Research ArticleSafety Evaluation of the Wear Life of High-Speed Railway BridgeBearings by Monitoring Train-Induced Dynamic Displacements

Gaoxin Wang 1 Youliang Ding 2 Hui Guo 3 and Xinxin Zhao3

1State Key Laboratory for Geomechanics and Deep Underground Engineering China University of Mining and TechnologyXuzhou China2ampe Key Laboratory of Concrete and Prestressed Concrete Structures of Ministry of Education Southeast UniversityNanjing 210096 China3Railway Engineering Research Institute China Academy of Railway Sciences Corporation Limited Beijing 100081 China

Correspondence should be addressed to Youliang Ding civilchinahotmailcom

Received 22 June 2018 Revised 30 September 2018 Accepted 14 October 2018 Published 1 November 2018

Academic Editor Mahdi Mohammadpour

Copyright copy 2018 GaoxinWang et alis 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

Bridge bearings experience numerous small-amplitude displacements under environmental loads e continuous cyclic accu-mulations of these small-amplitude displacements will result in severe wear on the poly-tetra-fluoro-ethylene (PTFE) plates in thebridge bearings which seriously endangers the service life of bearings Traditional method directly uses the linear wear rate ofcumulative displacements in a short period to evaluate the wearing life but the linear wear rate only in a short period such as severaldays may not represent the characteristics in the whole bridge service life Hence this research takes the spherical steel bearings of theNanjing Dashengguan Yangtze River Bridge as a study object e cumulative dynamic displacement (CDD) under the action ofa single train and the cumulative bearing travel (CBT) under the continual actions of many trains are studied using the monitoredlongitudinal displacement data from spherical steel bearings Furthermore the probability statistics and the Monte Carlo samplingsimulation for CDD are studied and the safety evaluation method for bearing wear life in the real environment is proposed usinga reliability index regarding the failure probability of monitored CBTover the wear limit during service lifetime In addition safetyevaluation on the bearing wear life was performed to assess the condition of spherical steel bearings in the real service environmente results can provide an important reference for analysis on the bearing wear life of long-span railway bridge structures

1 Introduction

As the service time of railway bridges increases some bridgecomponents may suffer from the aging and damage prob-lems due to the influence of external environment loadsAmong these bridge components the bridge bearing is animportant load-bearing component in large-span railwaybridge structures which can suffer from a series of diseasessuch as the disengagement of bearings the uneven com-pression the twisting and fracturing of the bearing plate andthe weld cracking which directly threaten the safety ofrailway bridges [1ndash5] For example Wei et al revealed thatthe fixed bearings of continuous high-speed railway bridgeswere vulnerable under earthquakes because they transferredmost of the seismic forces between superstructure and piers

and the sliding friction action of fixed bearings can protectother components from severe damage under earthquakes[2] the research results of Oh et al showed that the rubbermaterial characteristics and spherical shape of sphericalelastomeric bearing were important parameters in reducingvibrations of railway bridge [3] Huang et al emphasizedthat bearing is the important part that connects the su-perstructure and substructure of a railway bridge in whichthe occurrence of defects affects running safety [4] ere-fore it is necessary to further study the bearing damage ofhigh-speed railway bridges subjected to environmental loadsto ensure safe operation

Current research on bearing damage in high-speedrailway bridges mainly concentrated on the followingthree aspects (1) the causes of bearing damage and

HindawiShock and VibrationVolume 2018 Article ID 6479480 14 pageshttpsdoiorg10115520186479480

prevention methods [4 6 7] For example Yan conducteda detailed study on the cause analysis and remediationmeasures for the roller bearings of the Sunkou Yellow RiverBridge [7] and Huang et al researched a rapid treatmenttechnique for the damaged rocking axle bearing of a railwaysimply supported beam bridge [4] (2) e damage recog-nition methods of bridge bearings [1 8 9] For instanceZhan et al utilized the dynamic response of a bridge underan impact load to diagnose the bearing disengagement ofexisting rubber bearings [1] Qiao conducted an in-depthstudy on a damage identification technique for bridgebearings based on the change rate of natural frequency [9](3) e influence of bearing damage on the mechanicalproperties of bridge structure [310ndash12] For instance onestudy revealed that the increased vibration level of a railwaybridge could cause bridge bearing damage and the rubbermaterial characteristics and spherical shape were importantparameters in reducing bridge vibration [3] another studyshowed that the load-carrying fillet weld around the gussetplate of diagonal bracing at the bridge bearing was predictedto be the most fatigued critical detail among the variousstructural details and a fatigue assessment was carried outfor a composite railway bridge on which high-speed trainstraveled [12]

e latest studies revealed that bridge bearings experi-enced numerous small-amplitude dynamic displacementsunder dynamic loads [13 14] Continuous accumulations ofsmall-amplitude dynamic displacements can cause severewear to poly-tetra-fluoro-ethylene (PTFE) plates of bridgebearings which seriously decreases the service lifetime ofbearings [13 14] For instance according to the field resultsthe small but rapid girder movements contributed the mostto the large cumulative movements resulting in acceleratedfatigue damage and bearing wear [13 14] However currentresearch mainly focuses on highway bridge bearings ratherthan railway bridge bearings Large-span high-speed railwaybridges have the special characteristics of being designed forhigh-speed trains For example the Dashengguan YangtzeRiver Bridge is one large-span steel truss arch bridge witha 336-meter-long main span and six tracks and the designedhigh-speed train passed speed can reach 300 kmh [15]ese conditions can easily induce a large number of small-amplitude dynamic displacements which can result in se-rious wear damage However current research has notclearly revealed the cumulative wear characteristics and thesafety evaluation method for railway bridge bearings in theservice lifetime

erefore this research takes the spherical steel bearingsof the Nanjing Dashengguan Yangtze River Bridge as a studyobject as shown in Figure 1 to investigate the cumulativedynamic displacement (CDD) under the action of a singletrain and the cumulative bearing travel (CBT) under thecontinual actions of many trains using monitored longitu-dinal displacement data of spherical steel bearings Fur-thermore the probability statistics and the Monte Carlosampling simulation for CDD are studied and the safetyevaluation method for the bearing wear life is proposedusing a reliability index regarding the failure probability ofmonitored CBTover the wear limit in the service lifetime In

addition the safety evaluation for the bearing wear life wasperformed to assess the condition of spherical steel bearingsin real service environment Traditional method directly usesthe linear wear rate of cumulative displacements in a shortperiod to evaluate the wearing life and such linear wear rateonly in a short period such as several days may not representthe characteristics in the whole bridge service life so thenovelty of this paper is the safety evaluation method for thebearing wear life using a reliability index integrated withprobability statistics analysis and Monte Carlo samplingsimulation in the whole bridge life

2 Monitoring Data Analysis

21 Description of Bridge Health Monitoring e NanjingDashengguan Yangtze River Bridge is a large-span high-speed railway steel truss arch bridge It is designed for rivercrossing of the Beijing-Shanghai high-speed railway and theShanghai-Wuhan-Chengdu high-speed railway as shown inFigure 2 e upper structure of this bridge is a six-spancontinuous steel truss arch girder which is composed of steeltruss arch and steel bridge deck [16] In detail the steel trussarch is composed of chord members diagonal web mem-bers vertical web members and horizontal and verticalbracings the steel deck is composed of top plate andtransverse stiffening girder In addition a total of sevengroups of spherical steel bearings (ie bearings 1sim7) are usedto support the steel truss arch girder where the bearing 4 istotally fixed in all the three translational directions while theother six bearings allow longitudinal thermal expansion inthe steel truss arch girder

erefore 12 LVDT displacement sensors (namelylinear variable differential transformers) are installed at theupstream and downstream sides of the six groups ofbearings (ie B1usimB3u B5usimB7u B1dsimB3d and B5dsimB7d)for long-term monitoring of the longitudinal displacementdata where ldquourdquo indicates the upstream side and ldquodrdquo in-dicates the downstream side In detail the voltages in thedisplacement sensors which are termed as analog signalsare amplified filtered and then transformed to the digitalsignals in the data acquisition station at bridge site whichcan be remotely monitored displayed and downloadedthrough Internet in the health monitoring center far from

Top bearing plate

Bottom bearing plate

Seal ring

Spherical plate

Flat PTFE plate

Spherical PTFE plate

Figure 1 e components of spherical steel bearing

2 Shock and Vibration

the bridge site e sampling frequency of the 12 displace-ment sensors is 1Hz Because the Nanjing DashengguanYangtze River Bridge is one symmetrical structure only thebearings on the left half (ie B1usimB3u and B1dsimB3d) areselected for analysis

22 Cumulative Dynamic Displacement Caused by a PassedTrain e monitored longitudinal displacement data at theith spherical steel bearing Biu are denoted by Diu themonitored longitudinal displacement data at the ithspherical steel bearing Bid are denoted by Did i 1 2 3Because the time histories of longitudinal displacementshave similar change trends [16] only D1u is selected foranalysis e time history of D1u from March to October in2017 is shown in Figure 3(a) showing typical seasonalcharacteristics over eight months Furthermore the timehistory of D1u on May 20th 2017 is plotted in Figure 3(b)showing typical sinusoidal characteristics ese variationcharacteristics are very similar to those of temperature field[17] indicating that the displacement is mainly induced bytemperature field In addition Figure 3(b) shows the timehistory of D1u from 1600 to 1800 on May 20th 2017 It isclear that the longitudinal displacements contain numeroussmall-amplitude dynamic displacements with random var-iability that are mainly induced by dynamic loads

Because the monitored longitudinal displacement datacontain static longitudinal displacements induced by thetemperature field and dynamic longitudinal displacementsinduced by dynamic loads as well the dynamic longitudinaldisplacements need to be extracted from the monitored datafirst Note that the static and dynamic longitudinal dis-placements are in different frequency bands e staticlongitudinal displacement is greatly influenced by temper-ature changes so the cycle period is about one day (ie thecorresponding frequency is 186400Hz) To ensure that thefrequency of static longitudinal displacements is completelywithin the band range the upper limit of the frequency bandis designed to be 586400Hz namely (0 586400] [13 14]e frequency of dynamic longitudinal displacements isfar higher than the frequency of static longitudinal dis-placements and relevant research has indicated that the

frequency bandwidth of dynamic longitudinal displace-ments is within the range of (586400 05] [13 14] ereason for choosing 05Hz as the upper limit is that thedynamic displacements with frequencies over 05Hz aremainly induced by electromagnetic interference [13 14]which are not in the range of concern

e wavelet packet decomposition method can use a pairof filters to decompose the monitored longitudinal displace-ment into different frequency bands scale by scale [18ndash20]Figure 4 shows the tree structure of the wavelet packet co-efficients within 13 scales after layer-by-layer decompositionwhere xjm denotes the wavelet packet coefficient in the mthfrequency band of the jth scale (m 0 1 2 2j minus 1) andfsdenotes the sampling frequency with fs 1Hz in this re-search e wavelet packet coefficient in the first frequencyband (0 1214] which is close to (0 586400] is selected andreconstructed to obtain static displacements the waveletpacket coefficients in the other frequency bands (1214 12]which are close to (586400 05] are selected and recon-structed to obtain dynamic displacements

Taking Figure 3(b) as an example the time history of D1uon May 20th 2017 is decomposed by the wavelet packetmethod and the decomposition result is shown in Figure 5 Itis clear that the wavelet packet decomposition method caneffectively decompose the monitored longitudinal displace-ment data into two components the first component causedby temperature field and the second component (namelysmall-amplitude dynamic displacements) caused by dynamicloads Furthermore the cumulative displacements of twocomponents are calculated to investigate the contribution oftwo components to cumulative displacements which are471mm and 7734mm respectively Obviously the secondcomponent caused by dynamic loads contributes the most tocumulative displacements In detail the dynamic loads mainlyinclude train and wind loads What should be mentioned isthat there is no train passing the Dashengguan Yangtze RiverBridge from 0 am to 6 am for bridge inspection so the firstcomponent during this period ismainly induced by wind loadin addition the second component during the rest of time ismainly induced by both wind and train loads as shown inFigure 5(b) It can be inferred from Figure 5(b) that train loadis the major influence factor for cumulative displacements

108 192 336

1272

1

336 192 108

B6 uB6 d

B5 uB5 d

B4 uB4 d

B3 uB3 d

B2 uB2 d

B1 uB1 d B7 d

Bearing 7Bearing 6Bearing 5Bearing 4Bearing 3

1

1 2 3 4 5 6 7

Bearing 2Bearing 1

B7 u

Figure 2 Locations of the longitudinal displacement sensors m

Shock and Vibration 3

because the influence of train load is obviously higher thanwind load Hence the small-amplitude dynamic displacementcaused by train load is specifically studied in this research

e continuous accumulation of small-amplitude dy-namic displacements will induce serious fatigue wear whichcan severely decrease the service lifetime of bearingsus itis necessary to carry out study on the cumulative charac-teristics of small-amplitude dynamic displacements eCDD caused by a single passed train was calculated toevaluate the contribution of this train to the cumulativewear e CDD caused by the kth passed train which isdenoted byN(k) is calculated by referring to the equation inTable 1 where 1113957D(k) denotes the dynamic displacementcaused by the kth passed train 1113957Dj(k) and 1113957Dj+1(k) denote thejth and (j + 1)th values in 1113957D(k) respectively wherej 1 2 Uminus 1 U denotes the number of monitoredvalues in 1113957D(k) N(k) indicates the cumulative travel ofbearing caused by the kth passed train For example onetime history of dynamic displacement of B3u caused byone train is plotted in Figure 6 and the number of U in thetrain-passed segment of the plot is 48 which can be clearlycaptured so the value of N(k) is 203mm by calculating theCDD in the train-passed segment

e monitored data from August 10th to August 31st in2017 is selected for analysis A total of 3495 trains passedover the Dashengguan Yangtze River Bridge over this pe-riod and each train corresponds to one value of N(k)erefore 3495 values of N(k) for each bearing at upstreamand downstream sides (ie B1usimB3u and B1dsimB3d) arecalculated as shown in Figure 7 where Niu and Nid denotea set of N(1) N(2) N(3495) for the bearings Biu andBid respectively as shown in Table 1 It can be seen that allthe CDDs show uniform and random characteristics Fur-thermore the average value of 3495 CDDs for each Niu orNid which is denoted by mean(Niu) or mean(Nid) re-spectively is calculated by referring to the equations inTable 1 with the results shown in Figure 7 It can be seen thatmean(N3d) has the largest value about 7020mm whilemean(N1u) has the smallest value about 2904mm

23 Cumulative Bearing Travel Caused by Many PassedTrains Each value of Niu or Nid corresponds to one passedtrain and furthermore the 3495 values for each Biu or Bid areaccumulated to obtain CBT which indicates the influence ofthe continuous actions of passed trains on bearing wear ecalculation formula for CBT is expressed as follows

Scale

0

1

2

Signalx00

x10 x11

x20 x21 x22 x23

j xj0 xj1 xj2jndash2 xj2jndash1

13 x130 x131 x138191

Frequency band ]21420 fs(0 ]

2142fs

21420 fs(

]2

214

(213-1)fs( fs

Staticdisplacement

Dynamicdisplacement

Figure 4 Tree structure of the wavelet packet coefficients

3 4 5 6 7 8 9 10 11ndash200

ndash100

0

100

200

Month

Mon

itorin

g di

spla

cem

ent (

mm

)

(a)

0 4 8 12 16 20 2425

30

35

40

45

Hour

Disp

lace

men

t (m

m)

(b)

Figure 3 e monitored result of D1u (a) From March to October in 2017 (b) On May 20th 2017

4 Shock and Vibration

M(p) 1113944

p

k1N(k) (1)

where M(p) denotes the CBT caused by p passed trainsand p 1 2 3495 e M(p) values at the upstreamand downstream are shown in Figure 8 where Miu de-notes the CBT of Biu and Mid denotes the CBT ofBid(i 1 2 3) It can be seen that all the CBTs have ob-vious linear correlations with the number of passed trainsbut the linear growth rates are different for each Miu orMid where the linear growth rate of M3d is the fastestand its maximum CBT caused by 3495 passed trains is24360mm

3 Probability Statistics Analysis

e number of monitored CDDs are not enough to satisfythe requirement of large sample for safety evaluation ofbearing wear life erefore it is necessary to performsampling simulation of CDDs using probabilistic statistics ofthe monitored CDDs to obtain adequate data

31 Cumulative Probability Characteristics Figure 7 showsthat Niu and Nid contain obvious uniform and randomcharacteristics so Niu and Nid can be treated as statisticalvariables for probabilistic statistics analysis In detail thecumulative probability (CP) is used to describe the proba-bilistic statistics characteristics of CDDs First the cumu-lative probabilities of Niu and Nid are calculated based onprobability statistics theory and then the cumulative dis-tribution function is fitted in a least-squares manner usingthe cumulative probabilities of Niu and Nid which is il-lustrated in two steps as follows

(1) e cumulative probabilities of Niu and Nid arecalculated as follows

P Niu(j)11138731113872 NUM Niu ltNiu(j)11138731113872

NUM Niu1113872 1113873 (2a)

P Nid(j)11138731113872 NUM Nid ltNid(j)11138731113872

NUM Nid1113872 1113873 (2b)

where P(Niu(j)) denotes the CP of the jth valueNiu(j) in Niu P(Nid(j)) denotes the CP of the jth

0 4 8 12 16 20 2425

30

35

40

45

Hour

Disp

lace

men

t (m

m)

(a)

0 4 8 12 16 20 24ndash3

ndash1

1

3

Hour

Disp

lace

men

t (m

m)

Wind load Wind and train loads

The firstcomponent The second component

(b)

Figure 5 Decomposition result of monitored longitudinal displacement (a) Component caused by temperature field (b) Componentcaused by dynamical loads

Table 1 Calculation of variables N(k) Niu Nid mean(Niu) and mean(Nid)

Variable Calculation equation

N(k) N(k) 1113936Uminus1j1 | 1113957Dj+1(k)minus 1113957Dj(k)|

Niu NidNiu N(1) N(2) N(3495) for the bearings at upstream

Nid N(1) N(2) N(3495) for the bearings at downstream

mean(Niu) mean(Nid)mean(Niu) (N(1) + N(2) + middot middot middot + N(3495)3495) for the bearings at upstream

mean(Nid) (N(1) + N(2) + middot middot middot + N(3495)3495) for the bearings at downstream

0 40 80 120 160ndash3

ndash2

ndash1

0

1

2

3

Time (s)

Disp

lace

men

t (m

m)

No train No trainPassed train

Figure 6 e dynamic displacement of B3u caused by one train

Shock and Vibration 5

value Nid(j) in Nid NUM(Niu) denotes the totalnumber of Niu NUM(Nid) denotes the totalnumber of Nid NUM(Niu ltNiu(j)) denotes thenumber of CDDs in Niu that are smaller thanNiu(j) and NUM(Nid ltNid(j)) denotes thenumber of CDDs in Nid that are smaller thanNid(j) For instance the CP of N1u is shown inFigure 9

(2) e cumulative distribution function is fitted in a least-squares manner using the cumulative probabilities of

Niu and Nid Considering that the true cumulativedistribution function of Niu and Nid is unknownthree possible cumulative distribution functions(ie normal distribution (ND) Weibull distribution(WD) and general extreme value distribution(GEVD)) are chosen for least-square fitting and thenthe cumulative distribution function with the bestfitting effect is selected as the true cumulative distri-bution function of Niu and Nid Taking N1u as anexample the fitting results of the three cumulative

0 700 1400 2100 2800 35000

5

10

15

20

25

k

Cum

ulat

ive v

alue

(mm

)mean (N1u) = 2904mm

(a)

0 700 1400 2100 2800 35000

10

20

30

40

50

60

k

Cum

ulat

ive v

alue

(mm

)

mean (N1d) = 4067mm

(b)

0 700 1400 2100 2800 35000

5

10

15

20

25mean (N2u) = 3888mm

Cum

ulat

ive v

alue

(mm

)

k

(c)

0 700 1400 2100 2800 35000

5

10

15

Cum

ulat

ive v

alue

(mm

)

k

mean (N2d) = 2960mm

(d)

0 700 1400 2100 2800 35000

5

10

15

20

25

k

Cum

ulat

ive v

alue

(mm

)

mean (N3u) = 6559mm

(e)

