2 )e level-crossing principle

Shock and Vibration 3

distribution )erefore the probability of exceeding athreshold can be written as

Pr(b) exp minusvb middot T1113858 1113859 (6)

Note that the accuracy of the estimation mostly dependson the fitting to the curves as shown in Figure 2 In additiona higher threshold b will result in a more reliable estimationIn general the mean value of the traffic load effect on abridge is not zero which is different from seismic loadeffects)erefore this study developed the above proceduresto a nonzero mean value process which is rewritten as

vb 12π

σ _x

σx

exp minusb minus mx( 1113857

2

2σ2x1113890 1113891 (7)

where mx is the mean value of the traffic load effect)e density of traffic loads changes with time where the

traffic will be congested and free-flowing for the day andnight respectively )erefore the daily vehicle flow wasdivided into sparse and dense states based on the density ofstochastic vehicle model )e probability can be treated asthe superposition of the two types of level-crossing prob-ability which is written as

Pf(b T) 1 minus exp minus a1 middot vb1 + 1 minus a1( 1113857 middot vb2( 1113857T1113858 1113859 (8)

where vb1 and vb2 represent the level-crossing rates for thecongested and free-flowing vehicle flows respectively a1 anda2 represent the proportions of the congested and free-flowing vehicle flows respectively

)ere is a relationship between the structural reliabilityand the first-passage probability On this basis the dynamicreliability can be written as

β Φminus 1 1 minus Pf1113872 1113873 (9)

where β is the reliability index Φ() is the cumulative dis-tribution function (CDF) of the standard normal distribu-tion and Φminus1() is the corresponding inverse function

23 Computational Procedures Based on the derived for-mulations of the vehicle-bridge coupled vibration systemand the first-passage criterion a computational framework isessential to combine the components as a system )is studypresented a comprehensive framework for dynamic andprobabilistic analysis for traffic load effects on rigid-framebridges based on traffic monitoring data Figure 3 shows theflow chart of the procedures

As shown in Figure 3 the framework consists of threemain procedures the stochastic traffic modelling the traffic-bridge interaction analysis and the probability evaluationDetailed illustrations of the procedures are shown as follows

)e first step is the stochastic traffic modelling Initiallythe screened trafficmonitoring data should be collected fromsite-specific highways Subsequently all vehicles are classi-fied as 6 groups according to the configuration associatedwith the number of axles On this basis the statisticalanalysis can be conducted for probability modelling of thegross vehicle weight (GVW) the driving lanes and the

vehicle spacing Finally the stochastic traffic load model canbe simulated utilizing the Monte Carlo simulation (MCS)approach

)e second step is the traffic-bridge interaction analysis)e critical step is to model the dynamic parameters of thevehicle model and the modal parameters of the bridgeUsually a 2D vehicle model as shown in Figure 1 can beutilized for a relatively precise analysis )e dynamiccharacteristics of the bridge are considered as mode shapesand natural frequencies Since the long-span rigid-framebridge is more complex than a conventional-girder bridgethe numbers of mode shapes and frequencies should beconsidered as more as possible Subsequently the EDWLapproach is utilized to convert the moving stochastic trafficflow into time-varying forces Finally the time history oftraffic load effects can be simulated based on the pseudo-dynamic analysis

)e third step is the probability evaluation )is step isconducted based on Ricersquos level-crossing theory Based onthe time histories evaluated from the traffic-bridge inter-action analysis the number of level-crossings can becounted )e number of crossings is formed as histogramswhere the level-crossing rate can be fitted accurately asshown in Figure 2(b) Subsequently the maximum trafficload effect can be extrapolated with consideration of a returnperiod Finally the first-passage probability and reliabilityindex can be evaluated based on the assumption of Poissondistribution

3 Stochastic Vehicle Flow Simulation Based onWIM Data

31 Probabilistic Modelling of WIM Measurements Due tothe truck overloading behaviour the GVW usually follows amultimodal distribution In this regard this study utilized aGaussian mixture model (GMM) which is a superpositionof several Gaussian models In practice the GMMmethod iswidely used in the application of data clustering [28] but it isrelatively insufficient in the vehicle weight modelling

In general a GMM can be represented as

P w | ai μi σ2i1113872 11138731113966 1113967 1113944

M

i1ai middot g w | μi σ

2i1113872 1113873 (10)

where w is the GVW ai is a weight coefficient g is aGaussian distribution function and M is the number ofGaussian functions )e Gaussian is written as

g w | μi σ2i1113872 1113873

12πσi

1113968 exp minus12σ2i

w minus μi( 11138572

1113890 1113891 (11)

where μi represents mean value of the ith Gaussian distri-bution function and σi represents standard deviation of theith Gaussian distribution function )e parameters ofGMMs are usually estimated based on the maximum like-lihood optimization )is study utilized the Expectation-Maximization (EM) algorithm to evaluate these parameters


)enumber of components is estimated based on the Akaikeinformation criterion (AIC) [29]

)e authors have conducted extensive analyses on theprobability modelling of traffic parameters as shown in thedoctoral dissertation [30] Due to the limit of article only theprobability distributions for 6-axle trucks are providedFigure 3 plots the fitted probability distribution function(PDF) of the axle weight and the GVW respectively InFigure 4 W64 represents the second axle load of six-axlevehicle

It is observed that the axle weight and the GVW followbimodal distribution In addition the GMMs have ideallyfitted to the empirical histograms In other words the GMMhas captured the probability behaviour of truck overloadingNote that the threshold weight for 6-axle trucks is 550 kNaccording to the traffic law in China It can be observed thathalf of the 6-axle trucks are overloaded which is indeed arisk for existing bridges

32 StochasticTrafficFlowSimulation With the probabilitymodels established based on the weigh-in-motion (WIM)data this study utilized MCS to establish different typesof stochastic traffic flows )e detailed procedures aresummarized in Figure 5 where NADT is the total numberof daily traffic )e programming package MATLAB wasutilized to translate the traffic as a matrix that will becalled in the subsequent vehicle-bridge interactionanalysis

)e preliminary statistical analytical results of theactual traffic monitoring data indicate that the vehiclespacing between two following vehicles follows theWeibull distribution for congested traffic while it followsGamma distribution for free-flowing state Figure 6 plotsthe sample of a free-flowing stochastic traffic flow whereV1 denotes the lightweight cars and busses and V2simV6denote the 2-axle to 6-axle trucks respectively Details ofthe configuration of the vehicle types can be found in[28]

)e benefit of the stochastic traffic flow is the prob-ability distribution in accordance with the site-specifictraffic monitoring data In addition the stochastic trafficmodel can be updated with the consideration of the trafficvolume growth and the limit of truck overloading