0 700 1400 2100 2800 35000

10

20

30

40

k

Cum

ulat

ive v

alue

(mm

)

mean (N3d) = 7020mm

(f)

Figure 7 Niu and Nid (a) N1u (b) N1d (c) N2u (d) N2d (e) N3u (f ) N3d

6 Shock and Vibration

distribution functions are shown in Figure 10 It can beseen that GEVDhas the best fitting results so GEVD ischosen to describe the cumulative probability char-acteristics of N1u e formula is expressed as

G N1u1113872 1113873 exp minus 1 + rN1u minus b

a1113888 11138891113890 1113891

minus1r⎡⎣ ⎤⎦ (3)

where G(N1u) denotes the cumulative distributionfunction of N1u which follows the generalized ex-treme value distribution r denotes the shape pa-rameter a denotes the scale parameter and b

denotes the location parameter e fitted values ofthese three parameters for G(N1u) are r 03058a 08163 and b 20294

Based on the method above the best cumulative dis-tribution functions of N1u minusN3u and N1d minusN3d are allGEVD as shown in Figures 10(a)ndash10(f ) It can be seen thatGEVD can well describe the cumulative probability char-acteristics of N1u minusN3u and N1d minusN3d

32 Sampling SimulationMethod Since Niu and Nid followGEVDs their GEVDs can be utilized for Monte Carlosampling to obtain a demanded number of CDDs Specif-ically if the demanded number of CDDs is K K values are

0 1000 2000 3000 40000

5

10

15

M1uM1d

p

Cum

ulat

ive t

rave

l (m

m times

103 )

(a)

0 1000 2000 3000 40000

5

10

15

M2uM2d

p

Cum

ulat

ive t

rave

l (m

m times

103 )

(b)

0 1000 2000 3000 40000

5

10

15

20

25

M3uM3d

p

Cum

ulat

ive t

rave

l (m

m times

103 )

(c)

Figure 8 e M1u minusM3u and M1u minusM3u caused by many passed trains (a) M1u and M1d (b) M2u and M2d (c) M3u and M3d

5 10 15 20 25 300

02

04

06

08

1

Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Figure 9 e cumulative probability of N1u

Shock and Vibration 7

5 10 15 20 25 300

02

04

06

08

1

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

WD

ND

GEVD

GEVDND

WDMonitored curve

(a)

10 20 30 40 500

02

04

06

08

1

r = 04765a = 10599b = 25108

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored curve Fitted curve

(b)

5 10 15 20 25 300

02

04

06

08

1

r = 01926a = 12320b = 28604

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored curveFitted curve

(c)

2 4 6 8 10 12 140

02

04

06

08

1

r = 00493a = 08440b = 24302

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored curveFitted curve

(d)

5 10 15 20 250

02

04

06

08

1

r = 00830a = 18184b = 53395

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored curveFitted curve

(e)

5 10 15 20 250

02

04

06

08

1

r = 01834a = 19436b = 54517

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored curveFitted curve

(f )

Figure 10 GEVDs of N1u minusN6u and N1u minusN6u (a) N1u (b) N1d (c) N2u (d) N2d (e) N3u (f ) N3d

8 Shock and Vibration

uniformly and randomly sampled from the interval (0 1)and then the demanded number of CDDs can be simulatedby substituting the K values into the cumulative probabilityR in the following equations

R G Nius1113872 1113873 (4a)

R G Nids1113872 1113873 (4b)

G Nius1113872 1113873 exp minus 1 + rNius minus b

a1113888 11138891113890 1113891

minus1r⎡⎣ ⎤⎦ (4c)

G Nids1113872 1113873 exp minus 1 + rNids minus b

a1113888 11138891113890 1113891

minus1r⎡⎣ ⎤⎦ (4d)

where Nius and Nids denote the simulated CDDs of Niu andNid respectively e values of r a and b have been cal-culated in the Section 32 However Niu and Nid areembedded in G(Niu) and G(Nids) so they cannot be di-rectly solved from Equations (4a) and (4b) Considering thatG(Nius) and G(Nids) are monotonically increasing func-tions Nius and Nids can be determined by the Newtoniteration method as follows

Nn+1ius N

nius minus

G Nnius1113872 1113873minusR

Gprime Nnius1113872 1113873

(5a)

Nn+1ids N

nids minus

G Nnids1113872 1113873minusR

Gprime Nnids1113872 1113873

(5b)

where Nn+1ius and Nn

ius denote the (n + 1)th and nth iterationresults of Nius respectively Nn+1

ids andNnids denote the (n + 1)

th and nth iteration results of Nids respectively Gprime(Nnius)

and Gprime(Nnids) denote the first-order derivatives of GEVD

(ie the probability density functions of GEVD) e iter-ation will terminate when the absolute difference betweentwo adjacent iterations is less than 10minus2 e last iterationresult is treated as the values of Nius and Nids

33 ampe Sampling Simulation Result Based on the samplingsimulation method above the simulation results of 3495N1us are shown in Figure 11(a) e mean value of thesimulated result is 2820mm which is very close to the meanvalue of Niu ie 2904mm Furthermore the CP of thesimulated result is calculated and then compared with themonitored CP as shown in Figure 11(b) It can be seen thatthe simulated CP coincides with the monitored CP whichverifies the accuracy of the sampling simulation method

Furthermore the simulated 3495 values of Nius andNidsare accumulated using Equation (1) to obtain the simulatedCBT (ie Mius and Mids) as shown in Figure 12 It can beseen that the simulated Mius and Mids present good lineargrowth characteristics with the number of passed trains Toverify the accuracy of the simulated linear growth ratesa comparison between the monitored and simulated lineargrowth rates was conducted as shown in Figure 12 It can beseen that the simulated linear growth rates have agreement

with the monitored ones which further verifies the accuracyof the sampling simulation method

4 Safety Evaluation Analysis

e CBT relates to the bearing wear condition If the CBTsexceed the wear limit in the service lifetime the sphericalsteel bearing is totally damaged and cannot be used forlongitudinal thermal expansion any longer erefore it isnecessary to analyze whether the CBTs at bridge site canexceed the wear limit in the service lifetime

41 EvaluationMethod e failure probabilities of sphericalsteel bearings in the service lifetime (ie the probabilities ofCBTover the wear limit in the service lifetime) are calculatedas follows

Pf 1minusF Mius1113960 11139611113872 1113873 P Mius gt Mius1113960 11139611113872 1113873 (6a)

or

Pf 1minusF Mids1113960 11139611113872 1113873 P Mids gt Mids1113960 11139611113872 1113873 (6b)

Mius 1113944

J

j1Nius(j) (6c)

Mids 1113944

J

j1Nids(j) (6d)

where Pf denotes the failure probability [Mius] and [Mids]

denote the wear limits of Biu and Bid respectivelyF([Mius]) and F([Mids]) denote the cumulative distribu-tion functions with assigned values [Mius] and [Mids]respectively namely F(Mius [Mius]) and F(Mids

[Mids]) Nius(j) and Nids(j) denote the jth values of Niusand Nids in the service lifetime respectively which can beobtained through sampling simulation as introduced in theSection 32 and J denotes the total number of passed trainsin the service lifetime

F(Mius) and F(Mius) are unknown which should befirst determined by the following two steps

(1) By referring to the sampling simulation method inthe Section 32 the values of Nius or Nids in theservice life are simulated through Monte Carlosampling en the values of Nius or Nids aresummed using Equation (6c) or (6d) respectivelyto obtain one value of Mius or Mids is step isrepeated W times to obtain W values of Mius orMids

(2) e cumulative probability characteristics for the W

values of Mius or Mids are fitted using cumulativedistribution function e true cumulative distri-bution function of Mius and Mids is unknown sothree possible cumulative distribution functions(namely ND WD and GEVD) are considered andthe cumulative distribution function with the bestfitting effect is selected as the true cumulative

Shock and Vibration 9

0 1000 2000 3000 40000

4

8

12

p

Cum

ulat

ive t

rave

l (m

m times

103 )

Monitored result Simulated result

(a)

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

5

10

15

p

Monitored result Simulated result

(b)

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

5

10

15

p

Monitored result Simulated result

(c)

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

5

10

15

20

25

p

Monitored result Simulated result

(d)

Figure 12 Continued

0 700 1400 2100 2800 35000

5

10

15

20

25

30

35

k

Cum

ulat

ive v

alue

(mm

) mean (N1us) = 2820 mm

(a)

5 10 15 20 25 300

02

04

06

08

1

Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored result Simulated result

(b)

Figure 11 e simulated N1us and its CP (a) e simulated result (b) A comparison between the simulated and monitored results

10 Shock and Vibration

distribution function of Mius and Mids after least-square fitting

After F(Mius) and F(Mius) are determined [Mius] and[Mids] are substituted in Equations (6a) and (6b) to obtainthe value of Pf Furthermore the reliability level β can becalculated using Pf as follows

Pf 1113946+infin

β

12π

radic eminusx22

dx (7)

If the calculated reliability level β is larger than the targetreliability level [β] (ie βgt [β]) it can be inferred that thebearing is still in a good service state otherwise (ie βle [β])it can be inferred that the bearing is in a poor service stateand should be replaced in time According to the Eurocode[21] the target reliability level [β] for newly built bridgeswith a 100-year design lifetime is 38 which corresponds toa failure probability Pf 72 times 10minus5 However theDashengguan Railway Bridge has been in operation for 8years and it is not a newly built bridge From the viewpointof bridge reliability a reduction in the uncertainties of theload and resistance parameters decreases the failure prob-ability of existing bridge structures indicating the possibilityof adopting a lower reliability level for the evaluation ofexisting bridges rather than the reliability level that shouldbe used for newly built bridges [22] According to Kotesrsquoresearch [22] the target reliability level [β] with a 90-yearremaining lifetime is 3773 so the target reliability level [β]

of the Dashengguan Railway Bridge with a 92-yearremaining lifetime is 3778 using the linear interpolationcalculation namely [β] 3778

42 Evaluation Results Figure 8 reveals that M3d has thefastest growth so M3d is taken as an example to calculate itsfailure probability and evaluate the wear condition of B3d inthe service lifetime Before evaluation three parameters

should be determined namely J [M3ds] and W For theparameter J the designed service life of the NanjingDashengguan Yangtze River Bridge is 100 years and thenumber of passed trains in 2017 was 84343 so the value of J

is 8434300 for the parameter [M3ds] the China codespecifies that the linear wear rate of PTFE in a spherical steelbearing should be less than 15 μmkm and that the thicknessof PTFE should be more than 7mm and less than 8mm(ie only a 1mm wear thickness is allowed for an 8mmthickness of PTFE) [23] so the value of [M3ds] is 667 km forthe spherical steel bearing for the parameter W a larger W

can provide better evaluation results and the value of W is1000 for evaluation in this research

After the three parameters are determined the bearingwear life is evaluated by the following seven steps

(1) e values of N3d in 100-year service life are sim-ulated through Monte Carlo simulation usingEquation (4b) and 8434300 values of N3ds areobtained

(2) All the 8434300 values of N3ds are summed to obtainthe M3ds of B3d in 100-year service life usingEquation (6d)

(3) Steps (1) and (2) are repeated 1000 times to obtain1000 values of M3ds as shown in Figure 13(a) e1000 values of M3ds show good uniform and randomcharacteristics and these values can be used to an-alyze the cumulative probability characteristics

(4) e cumulative probability of M3ds is calculated andfitted by the ND WD and GEVD with the fittedresults shown in Figure 13(b) Both the ND andGEVD can well describe the cumulative probabilitycharacteristics of M3ds and GEVD is chosen for thetrue cumulative distribution function of M3ds

(5) e failure probability Pf of CBTover the upper limit[M3ds] for bearing B3d is about 10minus8 calculated by

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

5

10

15

20

25

p

Monitored result Simulated result

(e)

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

4

8

12

p

Monitored result Simulated result

(f )

Figure 12e simulated Mius and Mids (a) M1us and M1u (b) M1ds and M1d (c) M2us and M2u (d) M2ds and M2d (e) M3us and M3u(f ) M3ds and M3d

Shock and Vibration 11

Equation (6b) which means that M3ds has littleprobability of exceeding [M3ds] because the simu-lated 1000 values of M3ds are within a small interval[5874 km 5931 km]

(6) e reliability level β is 5612 calculated by Equation(7) which is larger than the target reliability level3778 indicating that bearing B3d does not reach thewear limit in the service life

(7) Because the linear growth rate of CBT for B3d isfaster than those of the other spherical steel bearingsit can be inferred that the other spherical steelbearings do not reach the wear limit as well in theservice life

5 Conclusions

Based on the monitored dynamic displacement data of steeltruss arch bridge bearings the CDD under the action ofa single train and the CBT under the continual actions ofmany passed trains are investigated Furthermore theprobability statistics and the Monte Carlo sampling simu-lationmethod for CDD are studied and the safety evaluationmethod for the bearing wear life is proposed using a re-liability index regarding the failure probability of monitoredCBT over the wear limit in the service lifetime e mainconclusions are drawn as follows

(1) rough monitored data analysis the monitoredlongitudinal displacement data mainly consist oftemperature-induced static displacements and train-induced dynamic displacements e wavelet packetdecomposition method can effectively extract thesmall-amplitude dynamic displacements caused bytrain loads from the monitored data All the CDDscontain uniform and random characteristics and allthe CBTs have an obvious linear correlation with the

number of passed trains but their linear growth ratesare different

(2) rough probability statistics analysis GEVD canwell describe the cumulative probability character-istics of CDDeMonte Carlo sampling simulationwas performed using GEVD to simulate CDD andCBT and the results show that the simulated cu-mulative probabilities of CDD have agreement withthemonitored ones of CDD and the simulated lineargrowth rates of CBT have agreement with themonitored ones of CBT

(3) rough safety evaluation analysis the safety eval-uation method for the bearing wear life is proposedusing a reliability index and it is finally judgedwhether the CBTs can exceed the wear limit in theservice lifetime e evaluation results show that allthe spherical steel bearings do not exceed the wearlimit in the service life

Notation

Biu ith spherical steel bearing at theupstream

Bid ith spherical steel bearing at thedownstream

CBT cumulative bearing travelCDD cumulative dynamic displacementCP cumulative probabilityDiu monitored longitudinal

displacement data at the ithspherical steel bearing Biu

Did monitored longitudinaldisplacement data at the ithspherical steel bearing Bid

1113957D(k) dynamic displacement caused by thekth passed train

0 200 400 600 800 1000585

5875

59

5925

595

The number of simulations

M3

ds (k

m)

(a)

M3ds (km)587 588 589 59 591 592 593 5940

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y

Cumulative probabilityGEVD

NDWD

(b)

Figure 13 e 1000 values of M3ds and their cumulative probabilities (a)1000M3ds (b) Cumulative probability of M3ds and its fittedresults

12 Shock and Vibration

1113957Dj(k) jth value in 1113957D(k)

F([Mius]) cumulative distribution functionwith assigned value [Mius]

F([Mids]) cumulative distribution functionwith assigned value [Mids]

GEVD general extreme value distributionG(Niu) cumulative distribution function of

NiuG(Nid) cumulative distribution function of

NidG(Nius) cumulative distribution function of

NiusG(Nids) cumulative distribution function of

NidsLVDT linear variable differential

transformermean(Niu) average value of 3495 CDDs for each

Niumean(Nid) average value of 3495 CDDs for each

NidM(p) CBT caused by p passed trainsMiu CBT of BiuMid CBT of Bid[Mius] wear limit of Biu[Mids] wear limit of BidN(k) cumulative travel of bearing caused

by the kth passed trainNiu set of N(1) N(2) N(3495)

for the bearings Biu

Nid set of N(1) N(2) N(3495)

for the bearings BidNius simulated CDDs of NiuNids simulated CDDs of NidNUM(Niu) total number of NiuNUM(Nid) total number of NidNUM(Niu ltNiu(j)) number of CDDs in Niu that are

smaller than Niu(j)

NUM(Nid ltNid(j)) number of CDDs in Nid that aresmaller than Nid(j)

ND normal distributionPTFE poly-tetra-fluoro-ethyleneP(Niu(j)) CP of the jth value Niu(j) in NiuP(Nid(j)) CP of the jth value Nid(j) in NidPf failure probabilityR value of cumulative probabilityWD Weibull distributionxjm wavelet packet coefficient in themth

frequency band of the jth scale[β] target reliability level

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors thank the Natural Science Foundation ofJiangsu Province of China (BK20180652) and the ChinaPostdoctoral Science Foundation (2017M621865)

References

[1] J W Zhan H Xia N Zhang and Y Lu ldquoA diagnosis methodfor bridge rubber support disease based on impact responsesrdquoJournal of Vibration and Shock vol 32 no 8 pp 153ndash1572013 in Chinese

[2] B Wei T H Yang L Z Jiang and X H He ldquoEffects offriction-based fixed bearings on the seismic vulnerability ofa high-speed railway continuous bridgerdquo Advances inStructural Engineering vol 21 no 5 pp 643ndash657 2018

[3] J Oh C Jang and J H Kim ldquoA study on the characteristics ofbridge bearings behavior by finite element analysis and modeltestrdquo Journal of Vibroengineering vol 17 no 5 pp 2559ndash2571 2015

[4] J J Huang Q Su L C Zhang Y J Li and B Liu ldquoRapidtreatment technique of diseases of the rocking axle bearing ofrailway simply supported beam bridgerdquo Applied Mechanicsand Materials vol 351-352 pp 1440ndash1444 2013

[5] A S Deshpande and J M C Kishen ldquoFatigue crack prop-agation in rocker and roller-rocker bearings of railway steelbridgesrdquo Engineering Fracture Mechanics vol 77 no 9pp 1454ndash1466 2010

[6] Q J Shi ldquoDefect analysis and reconstruction method forexisting railway bridge bearingrdquo Railway Engineering vol 57no 10 pp 12ndash14 2017

[7] S Y Yan ldquoAn analysis of causes for defects of roller bearingand treatment measuresrdquo Railway Standard Design vol 2002no 6 pp 30ndash32 2002 in Chinese

[8] H Wang Study on Damage Identification Method for RailwayBridge Rubber Bearings Based on Dynamic Responses BeijingJiaotong University Beijing China 2014 in Chinese

[9] Z Qiao Study on the Bridge Bearing Disease IdentificationTechnique Based on the Natural Frequency Change Rate EastChina Jiaotong University Nanchang China 2014 inChinese

[10] S L Chen Y Q Liu and Y B Zhang ldquoEffects of bearingdamage on vibration characteristics of a large span steel trussbridgerdquo Journal of Vibration and Shock vol 36 no 13pp 195ndash200 2017 in Chinese

[11] F Q Hu J Chen Y B Xu and X M Feng ldquoAnalysis ofmechanical properties of continuous T-beam bridge withdiseased bearingsrdquo Journal of Nanchang University Engi-neering and Technology vol 37 no 2 pp 123ndash127 2015 inChinese

[12] H Zhou K Liu G Shi Y Q Wang Y J Shi andG De Roeck ldquoFatigue assessment of a composite railwaybridge for high speed trains Part I modeling and fatiguecritical detailsrdquo Journal of Constructional Steel Researchvol 82 pp 234ndash245 2013

[13] T Guo J Liu and L Y Huang ldquoInvestigation and control ofexcessive cumulative girder movements of long-span steelsuspension bridgesrdquo Engineering Structures vol 125pp 217ndash226 2016

[14] 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 04014099 2015

Shock and Vibration 13

[15] Y L Ding and G X Wang ldquoEvaluation of dynamic loadfactors for a high-speed railway truss arch bridgerdquo Shock andVibration vol 2016 Article ID 5310769 15 pages 2016

[16] G X Wang Y L Ding Y S Song L Y Wu Q Yue andG H Mao ldquoDetection and location of the degraded bearingsbased on monitoring the longitudinal expansion performanceof the main girder of the Dashengguan Yangtze BridgerdquoJournal of Performance of Constructed Facilities vol 30 no 4article 04015074 2016

[17] G X Wang and Y L Ding ldquoResearch on monitoring tem-perature difference from cross sections of steel truss archgirder of Dashengguan Yangtze Bridgerdquo International Journalof Steel Structures vol 15 no 3 pp 647ndash660 2015

[18] M Y Gokhale and K D Kaur ldquoTime domain signal analysisusing wavelet packet decomposition approachrdquo InternationalJournal of communications Networks and System Sciencesvol 3 no 3 pp 321ndash329 2010

[19] T Figlus S Liscak A Wilk and B Aazarz ldquoConditionmonitoring of engine timing system by using wavelet packetdecomposition of a acoustic signalrdquo Journal of MechanicalScience and Technology vol 28 no 5 pp 1663ndash1671 2014

[20] R Wald T M Khoshgoftaar and J C Sloan ldquoFeature se-lection for optimization of wavelet packet decomposition inreliability analysis of systemsrdquo International Journal on Ar-tificial Intelligence Tools vol 22 no 5 article 1360011 2013

[21] EN 1991-1-5 Eurocode 1 Actions on StructuresmdashPart 1ndash5General Actionsmdashampermal Actions European Committee forStandardization (CEN) Brussels Belgium Europe 2004

[22] P Kotes and J Vican ldquoReliability levels for existing bridgesevaluation according to eurocodesrdquo Procedia Engineeringvol 40 pp 211ndash216 2012

[23] TBT 3320-2013 Spherical Bearings for Railway BridgesChina Railway Publishing House Beijing China 2013 inChinese

14 Shock and Vibration

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the bridge site e sampling frequency of the 12 displace-ment sensors is 1Hz Because the Nanjing DashengguanYangtze River Bridge is one symmetrical structure only thebearings on the left half (ie B1usimB3u and B1dsimB3d) areselected for analysis

22 Cumulative Dynamic Displacement Caused by a PassedTrain e monitored longitudinal displacement data at theith spherical steel bearing Biu are denoted by Diu themonitored longitudinal displacement data at the ithspherical steel bearing Bid are denoted by Did i 1 2 3Because the time histories of longitudinal displacementshave similar change trends [16] only D1u is selected foranalysis e time history of D1u from March to October in2017 is shown in Figure 3(a) showing typical seasonalcharacteristics over eight months Furthermore the timehistory of D1u on May 20th 2017 is plotted in Figure 3(b)showing typical sinusoidal characteristics ese variationcharacteristics are very similar to those of temperature field[17] indicating that the displacement is mainly induced bytemperature field In addition Figure 3(b) shows the timehistory of D1u from 1600 to 1800 on May 20th 2017 It isclear that the longitudinal displacements contain numeroussmall-amplitude dynamic displacements with random var-iability that are mainly induced by dynamic loads

Because the monitored longitudinal displacement datacontain static longitudinal displacements induced by thetemperature field and dynamic longitudinal displacementsinduced by dynamic loads as well the dynamic longitudinaldisplacements need to be extracted from the monitored datafirst Note that the static and dynamic longitudinal dis-placements are in different frequency bands e staticlongitudinal displacement is greatly influenced by temper-ature changes so the cycle period is about one day (ie thecorresponding frequency is 186400Hz) To ensure that thefrequency of static longitudinal displacements is completelywithin the band range the upper limit of the frequency bandis designed to be 586400Hz namely (0 586400] [13 14]e frequency of dynamic longitudinal displacements isfar higher than the frequency of static longitudinal dis-placements and relevant research has indicated that the

frequency bandwidth of dynamic longitudinal displace-ments is within the range of (586400 05] [13 14] ereason for choosing 05Hz as the upper limit is that thedynamic displacements with frequencies over 05Hz aremainly induced by electromagnetic interference [13 14]which are not in the range of concern

e wavelet packet decomposition method can use a pairof filters to decompose the monitored longitudinal displace-ment into different frequency bands scale by scale [18ndash20]Figure 4 shows the tree structure of the wavelet packet co-efficients within 13 scales after layer-by-layer decompositionwhere xjm denotes the wavelet packet coefficient in the mthfrequency band of the jth scale (m 0 1 2 2j minus 1) andfsdenotes the sampling frequency with fs 1Hz in this re-search e wavelet packet coefficient in the first frequencyband (0 1214] which is close to (0 586400] is selected andreconstructed to obtain static displacements the waveletpacket coefficients in the other frequency bands (1214 12]which are close to (586400 05] are selected and recon-structed to obtain dynamic displacements

Taking Figure 3(b) as an example the time history of D1uon May 20th 2017 is decomposed by the wavelet packetmethod and the decomposition result is shown in Figure 5 Itis clear that the wavelet packet decomposition method caneffectively decompose the monitored longitudinal displace-ment data into two components the first component causedby temperature field and the second component (namelysmall-amplitude dynamic displacements) caused by dynamicloads Furthermore the cumulative displacements of twocomponents are calculated to investigate the contribution oftwo components to cumulative displacements which are471mm and 7734mm respectively Obviously the secondcomponent caused by dynamic loads contributes the most tocumulative displacements In detail the dynamic loads mainlyinclude train and wind loads What should be mentioned isthat there is no train passing the Dashengguan Yangtze RiverBridge from 0 am to 6 am for bridge inspection so the firstcomponent during this period ismainly induced by wind loadin addition the second component during the rest of time ismainly induced by both wind and train loads as shown inFigure 5(b) It can be inferred from Figure 5(b) that train loadis the major influence factor for cumulative displacements

108 192 336

1272

1

336 192 108

B6 uB6 d

B5 uB5 d

B4 uB4 d

B3 uB3 d

B2 uB2 d

B1 uB1 d B7 d

Bearing 7Bearing 6Bearing 5Bearing 4Bearing 3

1

1 2 3 4 5 6 7

Bearing 2Bearing 1

B7 u

Figure 2 Locations of the longitudinal displacement sensors m

Shock and Vibration 3

because the influence of train load is obviously higher thanwind load Hence the small-amplitude dynamic displacementcaused by train load is specifically studied in this research

e continuous accumulation of small-amplitude dy-namic displacements will induce serious fatigue wear whichcan severely decrease the service lifetime of bearingsus itis necessary to carry out study on the cumulative charac-teristics of small-amplitude dynamic displacements eCDD caused by a single passed train was calculated toevaluate the contribution of this train to the cumulativewear e CDD caused by the kth passed train which isdenoted byN(k) is calculated by referring to the equation inTable 1 where 1113957D(k) denotes the dynamic displacementcaused by the kth passed train 1113957Dj(k) and 1113957Dj+1(k) denote thejth and (j + 1)th values in 1113957D(k) respectively wherej 1 2 Uminus 1 U denotes the number of monitoredvalues in 1113957D(k) N(k) indicates the cumulative travel ofbearing caused by the kth passed train For example onetime history of dynamic displacement of B3u caused byone train is plotted in Figure 6 and the number of U in thetrain-passed segment of the plot is 48 which can be clearlycaptured so the value of N(k) is 203mm by calculating theCDD in the train-passed segment

e monitored data from August 10th to August 31st in2017 is selected for analysis A total of 3495 trains passedover the Dashengguan Yangtze River Bridge over this pe-riod and each train corresponds to one value of N(k)erefore 3495 values of N(k) for each bearing at upstreamand downstream sides (ie B1usimB3u and B1dsimB3d) arecalculated as shown in Figure 7 where Niu and Nid denotea set of N(1) N(2) N(3495) for the bearings Biu andBid respectively as shown in Table 1 It can be seen that allthe CDDs show uniform and random characteristics Fur-thermore the average value of 3495 CDDs for each Niu orNid which is denoted by mean(Niu) or mean(Nid) re-spectively is calculated by referring to the equations inTable 1 with the results shown in Figure 7 It can be seen thatmean(N3d) has the largest value about 7020mm whilemean(N1u) has the smallest value about 2904mm

23 Cumulative Bearing Travel Caused by Many PassedTrains Each value of Niu or Nid corresponds to one passedtrain and furthermore the 3495 values for each Biu or Bid areaccumulated to obtain CBT which indicates the influence ofthe continuous actions of passed trains on bearing wear ecalculation formula for CBT is expressed as follows

Scale

0

1

2

Signalx00

x10 x11

x20 x21 x22 x23

j xj0 xj1 xj2jndash2 xj2jndash1

13 x130 x131 x138191

Frequency band ]21420 fs(0 ]

2142fs

21420 fs(

]2

214

(213-1)fs( fs

Staticdisplacement

Dynamicdisplacement

Figure 4 Tree structure of the wavelet packet coefficients

3 4 5 6 7 8 9 10 11ndash200

ndash100

0

100

200

Month

Mon

itorin

g di

spla

cem

ent (

mm

)

(a)

0 4 8 12 16 20 2425

30

35

40

45

Hour

Disp

lace

men

t (m

m)

(b)

Figure 3 e monitored result of D1u (a) From March to October in 2017 (b) On May 20th 2017

4 Shock and Vibration

M(p) 1113944

p

k1N(k) (1)

where M(p) denotes the CBT caused by p passed trainsand p 1 2 3495 e M(p) values at the upstreamand downstream are shown in Figure 8 where Miu de-notes the CBT of Biu and Mid denotes the CBT ofBid(i 1 2 3) It can be seen that all the CBTs have ob-vious linear correlations with the number of passed trainsbut the linear growth rates are different for each Miu orMid where the linear growth rate of M3d is the fastestand its maximum CBT caused by 3495 passed trains is24360mm

3 Probability Statistics Analysis

e number of monitored CDDs are not enough to satisfythe requirement of large sample for safety evaluation ofbearing wear life erefore it is necessary to performsampling simulation of CDDs using probabilistic statistics ofthe monitored CDDs to obtain adequate data

31 Cumulative Probability Characteristics Figure 7 showsthat Niu and Nid contain obvious uniform and randomcharacteristics so Niu and Nid can be treated as statisticalvariables for probabilistic statistics analysis In detail thecumulative probability (CP) is used to describe the proba-bilistic statistics characteristics of CDDs First the cumu-lative probabilities of Niu and Nid are calculated based onprobability statistics theory and then the cumulative dis-tribution function is fitted in a least-squares manner usingthe cumulative probabilities of Niu and Nid which is il-lustrated in two steps as follows

(1) e cumulative probabilities of Niu and Nid arecalculated as follows

P Niu(j)11138731113872 NUM Niu ltNiu(j)11138731113872

NUM Niu1113872 1113873 (2a)

P Nid(j)11138731113872 NUM Nid ltNid(j)11138731113872

NUM Nid1113872 1113873 (2b)

where P(Niu(j)) denotes the CP of the jth valueNiu(j) in Niu P(Nid(j)) denotes the CP of the jth

0 4 8 12 16 20 2425

30

35

40

45

Hour

Disp

lace

men

t (m

m)

(a)

0 4 8 12 16 20 24ndash3

ndash1

1

3

Hour

Disp

lace

men

t (m

m)

Wind load Wind and train loads

The firstcomponent The second component

(b)

Figure 5 Decomposition result of monitored longitudinal displacement (a) Component caused by temperature field (b) Componentcaused by dynamical loads

Table 1 Calculation of variables N(k) Niu Nid mean(Niu) and mean(Nid)

Variable Calculation equation

N(k) N(k) 1113936Uminus1j1 | 1113957Dj+1(k)minus 1113957Dj(k)|

Niu NidNiu N(1) N(2) N(3495) for the bearings at upstream

Nid N(1) N(2) N(3495) for the bearings at downstream

mean(Niu) mean(Nid)mean(Niu) (N(1) + N(2) + middot middot middot + N(3495)3495) for the bearings at upstream

mean(Nid) (N(1) + N(2) + middot middot middot + N(3495)3495) for the bearings at downstream

0 40 80 120 160ndash3

ndash2

ndash1

0

1

2

3

Time (s)

Disp

lace

men

t (m

m)

No train No trainPassed train

Figure 6 e dynamic displacement of B3u caused by one train

Shock and Vibration 5

value Nid(j) in Nid NUM(Niu) denotes the totalnumber of Niu NUM(Nid) denotes the totalnumber of Nid NUM(Niu ltNiu(j)) denotes thenumber of CDDs in Niu that are smaller thanNiu(j) and NUM(Nid ltNid(j)) denotes thenumber of CDDs in Nid that are smaller thanNid(j) For instance the CP of N1u is shown inFigure 9

(2) e cumulative distribution function is fitted in a least-squares manner using the cumulative probabilities of

Niu and Nid Considering that the true cumulativedistribution function of Niu and Nid is unknownthree possible cumulative distribution functions(ie normal distribution (ND) Weibull distribution(WD) and general extreme value distribution(GEVD)) are chosen for least-square fitting and thenthe cumulative distribution function with the bestfitting effect is selected as the true cumulative distri-bution function of Niu and Nid Taking N1u as anexample the fitting results of the three cumulative

0 700 1400 2100 2800 35000

5

10

15

20

25

k

Cum

ulat

ive v

alue

(mm

)mean (N1u) = 2904mm

(a)

0 700 1400 2100 2800 35000

10

20

30

40

50

60

k

Cum

ulat

ive v

alue

(mm

)

mean (N1d) = 4067mm

(b)

0 700 1400 2100 2800 35000

5

10

15

20

25mean (N2u) = 3888mm

Cum

ulat

ive v

alue

(mm

)

k

(c)

0 700 1400 2100 2800 35000

5

10

15

Cum

ulat

ive v

alue

(mm

)

k

mean (N2d) = 2960mm

(d)

0 700 1400 2100 2800 35000

5

10

15

20

25

k

Cum

ulat

ive v

alue

(mm

)

mean (N3u) = 6559mm

(e)

0 700 1400 2100 2800 35000

10

20

30

40

k

Cum

ulat

ive v

alue

(mm

)

mean (N3d) = 7020mm

(f)

Figure 7 Niu and Nid (a) N1u (b) N1d (c) N2u (d) N2d (e) N3u (f ) N3d

6 Shock and Vibration

distribution functions are shown in Figure 10 It can beseen that GEVDhas the best fitting results so GEVD ischosen to describe the cumulative probability char-acteristics of N1u e formula is expressed as

G N1u1113872 1113873 exp minus 1 + rN1u minus b

a1113888 11138891113890 1113891

minus1r⎡⎣ ⎤⎦ (3)

where G(N1u) denotes the cumulative distributionfunction of N1u which follows the generalized ex-treme value distribution r denotes the shape pa-rameter a denotes the scale parameter and b

denotes the location parameter e fitted values ofthese three parameters for G(N1u) are r 03058a 08163 and b 20294

Based on the method above the best cumulative dis-tribution functions of N1u minusN3u and N1d minusN3d are allGEVD as shown in Figures 10(a)ndash10(f ) It can be seen thatGEVD can well describe the cumulative probability char-acteristics of N1u minusN3u and N1d minusN3d

32 Sampling SimulationMethod Since Niu and Nid followGEVDs their GEVDs can be utilized for Monte Carlosampling to obtain a demanded number of CDDs Specif-ically if the demanded number of CDDs is K K values are

0 1000 2000 3000 40000

5

10

15

M1uM1d

p

Cum

ulat

ive t

rave

l (m

m times

103 )

(a)

0 1000 2000 3000 40000

5

10

15

M2uM2d

p

Cum

ulat

ive t

rave

l (m

m times

103 )

(b)

0 1000 2000 3000 40000

5

10

15

20

25

M3uM3d

p

Cum

ulat

ive t

rave

l (m

m times

103 )

(c)

Figure 8 e M1u minusM3u and M1u minusM3u caused by many passed trains (a) M1u and M1d (b) M2u and M2d (c) M3u and M3d

5 10 15 20 25 300

02

04

06

08

1

Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Figure 9 e cumulative probability of N1u

Shock and Vibration 7

5 10 15 20 25 300

02

04

06

08

1

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

WD

ND

GEVD

GEVDND

WDMonitored curve

(a)

10 20 30 40 500

02

04

06

08

1

r = 04765a = 10599b = 25108

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored curve Fitted curve

(b)

5 10 15 20 25 300

02

04

06

08

1

r = 01926a = 12320b = 28604

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored curveFitted curve

(c)

2 4 6 8 10 12 140

02

04

06

08

1

r = 00493a = 08440b = 24302

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored curveFitted curve

(d)

5 10 15 20 250

02

04

06

08

1

r = 00830a = 18184b = 53395

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored curveFitted curve

(e)

5 10 15 20 250

02

04

06

08

1

r = 01834a = 19436b = 54517

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored curveFitted curve

(f )

Figure 10 GEVDs of N1u minusN6u and N1u minusN6u (a) N1u (b) N1d (c) N2u (d) N2d (e) N3u (f ) N3d

8 Shock and Vibration

uniformly and randomly sampled from the interval (0 1)and then the demanded number of CDDs can be simulatedby substituting the K values into the cumulative probabilityR in the following equations

R G Nius1113872 1113873 (4a)

R G Nids1113872 1113873 (4b)

G Nius1113872 1113873 exp minus 1 + rNius minus b

a1113888 11138891113890 1113891

minus1r⎡⎣ ⎤⎦ (4c)

G Nids1113872 1113873 exp minus 1 + rNids minus b

a1113888 11138891113890 1113891

minus1r⎡⎣ ⎤⎦ (4d)

where Nius and Nids denote the simulated CDDs of Niu andNid respectively e values of r a and b have been cal-culated in the Section 32 However Niu and Nid areembedded in G(Niu) and G(Nids) so they cannot be di-rectly solved from Equations (4a) and (4b) Considering thatG(Nius) and G(Nids) are monotonically increasing func-tions Nius and Nids can be determined by the Newtoniteration method as follows

Nn+1ius N

nius minus

G Nnius1113872 1113873minusR

Gprime Nnius1113872 1113873

(5a)

Nn+1ids N

nids minus

G Nnids1113872 1113873minusR

Gprime Nnids1113872 1113873

(5b)

where Nn+1ius and Nn

ius denote the (n + 1)th and nth iterationresults of Nius respectively Nn+1

ids andNnids denote the (n + 1)

th and nth iteration results of Nids respectively Gprime(Nnius)

and Gprime(Nnids) denote the first-order derivatives of GEVD

(ie the probability density functions of GEVD) e iter-ation will terminate when the absolute difference betweentwo adjacent iterations is less than 10minus2 e last iterationresult is treated as the values of Nius and Nids

33 ampe Sampling Simulation Result Based on the samplingsimulation method above the simulation results of 3495N1us are shown in Figure 11(a) e mean value of thesimulated result is 2820mm which is very close to the meanvalue of Niu ie 2904mm Furthermore the CP of thesimulated result is calculated and then compared with themonitored CP as shown in Figure 11(b) It can be seen thatthe simulated CP coincides with the monitored CP whichverifies the accuracy of the sampling simulation method

Furthermore the simulated 3495 values of Nius andNidsare accumulated using Equation (1) to obtain the simulatedCBT (ie Mius and Mids) as shown in Figure 12 It can beseen that the simulated Mius and Mids present good lineargrowth characteristics with the number of passed trains Toverify the accuracy of the simulated linear growth ratesa comparison between the monitored and simulated lineargrowth rates was conducted as shown in Figure 12 It can beseen that the simulated linear growth rates have agreement

with the monitored ones which further verifies the accuracyof the sampling simulation method

4 Safety Evaluation Analysis

e CBT relates to the bearing wear condition If the CBTsexceed the wear limit in the service lifetime the sphericalsteel bearing is totally damaged and cannot be used forlongitudinal thermal expansion any longer erefore it isnecessary to analyze whether the CBTs at bridge site canexceed the wear limit in the service lifetime

41 EvaluationMethod e failure probabilities of sphericalsteel bearings in the service lifetime (ie the probabilities ofCBTover the wear limit in the service lifetime) are calculatedas follows

Pf 1minusF Mius1113960 11139611113872 1113873 P Mius gt Mius1113960 11139611113872 1113873 (6a)

or

Pf 1minusF Mids1113960 11139611113872 1113873 P Mids gt Mids1113960 11139611113872 1113873 (6b)

Mius 1113944

J

j1Nius(j) (6c)

Mids 1113944

J

j1Nids(j) (6d)

where Pf denotes the failure probability [Mius] and [Mids]

denote the wear limits of Biu and Bid respectivelyF([Mius]) and F([Mids]) denote the cumulative distribu-tion functions with assigned values [Mius] and [Mids]respectively namely F(Mius [Mius]) and F(Mids