4 Case Study

41 Backgrounds of the Prototype Bridge Labajin Bridge acontinuous rigid-frame bridge in Yalu highway of Sichuan inChina is selected as prototype to investigate the deflection offirst-passage probability Site photos of the bridge are shownin Figure 7 )e span length of the bridge is(105 + 200 + 200 + 105) m )e bridge is very famous due tothe highest piers in the area of Asia where the length of pierNo 10 is 182m)e dimensions of the cross sections of box-girders and piers are shown in Figure 8 )e material of thebox-girder is C60 concrete )e pier is a composite structurewith concrete filled steel tube )e internal and externalconcrete grades are C80 and C50 respectively

)e finite element model is shown in Figure 9 All of theelements were simulated with beam elements )e boundarycondition of the bridge piers is the constraint of all degrees offreedom at the bottom elements )e top nodes of mid-spanpiers are connected with the closest girder nodes in rigidconnection )e top nodes of side-span piers are connectedwith the closet nodes with the compressive only connection)e top 50 vertical modes of vibration were extracted for thefollowing vehicle-bridge interaction analysis

42 Influence of RRC on Probability Distribution Since theroad roughness condition (RRC) has a significant influenceon the vehicle-bridge interaction system this study selectedldquogoodrdquo and ldquopoorrdquo RRCs for the comparative study )eRRCs were simulated by an inverse Fourier transformationfunction and corresponding power spectral density )ecoefficients for ldquogoodrdquo and ldquopoorrdquo RRCs are 32times10minus6m3cycle and 512times10minus6m3cycle respectively Figure 10 plotsthe time histories of the two types of RRC samples It isobvious that the poor RRC has a wider fluctuation rangecompared to the one of good RRC

)e modal analysis of the bridge was conducted firstlyconsidering the modal switches in three space directions)e first 50-order modal shape of the bridge was extracted tosolve vehicle-bridge coupled vibration analysis )edamping constant of the concrete was taken as 005 Tstochastic vehicle flow samples were transformed asequivalent time-varying concentrated forces based onEDWL method Subsequently the time histories of thevehicle load effects were evaluated by utilizing the proposed

Vehicle dynamic model

Equivalent dynamicwheel load on the bridge

Collection of WIM data fromhighways

Stochastic traffic modeling

Classification of vehicle types

Statistical analysis of GVWsdriving lanes and spacings

Stochastic traffic simulationbased on MCS

Traffic-bridge interaction analysis

Mode shapes andfrequencies of the bridge

Traffic load effectcomputation

Probability evaluation

Extrapolation based on Ricersquosprinciple

First-passage evaluation basedon Poisson distribution

Counting the number oflevel-crossings

Fitting to the histograms oflevel-crossings

Figure 3 Flow chart of the proposed computational framework


pseudodynamic approach Figure 11 plots the vertical de-flection histories of the critical point in the mid-span of thebridge accounting for both good and poor RRCsrespectively

It is observed from Figure 10 that the poor RRC results inmore deviations compared with the good RRC )e maxi-mum deflection is located at 162 s where several heavytrucks load on the mid-span point simultaneously )emaximum deflections for good and poor RRCs are minus0196mand minus0246m respectively )us it is demonstrated that theRRC has a significant influence on the maximum load effect

Since the basis for extreme value extrapolation is theassumption of Gaussian distribution the probability dis-tribution should be fitted Figure 12 plots the histograms ofthe deflections and the fitted PDF and CDF curves ofGaussian distribution It is obvious that both good and poorRRCs follow Gaussian distribution In addition the shape ofhistograms for the poor RRC is more flat than that of thegood RRC

)e hypothesis testing was conducted to check the fittingto Gaussian distribution)e fitting variances are 00028 and00019 for the good and poor RRCs respectively It isconfirmed that the samples do not refuse the Gaussiandistribution In order to capture the influence of the twotypes of RRCs on the probabilistic load effect the histogramsof numbers of level-crossings were counted as shown inFigure 13

As shown in Figure 13 Ricersquos level-crossings were fittedto the top 30 samples It is observed that the poor RRCleads to larger number of level-crossings compared to thegood RRC Based on the above analysis it can be concludedthat the RRC can affect the probabilistic characteristics oftraffic load effects and thus a poor RRC will amplify themaximum traffic load effect

43 Influence of Bridge Span Length on the DynamicDeflection In order to investigate the influence of bridgespan length on the deflection both shorter-span and longer-span continuous rigid-frame bridges were selected forcomparison study )e main span lengths of the threebridges are 140m 200m and 248m respectively In orderto make the comparison of general significance the sametraffic flow as shown in Figure 6 was utilized for the threebridges )e bridge parameters and the maximum deflec-tions are shown in Table 1

As observed from Table 1 the maximum deflectionincreases with increase of the bridge main span length

Empirical dataEstimated curve

0

002

004

006

008

01

Prob

abili

ty d

ensit

y

100 150 200 25050 Axle weight AV64 (kN)

(a)

0

0005

001

0015

002

Prob

abili

ty d

ensit

y

500 1000 1500 20000GVW (kN)


w1 = 024μ1 = 39σ1 = 74

w2 = 076μ2 = 86σ2 = 14

(b)

Figure 4 Probabilistic distribution of axle load of 6-axle trucks (a) axle weight (b) gross weight

Collection and probabilisticanalysis of traffic WIM data

Initializing the ith vehicle

Sampling the vehicle configuration

Sampling thedriving lane

Sampling the GVW andassigning axle weights

Sampling thedriving speed

Sampling the vehicle spacing

Determining drivingbehavior via CA

No i = i + 1

A multiscale daily trafficload model

i = NADT

Yes

Figure 5 Procedures of stochastic vehicle flow simulation


However the ratio between the good and poor RRCeffects decreases with the increase of the bridge spanlength )is phenomenon can be explained by the the-oretical basis that the bridge with a longer span length hasa lower frequency which will weaken the vehicle-bridgeinteraction )erefore the influence of the RRC on thedynamic effect is weakened with the increase of the bridgespan length

44 Probability Estimation It was concluded by many re-searchers that dynamic response of bridges under stochasticvehicle flow can be assumed as stationary random process[31] In order to investigate the influence of random processsamples on the characteristics of stationary random pro-cesses the mean value and the correlation coefficient inrandom process were analysed Figure 14(a) plots the de-flection spectrum density of the bridge girder under

stochastic traffic flow load Figure 14(b) plots the autocor-relation coefficient of the samples in 100 s 200 s and 300 srespectively