[Mids]) Nius(j) and Nids(j) denote the jth values of Niusand Nids in the service lifetime respectively which can beobtained through sampling simulation as introduced in theSection 32 and J denotes the total number of passed trainsin the service lifetime

F(Mius) and F(Mius) are unknown which should befirst determined by the following two steps

(1) By referring to the sampling simulation method inthe Section 32 the values of Nius or Nids in theservice life are simulated through Monte Carlosampling en the values of Nius or Nids aresummed using Equation (6c) or (6d) respectivelyto obtain one value of Mius or Mids is step isrepeated W times to obtain W values of Mius orMids

(2) e cumulative probability characteristics for the W

values of Mius or Mids are fitted using cumulativedistribution function e true cumulative distri-bution function of Mius and Mids is unknown sothree possible cumulative distribution functions(namely ND WD and GEVD) are considered andthe cumulative distribution function with the bestfitting effect is selected as the true cumulative

Shock and Vibration 9

0 1000 2000 3000 40000

4

8

12

p

Cum

ulat

ive t

rave

l (m

m times

103 )

Monitored result Simulated result

(a)

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

5

10

15

p

Monitored result Simulated result

(b)

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

5

10

15

p

Monitored result Simulated result

(c)

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

5

10

15

20

25

p

Monitored result Simulated result

(d)

Figure 12 Continued

0 700 1400 2100 2800 35000

5

10

15

20

25

30

35

k

Cum

ulat

ive v

alue

(mm

) mean (N1us) = 2820 mm

(a)

5 10 15 20 25 300

02

04

06

08

1

Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored result Simulated result

(b)

Figure 11 e simulated N1us and its CP (a) e simulated result (b) A comparison between the simulated and monitored results

10 Shock and Vibration

distribution function of Mius and Mids after least-square fitting

After F(Mius) and F(Mius) are determined [Mius] and[Mids] are substituted in Equations (6a) and (6b) to obtainthe value of Pf Furthermore the reliability level β can becalculated using Pf as follows

Pf 1113946+infin

β

12π

radic eminusx22

dx (7)

If the calculated reliability level β is larger than the targetreliability level [β] (ie βgt [β]) it can be inferred that thebearing is still in a good service state otherwise (ie βle [β])it can be inferred that the bearing is in a poor service stateand should be replaced in time According to the Eurocode[21] the target reliability level [β] for newly built bridgeswith a 100-year design lifetime is 38 which corresponds toa failure probability Pf 72 times 10minus5 However theDashengguan Railway Bridge has been in operation for 8years and it is not a newly built bridge From the viewpointof bridge reliability a reduction in the uncertainties of theload and resistance parameters decreases the failure prob-ability of existing bridge structures indicating the possibilityof adopting a lower reliability level for the evaluation ofexisting bridges rather than the reliability level that shouldbe used for newly built bridges [22] According to Kotesrsquoresearch [22] the target reliability level [β] with a 90-yearremaining lifetime is 3773 so the target reliability level [β]

of the Dashengguan Railway Bridge with a 92-yearremaining lifetime is 3778 using the linear interpolationcalculation namely [β] 3778

42 Evaluation Results Figure 8 reveals that M3d has thefastest growth so M3d is taken as an example to calculate itsfailure probability and evaluate the wear condition of B3d inthe service lifetime Before evaluation three parameters

should be determined namely J [M3ds] and W For theparameter J the designed service life of the NanjingDashengguan Yangtze River Bridge is 100 years and thenumber of passed trains in 2017 was 84343 so the value of J

is 8434300 for the parameter [M3ds] the China codespecifies that the linear wear rate of PTFE in a spherical steelbearing should be less than 15 μmkm and that the thicknessof PTFE should be more than 7mm and less than 8mm(ie only a 1mm wear thickness is allowed for an 8mmthickness of PTFE) [23] so the value of [M3ds] is 667 km forthe spherical steel bearing for the parameter W a larger W

can provide better evaluation results and the value of W is1000 for evaluation in this research

After the three parameters are determined the bearingwear life is evaluated by the following seven steps

(1) e values of N3d in 100-year service life are sim-ulated through Monte Carlo simulation usingEquation (4b) and 8434300 values of N3ds areobtained

(2) All the 8434300 values of N3ds are summed to obtainthe M3ds of B3d in 100-year service life usingEquation (6d)

(3) Steps (1) and (2) are repeated 1000 times to obtain1000 values of M3ds as shown in Figure 13(a) e1000 values of M3ds show good uniform and randomcharacteristics and these values can be used to an-alyze the cumulative probability characteristics

(4) e cumulative probability of M3ds is calculated andfitted by the ND WD and GEVD with the fittedresults shown in Figure 13(b) Both the ND andGEVD can well describe the cumulative probabilitycharacteristics of M3ds and GEVD is chosen for thetrue cumulative distribution function of M3ds

(5) e failure probability Pf of CBTover the upper limit[M3ds] for bearing B3d is about 10minus8 calculated by

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

5

10

15

20

25

p

Monitored result Simulated result

(e)

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

4

8

12

p

Monitored result Simulated result

(f )

Figure 12e simulated Mius and Mids (a) M1us and M1u (b) M1ds and M1d (c) M2us and M2u (d) M2ds and M2d (e) M3us and M3u(f ) M3ds and M3d

Shock and Vibration 11

Equation (6b) which means that M3ds has littleprobability of exceeding [M3ds] because the simu-lated 1000 values of M3ds are within a small interval[5874 km 5931 km]

(6) e reliability level β is 5612 calculated by Equation(7) which is larger than the target reliability level3778 indicating that bearing B3d does not reach thewear limit in the service life

(7) Because the linear growth rate of CBT for B3d isfaster than those of the other spherical steel bearingsit can be inferred that the other spherical steelbearings do not reach the wear limit as well in theservice life

5 Conclusions

Based on the monitored dynamic displacement data of steeltruss arch bridge bearings the CDD under the action ofa single train and the CBT under the continual actions ofmany passed trains are investigated Furthermore theprobability statistics and the Monte Carlo sampling simu-lationmethod for CDD are studied and the safety evaluationmethod for the bearing wear life is proposed using a re-liability index regarding the failure probability of monitoredCBT over the wear limit in the service lifetime e mainconclusions are drawn as follows

(1) rough monitored data analysis the monitoredlongitudinal displacement data mainly consist oftemperature-induced static displacements and train-induced dynamic displacements e wavelet packetdecomposition method can effectively extract thesmall-amplitude dynamic displacements caused bytrain loads from the monitored data All the CDDscontain uniform and random characteristics and allthe CBTs have an obvious linear correlation with the

number of passed trains but their linear growth ratesare different

(2) rough probability statistics analysis GEVD canwell describe the cumulative probability character-istics of CDDeMonte Carlo sampling simulationwas performed using GEVD to simulate CDD andCBT and the results show that the simulated cu-mulative probabilities of CDD have agreement withthemonitored ones of CDD and the simulated lineargrowth rates of CBT have agreement with themonitored ones of CBT

(3) rough safety evaluation analysis the safety eval-uation method for the bearing wear life is proposedusing a reliability index and it is finally judgedwhether the CBTs can exceed the wear limit in theservice lifetime e evaluation results show that allthe spherical steel bearings do not exceed the wearlimit in the service life

Notation

Biu ith spherical steel bearing at theupstream

Bid ith spherical steel bearing at thedownstream

CBT cumulative bearing travelCDD cumulative dynamic displacementCP cumulative probabilityDiu monitored longitudinal

displacement data at the ithspherical steel bearing Biu

Did monitored longitudinaldisplacement data at the ithspherical steel bearing Bid

1113957D(k) dynamic displacement caused by thekth passed train

0 200 400 600 800 1000585

5875

59

5925

595

The number of simulations

M3

ds (k

m)

(a)

M3ds (km)587 588 589 59 591 592 593 5940

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y

Cumulative probabilityGEVD

NDWD

(b)

Figure 13 e 1000 values of M3ds and their cumulative probabilities (a)1000M3ds (b) Cumulative probability of M3ds and its fittedresults

12 Shock and Vibration

1113957Dj(k) jth value in 1113957D(k)

F([Mius]) cumulative distribution functionwith assigned value [Mius]

F([Mids]) cumulative distribution functionwith assigned value [Mids]

GEVD general extreme value distributionG(Niu) cumulative distribution function of

NiuG(Nid) cumulative distribution function of

NidG(Nius) cumulative distribution function of

NiusG(Nids) cumulative distribution function of

NidsLVDT linear variable differential

transformermean(Niu) average value of 3495 CDDs for each

Niumean(Nid) average value of 3495 CDDs for each

NidM(p) CBT caused by p passed trainsMiu CBT of BiuMid CBT of Bid[Mius] wear limit of Biu[Mids] wear limit of BidN(k) cumulative travel of bearing caused

by the kth passed trainNiu set of N(1) N(2) N(3495)

for the bearings Biu

Nid set of N(1) N(2) N(3495)

for the bearings BidNius simulated CDDs of NiuNids simulated CDDs of NidNUM(Niu) total number of NiuNUM(Nid) total number of NidNUM(Niu ltNiu(j)) number of CDDs in Niu that are

smaller than Niu(j)

NUM(Nid ltNid(j)) number of CDDs in Nid that aresmaller than Nid(j)

ND normal distributionPTFE poly-tetra-fluoro-ethyleneP(Niu(j)) CP of the jth value Niu(j) in NiuP(Nid(j)) CP of the jth value Nid(j) in NidPf failure probabilityR value of cumulative probabilityWD Weibull distributionxjm wavelet packet coefficient in themth

frequency band of the jth scale[β] target reliability level

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors thank the Natural Science Foundation ofJiangsu Province of China (BK20180652) and the ChinaPostdoctoral Science Foundation (2017M621865)

References

[1] J W Zhan H Xia N Zhang and Y Lu ldquoA diagnosis methodfor bridge rubber support disease based on impact responsesrdquoJournal of Vibration and Shock vol 32 no 8 pp 153ndash1572013 in Chinese

[2] B Wei T H Yang L Z Jiang and X H He ldquoEffects offriction-based fixed bearings on the seismic vulnerability ofa high-speed railway continuous bridgerdquo Advances inStructural Engineering vol 21 no 5 pp 643ndash657 2018

[3] J Oh C Jang and J H Kim ldquoA study on the characteristics ofbridge bearings behavior by finite element analysis and modeltestrdquo Journal of Vibroengineering vol 17 no 5 pp 2559ndash2571 2015

[4] J J Huang Q Su L C Zhang Y J Li and B Liu ldquoRapidtreatment technique of diseases of the rocking axle bearing ofrailway simply supported beam bridgerdquo Applied Mechanicsand Materials vol 351-352 pp 1440ndash1444 2013

[5] A S Deshpande and J M C Kishen ldquoFatigue crack prop-agation in rocker and roller-rocker bearings of railway steelbridgesrdquo Engineering Fracture Mechanics vol 77 no 9pp 1454ndash1466 2010

[6] Q J Shi ldquoDefect analysis and reconstruction method forexisting railway bridge bearingrdquo Railway Engineering vol 57no 10 pp 12ndash14 2017

[7] S Y Yan ldquoAn analysis of causes for defects of roller bearingand treatment measuresrdquo Railway Standard Design vol 2002no 6 pp 30ndash32 2002 in Chinese

[8] H Wang Study on Damage Identification Method for RailwayBridge Rubber Bearings Based on Dynamic Responses BeijingJiaotong University Beijing China 2014 in Chinese

[9] Z Qiao Study on the Bridge Bearing Disease IdentificationTechnique Based on the Natural Frequency Change Rate EastChina Jiaotong University Nanchang China 2014 inChinese

[10] S L Chen Y Q Liu and Y B Zhang ldquoEffects of bearingdamage on vibration characteristics of a large span steel trussbridgerdquo Journal of Vibration and Shock vol 36 no 13pp 195ndash200 2017 in Chinese

[11] F Q Hu J Chen Y B Xu and X M Feng ldquoAnalysis ofmechanical properties of continuous T-beam bridge withdiseased bearingsrdquo Journal of Nanchang University Engi-neering and Technology vol 37 no 2 pp 123ndash127 2015 inChinese

[12] H Zhou K Liu G Shi Y Q Wang Y J Shi andG De Roeck ldquoFatigue assessment of a composite railwaybridge for high speed trains Part I modeling and fatiguecritical detailsrdquo Journal of Constructional Steel Researchvol 82 pp 234ndash245 2013

[13] T Guo J Liu and L Y Huang ldquoInvestigation and control ofexcessive cumulative girder movements of long-span steelsuspension bridgesrdquo Engineering Structures vol 125pp 217ndash226 2016

[14] 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 04014099 2015

Shock and Vibration 13

[15] Y L Ding and G X Wang ldquoEvaluation of dynamic loadfactors for a high-speed railway truss arch bridgerdquo Shock andVibration vol 2016 Article ID 5310769 15 pages 2016

[16] G X Wang Y L Ding Y S Song L Y Wu Q Yue andG H Mao ldquoDetection and location of the degraded bearingsbased on monitoring the longitudinal expansion performanceof the main girder of the Dashengguan Yangtze BridgerdquoJournal of Performance of Constructed Facilities vol 30 no 4article 04015074 2016

[17] G X Wang and Y L Ding ldquoResearch on monitoring tem-perature difference from cross sections of steel truss archgirder of Dashengguan Yangtze Bridgerdquo International Journalof Steel Structures vol 15 no 3 pp 647ndash660 2015

[18] M Y Gokhale and K D Kaur ldquoTime domain signal analysisusing wavelet packet decomposition approachrdquo InternationalJournal of communications Networks and System Sciencesvol 3 no 3 pp 321ndash329 2010

[19] T Figlus S Liscak A Wilk and B Aazarz ldquoConditionmonitoring of engine timing system by using wavelet packetdecomposition of a acoustic signalrdquo Journal of MechanicalScience and Technology vol 28 no 5 pp 1663ndash1671 2014

[20] R Wald T M Khoshgoftaar and J C Sloan ldquoFeature se-lection for optimization of wavelet packet decomposition inreliability analysis of systemsrdquo International Journal on Ar-tificial Intelligence Tools vol 22 no 5 article 1360011 2013

[21] EN 1991-1-5 Eurocode 1 Actions on StructuresmdashPart 1ndash5General Actionsmdashampermal Actions European Committee forStandardization (CEN) Brussels Belgium Europe 2004

[22] P Kotes and J Vican ldquoReliability levels for existing bridgesevaluation according to eurocodesrdquo Procedia Engineeringvol 40 pp 211ndash216 2012

[23] TBT 3320-2013 Spherical Bearings for Railway BridgesChina Railway Publishing House Beijing China 2013 inChinese

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because the influence of train load is obviously higher thanwind load Hence the small-amplitude dynamic displacementcaused by train load is specifically studied in this research

e continuous accumulation of small-amplitude dy-namic displacements will induce serious fatigue wear whichcan severely decrease the service lifetime of bearingsus itis necessary to carry out study on the cumulative charac-teristics of small-amplitude dynamic displacements eCDD caused by a single passed train was calculated toevaluate the contribution of this train to the cumulativewear e CDD caused by the kth passed train which isdenoted byN(k) is calculated by referring to the equation inTable 1 where 1113957D(k) denotes the dynamic displacementcaused by the kth passed train 1113957Dj(k) and 1113957Dj+1(k) denote thejth and (j + 1)th values in 1113957D(k) respectively wherej 1 2 Uminus 1 U denotes the number of monitoredvalues in 1113957D(k) N(k) indicates the cumulative travel ofbearing caused by the kth passed train For example onetime history of dynamic displacement of B3u caused byone train is plotted in Figure 6 and the number of U in thetrain-passed segment of the plot is 48 which can be clearlycaptured so the value of N(k) is 203mm by calculating theCDD in the train-passed segment

e monitored data from August 10th to August 31st in2017 is selected for analysis A total of 3495 trains passedover the Dashengguan Yangtze River Bridge over this pe-riod and each train corresponds to one value of N(k)erefore 3495 values of N(k) for each bearing at upstreamand downstream sides (ie B1usimB3u and B1dsimB3d) arecalculated as shown in Figure 7 where Niu and Nid denotea set of N(1) N(2) N(3495) for the bearings Biu andBid respectively as shown in Table 1 It can be seen that allthe CDDs show uniform and random characteristics Fur-thermore the average value of 3495 CDDs for each Niu orNid which is denoted by mean(Niu) or mean(Nid) re-spectively is calculated by referring to the equations inTable 1 with the results shown in Figure 7 It can be seen thatmean(N3d) has the largest value about 7020mm whilemean(N1u) has the smallest value about 2904mm

23 Cumulative Bearing Travel Caused by Many PassedTrains Each value of Niu or Nid corresponds to one passedtrain and furthermore the 3495 values for each Biu or Bid areaccumulated to obtain CBT which indicates the influence ofthe continuous actions of passed trains on bearing wear ecalculation formula for CBT is expressed as follows

Scale

0

1

2

Signalx00

x10 x11

x20 x21 x22 x23

j xj0 xj1 xj2jndash2 xj2jndash1

13 x130 x131 x138191

Frequency band ]21420 fs(0 ]

2142fs

21420 fs(

]2

214

(213-1)fs( fs

Staticdisplacement

Dynamicdisplacement

Figure 4 Tree structure of the wavelet packet coefficients

3 4 5 6 7 8 9 10 11ndash200

ndash100

0

100

200

Month

Mon

itorin

g di

spla

cem

ent (

mm

)

(a)

0 4 8 12 16 20 2425

30

35

40

45

Hour

Disp

lace

men

t (m

m)

(b)

Figure 3 e monitored result of D1u (a) From March to October in 2017 (b) On May 20th 2017

4 Shock and Vibration

M(p) 1113944

p

k1N(k) (1)

where M(p) denotes the CBT caused by p passed trainsand p 1 2 3495 e M(p) values at the upstreamand downstream are shown in Figure 8 where Miu de-notes the CBT of Biu and Mid denotes the CBT ofBid(i 1 2 3) It can be seen that all the CBTs have ob-vious linear correlations with the number of passed trainsbut the linear growth rates are different for each Miu orMid where the linear growth rate of M3d is the fastestand its maximum CBT caused by 3495 passed trains is24360mm

3 Probability Statistics Analysis

e number of monitored CDDs are not enough to satisfythe requirement of large sample for safety evaluation ofbearing wear life erefore it is necessary to performsampling simulation of CDDs using probabilistic statistics ofthe monitored CDDs to obtain adequate data

31 Cumulative Probability Characteristics Figure 7 showsthat Niu and Nid contain obvious uniform and randomcharacteristics so Niu and Nid can be treated as statisticalvariables for probabilistic statistics analysis In detail thecumulative probability (CP) is used to describe the proba-bilistic statistics characteristics of CDDs First the cumu-lative probabilities of Niu and Nid are calculated based onprobability statistics theory and then the cumulative dis-tribution function is fitted in a least-squares manner usingthe cumulative probabilities of Niu and Nid which is il-lustrated in two steps as follows

(1) e cumulative probabilities of Niu and Nid arecalculated as follows

P Niu(j)11138731113872 NUM Niu ltNiu(j)11138731113872

NUM Niu1113872 1113873 (2a)

P Nid(j)11138731113872 NUM Nid ltNid(j)11138731113872

NUM Nid1113872 1113873 (2b)

where P(Niu(j)) denotes the CP of the jth valueNiu(j) in Niu P(Nid(j)) denotes the CP of the jth

0 4 8 12 16 20 2425

30

35

40

45

Hour

Disp

lace

men

t (m

m)

(a)

0 4 8 12 16 20 24ndash3

ndash1

1

3

Hour

Disp

lace

men

t (m

m)

Wind load Wind and train loads

The firstcomponent The second component

(b)

Figure 5 Decomposition result of monitored longitudinal displacement (a) Component caused by temperature field (b) Componentcaused by dynamical loads

Table 1 Calculation of variables N(k) Niu Nid mean(Niu) and mean(Nid)

Variable Calculation equation

N(k) N(k) 1113936Uminus1j1 | 1113957Dj+1(k)minus 1113957Dj(k)|

Niu NidNiu N(1) N(2) N(3495) for the bearings at upstream

Nid N(1) N(2) N(3495) for the bearings at downstream

mean(Niu) mean(Nid)mean(Niu) (N(1) + N(2) + middot middot middot + N(3495)3495) for the bearings at upstream

mean(Nid) (N(1) + N(2) + middot middot middot + N(3495)3495) for the bearings at downstream

0 40 80 120 160ndash3

ndash2

ndash1

0

1

2

3

Time (s)

Disp

lace

men

t (m

m)

No train No trainPassed train

Figure 6 e dynamic displacement of B3u caused by one train

Shock and Vibration 5

value Nid(j) in Nid NUM(Niu) denotes the totalnumber of Niu NUM(Nid) denotes the totalnumber of Nid NUM(Niu ltNiu(j)) denotes thenumber of CDDs in Niu that are smaller thanNiu(j) and NUM(Nid ltNid(j)) denotes thenumber of CDDs in Nid that are smaller thanNid(j) For instance the CP of N1u is shown inFigure 9

(2) e cumulative distribution function is fitted in a least-squares manner using the cumulative probabilities of

Niu and Nid Considering that the true cumulativedistribution function of Niu and Nid is unknownthree possible cumulative distribution functions(ie normal distribution (ND) Weibull distribution(WD) and general extreme value distribution(GEVD)) are chosen for least-square fitting and thenthe cumulative distribution function with the bestfitting effect is selected as the true cumulative distri-bution function of Niu and Nid Taking N1u as anexample the fitting results of the three cumulative

0 700 1400 2100 2800 35000

5

10

15

20

25

k

Cum

ulat

ive v

alue

(mm

)mean (N1u) = 2904mm

(a)

0 700 1400 2100 2800 35000

10

20

30

40

50

60

k

Cum

ulat

ive v

alue

(mm

)

mean (N1d) = 4067mm

(b)

0 700 1400 2100 2800 35000

5

10

15

20

25mean (N2u) = 3888mm

Cum

ulat

ive v

alue

(mm

)

k

(c)

0 700 1400 2100 2800 35000

5

10

15

Cum

ulat

ive v

alue

(mm

)

k

mean (N2d) = 2960mm

(d)

0 700 1400 2100 2800 35000

5

10

15

20

25

k

Cum

ulat

ive v

alue

(mm

)

mean (N3u) = 6559mm

(e)

0 700 1400 2100 2800 35000

10

20

30

40

k

Cum

ulat

ive v

alue

(mm

)

mean (N3d) = 7020mm

(f)

Figure 7 Niu and Nid (a) N1u (b) N1d (c) N2u (d) N2d (e) N3u (f ) N3d

6 Shock and Vibration

distribution functions are shown in Figure 10 It can beseen that GEVDhas the best fitting results so GEVD ischosen to describe the cumulative probability char-acteristics of N1u e formula is expressed as

G N1u1113872 1113873 exp minus 1 + rN1u minus b

a1113888 11138891113890 1113891

minus1r⎡⎣ ⎤⎦ (3)

where G(N1u) denotes the cumulative distributionfunction of N1u which follows the generalized ex-treme value distribution r denotes the shape pa-rameter a denotes the scale parameter and b

denotes the location parameter e fitted values ofthese three parameters for G(N1u) are r 03058a 08163 and b 20294

Based on the method above the best cumulative dis-tribution functions of N1u minusN3u and N1d minusN3d are allGEVD as shown in Figures 10(a)ndash10(f ) It can be seen thatGEVD can well describe the cumulative probability char-acteristics of N1u minusN3u and N1d minusN3d

32 Sampling SimulationMethod Since Niu and Nid followGEVDs their GEVDs can be utilized for Monte Carlosampling to obtain a demanded number of CDDs Specif-ically if the demanded number of CDDs is K K values are

0 1000 2000 3000 40000

5

10

15

M1uM1d

p

Cum

ulat

ive t

rave

l (m

m times

103 )

(a)

0 1000 2000 3000 40000

5

10

15

M2uM2d

p

Cum

ulat

ive t

rave

l (m

m times

103 )

(b)

0 1000 2000 3000 40000

5

10

15

20

25

M3uM3d

p

Cum

ulat

ive t

rave

l (m

m times

103 )

(c)

Figure 8 e M1u minusM3u and M1u minusM3u caused by many passed trains (a) M1u and M1d (b) M2u and M2d (c) M3u and M3d

5 10 15 20 25 300

02

04

06

08

1

Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Figure 9 e cumulative probability of N1u

Shock and Vibration 7

5 10 15 20 25 300

02

04

06

08

1

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

WD

ND

GEVD

GEVDND

WDMonitored curve

(a)

10 20 30 40 500

02

04

06

08

1

r = 04765a = 10599b = 25108

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored curve Fitted curve

(b)

5 10 15 20 25 300

02

04

06

08

1

r = 01926a = 12320b = 28604

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored curveFitted curve

(c)

2 4 6 8 10 12 140

02

04

06

08

1

r = 00493a = 08440b = 24302

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored curveFitted curve

(d)

5 10 15 20 250

02

04

06

08

1

r = 00830a = 18184b = 53395

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored curveFitted curve

(e)

5 10 15 20 250

02

04

06

08

1

r = 01834a = 19436b = 54517

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored curveFitted curve

(f )

Figure 10 GEVDs of N1u minusN6u and N1u minusN6u (a) N1u (b) N1d (c) N2u (d) N2d (e) N3u (f ) N3d

8 Shock and Vibration

uniformly and randomly sampled from the interval (0 1)and then the demanded number of CDDs can be simulatedby substituting the K values into the cumulative probabilityR in the following equations

R G Nius1113872 1113873 (4a)

R G Nids1113872 1113873 (4b)

G Nius1113872 1113873 exp minus 1 + rNius minus b

a1113888 11138891113890 1113891

minus1r⎡⎣ ⎤⎦ (4c)

G Nids1113872 1113873 exp minus 1 + rNids minus b

a1113888 11138891113890 1113891

minus1r⎡⎣ ⎤⎦ (4d)

where Nius and Nids denote the simulated CDDs of Niu andNid respectively e values of r a and b have been cal-culated in the Section 32 However Niu and Nid areembedded in G(Niu) and G(Nids) so they cannot be di-rectly solved from Equations (4a) and (4b) Considering thatG(Nius) and G(Nids) are monotonically increasing func-tions Nius and Nids can be determined by the Newtoniteration method as follows

Nn+1ius N

nius minus

G Nnius1113872 1113873minusR

Gprime Nnius1113872 1113873

(5a)

Nn+1ids N

nids minus

G Nnids1113872 1113873minusR

Gprime Nnids1113872 1113873

(5b)

where Nn+1ius and Nn

ius denote the (n + 1)th and nth iterationresults of Nius respectively Nn+1

ids andNnids denote the (n + 1)

th and nth iteration results of Nids respectively Gprime(Nnius)

and Gprime(Nnids) denote the first-order derivatives of GEVD

(ie the probability density functions of GEVD) e iter-ation will terminate when the absolute difference betweentwo adjacent iterations is less than 10minus2 e last iterationresult is treated as the values of Nius and Nids

33 ampe Sampling Simulation Result Based on the samplingsimulation method above the simulation results of 3495N1us are shown in Figure 11(a) e mean value of thesimulated result is 2820mm which is very close to the meanvalue of Niu ie 2904mm Furthermore the CP of thesimulated result is calculated and then compared with themonitored CP as shown in Figure 11(b) It can be seen thatthe simulated CP coincides with the monitored CP whichverifies the accuracy of the sampling simulation method

Furthermore the simulated 3495 values of Nius andNidsare accumulated using Equation (1) to obtain the simulatedCBT (ie Mius and Mids) as shown in Figure 12 It can beseen that the simulated Mius and Mids present good lineargrowth characteristics with the number of passed trains Toverify the accuracy of the simulated linear growth ratesa comparison between the monitored and simulated lineargrowth rates was conducted as shown in Figure 12 It can beseen that the simulated linear growth rates have agreement

with the monitored ones which further verifies the accuracyof the sampling simulation method

4 Safety Evaluation Analysis

e CBT relates to the bearing wear condition If the CBTsexceed the wear limit in the service lifetime the sphericalsteel bearing is totally damaged and cannot be used forlongitudinal thermal expansion any longer erefore it isnecessary to analyze whether the CBTs at bridge site canexceed the wear limit in the service lifetime

41 EvaluationMethod e failure probabilities of sphericalsteel bearings in the service lifetime (ie the probabilities ofCBTover the wear limit in the service lifetime) are calculatedas follows

Pf 1minusF Mius1113960 11139611113872 1113873 P Mius gt Mius1113960 11139611113872 1113873 (6a)

or

Pf 1minusF Mids1113960 11139611113872 1113873 P Mids gt Mids1113960 11139611113872 1113873 (6b)

Mius 1113944

J

j1Nius(j) (6c)

Mids 1113944

J

j1Nids(j) (6d)

where Pf denotes the failure probability [Mius] and [Mids]

denote the wear limits of Biu and Bid respectivelyF([Mius]) and F([Mids]) denote the cumulative distribu-tion functions with assigned values [Mius] and [Mids]respectively namely F(Mius [Mius]) and F(Mids

[Mids]) Nius(j) and Nids(j) denote the jth values of Niusand Nids in the service lifetime respectively which can beobtained through sampling simulation as introduced in theSection 32 and J denotes the total number of passed trainsin the service lifetime

F(Mius) and F(Mius) are unknown which should befirst determined by the following two steps

(1) By referring to the sampling simulation method inthe Section 32 the values of Nius or Nids in theservice life are simulated through Monte Carlosampling en the values of Nius or Nids aresummed using Equation (6c) or (6d) respectivelyto obtain one value of Mius or Mids is step isrepeated W times to obtain W values of Mius orMids

(2) e cumulative probability characteristics for the W

values of Mius or Mids are fitted using cumulativedistribution function e true cumulative distri-bution function of Mius and Mids is unknown sothree possible cumulative distribution functions(namely ND WD and GEVD) are considered andthe cumulative distribution function with the bestfitting effect is selected as the true cumulative

Shock and Vibration 9

0 1000 2000 3000 40000

4

8

12

p

Cum

ulat

ive t

rave

l (m

m times

103 )

Monitored result Simulated result

(a)

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

5

10

15

p

Monitored result Simulated result

(b)

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

5

10

15

p

Monitored result Simulated result

(c)

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

5

10

15

20

25

p

Monitored result Simulated result

(d)

Figure 12 Continued

0 700 1400 2100 2800 35000

5

10

15

20

25

30

35

k

Cum

ulat

ive v

alue

(mm

) mean (N1us) = 2820 mm

(a)

5 10 15 20 25 300

02

04

06

08

1

Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored result Simulated result

(b)

Figure 11 e simulated N1us and its CP (a) e simulated result (b) A comparison between the simulated and monitored results

10 Shock and Vibration

distribution function of Mius and Mids after least-square fitting

After F(Mius) and F(Mius) are determined [Mius] and[Mids] are substituted in Equations (6a) and (6b) to obtainthe value of Pf Furthermore the reliability level β can becalculated using Pf as follows

Pf 1113946+infin

β

12π

radic eminusx22

dx (7)

If the calculated reliability level β is larger than the targetreliability level [β] (ie βgt [β]) it can be inferred that thebearing is still in a good service state otherwise (ie βle [β])it can be inferred that the bearing is in a poor service stateand should be replaced in time According to the Eurocode[21] the target reliability level [β] for newly built bridgeswith a 100-year design lifetime is 38 which corresponds toa failure probability Pf 72 times 10minus5 However theDashengguan Railway Bridge has been in operation for 8years and it is not a newly built bridge From the viewpointof bridge reliability a reduction in the uncertainties of theload and resistance parameters decreases the failure prob-ability of existing bridge structures indicating the possibilityof adopting a lower reliability level for the evaluation ofexisting bridges rather than the reliability level that shouldbe used for newly built bridges [22] According to Kotesrsquoresearch [22] the target reliability level [β] with a 90-yearremaining lifetime is 3773 so the target reliability level [β]

of the Dashengguan Railway Bridge with a 92-yearremaining lifetime is 3778 using the linear interpolationcalculation namely [β] 3778

42 Evaluation Results Figure 8 reveals that M3d has thefastest growth so M3d is taken as an example to calculate itsfailure probability and evaluate the wear condition of B3d inthe service lifetime Before evaluation three parameters

should be determined namely J [M3ds] and W For theparameter J the designed service life of the NanjingDashengguan Yangtze River Bridge is 100 years and thenumber of passed trains in 2017 was 84343 so the value of J

is 8434300 for the parameter [M3ds] the China codespecifies that the linear wear rate of PTFE in a spherical steelbearing should be less than 15 μmkm and that the thicknessof PTFE should be more than 7mm and less than 8mm(ie only a 1mm wear thickness is allowed for an 8mmthickness of PTFE) [23] so the value of [M3ds] is 667 km forthe spherical steel bearing for the parameter W a larger W

can provide better evaluation results and the value of W is1000 for evaluation in this research

After the three parameters are determined the bearingwear life is evaluated by the following seven steps

(1) e values of N3d in 100-year service life are sim-ulated through Monte Carlo simulation usingEquation (4b) and 8434300 values of N3ds areobtained

(2) All the 8434300 values of N3ds are summed to obtainthe M3ds of B3d in 100-year service life usingEquation (6d)

(3) Steps (1) and (2) are repeated 1000 times to obtain1000 values of M3ds as shown in Figure 13(a) e1000 values of M3ds show good uniform and randomcharacteristics and these values can be used to an-alyze the cumulative probability characteristics

(4) e cumulative probability of M3ds is calculated andfitted by the ND WD and GEVD with the fittedresults shown in Figure 13(b) Both the ND andGEVD can well describe the cumulative probabilitycharacteristics of M3ds and GEVD is chosen for thetrue cumulative distribution function of M3ds

(5) e failure probability Pf of CBTover the upper limit[M3ds] for bearing B3d is about 10minus8 calculated by

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

5

10

15

20

25

p

Monitored result Simulated result

(e)

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

4

8

12

p

Monitored result Simulated result

(f )

Figure 12e simulated Mius and Mids (a) M1us and M1u (b) M1ds and M1d (c) M2us and M2u (d) M2ds and M2d (e) M3us and M3u(f ) M3ds and M3d

Shock and Vibration 11

Equation (6b) which means that M3ds has littleprobability of exceeding [M3ds] because the simu-lated 1000 values of M3ds are within a small interval[5874 km 5931 km]

(6) e reliability level β is 5612 calculated by Equation(7) which is larger than the target reliability level3778 indicating that bearing B3d does not reach thewear limit in the service life

(7) Because the linear growth rate of CBT for B3d isfaster than those of the other spherical steel bearingsit can be inferred that the other spherical steelbearings do not reach the wear limit as well in theservice life

5 Conclusions

Based on the monitored dynamic displacement data of steeltruss arch bridge bearings the CDD under the action ofa single train and the CBT under the continual actions ofmany passed trains are investigated Furthermore theprobability statistics and the Monte Carlo sampling simu-lationmethod for CDD are studied and the safety evaluationmethod for the bearing wear life is proposed using a re-liability index regarding the failure probability of monitoredCBT over the wear limit in the service lifetime e mainconclusions are drawn as follows

(1) rough monitored data analysis the monitoredlongitudinal displacement data mainly consist oftemperature-induced static displacements and train-induced dynamic displacements e wavelet packetdecomposition method can effectively extract thesmall-amplitude dynamic displacements caused bytrain loads from the monitored data All the CDDscontain uniform and random characteristics and allthe CBTs have an obvious linear correlation with the

number of passed trains but their linear growth ratesare different

(2) rough probability statistics analysis GEVD canwell describe the cumulative probability character-istics of CDDeMonte Carlo sampling simulationwas performed using GEVD to simulate CDD andCBT and the results show that the simulated cu-mulative probabilities of CDD have agreement withthemonitored ones of CDD and the simulated lineargrowth rates of CBT have agreement with themonitored ones of CBT

(3) rough safety evaluation analysis the safety eval-uation method for the bearing wear life is proposedusing a reliability index and it is finally judgedwhether the CBTs can exceed the wear limit in theservice lifetime e evaluation results show that allthe spherical steel bearings do not exceed the wearlimit in the service life

Notation

Biu ith spherical steel bearing at theupstream

Bid ith spherical steel bearing at thedownstream

CBT cumulative bearing travelCDD cumulative dynamic displacementCP cumulative probabilityDiu monitored longitudinal

displacement data at the ithspherical steel bearing Biu

Did monitored longitudinaldisplacement data at the ithspherical steel bearing Bid

1113957D(k) dynamic displacement caused by thekth passed train

0 200 400 600 800 1000585

5875

59

5925

595

The number of simulations

M3

ds (k

m)

(a)

M3ds (km)587 588 589 59 591 592 593 5940

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y

Cumulative probabilityGEVD

NDWD

(b)

Figure 13 e 1000 values of M3ds and their cumulative probabilities (a)1000M3ds (b) Cumulative probability of M3ds and its fittedresults

12 Shock and Vibration

1113957Dj(k) jth value in 1113957D(k)

F([Mius]) cumulative distribution functionwith assigned value [Mius]

F([Mids]) cumulative distribution functionwith assigned value [Mids]

GEVD general extreme value distributionG(Niu) cumulative distribution function of

NiuG(Nid) cumulative distribution function of

NidG(Nius) cumulative distribution function of

NiusG(Nids) cumulative distribution function of

NidsLVDT linear variable differential

transformermean(Niu) average value of 3495 CDDs for each

Niumean(Nid) average value of 3495 CDDs for each

NidM(p) CBT caused by p passed trainsMiu CBT of BiuMid CBT of Bid[Mius] wear limit of Biu[Mids] wear limit of BidN(k) cumulative travel of bearing caused

by the kth passed trainNiu set of N(1) N(2) N(3495)

for the bearings Biu

Nid set of N(1) N(2) N(3495)

for the bearings BidNius simulated CDDs of NiuNids simulated CDDs of NidNUM(Niu) total number of NiuNUM(Nid) total number of NidNUM(Niu ltNiu(j)) number of CDDs in Niu that are

smaller than Niu(j)

NUM(Nid ltNid(j)) number of CDDs in Nid that aresmaller than Nid(j)

ND normal distributionPTFE poly-tetra-fluoro-ethyleneP(Niu(j)) CP of the jth value Niu(j) in NiuP(Nid(j)) CP of the jth value Nid(j) in NidPf failure probabilityR value of cumulative probabilityWD Weibull distributionxjm wavelet packet coefficient in themth

frequency band of the jth scale[β] target reliability level

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors thank the Natural Science Foundation ofJiangsu Province of China (BK20180652) and the ChinaPostdoctoral Science Foundation (2017M621865)

References

[1] J W Zhan H Xia N Zhang and Y Lu ldquoA diagnosis methodfor bridge rubber support disease based on impact responsesrdquoJournal of Vibration and Shock vol 32 no 8 pp 153ndash1572013 in Chinese

[2] B Wei T H Yang L Z Jiang and X H He ldquoEffects offriction-based fixed bearings on the seismic vulnerability ofa high-speed railway continuous bridgerdquo Advances inStructural Engineering vol 21 no 5 pp 643ndash657 2018

[3] J Oh C Jang and J H Kim ldquoA study on the characteristics ofbridge bearings behavior by finite element analysis and modeltestrdquo Journal of Vibroengineering vol 17 no 5 pp 2559ndash2571 2015

[4] J J Huang Q Su L C Zhang Y J Li and B Liu ldquoRapidtreatment technique of diseases of the rocking axle bearing ofrailway simply supported beam bridgerdquo Applied Mechanicsand Materials vol 351-352 pp 1440ndash1444 2013

[5] A S Deshpande and J M C Kishen ldquoFatigue crack prop-agation in rocker and roller-rocker bearings of railway steelbridgesrdquo Engineering Fracture Mechanics vol 77 no 9pp 1454ndash1466 2010

[6] Q J Shi ldquoDefect analysis and reconstruction method forexisting railway bridge bearingrdquo Railway Engineering vol 57no 10 pp 12ndash14 2017

[7] S Y Yan ldquoAn analysis of causes for defects of roller bearingand treatment measuresrdquo Railway Standard Design vol 2002no 6 pp 30ndash32 2002 in Chinese