It is observed that the mean value and standard valueof the initial position are greatly affected by 7 seconds Infact the mean value and correlation coefficient tend to beconstants with the increase of the samplings number)erefore if the sample is large enough this process canbe assumed as stationary random process )e randomprocess under stochastic vehicle loads can be representedby the load effects which was calculated by a large sampleof vehicle flows

)e mean value and root mean square deflection of thecontinuous rigid-frame bridge were obtained by the samemethod Figure 15 shows the distribution of mean value androot mean square deflection along the girders

As observed in Figure 15 the maximum mean value andthe root mean square deflection of a continuous rigid-frame

Slow laneFast lane Fast lane

Slow lane0

1530

4560

Arrival tim

e (min)

Dirving lane

V1V2V3

V4V5V6

0

500

1000

1500

GV

W (t

)

Figure 6 A stochastic vehicle flow sample

(a) (b)

Figure 7 In-site photos of the Labajin bridge in Yalu highway (a) final state (b) construction state


605 60525 85 50

340

340

380290

50

220120

1275

(a)

132170

680

C50Steel tube

C80

1000

(b)

Figure 8 Dimensions of cross-sections of Labajin Bridge (a) girders (b) piers

(a) (b)

Figure 9 Continued


(c)

Figure 9 Finite element model of Labajin Bridge (a) overall model (b) pier element (c) girder element

RRC = goodRRC = poor

100 200 300 400 500 6000Location on the bridge (m)

ndash04

ndash02

0

02

04

Roug

hnes

s (m

)

Figure 10 Samples of two types of road roughness conditions

ndash03

ndash02

ndash01

0

01

02

Vert

ical

def

lect

ion

(m)

100 200 300 400 500 6000Time (s)


Figure 11 Vertical deflection histories of the critical point in the mid-span of the bridge girders in 10min


bridge are minus0071m and 0088m respectively )ereforethis study selected the mid-span point as representation)efollowing investigation will focus on the deflection of thecritical point under stochastic traffic loads

45 First-Passage Reliability Evaluation In general the first-passage reliability analysis can be conducted based onprobabilistic model associated withmaximum extrapolation)e upper threshold deflection of a continuous rigid-framebridge is a L500 04m according to designed codes inChina Figure 16 shows the time-varying reliability index

based on the first-passage criterion of the maximum girderdeflection

As shown in Figure 16 the initial reliability index ofthe bridge is 645 However the reliability index decreasessignificantly with the time and the proportion of densetraffic flow With consideration of the proportion ofdense traffic flow as 12 24 and 36 the corre-sponding reliability indexes of the bridge in 100 years are576 562 and 560 respectively )erefore it is con-cluded that the proportion of dense vehicle flow hassignificant influence on the first-passage probability ofthe bridge deflection )us the control of dense vehicle

RRC = goodGaussian fitting

RRC = poorGaussian fitting

ndash02 ndash01 0 01ndash03Deflection (m)

0

002

004

006

008

01

012

014Pr

obab

ility

den

sity

(a)

0

02

04

06

08

1

CDF




(b)

Figure 12 Probability distribution of the critical deflection (a) PDF (b) CDF

RRC = poorRRC = good

Ricersquos fittingRicersquos fitting

ndash02 ndash015 ndash01 ndash005 0 005ndash025Deflection threshold (m)

0

50

100

150

200

Num

ber o

f cro

ssin

gs

Figure 13 Histograms and fitting to the number of level-crossings


02 04 06 08 10Frequency (Hz)

0

002

004

006

008

01

Disp

lace

men

t spe

ctru

m d

ensit

y

Free trafficBusy traffic

(a)

t = 100st = 200st = 300s

1 2 3 4 5 6 70Time (s)

0

02

04

06

08

1

Cor

relat

ion

coef

ficie

nt

(b)

Figure 14 Bride dynamic deflections in frequency domain (a) spectrum density (b) correlation coefficient

Table 1 Critical parameters of three continuous rigid-frame bridges

Bridge name Span combination (m) Height of the 0 segment (m)Maximum deflection (m) Dynamic ratio between poor

and good RRCsRRC good RRC poorShizijing River Bridge 74 + 140 + 74 80 minus0137 minus0182 132Labajin Bridge 105 + 200 + 200 + 105 125 minus0196 minus0246 125Lengshui River Bridge 130 + 248 + 130 160 minus0254 minus0273 107

105 305 505 6100Location on girder (m)

ndash015

ndash01

ndash005

0

005

01

015

Disp

lace

men

t (m

)

Mean valueStandard deviation

Figure 15 Trends of mean value and root mean square of displacement along the girders


flow is essential for ensuring the bridge safety in serviceperiod

5 Conclusions

)is study developed the vehicle-bridge coupled vibrationmodel to the dynamic reliability analysis of long-span bridges Asimplified analytical method was presented for evaluatingprobabilistic dynamic load effect induced by stochastic vehicleloads )e effectiveness of the proposed computationalframework was demonstrated in the case study of a continuousrigid-frame bridge Influence of the RRC the bridge spanlength and the traffic growth on the bridge deflection wasinvestigated )e conclusions are summarised as follows

(1) )e mean value and the root mean square deflectionsinduced by stochastic traffic loads are constants whichare uncorrelated with time )erefore the stochastictraffic-bridge coupled vibration is demonstrated as astationary Gaussian random process which can betreated as a long time history

(2) )e level-crossing is able to capture the probabilitycharacteristics of the dynamic traffic load effect andthus provides a connection between the dynamiceffect and the probability model

(3) )e RRC will impact the probabilistic characteristicsof traffic load effects A poor RRC leads to largernumber of level-crossings compared to a good RRCand thus a poor RRC will amplify the maximumtraffic load effect

(4) )e influence of the RRC on the dynamic effect isweakened with the increase of the bridge span length)is phenomenon can be explained by the theoreticalbasis that the bridge with a longer span length has alower frequency which will weaken the vehicle-bridgeinteraction

(5) With consideration of the proportion of dense trafficflow as 12 24 and 36 the corresponding reli-ability indexes of the bridge in 100 years are 576 562and 560 respectively )e proportion of dense vehicleflow has significant influence on the first-passageprobability of the bridge deflection)us the control ofdense vehicle flow is essential for ensuring the bridgesafety in service period

Even though the proposed framework is verified for a rigid-frame bridge it can be applied for more types of bridgesHowever more studies are necessary to improve the compu-tational efficiency and accuracy A more refined traffic-bridgeinteraction analysis approach instead of the simplified EDWLapproach is critical to make the simulation more reasonable Inaddition the simulated traffic load effect needs to be comparedwith structural health monitoring data