[8] H Wang Study on Damage Identification Method for RailwayBridge Rubber Bearings Based on Dynamic Responses BeijingJiaotong University Beijing China 2014 in Chinese

[9] Z Qiao Study on the Bridge Bearing Disease IdentificationTechnique Based on the Natural Frequency Change Rate EastChina Jiaotong University Nanchang China 2014 inChinese

[10] S L Chen Y Q Liu and Y B Zhang ldquoEffects of bearingdamage on vibration characteristics of a large span steel trussbridgerdquo Journal of Vibration and Shock vol 36 no 13pp 195ndash200 2017 in Chinese

[11] F Q Hu J Chen Y B Xu and X M Feng ldquoAnalysis ofmechanical properties of continuous T-beam bridge withdiseased bearingsrdquo Journal of Nanchang University Engi-neering and Technology vol 37 no 2 pp 123ndash127 2015 inChinese

[12] H Zhou K Liu G Shi Y Q Wang Y J Shi andG De Roeck ldquoFatigue assessment of a composite railwaybridge for high speed trains Part I modeling and fatiguecritical detailsrdquo Journal of Constructional Steel Researchvol 82 pp 234ndash245 2013

[13] T Guo J Liu and L Y Huang ldquoInvestigation and control ofexcessive cumulative girder movements of long-span steelsuspension bridgesrdquo Engineering Structures vol 125pp 217ndash226 2016

[14] 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 04014099 2015

Shock and Vibration 13

[15] Y L Ding and G X Wang ldquoEvaluation of dynamic loadfactors for a high-speed railway truss arch bridgerdquo Shock andVibration vol 2016 Article ID 5310769 15 pages 2016

[16] G X Wang Y L Ding Y S Song L Y Wu Q Yue andG H Mao ldquoDetection and location of the degraded bearingsbased on monitoring the longitudinal expansion performanceof the main girder of the Dashengguan Yangtze BridgerdquoJournal of Performance of Constructed Facilities vol 30 no 4article 04015074 2016

[17] G X Wang and Y L Ding ldquoResearch on monitoring tem-perature difference from cross sections of steel truss archgirder of Dashengguan Yangtze Bridgerdquo International Journalof Steel Structures vol 15 no 3 pp 647ndash660 2015

[18] M Y Gokhale and K D Kaur ldquoTime domain signal analysisusing wavelet packet decomposition approachrdquo InternationalJournal of communications Networks and System Sciencesvol 3 no 3 pp 321ndash329 2010

[19] T Figlus S Liscak A Wilk and B Aazarz ldquoConditionmonitoring of engine timing system by using wavelet packetdecomposition of a acoustic signalrdquo Journal of MechanicalScience and Technology vol 28 no 5 pp 1663ndash1671 2014

[20] R Wald T M Khoshgoftaar and J C Sloan ldquoFeature se-lection for optimization of wavelet packet decomposition inreliability analysis of systemsrdquo International Journal on Ar-tificial Intelligence Tools vol 22 no 5 article 1360011 2013

[21] EN 1991-1-5 Eurocode 1 Actions on StructuresmdashPart 1ndash5General Actionsmdashampermal Actions European Committee forStandardization (CEN) Brussels Belgium Europe 2004

[22] P Kotes and J Vican ldquoReliability levels for existing bridgesevaluation according to eurocodesrdquo Procedia Engineeringvol 40 pp 211ndash216 2012

[23] TBT 3320-2013 Spherical Bearings for Railway BridgesChina Railway Publishing House Beijing China 2013 inChinese

14 Shock and Vibration

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Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

M(p) 1113944

p

k1N(k) (1)

where M(p) denotes the CBT caused by p passed trainsand p 1 2 3495 e M(p) values at the upstreamand downstream are shown in Figure 8 where Miu de-notes the CBT of Biu and Mid denotes the CBT ofBid(i 1 2 3) It can be seen that all the CBTs have ob-vious linear correlations with the number of passed trainsbut the linear growth rates are different for each Miu orMid where the linear growth rate of M3d is the fastestand its maximum CBT caused by 3495 passed trains is24360mm

3 Probability Statistics Analysis

e number of monitored CDDs are not enough to satisfythe requirement of large sample for safety evaluation ofbearing wear life erefore it is necessary to performsampling simulation of CDDs using probabilistic statistics ofthe monitored CDDs to obtain adequate data

31 Cumulative Probability Characteristics Figure 7 showsthat Niu and Nid contain obvious uniform and randomcharacteristics so Niu and Nid can be treated as statisticalvariables for probabilistic statistics analysis In detail thecumulative probability (CP) is used to describe the proba-bilistic statistics characteristics of CDDs First the cumu-lative probabilities of Niu and Nid are calculated based onprobability statistics theory and then the cumulative dis-tribution function is fitted in a least-squares manner usingthe cumulative probabilities of Niu and Nid which is il-lustrated in two steps as follows

(1) e cumulative probabilities of Niu and Nid arecalculated as follows

P Niu(j)11138731113872 NUM Niu ltNiu(j)11138731113872

NUM Niu1113872 1113873 (2a)

P Nid(j)11138731113872 NUM Nid ltNid(j)11138731113872

NUM Nid1113872 1113873 (2b)

where P(Niu(j)) denotes the CP of the jth valueNiu(j) in Niu P(Nid(j)) denotes the CP of the jth

0 4 8 12 16 20 2425

30

35

40

45

Hour

Disp

lace

men

t (m

m)

(a)

0 4 8 12 16 20 24ndash3

ndash1

1

3

Hour

Disp

lace

men

t (m

m)

Wind load Wind and train loads

The firstcomponent The second component

(b)

Figure 5 Decomposition result of monitored longitudinal displacement (a) Component caused by temperature field (b) Componentcaused by dynamical loads

Table 1 Calculation of variables N(k) Niu Nid mean(Niu) and mean(Nid)

Variable Calculation equation

N(k) N(k) 1113936Uminus1j1 | 1113957Dj+1(k)minus 1113957Dj(k)|

Niu NidNiu N(1) N(2) N(3495) for the bearings at upstream

Nid N(1) N(2) N(3495) for the bearings at downstream

mean(Niu) mean(Nid)mean(Niu) (N(1) + N(2) + middot middot middot + N(3495)3495) for the bearings at upstream

mean(Nid) (N(1) + N(2) + middot middot middot + N(3495)3495) for the bearings at downstream

0 40 80 120 160ndash3

ndash2

ndash1

0

1

2

3

Time (s)

Disp

lace

men

t (m

m)

No train No trainPassed train

Figure 6 e dynamic displacement of B3u caused by one train

Shock and Vibration 5

value Nid(j) in Nid NUM(Niu) denotes the totalnumber of Niu NUM(Nid) denotes the totalnumber of Nid NUM(Niu ltNiu(j)) denotes thenumber of CDDs in Niu that are smaller thanNiu(j) and NUM(Nid ltNid(j)) denotes thenumber of CDDs in Nid that are smaller thanNid(j) For instance the CP of N1u is shown inFigure 9

(2) e cumulative distribution function is fitted in a least-squares manner using the cumulative probabilities of

Niu and Nid Considering that the true cumulativedistribution function of Niu and Nid is unknownthree possible cumulative distribution functions(ie normal distribution (ND) Weibull distribution(WD) and general extreme value distribution(GEVD)) are chosen for least-square fitting and thenthe cumulative distribution function with the bestfitting effect is selected as the true cumulative distri-bution function of Niu and Nid Taking N1u as anexample the fitting results of the three cumulative

0 700 1400 2100 2800 35000

5

10

15

20

25

k

Cum

ulat

ive v

alue

(mm

)mean (N1u) = 2904mm

(a)

0 700 1400 2100 2800 35000

10

20

30

40

50

60

k

Cum

ulat

ive v

alue

(mm

)

mean (N1d) = 4067mm

(b)

0 700 1400 2100 2800 35000

5

10

15

20

25mean (N2u) = 3888mm

Cum

ulat

ive v

alue

(mm

)

k

(c)

0 700 1400 2100 2800 35000

5

10

15

Cum

ulat

ive v

alue

(mm

)

k

mean (N2d) = 2960mm

(d)

0 700 1400 2100 2800 35000

5

10

15

20

25

k

Cum

ulat

ive v

alue

(mm

)

mean (N3u) = 6559mm

(e)

0 700 1400 2100 2800 35000

10

20

30

40

k

Cum

ulat

ive v

alue

(mm

)

mean (N3d) = 7020mm

(f)

Figure 7 Niu and Nid (a) N1u (b) N1d (c) N2u (d) N2d (e) N3u (f ) N3d

6 Shock and Vibration

distribution functions are shown in Figure 10 It can beseen that GEVDhas the best fitting results so GEVD ischosen to describe the cumulative probability char-acteristics of N1u e formula is expressed as

G N1u1113872 1113873 exp minus 1 + rN1u minus b

a1113888 11138891113890 1113891

minus1r⎡⎣ ⎤⎦ (3)

where G(N1u) denotes the cumulative distributionfunction of N1u which follows the generalized ex-treme value distribution r denotes the shape pa-rameter a denotes the scale parameter and b

denotes the location parameter e fitted values ofthese three parameters for G(N1u) are r 03058a 08163 and b 20294

Based on the method above the best cumulative dis-tribution functions of N1u minusN3u and N1d minusN3d are allGEVD as shown in Figures 10(a)ndash10(f ) It can be seen thatGEVD can well describe the cumulative probability char-acteristics of N1u minusN3u and N1d minusN3d

32 Sampling SimulationMethod Since Niu and Nid followGEVDs their GEVDs can be utilized for Monte Carlosampling to obtain a demanded number of CDDs Specif-ically if the demanded number of CDDs is K K values are

0 1000 2000 3000 40000

5

10

15

M1uM1d

p

Cum

ulat

ive t

rave

l (m

m times

103 )

(a)

0 1000 2000 3000 40000

5

10

15

M2uM2d

p

Cum

ulat

ive t

rave

l (m

m times

103 )

(b)

0 1000 2000 3000 40000

5

10

15

20

25

M3uM3d

p

Cum

ulat

ive t

rave

l (m

m times

103 )

(c)

Figure 8 e M1u minusM3u and M1u minusM3u caused by many passed trains (a) M1u and M1d (b) M2u and M2d (c) M3u and M3d

5 10 15 20 25 300

02

04

06

08

1

Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Figure 9 e cumulative probability of N1u

Shock and Vibration 7

5 10 15 20 25 300

02

04

06

08

1

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

WD

ND

GEVD

GEVDND

WDMonitored curve

(a)

10 20 30 40 500

02

04

06

08

1

r = 04765a = 10599b = 25108

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored curve Fitted curve

(b)

5 10 15 20 25 300

02

04

06

08

1

r = 01926a = 12320b = 28604

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored curveFitted curve

(c)

2 4 6 8 10 12 140

02

04

06

08

1

r = 00493a = 08440b = 24302

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored curveFitted curve

(d)

5 10 15 20 250

02

04

06

08

1

r = 00830a = 18184b = 53395

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored curveFitted curve

(e)

5 10 15 20 250

02

04

06

08

1

r = 01834a = 19436b = 54517

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored curveFitted curve

(f )

Figure 10 GEVDs of N1u minusN6u and N1u minusN6u (a) N1u (b) N1d (c) N2u (d) N2d (e) N3u (f ) N3d

8 Shock and Vibration

uniformly and randomly sampled from the interval (0 1)and then the demanded number of CDDs can be simulatedby substituting the K values into the cumulative probabilityR in the following equations

R G Nius1113872 1113873 (4a)

R G Nids1113872 1113873 (4b)

G Nius1113872 1113873 exp minus 1 + rNius minus b

a1113888 11138891113890 1113891

minus1r⎡⎣ ⎤⎦ (4c)

G Nids1113872 1113873 exp minus 1 + rNids minus b

a1113888 11138891113890 1113891

minus1r⎡⎣ ⎤⎦ (4d)

where Nius and Nids denote the simulated CDDs of Niu andNid respectively e values of r a and b have been cal-culated in the Section 32 However Niu and Nid areembedded in G(Niu) and G(Nids) so they cannot be di-rectly solved from Equations (4a) and (4b) Considering thatG(Nius) and G(Nids) are monotonically increasing func-tions Nius and Nids can be determined by the Newtoniteration method as follows

Nn+1ius N

nius minus

G Nnius1113872 1113873minusR

Gprime Nnius1113872 1113873

(5a)

Nn+1ids N

nids minus

G Nnids1113872 1113873minusR

Gprime Nnids1113872 1113873

(5b)

where Nn+1ius and Nn

ius denote the (n + 1)th and nth iterationresults of Nius respectively Nn+1

ids andNnids denote the (n + 1)

th and nth iteration results of Nids respectively Gprime(Nnius)

and Gprime(Nnids) denote the first-order derivatives of GEVD

(ie the probability density functions of GEVD) e iter-ation will terminate when the absolute difference betweentwo adjacent iterations is less than 10minus2 e last iterationresult is treated as the values of Nius and Nids

33 ampe Sampling Simulation Result Based on the samplingsimulation method above the simulation results of 3495N1us are shown in Figure 11(a) e mean value of thesimulated result is 2820mm which is very close to the meanvalue of Niu ie 2904mm Furthermore the CP of thesimulated result is calculated and then compared with themonitored CP as shown in Figure 11(b) It can be seen thatthe simulated CP coincides with the monitored CP whichverifies the accuracy of the sampling simulation method

Furthermore the simulated 3495 values of Nius andNidsare accumulated using Equation (1) to obtain the simulatedCBT (ie Mius and Mids) as shown in Figure 12 It can beseen that the simulated Mius and Mids present good lineargrowth characteristics with the number of passed trains Toverify the accuracy of the simulated linear growth ratesa comparison between the monitored and simulated lineargrowth rates was conducted as shown in Figure 12 It can beseen that the simulated linear growth rates have agreement

with the monitored ones which further verifies the accuracyof the sampling simulation method

4 Safety Evaluation Analysis

e CBT relates to the bearing wear condition If the CBTsexceed the wear limit in the service lifetime the sphericalsteel bearing is totally damaged and cannot be used forlongitudinal thermal expansion any longer erefore it isnecessary to analyze whether the CBTs at bridge site canexceed the wear limit in the service lifetime

41 EvaluationMethod e failure probabilities of sphericalsteel bearings in the service lifetime (ie the probabilities ofCBTover the wear limit in the service lifetime) are calculatedas follows

Pf 1minusF Mius1113960 11139611113872 1113873 P Mius gt Mius1113960 11139611113872 1113873 (6a)

or

Pf 1minusF Mids1113960 11139611113872 1113873 P Mids gt Mids1113960 11139611113872 1113873 (6b)

Mius 1113944

J

j1Nius(j) (6c)

Mids 1113944

J

j1Nids(j) (6d)

where Pf denotes the failure probability [Mius] and [Mids]

denote the wear limits of Biu and Bid respectivelyF([Mius]) and F([Mids]) denote the cumulative distribu-tion functions with assigned values [Mius] and [Mids]respectively namely F(Mius [Mius]) and F(Mids

[Mids]) Nius(j) and Nids(j) denote the jth values of Niusand Nids in the service lifetime respectively which can beobtained through sampling simulation as introduced in theSection 32 and J denotes the total number of passed trainsin the service lifetime

F(Mius) and F(Mius) are unknown which should befirst determined by the following two steps

(1) By referring to the sampling simulation method inthe Section 32 the values of Nius or Nids in theservice life are simulated through Monte Carlosampling en the values of Nius or Nids aresummed using Equation (6c) or (6d) respectivelyto obtain one value of Mius or Mids is step isrepeated W times to obtain W values of Mius orMids

(2) e cumulative probability characteristics for the W

values of Mius or Mids are fitted using cumulativedistribution function e true cumulative distri-bution function of Mius and Mids is unknown sothree possible cumulative distribution functions(namely ND WD and GEVD) are considered andthe cumulative distribution function with the bestfitting effect is selected as the true cumulative

Shock and Vibration 9

0 1000 2000 3000 40000

4

8

12

p

Cum

ulat

ive t

rave

l (m

m times

103 )

Monitored result Simulated result

(a)

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

5

10

15

p

Monitored result Simulated result

(b)

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

5

10

15

p

Monitored result Simulated result

(c)

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

5

10

15

20

25

p

Monitored result Simulated result

(d)

Figure 12 Continued

0 700 1400 2100 2800 35000

5

10

15

20

25

30

35

k

Cum

ulat

ive v

alue

(mm

) mean (N1us) = 2820 mm

(a)

5 10 15 20 25 300

02

04

06

08

1

Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored result Simulated result

(b)

Figure 11 e simulated N1us and its CP (a) e simulated result (b) A comparison between the simulated and monitored results

10 Shock and Vibration

distribution function of Mius and Mids after least-square fitting

After F(Mius) and F(Mius) are determined [Mius] and[Mids] are substituted in Equations (6a) and (6b) to obtainthe value of Pf Furthermore the reliability level β can becalculated using Pf as follows

Pf 1113946+infin

β

12π

radic eminusx22

dx (7)

If the calculated reliability level β is larger than the targetreliability level [β] (ie βgt [β]) it can be inferred that thebearing is still in a good service state otherwise (ie βle [β])it can be inferred that the bearing is in a poor service stateand should be replaced in time According to the Eurocode[21] the target reliability level [β] for newly built bridgeswith a 100-year design lifetime is 38 which corresponds toa failure probability Pf 72 times 10minus5 However theDashengguan Railway Bridge has been in operation for 8years and it is not a newly built bridge From the viewpointof bridge reliability a reduction in the uncertainties of theload and resistance parameters decreases the failure prob-ability of existing bridge structures indicating the possibilityof adopting a lower reliability level for the evaluation ofexisting bridges rather than the reliability level that shouldbe used for newly built bridges [22] According to Kotesrsquoresearch [22] the target reliability level [β] with a 90-yearremaining lifetime is 3773 so the target reliability level [β]

of the Dashengguan Railway Bridge with a 92-yearremaining lifetime is 3778 using the linear interpolationcalculation namely [β] 3778

42 Evaluation Results Figure 8 reveals that M3d has thefastest growth so M3d is taken as an example to calculate itsfailure probability and evaluate the wear condition of B3d inthe service lifetime Before evaluation three parameters

should be determined namely J [M3ds] and W For theparameter J the designed service life of the NanjingDashengguan Yangtze River Bridge is 100 years and thenumber of passed trains in 2017 was 84343 so the value of J

is 8434300 for the parameter [M3ds] the China codespecifies that the linear wear rate of PTFE in a spherical steelbearing should be less than 15 μmkm and that the thicknessof PTFE should be more than 7mm and less than 8mm(ie only a 1mm wear thickness is allowed for an 8mmthickness of PTFE) [23] so the value of [M3ds] is 667 km forthe spherical steel bearing for the parameter W a larger W

can provide better evaluation results and the value of W is1000 for evaluation in this research

After the three parameters are determined the bearingwear life is evaluated by the following seven steps

(1) e values of N3d in 100-year service life are sim-ulated through Monte Carlo simulation usingEquation (4b) and 8434300 values of N3ds areobtained

(2) All the 8434300 values of N3ds are summed to obtainthe M3ds of B3d in 100-year service life usingEquation (6d)

(3) Steps (1) and (2) are repeated 1000 times to obtain1000 values of M3ds as shown in Figure 13(a) e1000 values of M3ds show good uniform and randomcharacteristics and these values can be used to an-alyze the cumulative probability characteristics

(4) e cumulative probability of M3ds is calculated andfitted by the ND WD and GEVD with the fittedresults shown in Figure 13(b) Both the ND andGEVD can well describe the cumulative probabilitycharacteristics of M3ds and GEVD is chosen for thetrue cumulative distribution function of M3ds

(5) e failure probability Pf of CBTover the upper limit[M3ds] for bearing B3d is about 10minus8 calculated by

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

5

10

15

20

25

p

Monitored result Simulated result

(e)

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

4

8

12

p

Monitored result Simulated result

(f )

Figure 12e simulated Mius and Mids (a) M1us and M1u (b) M1ds and M1d (c) M2us and M2u (d) M2ds and M2d (e) M3us and M3u(f ) M3ds and M3d