Data Availability

)e original traffic monitoring data are available from theauthors if necessary

Conflicts of Interest

)e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)e authors would like to acknowledge the Management De-partment of the Nanxi Yangtze River Bridge duo for the helpwith the traffic data collection )is research was funded by theNational Natural Science Foundation of China (Grant no51908068) the Open Fund of Engineering Research Center ofCatastrophic Prophylaxis and Treatment of Road and TrafficSafety of Ministry of Education (Grant no KFJ190403)

10ndash12

10ndash10

10ndash8

10ndash6Pr

obab

ility

of e

xcee

danc

e

20 40 60 80 1000Time (year)

ρ = 12ρ = 24ρ = 36

(a)

40 60 80 10020Time (year)

5

55

6

65

7

Relia

bilit

y in

dex

ρ = 12ρ = 24ρ = 36

(b)

Figure 16 Reliability index of the continuous rigid-frame bridge based on the first-passage criterion of the girder deflection (a) failureprobability (b) reliability index


Industry Key Laboratory of Traffic Infrastructure Security RiskManagement in Changsha University of Science and Tech-nology (Grant no 19KF03) Open Fund of Key Laboratory ofBridge Engineering Safety Control in Department of Educationin Changsha University of Science and Technology (Grant no19KB02) National Science Foundation of Hunan Province(Grant nos 2020JJ5140 and 2020JJ5589) and the InnovationPlatform Open Fund Project of Hunan Education Department(19K002)

References

[1] A S Nowak ldquoCalibration of LRFD bridge coderdquo Journal ofStructural Engineering vol 121 no 8 pp 1245ndash1251 1995

[2] L Deng W Yan and L Nie ldquoA simple corrosion fatiguedesign method for bridges considering the coupled corrosion-overloading effectrdquo Engineering Structures vol 178pp 309ndash317 2019

[3] Y Liu X Xiao N Lu and Y Deng ldquoFatigue reliability as-sessment of orthotropic bridge decks under stochastic truckloadingrdquo Shock and Vibration vol 2016 Article ID 471259310 pages 2016

[4] Y Ma Yafei Z Guo L Wang Lei and J Zhang ldquoProbabilisticlife prediction for reinforced concrete structures subjected toseasonal corrosion-fatigue damagerdquo Journal of StructuralEngineering vol 146 no 7 2020

[5] Z Guo Y Ma L Wang et al ldquoCrack propagation-basedfatigue life prediction of corroded RC beams consideringbond degradationrdquo Journal of Bridge Engineering vol 25no 8 2020

[6] D Skokandic A M Ivankovic A Znidaric and M SrbicldquoModelling of traffic load effects in the assessment of existingroad bridgesrdquo GraCevinar vol 71 no 12 pp 1153ndash11652019

[7] X Q Zhu and S S Law ldquoRecent developments in inverseproblems of vehicle-bridge interaction dynamicsrdquo Journal ofCivil Structural Health Monitoring vol 6 no 1 pp 107ndash1282016

[8] Y Zhou and S Chen ldquoDynamic simulation of a long-spanbridge-traffic system subjected to combined service and ex-treme loadsrdquo Journal of Structural Engineering vol 141 no 9pp 1ndash18 2015

[9] S R Chen and J Wu ldquoDynamic performance simulation oflong-span bridge under combined loads of stochastic trafficand windrdquo Journal of Bridge Engineering vol 15 no 3pp 219ndash230 2009

[10] N Lu M Noori Y Liu et al ldquoLifetime deflections of long-span bridges under dynamic and growing traffic loadrdquo Journalof Bridge Engineering vol 22 no 11 2017

[11] N Lu M Noori and Y Liu ldquoFirst-passage probability of thedeflection of a cable-stayed bridge under long-term site-specific traffic loadingrdquo Advances in Mechanical Engineeringvol 9 no 1 pp 1ndash10 2017

[12] E J Obrien D Cantero B Enright and A GonzalezldquoCharacteristic dynamic increment for extreme traffic loadingevents on short and medium span highway bridgesrdquo Engi-neering Structures vol 32 no 12 pp 3827ndash3835 2010

[13] C Jiang C Wu C S Cai and W Xiong ldquoFatigue analysis ofstay cables on the long-span bridges under combined action oftraffic and windrdquo Engineering Structures vol 207 Article ID110212 2020

[14] S Li S Zhu Y-L Xu Z-W Chen and H Li ldquoLong-termcondition assessment of suspenders under traffic loads based onstructuralmonitoring system application to the Tsingma bridgerdquo

Structural Control and Health Monitoring vol 19 no 1pp 82ndash101 2012

[15] C C Caprani ldquoLifetime highway bridge traffic load effectfrom a combination of traffic states allowing for dynamicamplificationrdquo Journal of Bridge Engineering vol 18 no 9pp 901ndash909 2013

[16] L Shen Y Han C S Cai et al ldquoExceedance probabilityassessment of pedestrian wind environment based on mul-tiscale coupling numerical simulationrdquo Journal of AerospaceEngineering vol 33 no 4 2020

[17] P D Spanos and I A Kougioumtzoglou ldquoGalerkin schemebased determination of first-passage probability of nonlinearsystem responserdquo Structure and Infrastructure Engineeringvol 10 no 10 pp 1285ndash1294 2014

[18] F Sloothaak V Wachtel and B Zwart ldquoFirst-passage timeasymptotics over moving boundaries for random walkbridgesrdquo Journal of Applied Probability vol 55 no 2pp 627ndash651 2018

[19] A Fenerci and O Oslashiseth ldquoStrong wind characteristics anddynamic response of a long-span suspension bridge during astormrdquo Journal of Wind Engineering and Industrial Aerody-namics vol 172 pp 116ndash138 2018

[20] D Y Zhu Y H Zhang D Kennedy and F W WilliamsldquoStochastic vibration of the vehicle-bridge system subject tonon-uniform ground motionsrdquo Vehicle System Dynamicsvol 52 no 3 pp 410ndash428 2014

[21] X Yin Y Liu L Deng et al ldquoDynamic behavior of damagedbridge with multi-cracks under moving vehicular loadsrdquoInternational Journal of Structural Stability and Dynamicsvol 17 no 2 2017

[22] S R Chen and C S Cai ldquoEquivalent wheel load approach forslender cable-stayed bridge fatigue assessment under trafficand wind feasibility studyrdquo Journal of Bridge Engineeringvol 12 no 6 pp 755ndash764 2007