Shock and Vibration 11

Equation (6b) which means that M3ds has littleprobability of exceeding [M3ds] because the simu-lated 1000 values of M3ds are within a small interval[5874 km 5931 km]

(6) e reliability level β is 5612 calculated by Equation(7) which is larger than the target reliability level3778 indicating that bearing B3d does not reach thewear limit in the service life

(7) Because the linear growth rate of CBT for B3d isfaster than those of the other spherical steel bearingsit can be inferred that the other spherical steelbearings do not reach the wear limit as well in theservice life

5 Conclusions

Based on the monitored dynamic displacement data of steeltruss arch bridge bearings the CDD under the action ofa single train and the CBT under the continual actions ofmany passed trains are investigated Furthermore theprobability statistics and the Monte Carlo sampling simu-lationmethod for CDD are studied and the safety evaluationmethod for the bearing wear life is proposed using a re-liability index regarding the failure probability of monitoredCBT over the wear limit in the service lifetime e mainconclusions are drawn as follows

(1) rough monitored data analysis the monitoredlongitudinal displacement data mainly consist oftemperature-induced static displacements and train-induced dynamic displacements e wavelet packetdecomposition method can effectively extract thesmall-amplitude dynamic displacements caused bytrain loads from the monitored data All the CDDscontain uniform and random characteristics and allthe CBTs have an obvious linear correlation with the

number of passed trains but their linear growth ratesare different

(2) rough probability statistics analysis GEVD canwell describe the cumulative probability character-istics of CDDeMonte Carlo sampling simulationwas performed using GEVD to simulate CDD andCBT and the results show that the simulated cu-mulative probabilities of CDD have agreement withthemonitored ones of CDD and the simulated lineargrowth rates of CBT have agreement with themonitored ones of CBT

(3) rough safety evaluation analysis the safety eval-uation method for the bearing wear life is proposedusing a reliability index and it is finally judgedwhether the CBTs can exceed the wear limit in theservice lifetime e evaluation results show that allthe spherical steel bearings do not exceed the wearlimit in the service life

Notation

Biu ith spherical steel bearing at theupstream

Bid ith spherical steel bearing at thedownstream

CBT cumulative bearing travelCDD cumulative dynamic displacementCP cumulative probabilityDiu monitored longitudinal

displacement data at the ithspherical steel bearing Biu

Did monitored longitudinaldisplacement data at the ithspherical steel bearing Bid

1113957D(k) dynamic displacement caused by thekth passed train

0 200 400 600 800 1000585

5875

59

5925

595

The number of simulations

M3

ds (k

m)

(a)

M3ds (km)587 588 589 59 591 592 593 5940

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y

Cumulative probabilityGEVD

NDWD

(b)

Figure 13 e 1000 values of M3ds and their cumulative probabilities (a)1000M3ds (b) Cumulative probability of M3ds and its fittedresults

12 Shock and Vibration

1113957Dj(k) jth value in 1113957D(k)

F([Mius]) cumulative distribution functionwith assigned value [Mius]

F([Mids]) cumulative distribution functionwith assigned value [Mids]

GEVD general extreme value distributionG(Niu) cumulative distribution function of

NiuG(Nid) cumulative distribution function of

NidG(Nius) cumulative distribution function of

NiusG(Nids) cumulative distribution function of

NidsLVDT linear variable differential

transformermean(Niu) average value of 3495 CDDs for each

Niumean(Nid) average value of 3495 CDDs for each

NidM(p) CBT caused by p passed trainsMiu CBT of BiuMid CBT of Bid[Mius] wear limit of Biu[Mids] wear limit of BidN(k) cumulative travel of bearing caused

by the kth passed trainNiu set of N(1) N(2) N(3495)

for the bearings Biu

Nid set of N(1) N(2) N(3495)

for the bearings BidNius simulated CDDs of NiuNids simulated CDDs of NidNUM(Niu) total number of NiuNUM(Nid) total number of NidNUM(Niu ltNiu(j)) number of CDDs in Niu that are

smaller than Niu(j)

NUM(Nid ltNid(j)) number of CDDs in Nid that aresmaller than Nid(j)

ND normal distributionPTFE poly-tetra-fluoro-ethyleneP(Niu(j)) CP of the jth value Niu(j) in NiuP(Nid(j)) CP of the jth value Nid(j) in NidPf failure probabilityR value of cumulative probabilityWD Weibull distributionxjm wavelet packet coefficient in themth

frequency band of the jth scale[β] target reliability level

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors thank the Natural Science Foundation ofJiangsu Province of China (BK20180652) and the ChinaPostdoctoral Science Foundation (2017M621865)

References

[1] J W Zhan H Xia N Zhang and Y Lu ldquoA diagnosis methodfor bridge rubber support disease based on impact responsesrdquoJournal of Vibration and Shock vol 32 no 8 pp 153ndash1572013 in Chinese

[2] B Wei T H Yang L Z Jiang and X H He ldquoEffects offriction-based fixed bearings on the seismic vulnerability ofa high-speed railway continuous bridgerdquo Advances inStructural Engineering vol 21 no 5 pp 643ndash657 2018

[3] J Oh C Jang and J H Kim ldquoA study on the characteristics ofbridge bearings behavior by finite element analysis and modeltestrdquo Journal of Vibroengineering vol 17 no 5 pp 2559ndash2571 2015

[4] J J Huang Q Su L C Zhang Y J Li and B Liu ldquoRapidtreatment technique of diseases of the rocking axle bearing ofrailway simply supported beam bridgerdquo Applied Mechanicsand Materials vol 351-352 pp 1440ndash1444 2013

[5] A S Deshpande and J M C Kishen ldquoFatigue crack prop-agation in rocker and roller-rocker bearings of railway steelbridgesrdquo Engineering Fracture Mechanics vol 77 no 9pp 1454ndash1466 2010

[6] Q J Shi ldquoDefect analysis and reconstruction method forexisting railway bridge bearingrdquo Railway Engineering vol 57no 10 pp 12ndash14 2017

[7] S Y Yan ldquoAn analysis of causes for defects of roller bearingand treatment measuresrdquo Railway Standard Design vol 2002no 6 pp 30ndash32 2002 in Chinese

[8] H Wang Study on Damage Identification Method for RailwayBridge Rubber Bearings Based on Dynamic Responses BeijingJiaotong University Beijing China 2014 in Chinese

[9] Z Qiao Study on the Bridge Bearing Disease IdentificationTechnique Based on the Natural Frequency Change Rate EastChina Jiaotong University Nanchang China 2014 inChinese

[10] S L Chen Y Q Liu and Y B Zhang ldquoEffects of bearingdamage on vibration characteristics of a large span steel trussbridgerdquo Journal of Vibration and Shock vol 36 no 13pp 195ndash200 2017 in Chinese

[11] F Q Hu J Chen Y B Xu and X M Feng ldquoAnalysis ofmechanical properties of continuous T-beam bridge withdiseased bearingsrdquo Journal of Nanchang University Engi-neering and Technology vol 37 no 2 pp 123ndash127 2015 inChinese

[12] H Zhou K Liu G Shi Y Q Wang Y J Shi andG De Roeck ldquoFatigue assessment of a composite railwaybridge for high speed trains Part I modeling and fatiguecritical detailsrdquo Journal of Constructional Steel Researchvol 82 pp 234ndash245 2013

[13] T Guo J Liu and L Y Huang ldquoInvestigation and control ofexcessive cumulative girder movements of long-span steelsuspension bridgesrdquo Engineering Structures vol 125pp 217ndash226 2016

[14] 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 04014099 2015

Shock and Vibration 13

[15] Y L Ding and G X Wang ldquoEvaluation of dynamic loadfactors for a high-speed railway truss arch bridgerdquo Shock andVibration vol 2016 Article ID 5310769 15 pages 2016

[16] G X Wang Y L Ding Y S Song L Y Wu Q Yue andG H Mao ldquoDetection and location of the degraded bearingsbased on monitoring the longitudinal expansion performanceof the main girder of the Dashengguan Yangtze BridgerdquoJournal of Performance of Constructed Facilities vol 30 no 4article 04015074 2016

[17] G X Wang and Y L Ding ldquoResearch on monitoring tem-perature difference from cross sections of steel truss archgirder of Dashengguan Yangtze Bridgerdquo International Journalof Steel Structures vol 15 no 3 pp 647ndash660 2015

[18] M Y Gokhale and K D Kaur ldquoTime domain signal analysisusing wavelet packet decomposition approachrdquo InternationalJournal of communications Networks and System Sciencesvol 3 no 3 pp 321ndash329 2010

[19] T Figlus S Liscak A Wilk and B Aazarz ldquoConditionmonitoring of engine timing system by using wavelet packetdecomposition of a acoustic signalrdquo Journal of MechanicalScience and Technology vol 28 no 5 pp 1663ndash1671 2014

[20] R Wald T M Khoshgoftaar and J C Sloan ldquoFeature se-lection for optimization of wavelet packet decomposition inreliability analysis of systemsrdquo International Journal on Ar-tificial Intelligence Tools vol 22 no 5 article 1360011 2013

[21] EN 1991-1-5 Eurocode 1 Actions on StructuresmdashPart 1ndash5General Actionsmdashampermal Actions European Committee forStandardization (CEN) Brussels Belgium Europe 2004

[22] P Kotes and J Vican ldquoReliability levels for existing bridgesevaluation according to eurocodesrdquo Procedia Engineeringvol 40 pp 211ndash216 2012

[23] TBT 3320-2013 Spherical Bearings for Railway BridgesChina Railway Publishing House Beijing China 2013 inChinese

14 Shock and Vibration

International Journal of

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Submit your manuscripts atwwwhindawicom

value Nid(j) in Nid NUM(Niu) denotes the totalnumber of Niu NUM(Nid) denotes the totalnumber of Nid NUM(Niu ltNiu(j)) denotes thenumber of CDDs in Niu that are smaller thanNiu(j) and NUM(Nid ltNid(j)) denotes thenumber of CDDs in Nid that are smaller thanNid(j) For instance the CP of N1u is shown inFigure 9

(2) e cumulative distribution function is fitted in a least-squares manner using the cumulative probabilities of

Niu and Nid Considering that the true cumulativedistribution function of Niu and Nid is unknownthree possible cumulative distribution functions(ie normal distribution (ND) Weibull distribution(WD) and general extreme value distribution(GEVD)) are chosen for least-square fitting and thenthe cumulative distribution function with the bestfitting effect is selected as the true cumulative distri-bution function of Niu and Nid Taking N1u as anexample the fitting results of the three cumulative

0 700 1400 2100 2800 35000

5

10

15

20

25

k

Cum

ulat

ive v

alue

(mm

)mean (N1u) = 2904mm

(a)

0 700 1400 2100 2800 35000

10

20

30

40

50

60

k

Cum

ulat

ive v

alue

(mm

)

mean (N1d) = 4067mm

(b)

0 700 1400 2100 2800 35000

5

10

15

20

25mean (N2u) = 3888mm

Cum

ulat

ive v

alue

(mm

)

k

(c)

0 700 1400 2100 2800 35000

5

10

15

Cum

ulat

ive v

alue

(mm

)

k

mean (N2d) = 2960mm

(d)

0 700 1400 2100 2800 35000

5

10

15

20

25

k

Cum

ulat

ive v

alue

(mm

)

mean (N3u) = 6559mm

(e)

0 700 1400 2100 2800 35000

10

20

30

40

k

Cum

ulat

ive v

alue

(mm

)

mean (N3d) = 7020mm

(f)

Figure 7 Niu and Nid (a) N1u (b) N1d (c) N2u (d) N2d (e) N3u (f ) N3d

6 Shock and Vibration

distribution functions are shown in Figure 10 It can beseen that GEVDhas the best fitting results so GEVD ischosen to describe the cumulative probability char-acteristics of N1u e formula is expressed as

G N1u1113872 1113873 exp minus 1 + rN1u minus b

a1113888 11138891113890 1113891

minus1r⎡⎣ ⎤⎦ (3)

where G(N1u) denotes the cumulative distributionfunction of N1u which follows the generalized ex-treme value distribution r denotes the shape pa-rameter a denotes the scale parameter and b

denotes the location parameter e fitted values ofthese three parameters for G(N1u) are r 03058a 08163 and b 20294

Based on the method above the best cumulative dis-tribution functions of N1u minusN3u and N1d minusN3d are allGEVD as shown in Figures 10(a)ndash10(f ) It can be seen thatGEVD can well describe the cumulative probability char-acteristics of N1u minusN3u and N1d minusN3d

32 Sampling SimulationMethod Since Niu and Nid followGEVDs their GEVDs can be utilized for Monte Carlosampling to obtain a demanded number of CDDs Specif-ically if the demanded number of CDDs is K K values are

0 1000 2000 3000 40000

5

10

15

M1uM1d

p

Cum

ulat

ive t

rave

l (m

m times

103 )

(a)

0 1000 2000 3000 40000

5

10

15

M2uM2d

p

Cum

ulat

ive t

rave

l (m

m times

103 )

(b)

0 1000 2000 3000 40000

5

10

15

20

25

M3uM3d

p

Cum

ulat

ive t

rave

l (m

m times

103 )

(c)

Figure 8 e M1u minusM3u and M1u minusM3u caused by many passed trains (a) M1u and M1d (b) M2u and M2d (c) M3u and M3d

5 10 15 20 25 300

02

04

06

08

1

Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Figure 9 e cumulative probability of N1u

Shock and Vibration 7

5 10 15 20 25 300

02

04

06

08

1

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

WD

ND

GEVD

GEVDND

WDMonitored curve

(a)

10 20 30 40 500

02

04

06

08

1

r = 04765a = 10599b = 25108

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored curve Fitted curve

(b)

5 10 15 20 25 300

02

04

06

08

1

r = 01926a = 12320b = 28604

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored curveFitted curve

(c)

2 4 6 8 10 12 140

02

04

06

08

1

r = 00493a = 08440b = 24302

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored curveFitted curve

(d)

5 10 15 20 250

02

04

06

08

1

r = 00830a = 18184b = 53395

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored curveFitted curve

(e)

5 10 15 20 250

02

04

06

08

1

r = 01834a = 19436b = 54517

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored curveFitted curve

(f )

Figure 10 GEVDs of N1u minusN6u and N1u minusN6u (a) N1u (b) N1d (c) N2u (d) N2d (e) N3u (f ) N3d

8 Shock and Vibration

uniformly and randomly sampled from the interval (0 1)and then the demanded number of CDDs can be simulatedby substituting the K values into the cumulative probabilityR in the following equations

R G Nius1113872 1113873 (4a)

R G Nids1113872 1113873 (4b)

G Nius1113872 1113873 exp minus 1 + rNius minus b

a1113888 11138891113890 1113891

minus1r⎡⎣ ⎤⎦ (4c)

G Nids1113872 1113873 exp minus 1 + rNids minus b

a1113888 11138891113890 1113891

minus1r⎡⎣ ⎤⎦ (4d)

where Nius and Nids denote the simulated CDDs of Niu andNid respectively e values of r a and b have been cal-culated in the Section 32 However Niu and Nid areembedded in G(Niu) and G(Nids) so they cannot be di-rectly solved from Equations (4a) and (4b) Considering thatG(Nius) and G(Nids) are monotonically increasing func-tions Nius and Nids can be determined by the Newtoniteration method as follows

Nn+1ius N

nius minus

G Nnius1113872 1113873minusR

Gprime Nnius1113872 1113873

(5a)

Nn+1ids N

nids minus

G Nnids1113872 1113873minusR

Gprime Nnids1113872 1113873

(5b)

where Nn+1ius and Nn

ius denote the (n + 1)th and nth iterationresults of Nius respectively Nn+1

ids andNnids denote the (n + 1)

th and nth iteration results of Nids respectively Gprime(Nnius)

and Gprime(Nnids) denote the first-order derivatives of GEVD

(ie the probability density functions of GEVD) e iter-ation will terminate when the absolute difference betweentwo adjacent iterations is less than 10minus2 e last iterationresult is treated as the values of Nius and Nids

33 ampe Sampling Simulation Result Based on the samplingsimulation method above the simulation results of 3495N1us are shown in Figure 11(a) e mean value of thesimulated result is 2820mm which is very close to the meanvalue of Niu ie 2904mm Furthermore the CP of thesimulated result is calculated and then compared with themonitored CP as shown in Figure 11(b) It can be seen thatthe simulated CP coincides with the monitored CP whichverifies the accuracy of the sampling simulation method

Furthermore the simulated 3495 values of Nius andNidsare accumulated using Equation (1) to obtain the simulatedCBT (ie Mius and Mids) as shown in Figure 12 It can beseen that the simulated Mius and Mids present good lineargrowth characteristics with the number of passed trains Toverify the accuracy of the simulated linear growth ratesa comparison between the monitored and simulated lineargrowth rates was conducted as shown in Figure 12 It can beseen that the simulated linear growth rates have agreement

with the monitored ones which further verifies the accuracyof the sampling simulation method

4 Safety Evaluation Analysis

e CBT relates to the bearing wear condition If the CBTsexceed the wear limit in the service lifetime the sphericalsteel bearing is totally damaged and cannot be used forlongitudinal thermal expansion any longer erefore it isnecessary to analyze whether the CBTs at bridge site canexceed the wear limit in the service lifetime

41 EvaluationMethod e failure probabilities of sphericalsteel bearings in the service lifetime (ie the probabilities ofCBTover the wear limit in the service lifetime) are calculatedas follows

Pf 1minusF Mius1113960 11139611113872 1113873 P Mius gt Mius1113960 11139611113872 1113873 (6a)

or

Pf 1minusF Mids1113960 11139611113872 1113873 P Mids gt Mids1113960 11139611113872 1113873 (6b)

Mius 1113944

J

j1Nius(j) (6c)

Mids 1113944

J

j1Nids(j) (6d)

where Pf denotes the failure probability [Mius] and [Mids]

denote the wear limits of Biu and Bid respectivelyF([Mius]) and F([Mids]) denote the cumulative distribu-tion functions with assigned values [Mius] and [Mids]respectively namely F(Mius [Mius]) and F(Mids

[Mids]) Nius(j) and Nids(j) denote the jth values of Niusand Nids in the service lifetime respectively which can beobtained through sampling simulation as introduced in theSection 32 and J denotes the total number of passed trainsin the service lifetime

F(Mius) and F(Mius) are unknown which should befirst determined by the following two steps

(1) By referring to the sampling simulation method inthe Section 32 the values of Nius or Nids in theservice life are simulated through Monte Carlosampling en the values of Nius or Nids aresummed using Equation (6c) or (6d) respectivelyto obtain one value of Mius or Mids is step isrepeated W times to obtain W values of Mius orMids

(2) e cumulative probability characteristics for the W

values of Mius or Mids are fitted using cumulativedistribution function e true cumulative distri-bution function of Mius and Mids is unknown sothree possible cumulative distribution functions(namely ND WD and GEVD) are considered andthe cumulative distribution function with the bestfitting effect is selected as the true cumulative

Shock and Vibration 9

0 1000 2000 3000 40000

4

8

12

p

Cum

ulat

ive t

rave

l (m

m times

103 )

Monitored result Simulated result

(a)

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

5

10

15

p

Monitored result Simulated result

(b)

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

5

10

15

p

Monitored result Simulated result

(c)

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

5

10

15

20

25

p

Monitored result Simulated result

(d)

Figure 12 Continued

0 700 1400 2100 2800 35000

5

10

15

20

25

30

35

k

Cum

ulat

ive v

alue

(mm

) mean (N1us) = 2820 mm

(a)

5 10 15 20 25 300

02

04

06

08

1

Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored result Simulated result

(b)

Figure 11 e simulated N1us and its CP (a) e simulated result (b) A comparison between the simulated and monitored results

10 Shock and Vibration

distribution function of Mius and Mids after least-square fitting

After F(Mius) and F(Mius) are determined [Mius] and[Mids] are substituted in Equations (6a) and (6b) to obtainthe value of Pf Furthermore the reliability level β can becalculated using Pf as follows

Pf 1113946+infin

β

12π

radic eminusx22

dx (7)