[23] J Wu and S R Chen ldquoProbabilistic dynamic behavior of along-span bridge under extreme eventsrdquo Engineering Struc-tures vol 33 no 5 pp 1657ndash1665 2011

[24] N Lu Y Ma and Y Liu ldquoEvaluating probabilistic traffic loadeffects on large bridges using long-term traffic monitoringdatardquo Sensors vol 19 no 22 2019

[25] S O Rice ldquoMathematical analysis of random noiserdquo BellSystem Technical Journal vol 24 no 1 pp 46ndash156 1945

[26] L Deng W Yan and S Li ldquoComputer modeling and weightlimit analysis for bridge structure fatigue using OpenSEESrdquoJournal of Bridge Engineering vol 24 no 8 2019

[27] W He T Ling E J OrsquoBrien and L Deng ldquoVirtual axlemethod for bridge weigh-in-motion systems requiring no axledetectorrdquo Journal of Bridge Engineering vol 24 no 9 2019

[28] M OBrien A Khan D McCrum et al ldquoUsing statisticalanalysis of an acceleration-based bridge weigh-in-motionsystem for damage detectionrdquo Applied Sciences vol 10 no 22020

[29] N Lu Y Liu and Y Deng ldquoFatigue reliability evaluationof orthotropic steel bridge decks based on site-specificweigh-in-motion measurementsrdquo International Journal ofSteel Structures vol 19 no 1 pp 181ndash192 2019

[30] N Lu Probability model of dynamic responses and reliabilityassessment for stiffening girders of suspension bridges underrandom traffic flow PhD thesis Changsha University ofScience and Technology Changsha China 2014

[31] X Yin Y Liu G Song et al ldquoSuppression of bridge vi-bration induced by moving vehicles using pounding tunedmass dampersrdquo Journal of Bridge Engineering vol 23no 7 pp 1ndash14 2018


and Chen [8] investigated probabilistic characteristics of thedynamic response of a cable-stayed bridge under wind loadand stochastic vehicle load Lu et al [11] investigated thefirst-passage probability of a cable-stayed bridge understochastic vehicle loads Obrien et al [12] investigated thedynamic increment of extreme traffic loading on short-spanbridges Jiang et al [13] investigated the fatigue damage ofcables of cable-stayed bridge under combined effect of trafficand wind Li et al [14] investigated the safety of suspendersof Tsing Ma Bridge under traffic loads However there arerelatively few studies on probabilistic dynamic traffic loadeffects on long-span continuous rigid-frame bridges )eunique feature for the continuous rigid-frame bridge is thehigh-rise piers which are usually space-flexible affecting thedynamic behaviour of the bridge )us the probabilisticdynamic traffic load effects on continuous rigid-framebridges deserve more investigation

Most studies treated the traffic load effect on a bridge as arandom variable In practice the load effect on a bridge is atime-varying random process [15] )e commonly usedtheoretical basis for the probabilistic dynamic analysis ofbridges under vehicle load is the first-passage criterion[16 17] )e first-passage failure probability is usually es-timated based on Ricersquos level-crossing principle [18]However applications of the first-passage theory are mostlyconcentrated in seismic engineering while the application totraffic load analysis is relatively insufficient)e bottleneck isthat the root mean square of traffic load effects is usuallyevaluated in frequency domain while the vehicle-bridgecoupled vibration is mostly analysed in time domain [19])erefore the key point is how to extract the probabilityparameters from the dynamic load effect

)is study evaluated probabilistic dynamic traffic loadeffects on rigid-frame bridges under highway traffic moni-toring loads )e site-specific traffic monitoring data wereutilized to establish stochastic traffic models )e dynamiceffect was considered in a vehicle-bridge coupled vibrationmodel and the probability estimation was conducted basedon the first-passage criterion of the girder deflection )eprototype bridge is a continuous rigid-frame bridge withmid-span length of 200m and pier height of 182m It isdemonstrated that the dynamic traffic load effect followsGaussian distribution and thus can be treated as a stationaryrandom process Parametric studies were conducted ac-counting for the road roughness condition the bridge spanlength and the traffic growth ratio

2 Theoretical Basis of Probabilistic Traffic-Bridge Interaction Analysis

21 Traffic-Bridge InteractionModel A vehicle passing on abridge leads to the vibration of the bridge due to the roadsurface roughness [20] Meanwhile the vibration of thebridge impacts the vibration of the vehicle )us thevehicle-bridge interaction is a coupled vibration systemFigure 1 shows the plane motion model of two-axle ve-hicle where x represents the road roughness and C and Krepresent the damping and stiffness parameters re-spectively )e vehicle model has four degrees of freedom

including vertical motion of two axles and the motion androtation of the vehicle body

Based on the simplified model as shown in Figure 1 thedifferential equation of motion of the vehicle-bridge coupledsystem can be written as [21]

Mv eurouv + Cv _uv + Kvuv Fvg + Fvb (1a)

Mb euroub + Cb _ub + Kbub Fbg + Fbv (1b)

where Mv and Mb represent the mass matrix of the vehicleand that of the bridge respectively Cv and Cb represent thedamping matrix of the vehicle and that of the bridge re-spectively Kv and Kb represent the stiffness matrix of thevehicle and that of the bridge respectively uv and ub rep-resent the displacement vector of the vehicle and that of thebridge respectively Fvb and Fbv represent the interactionforces between the vehicle and the bridge Fvg and Fbgrepresent the weight of the vehicle and that of the bridgerespectively

In order to consider the multiple-vehicle effect on thebridge response this study utilized an equivalent dynamicwheel load (EDWL) approach proposed by Chen and Cai[22] Based on the EDWL approach the force of the jthvehicle on a bridge can be simplified as a time-varying forcewhich can be written as

F(t) wheelEq 1113944

nv

j11 minus Rj(t)1113960 1113961Gj middot 1113944

n

k1hk xj(t) + αk xj(t)dj(t)1113960 11139611113960 11139611113966 1113967

⎧⎨

⎩

⎫⎬

⎭

(2a)

Rj(t) EDWLj(t)

Gj

(2b)

EDWLj(t) 1113944

na

i1K

ivlY

ivl + C

ivl

_Yi

vl1113874 1113875 (2c)

where Rj represents wheel weight ratio of the jth vehicle Gjrepresents the GVW of the jth vehicle xj and dj representlongitudinal and lateral positions on the bridge respectivelyhk and ak represent the kth-order vertical and torsionalbending modes of the bridge n and nv respectively rep-resent modal orders of bridges and the number of vehicleson the bridges Ki

vl and Civl represent stiffness and damping

matrixes of the vehicle respectively Yivl and _Y

i

vl respectivelyrepresent the vertical displacement and the velocity of the ithaxle EDWLj(t) represents the equivalent dynamic axis loadof the jth vehicle at time t

)e effectiveness of EDWL approach has been verified byWu and Chen [23] and Lu et al [24] for long-span cable-stayed bridges and suspension bridges respectively )isstudy develops the EDWL approach in probability analysisand reliability evaluation for continuous rigid-frame bridges

22 First-Passage Criterion )e first-passage criterion isusually utilized for Ricersquos level-crossing probability analysisof a random process )e principle of Ricersquos level-crossing isshown in Figure 2 where t is time x is a random process




Nb(T) 1113946T

01113946

+infin




vb(t) 1113946infin



vb 12π

σ _x

σx

exp minusb2

2σ2x1113890 1113891 (5)


brvbarEgraver1

x

lv

b2lv

Ks2 Cs2

Kt2 Ct2

y2 y1

Ks1 Cs1

Kt1 Ct1

b1lv

Zv


t

Upcrossing

Downcrossing

Level 1

Level 2

x

(a)

x

v(x)

Ricersquos fitting

Interval length


(b)






vb 12π

σ _x

σx


2

2σ2x1113890 1113891 (7)






β Φminus 1 1 minus Pf1113872 1113873 (9)











P w | ai μi σ2i1113872 11138731113966 1113967 1113944

M


2i1113872 1113873 (10)


g w | μi σ2i1113872 1113873

12πσi



1113890 1113891 (11)









4 Case Study






























0

002

004

006

008

01

Prob

abili

ty d

ensit

y

100 150 200 25050 Axle weight AV64 (kN)

(a)

0

0005

001

0015

002

Prob

abili

ty d

ensit

y

500 1000 1500 20000GVW (kN)


w1 = 024μ1 = 39σ1 = 74

w2 = 076μ2 = 86σ2 = 14

(b)










No i = i + 1


i = NADT

Yes










Slow lane0

1530

4560

Arrival tim

e (min)

Dirving lane

V1V2V3

V4V5V6

0

500

1000

1500

GV

W (t

)


(a) (b)



605 60525 85 50

340

340

380290

50

220120

1275

(a)

132170

680

C50Steel tube

C80

1000

(b)


(a) (b)

Figure 9 Continued


(c)




ndash04

ndash02

0

02

04

Roug

hnes

s (m

)


ndash03

ndash02

ndash01

0

01

02

Vert

ical

def

lect

ion

(m)

100 200 300 400 500 6000Time (s)











0

002

004

006

008

01

012

014Pr

obab

ility

den

sity

(a)

0

02

04

06

08

1

CDF




(b)





0

50

100

150

200

Num

ber o

f cro

ssin

gs



02 04 06 08 10Frequency (Hz)

0

002

004

006

008

01

Disp

lace

men

t spe

ctru

m d

ensit

y


(a)

t = 100st = 200st = 300s

1 2 3 4 5 6 70Time (s)

0

02

04

06

08

1

Cor

relat

ion

coef

ficie

nt

(b)






ndash015

ndash01

ndash005

0

005

01

015

Disp

lace

men

t (m

)





5 Conclusions








Data Availability




Acknowledgments


10ndash12

10ndash10

10ndash8

10ndash6Pr

obab

ility

of e

xcee

danc

e

20 40 60 80 1000Time (year)

ρ = 12ρ = 24ρ = 36

(a)

40 60 80 10020Time (year)

5

55

6

65

7

Relia

bilit

y in

dex

ρ = 12ρ = 24ρ = 36

(b)




References




































Nb(T) 1113946T

01113946

+infin




vb(t) 1113946infin



vb 12π

σ _x

σx

exp minusb2

2σ2x1113890 1113891 (5)


brvbarEgraver1

x

lv

b2lv

Ks2 Cs2

Kt2 Ct2

y2 y1

Ks1 Cs1

Kt1 Ct1

b1lv

Zv


t

Upcrossing

Downcrossing

Level 1

Level 2

x

(a)

x

v(x)

Ricersquos fitting

Interval length


(b)






vb 12π

σ _x

σx


2

2σ2x1113890 1113891 (7)






β Φminus 1 1 minus Pf1113872 1113873 (9)











P w | ai μi σ2i1113872 11138731113966 1113967 1113944

M


2i1113872 1113873 (10)


g w | μi σ2i1113872 1113873

12πσi



1113890 1113891 (11)









4 Case Study






























0

002

004

006

008

01

Prob

abili

ty d

ensit

y

100 150 200 25050 Axle weight AV64 (kN)

(a)

0

0005

001

0015

002

Prob

abili

ty d

ensit

y

500 1000 1500 20000GVW (kN)


w1 = 024μ1 = 39σ1 = 74

w2 = 076μ2 = 86σ2 = 14

(b)










No i = i + 1


i = NADT

Yes










Slow lane0

1530

4560

Arrival tim

e (min)

Dirving lane

V1V2V3

V4V5V6

0

500

1000

1500

GV

W (t

)


(a) (b)



605 60525 85 50

340

340

380290

50

220120

1275

(a)

132170

680

C50Steel tube

C80

1000

(b)


(a) (b)

Figure 9 Continued


(c)




ndash04

ndash02

0

02

04

Roug

hnes

s (m

)


ndash03

ndash02

ndash01

0

01

02

Vert

ical

def

lect

ion

(m)

100 200 300 400 500 6000Time (s)











0

002

004

006

008

01

012

014Pr

obab

ility

den

sity

(a)

0

02

04

06

08

1

CDF




(b)





0

50

100

150

200

Num

ber o

f cro

ssin

gs



02 04 06 08 10Frequency (Hz)

0

002

004

006

008

01

Disp

lace

men

t spe

ctru

m d

ensit

y


(a)

t = 100st = 200st = 300s

1 2 3 4 5 6 70Time (s)

0

02

04

06

08

1

Cor

relat

ion

coef

ficie

nt

(b)






ndash015

ndash01

ndash005

0

005

01

015

Disp

lace

men

t (m

)





5 Conclusions








Data Availability




Acknowledgments


10ndash12

10ndash10

10ndash8

10ndash6Pr

obab

ility

of e

xcee

danc

e

20 40 60 80 1000Time (year)

ρ = 12ρ = 24ρ = 36

(a)

40 60 80 10020Time (year)

5

55

6

65

7

Relia

bilit

y in

dex

ρ = 12ρ = 24ρ = 36

(b)




References





































vb 12π

σ _x

σx


2

2σ2x1113890 1113891 (7)






β Φminus 1 1 minus Pf1113872 1113873 (9)











P w | ai μi σ2i1113872 11138731113966 1113967 1113944

M


2i1113872 1113873 (10)


g w | μi σ2i1113872 1113873

12πσi



1113890 1113891 (11)









4 Case Study






























0

002

004

006

008

01

Prob

abili

ty d

ensit

y

100 150 200 25050 Axle weight AV64 (kN)

(a)

0

0005

001

0015

002

Prob

abili

ty d

ensit

y

500 1000 1500 20000GVW (kN)


w1 = 024μ1 = 39σ1 = 74

w2 = 076μ2 = 86σ2 = 14

(b)










No i = i + 1


i = NADT

Yes










Slow lane0

1530

4560

Arrival tim

e (min)

Dirving lane

V1V2V3

V4V5V6

0

500

1000

1500

GV

W (t

)


(a) (b)



605 60525 85 50

340

340

380290

50

220120

1275

(a)

132170

680

C50Steel tube

C80

1000

(b)


(a) (b)

Figure 9 Continued


(c)




ndash04

ndash02

0

02

04

Roug

hnes

s (m

)


ndash03

ndash02

ndash01

0

01

02

Vert

ical

def

lect

ion

(m)

100 200 300 400 500 6000Time (s)











0

002

004

006

008

01

012

014Pr

obab

ility

den

sity

(a)

0

02

04

06

08

1

CDF




(b)





0

50

100

150

200

Num

ber o

f cro

ssin

gs



02 04 06 08 10Frequency (Hz)

0

002

004

006

008

01

Disp

lace

men

t spe

ctru

m d

ensit

y


(a)

t = 100st = 200st = 300s

1 2 3 4 5 6 70Time (s)

0

02

04

06

08

1

Cor

relat

ion

coef

ficie

nt

(b)






ndash015

ndash01

ndash005

0

005

01

015

Disp

lace

men

t (m

)





5 Conclusions








Data Availability




Acknowledgments


10ndash12

10ndash10

10ndash8

10ndash6Pr

obab

ility

of e

xcee

danc

e

20 40 60 80 1000Time (year)

ρ = 12ρ = 24ρ = 36

(a)

40 60 80 10020Time (year)

5

55

6

65

7

Relia

bilit

y in

dex

ρ = 12ρ = 24ρ = 36

(b)




References








































4 Case Study






























0

002

004

006

008

01

Prob

abili

ty d

ensit

y

100 150 200 25050 Axle weight AV64 (kN)

(a)

0

0005

001

0015

002

Prob

abili

ty d

ensit

y

500 1000 1500 20000GVW (kN)


w1 = 024μ1 = 39σ1 = 74

w2 = 076μ2 = 86σ2 = 14

(b)










No i = i + 1


i = NADT

Yes










Slow lane0

1530

4560

Arrival tim

e (min)

Dirving lane

V1V2V3

V4V5V6

0

500

1000

1500

GV

W (t

)


(a) (b)



605 60525 85 50

340

340

380290

50

220120

1275

(a)

132170

680

C50Steel tube

C80

1000

(b)


(a) (b)

Figure 9 Continued


(c)




ndash04

ndash02

0

02

04

Roug

hnes

s (m

)


ndash03

ndash02

ndash01

0

01

02

Vert

ical

def

lect

ion

(m)

100 200 300 400 500 6000Time (s)











0

002

004

006

008

01

012

014Pr

obab

ility

den

sity

(a)

0

02

04

06

08

1

CDF




(b)





0

50

100

150

200

Num

ber o

f cro

ssin

gs



02 04 06 08 10Frequency (Hz)

0

002

004

006

008

01

Disp

lace

men

t spe

ctru

m d

ensit

y


(a)

t = 100st = 200st = 300s

1 2 3 4 5 6 70Time (s)

0

02

04

06

08

1

Cor

relat

ion

coef

ficie

nt

(b)






ndash015

ndash01

ndash005

0

005

01

015

Disp

lace

men

t (m

)





5 Conclusions








Data Availability




Acknowledgments


10ndash12

10ndash10

10ndash8

10ndash6Pr

obab

ility

of e

xcee

danc

e

20 40 60 80 1000Time (year)

ρ = 12ρ = 24ρ = 36

(a)

40 60 80 10020Time (year)

5

55

6

65

7

Relia

bilit

y in

dex

ρ = 12ρ = 24ρ = 36

(b)




References










































0

002

004

006

008

01

Prob

abili

ty d

ensit

y

100 150 200 25050 Axle weight AV64 (kN)

(a)

0

0005

001

0015

002

Prob

abili

ty d

ensit

y

500 1000 1500 20000GVW (kN)


w1 = 024μ1 = 39σ1 = 74

w2 = 076μ2 = 86σ2 = 14

(b)










No i = i + 1


i = NADT

Yes










Slow lane0

1530

4560

Arrival tim

e (min)

Dirving lane

V1V2V3

V4V5V6

0

500

1000

1500

GV

W (t

)


(a) (b)



605 60525 85 50

340

340

380290

50

220120

1275

(a)

132170

680

C50Steel tube

C80

1000

(b)


(a) (b)

Figure 9 Continued


(c)




ndash04

ndash02

0

02

04

Roug

hnes

s (m

)


ndash03

ndash02

ndash01

0

01

02

Vert

ical

def

lect

ion

(m)

100 200 300 400 500 6000Time (s)











0

002

004

006

008

01

012

014Pr

obab

ility

den

sity

(a)

0

02

04

06

08

1

CDF




(b)





0

50

100

150

200

Num

ber o

f cro

ssin

gs



02 04 06 08 10Frequency (Hz)

0

002

004

006

008

01

Disp

lace

men

t spe

ctru

m d

ensit

y


(a)

t = 100st = 200st = 300s

1 2 3 4 5 6 70Time (s)

0

02

04

06

08

1

Cor

relat

ion

coef

ficie

nt

(b)






ndash015

ndash01

ndash005

0

005

01

015

Disp

lace

men

t (m

)





5 Conclusions








Data Availability




Acknowledgments


10ndash12

10ndash10

10ndash8

10ndash6Pr

obab

ility

of e

xcee

danc

e

20 40 60 80 1000Time (year)

ρ = 12ρ = 24ρ = 36

(a)

40 60 80 10020Time (year)

5

55

6

65

7

Relia

bilit

y in

dex

ρ = 12ρ = 24ρ = 36

(b)




References









































Slow lane0

1530

4560

Arrival tim

e (min)

Dirving lane

V1V2V3

V4V5V6

0

500

1000

1500

GV

W (t

)


(a) (b)



605 60525 85 50

340

340

380290

50

220120

1275

(a)

132170

680

C50Steel tube

C80

1000

(b)


(a) (b)

Figure 9 Continued


(c)




ndash04

ndash02

0

02

04

Roug

hnes

s (m

)


ndash03

ndash02

ndash01

0

01

02

Vert

ical

def

lect

ion

(m)

100 200 300 400 500 6000Time (s)











0

002

004

006

008

01

012

014Pr

obab

ility

den

sity

(a)

0

02

04

06

08

1

CDF




(b)





0

50

100

150

200

Num

ber o

f cro

ssin

gs



02 04 06 08 10Frequency (Hz)

0

002

004

006

008

01

Disp

lace

men

t spe

ctru

m d

ensit

y


(a)

t = 100st = 200st = 300s

1 2 3 4 5 6 70Time (s)

0

02

04

06

08

1

Cor

relat

ion

coef

ficie

nt

(b)






ndash015

ndash01

ndash005

0

005

01

015

Disp

lace

men

t (m

)





5 Conclusions








Data Availability




Acknowledgments


10ndash12

10ndash10

10ndash8

10ndash6Pr

obab

ility

of e

xcee

danc

e

20 40 60 80 1000Time (year)

ρ = 12ρ = 24ρ = 36

(a)

40 60 80 10020Time (year)

5

55

6

65

7

Relia

bilit

y in

dex

ρ = 12ρ = 24ρ = 36

(b)




References


































605 60525 85 50

340

340

380290

50

220120

1275

(a)

132170

680

C50Steel tube

C80

1000

(b)


(a) (b)

Figure 9 Continued


(c)




ndash04

ndash02

0

02

04

Roug

hnes

s (m

)


ndash03

ndash02

ndash01

0

01

02

Vert

ical

def

lect

ion

(m)

100 200 300 400 500 6000Time (s)











0

002

004

006

008

01

012

014Pr

obab

ility

den

sity

(a)

0

02

04

06

08

1

CDF




(b)





0

50

100

150

200

Num

ber o

f cro

ssin

gs



02 04 06 08 10Frequency (Hz)

0

002

004

006

008

01

Disp

lace

men

t spe

ctru

m d

ensit

y


(a)

t = 100st = 200st = 300s

1 2 3 4 5 6 70Time (s)

0

02

04

06

08

1

Cor

relat

ion

coef

ficie

nt

(b)






ndash015

ndash01

ndash005

0

005

01

015

Disp

lace

men

t (m

)





5 Conclusions








Data Availability




Acknowledgments


10ndash12

10ndash10

10ndash8

10ndash6Pr

obab

ility

of e

xcee

danc

e

20 40 60 80 1000Time (year)

ρ = 12ρ = 24ρ = 36

(a)

40 60 80 10020Time (year)

5

55

6

65

7

Relia

bilit

y in

dex

ρ = 12ρ = 24ρ = 36

(b)




References


































(c)




ndash04

ndash02

0

02

04

Roug

hnes

s (m

)


ndash03

ndash02

ndash01

0

01

02

Vert

ical

def

lect

ion

(m)

100 200 300 400 500 6000Time (s)











0

002

004

006

008

01

012

014Pr

obab

ility

den

sity

(a)

0

02

04

06

08

1

CDF




(b)





0

50

100

150

200

Num

ber o

f cro

ssin

gs



02 04 06 08 10Frequency (Hz)

0

002

004

006

008

01

Disp

lace

men

t spe

ctru

m d

ensit

y


(a)

t = 100st = 200st = 300s

1 2 3 4 5 6 70Time (s)

0

02

04

06

08

1

Cor

relat

ion

coef

ficie

nt

(b)






ndash015

ndash01

ndash005

0

005

01

015

Disp

lace

men

t (m

)





5 Conclusions








Data Availability




Acknowledgments


10ndash12

10ndash10

10ndash8

10ndash6Pr

obab

ility

of e

xcee

danc

e

20 40 60 80 1000Time (year)

ρ = 12ρ = 24ρ = 36

(a)

40 60 80 10020Time (year)

5

55

6

65

7

Relia

bilit

y in

dex

ρ = 12ρ = 24ρ = 36

(b)




References









































0

002

004

006

008

01

012

014Pr

obab

ility

den

sity

(a)

0

02

04

06

08

1

CDF




(b)





0

50

100

150

200

Num

ber o

f cro

ssin

gs



02 04 06 08 10Frequency (Hz)

0

002

004

006

008

01

Disp

lace

men

t spe

ctru

m d

ensit

y


(a)

t = 100st = 200st = 300s

1 2 3 4 5 6 70Time (s)

0

02

04

06

08

1

Cor

relat

ion

coef

ficie

nt

(b)






ndash015

ndash01

ndash005

0

005

01

015

Disp

lace

men

t (m

)





5 Conclusions








Data Availability




Acknowledgments


10ndash12

10ndash10

10ndash8

10ndash6Pr

obab

ility

of e

xcee

danc

e

20 40 60 80 1000Time (year)

ρ = 12ρ = 24ρ = 36

(a)

40 60 80 10020Time (year)

5

55

6

65

7

Relia

bilit

y in

dex

ρ = 12ρ = 24ρ = 36

(b)




References


































02 04 06 08 10Frequency (Hz)

0

002

004

006

008

01

Disp

lace

men

t spe

ctru

m d

ensit

y


(a)

t = 100st = 200st = 300s

1 2 3 4 5 6 70Time (s)

0

02

04

06

08

1

Cor

relat

ion

coef

ficie

nt

(b)






ndash015

ndash01

ndash005

0

005

01

015

Disp

lace

men

t (m

)





5 Conclusions








Data Availability




Acknowledgments


10ndash12

10ndash10

10ndash8

10ndash6Pr

obab

ility

of e

xcee

danc

e

20 40 60 80 1000Time (year)

ρ = 12ρ = 24ρ = 36

(a)

40 60 80 10020Time (year)

5

55

6

65

7

Relia

bilit

y in

dex

ρ = 12ρ = 24ρ = 36

(b)




References



































5 Conclusions








Data Availability




Acknowledgments


10ndash12

10ndash10

10ndash8

10ndash6Pr

obab

ility

of e

xcee

danc

e

20 40 60 80 1000Time (year)

ρ = 12ρ = 24ρ = 36

(a)

40 60 80 10020Time (year)

5

55

6

65

7

Relia

bilit

y in

dex

ρ = 12ρ = 24ρ = 36

(b)




References



































References


































Dynamic Reliability of Continuous Rigid-Frame Bridges...

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