If the calculated reliability level β is larger than the targetreliability level [β] (ie βgt [β]) it can be inferred that thebearing is still in a good service state otherwise (ie βle [β])it can be inferred that the bearing is in a poor service stateand should be replaced in time According to the Eurocode[21] the target reliability level [β] for newly built bridgeswith a 100-year design lifetime is 38 which corresponds toa failure probability Pf 72 times 10minus5 However theDashengguan Railway Bridge has been in operation for 8years and it is not a newly built bridge From the viewpointof bridge reliability a reduction in the uncertainties of theload and resistance parameters decreases the failure prob-ability of existing bridge structures indicating the possibilityof adopting a lower reliability level for the evaluation ofexisting bridges rather than the reliability level that shouldbe used for newly built bridges [22] According to Kotesrsquoresearch [22] the target reliability level [β] with a 90-yearremaining lifetime is 3773 so the target reliability level [β]

of the Dashengguan Railway Bridge with a 92-yearremaining lifetime is 3778 using the linear interpolationcalculation namely [β] 3778

42 Evaluation Results Figure 8 reveals that M3d has thefastest growth so M3d is taken as an example to calculate itsfailure probability and evaluate the wear condition of B3d inthe service lifetime Before evaluation three parameters

should be determined namely J [M3ds] and W For theparameter J the designed service life of the NanjingDashengguan Yangtze River Bridge is 100 years and thenumber of passed trains in 2017 was 84343 so the value of J

is 8434300 for the parameter [M3ds] the China codespecifies that the linear wear rate of PTFE in a spherical steelbearing should be less than 15 μmkm and that the thicknessof PTFE should be more than 7mm and less than 8mm(ie only a 1mm wear thickness is allowed for an 8mmthickness of PTFE) [23] so the value of [M3ds] is 667 km forthe spherical steel bearing for the parameter W a larger W

can provide better evaluation results and the value of W is1000 for evaluation in this research

After the three parameters are determined the bearingwear life is evaluated by the following seven steps

(1) e values of N3d in 100-year service life are sim-ulated through Monte Carlo simulation usingEquation (4b) and 8434300 values of N3ds areobtained

(2) All the 8434300 values of N3ds are summed to obtainthe M3ds of B3d in 100-year service life usingEquation (6d)

(3) Steps (1) and (2) are repeated 1000 times to obtain1000 values of M3ds as shown in Figure 13(a) e1000 values of M3ds show good uniform and randomcharacteristics and these values can be used to an-alyze the cumulative probability characteristics

(4) e cumulative probability of M3ds is calculated andfitted by the ND WD and GEVD with the fittedresults shown in Figure 13(b) Both the ND andGEVD can well describe the cumulative probabilitycharacteristics of M3ds and GEVD is chosen for thetrue cumulative distribution function of M3ds

(5) e failure probability Pf of CBTover the upper limit[M3ds] for bearing B3d is about 10minus8 calculated by

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

5

10

15

20

25

p

Monitored result Simulated result

(e)

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

4

8

12

p

Monitored result Simulated result

(f )

Figure 12e simulated Mius and Mids (a) M1us and M1u (b) M1ds and M1d (c) M2us and M2u (d) M2ds and M2d (e) M3us and M3u(f ) M3ds and M3d

Shock and Vibration 11

Equation (6b) which means that M3ds has littleprobability of exceeding [M3ds] because the simu-lated 1000 values of M3ds are within a small interval[5874 km 5931 km]

(6) e reliability level β is 5612 calculated by Equation(7) which is larger than the target reliability level3778 indicating that bearing B3d does not reach thewear limit in the service life

(7) Because the linear growth rate of CBT for B3d isfaster than those of the other spherical steel bearingsit can be inferred that the other spherical steelbearings do not reach the wear limit as well in theservice life

5 Conclusions

Based on the monitored dynamic displacement data of steeltruss arch bridge bearings the CDD under the action ofa single train and the CBT under the continual actions ofmany passed trains are investigated Furthermore theprobability statistics and the Monte Carlo sampling simu-lationmethod for CDD are studied and the safety evaluationmethod for the bearing wear life is proposed using a re-liability index regarding the failure probability of monitoredCBT over the wear limit in the service lifetime e mainconclusions are drawn as follows

(1) rough monitored data analysis the monitoredlongitudinal displacement data mainly consist oftemperature-induced static displacements and train-induced dynamic displacements e wavelet packetdecomposition method can effectively extract thesmall-amplitude dynamic displacements caused bytrain loads from the monitored data All the CDDscontain uniform and random characteristics and allthe CBTs have an obvious linear correlation with the

number of passed trains but their linear growth ratesare different

(2) rough probability statistics analysis GEVD canwell describe the cumulative probability character-istics of CDDeMonte Carlo sampling simulationwas performed using GEVD to simulate CDD andCBT and the results show that the simulated cu-mulative probabilities of CDD have agreement withthemonitored ones of CDD and the simulated lineargrowth rates of CBT have agreement with themonitored ones of CBT

(3) rough safety evaluation analysis the safety eval-uation method for the bearing wear life is proposedusing a reliability index and it is finally judgedwhether the CBTs can exceed the wear limit in theservice lifetime e evaluation results show that allthe spherical steel bearings do not exceed the wearlimit in the service life

Notation

Biu ith spherical steel bearing at theupstream

Bid ith spherical steel bearing at thedownstream

CBT cumulative bearing travelCDD cumulative dynamic displacementCP cumulative probabilityDiu monitored longitudinal

displacement data at the ithspherical steel bearing Biu

Did monitored longitudinaldisplacement data at the ithspherical steel bearing Bid

1113957D(k) dynamic displacement caused by thekth passed train

0 200 400 600 800 1000585

5875

59

5925

595

The number of simulations

M3

ds (k

m)

(a)

M3ds (km)587 588 589 59 591 592 593 5940

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y

Cumulative probabilityGEVD

NDWD

(b)

Figure 13 e 1000 values of M3ds and their cumulative probabilities (a)1000M3ds (b) Cumulative probability of M3ds and its fittedresults

12 Shock and Vibration

1113957Dj(k) jth value in 1113957D(k)

F([Mius]) cumulative distribution functionwith assigned value [Mius]

F([Mids]) cumulative distribution functionwith assigned value [Mids]

GEVD general extreme value distributionG(Niu) cumulative distribution function of

NiuG(Nid) cumulative distribution function of

NidG(Nius) cumulative distribution function of

NiusG(Nids) cumulative distribution function of

NidsLVDT linear variable differential

transformermean(Niu) average value of 3495 CDDs for each

Niumean(Nid) average value of 3495 CDDs for each

NidM(p) CBT caused by p passed trainsMiu CBT of BiuMid CBT of Bid[Mius] wear limit of Biu[Mids] wear limit of BidN(k) cumulative travel of bearing caused

by the kth passed trainNiu set of N(1) N(2) N(3495)

for the bearings Biu

Nid set of N(1) N(2) N(3495)

for the bearings BidNius simulated CDDs of NiuNids simulated CDDs of NidNUM(Niu) total number of NiuNUM(Nid) total number of NidNUM(Niu ltNiu(j)) number of CDDs in Niu that are

smaller than Niu(j)

NUM(Nid ltNid(j)) number of CDDs in Nid that aresmaller than Nid(j)

ND normal distributionPTFE poly-tetra-fluoro-ethyleneP(Niu(j)) CP of the jth value Niu(j) in NiuP(Nid(j)) CP of the jth value Nid(j) in NidPf failure probabilityR value of cumulative probabilityWD Weibull distributionxjm wavelet packet coefficient in themth

frequency band of the jth scale[β] target reliability level

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors thank the Natural Science Foundation ofJiangsu Province of China (BK20180652) and the ChinaPostdoctoral Science Foundation (2017M621865)

References

[1] J W Zhan H Xia N Zhang and Y Lu ldquoA diagnosis methodfor bridge rubber support disease based on impact responsesrdquoJournal of Vibration and Shock vol 32 no 8 pp 153ndash1572013 in Chinese

[2] B Wei T H Yang L Z Jiang and X H He ldquoEffects offriction-based fixed bearings on the seismic vulnerability ofa high-speed railway continuous bridgerdquo Advances inStructural Engineering vol 21 no 5 pp 643ndash657 2018

[3] J Oh C Jang and J H Kim ldquoA study on the characteristics ofbridge bearings behavior by finite element analysis and modeltestrdquo Journal of Vibroengineering vol 17 no 5 pp 2559ndash2571 2015

[4] J J Huang Q Su L C Zhang Y J Li and B Liu ldquoRapidtreatment technique of diseases of the rocking axle bearing ofrailway simply supported beam bridgerdquo Applied Mechanicsand Materials vol 351-352 pp 1440ndash1444 2013

[5] A S Deshpande and J M C Kishen ldquoFatigue crack prop-agation in rocker and roller-rocker bearings of railway steelbridgesrdquo Engineering Fracture Mechanics vol 77 no 9pp 1454ndash1466 2010

[6] Q J Shi ldquoDefect analysis and reconstruction method forexisting railway bridge bearingrdquo Railway Engineering vol 57no 10 pp 12ndash14 2017

[7] S Y Yan ldquoAn analysis of causes for defects of roller bearingand treatment measuresrdquo Railway Standard Design vol 2002no 6 pp 30ndash32 2002 in Chinese

[8] H Wang Study on Damage Identification Method for RailwayBridge Rubber Bearings Based on Dynamic Responses BeijingJiaotong University Beijing China 2014 in Chinese

[9] Z Qiao Study on the Bridge Bearing Disease IdentificationTechnique Based on the Natural Frequency Change Rate EastChina Jiaotong University Nanchang China 2014 inChinese

[10] S L Chen Y Q Liu and Y B Zhang ldquoEffects of bearingdamage on vibration characteristics of a large span steel trussbridgerdquo Journal of Vibration and Shock vol 36 no 13pp 195ndash200 2017 in Chinese

[11] F Q Hu J Chen Y B Xu and X M Feng ldquoAnalysis ofmechanical properties of continuous T-beam bridge withdiseased bearingsrdquo Journal of Nanchang University Engi-neering and Technology vol 37 no 2 pp 123ndash127 2015 inChinese

[12] H Zhou K Liu G Shi Y Q Wang Y J Shi andG De Roeck ldquoFatigue assessment of a composite railwaybridge for high speed trains Part I modeling and fatiguecritical detailsrdquo Journal of Constructional Steel Researchvol 82 pp 234ndash245 2013

[13] T Guo J Liu and L Y Huang ldquoInvestigation and control ofexcessive cumulative girder movements of long-span steelsuspension bridgesrdquo Engineering Structures vol 125pp 217ndash226 2016

[14] 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 04014099 2015

Shock and Vibration 13

[15] Y L Ding and G X Wang ldquoEvaluation of dynamic loadfactors for a high-speed railway truss arch bridgerdquo Shock andVibration vol 2016 Article ID 5310769 15 pages 2016

[16] G X Wang Y L Ding Y S Song L Y Wu Q Yue andG H Mao ldquoDetection and location of the degraded bearingsbased on monitoring the longitudinal expansion performanceof the main girder of the Dashengguan Yangtze BridgerdquoJournal of Performance of Constructed Facilities vol 30 no 4article 04015074 2016

[17] G X Wang and Y L Ding ldquoResearch on monitoring tem-perature difference from cross sections of steel truss archgirder of Dashengguan Yangtze Bridgerdquo International Journalof Steel Structures vol 15 no 3 pp 647ndash660 2015

[18] M Y Gokhale and K D Kaur ldquoTime domain signal analysisusing wavelet packet decomposition approachrdquo InternationalJournal of communications Networks and System Sciencesvol 3 no 3 pp 321ndash329 2010

[19] T Figlus S Liscak A Wilk and B Aazarz ldquoConditionmonitoring of engine timing system by using wavelet packetdecomposition of a acoustic signalrdquo Journal of MechanicalScience and Technology vol 28 no 5 pp 1663ndash1671 2014

[20] R Wald T M Khoshgoftaar and J C Sloan ldquoFeature se-lection for optimization of wavelet packet decomposition inreliability analysis of systemsrdquo International Journal on Ar-tificial Intelligence Tools vol 22 no 5 article 1360011 2013

[21] EN 1991-1-5 Eurocode 1 Actions on StructuresmdashPart 1ndash5General Actionsmdashampermal Actions European Committee forStandardization (CEN) Brussels Belgium Europe 2004

[22] P Kotes and J Vican ldquoReliability levels for existing bridgesevaluation according to eurocodesrdquo Procedia Engineeringvol 40 pp 211ndash216 2012

[23] TBT 3320-2013 Spherical Bearings for Railway BridgesChina Railway Publishing House Beijing China 2013 inChinese

14 Shock and Vibration

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

distribution functions are shown in Figure 10 It can beseen that GEVDhas the best fitting results so GEVD ischosen to describe the cumulative probability char-acteristics of N1u e formula is expressed as

G N1u1113872 1113873 exp minus 1 + rN1u minus b

a1113888 11138891113890 1113891

minus1r⎡⎣ ⎤⎦ (3)

where G(N1u) denotes the cumulative distributionfunction of N1u which follows the generalized ex-treme value distribution r denotes the shape pa-rameter a denotes the scale parameter and b

denotes the location parameter e fitted values ofthese three parameters for G(N1u) are r 03058a 08163 and b 20294

Based on the method above the best cumulative dis-tribution functions of N1u minusN3u and N1d minusN3d are allGEVD as shown in Figures 10(a)ndash10(f ) It can be seen thatGEVD can well describe the cumulative probability char-acteristics of N1u minusN3u and N1d minusN3d

32 Sampling SimulationMethod Since Niu and Nid followGEVDs their GEVDs can be utilized for Monte Carlosampling to obtain a demanded number of CDDs Specif-ically if the demanded number of CDDs is K K values are

0 1000 2000 3000 40000

5

10

15

M1uM1d

p

Cum

ulat

ive t

rave

l (m

m times

103 )

(a)

0 1000 2000 3000 40000

5

10

15

M2uM2d

p

Cum

ulat

ive t

rave

l (m

m times

103 )

(b)

0 1000 2000 3000 40000

5

10

15

20

25

M3uM3d

p

Cum

ulat

ive t

rave

l (m

m times

103 )

(c)

Figure 8 e M1u minusM3u and M1u minusM3u caused by many passed trains (a) M1u and M1d (b) M2u and M2d (c) M3u and M3d

5 10 15 20 25 300

02

04

06

08

1

Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Figure 9 e cumulative probability of N1u

Shock and Vibration 7

5 10 15 20 25 300

02

04

06

08

1

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

WD

ND

GEVD

GEVDND

WDMonitored curve

(a)

10 20 30 40 500

02

04

06

08

1

r = 04765a = 10599b = 25108

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored curve Fitted curve

(b)

5 10 15 20 25 300

02

04

06

08

1

r = 01926a = 12320b = 28604

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored curveFitted curve

(c)

2 4 6 8 10 12 140

02

04

06

08

1

r = 00493a = 08440b = 24302

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored curveFitted curve

(d)

5 10 15 20 250

02

04

06

08

1

r = 00830a = 18184b = 53395

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored curveFitted curve

(e)

5 10 15 20 250

02

04

06

08

1

r = 01834a = 19436b = 54517

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored curveFitted curve

(f )

Figure 10 GEVDs of N1u minusN6u and N1u minusN6u (a) N1u (b) N1d (c) N2u (d) N2d (e) N3u (f ) N3d

8 Shock and Vibration

uniformly and randomly sampled from the interval (0 1)and then the demanded number of CDDs can be simulatedby substituting the K values into the cumulative probabilityR in the following equations

R G Nius1113872 1113873 (4a)

R G Nids1113872 1113873 (4b)

G Nius1113872 1113873 exp minus 1 + rNius minus b

a1113888 11138891113890 1113891

minus1r⎡⎣ ⎤⎦ (4c)

G Nids1113872 1113873 exp minus 1 + rNids minus b

a1113888 11138891113890 1113891

minus1r⎡⎣ ⎤⎦ (4d)

where Nius and Nids denote the simulated CDDs of Niu andNid respectively e values of r a and b have been cal-culated in the Section 32 However Niu and Nid areembedded in G(Niu) and G(Nids) so they cannot be di-rectly solved from Equations (4a) and (4b) Considering thatG(Nius) and G(Nids) are monotonically increasing func-tions Nius and Nids can be determined by the Newtoniteration method as follows

Nn+1ius N

nius minus

G Nnius1113872 1113873minusR

Gprime Nnius1113872 1113873

(5a)

Nn+1ids N

nids minus

G Nnids1113872 1113873minusR

Gprime Nnids1113872 1113873

(5b)

where Nn+1ius and Nn

ius denote the (n + 1)th and nth iterationresults of Nius respectively Nn+1

ids andNnids denote the (n + 1)

th and nth iteration results of Nids respectively Gprime(Nnius)

and Gprime(Nnids) denote the first-order derivatives of GEVD

(ie the probability density functions of GEVD) e iter-ation will terminate when the absolute difference betweentwo adjacent iterations is less than 10minus2 e last iterationresult is treated as the values of Nius and Nids

33 ampe Sampling Simulation Result Based on the samplingsimulation method above the simulation results of 3495N1us are shown in Figure 11(a) e mean value of thesimulated result is 2820mm which is very close to the meanvalue of Niu ie 2904mm Furthermore the CP of thesimulated result is calculated and then compared with themonitored CP as shown in Figure 11(b) It can be seen thatthe simulated CP coincides with the monitored CP whichverifies the accuracy of the sampling simulation method

Furthermore the simulated 3495 values of Nius andNidsare accumulated using Equation (1) to obtain the simulatedCBT (ie Mius and Mids) as shown in Figure 12 It can beseen that the simulated Mius and Mids present good lineargrowth characteristics with the number of passed trains Toverify the accuracy of the simulated linear growth ratesa comparison between the monitored and simulated lineargrowth rates was conducted as shown in Figure 12 It can beseen that the simulated linear growth rates have agreement

with the monitored ones which further verifies the accuracyof the sampling simulation method

4 Safety Evaluation Analysis

e CBT relates to the bearing wear condition If the CBTsexceed the wear limit in the service lifetime the sphericalsteel bearing is totally damaged and cannot be used forlongitudinal thermal expansion any longer erefore it isnecessary to analyze whether the CBTs at bridge site canexceed the wear limit in the service lifetime

41 EvaluationMethod e failure probabilities of sphericalsteel bearings in the service lifetime (ie the probabilities ofCBTover the wear limit in the service lifetime) are calculatedas follows

Pf 1minusF Mius1113960 11139611113872 1113873 P Mius gt Mius1113960 11139611113872 1113873 (6a)

or

Pf 1minusF Mids1113960 11139611113872 1113873 P Mids gt Mids1113960 11139611113872 1113873 (6b)

Mius 1113944

J

j1Nius(j) (6c)

Mids 1113944

J

j1Nids(j) (6d)

where Pf denotes the failure probability [Mius] and [Mids]

denote the wear limits of Biu and Bid respectivelyF([Mius]) and F([Mids]) denote the cumulative distribu-tion functions with assigned values [Mius] and [Mids]respectively namely F(Mius [Mius]) and F(Mids

[Mids]) Nius(j) and Nids(j) denote the jth values of Niusand Nids in the service lifetime respectively which can beobtained through sampling simulation as introduced in theSection 32 and J denotes the total number of passed trainsin the service lifetime

F(Mius) and F(Mius) are unknown which should befirst determined by the following two steps

(1) By referring to the sampling simulation method inthe Section 32 the values of Nius or Nids in theservice life are simulated through Monte Carlosampling en the values of Nius or Nids aresummed using Equation (6c) or (6d) respectivelyto obtain one value of Mius or Mids is step isrepeated W times to obtain W values of Mius orMids

(2) e cumulative probability characteristics for the W

values of Mius or Mids are fitted using cumulativedistribution function e true cumulative distri-bution function of Mius and Mids is unknown sothree possible cumulative distribution functions(namely ND WD and GEVD) are considered andthe cumulative distribution function with the bestfitting effect is selected as the true cumulative

Shock and Vibration 9

0 1000 2000 3000 40000

4

8

12

p

Cum

ulat

ive t

rave

l (m

m times

103 )

Monitored result Simulated result

(a)

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

5

10

15

p

Monitored result Simulated result

(b)

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

5

10

15

p

Monitored result Simulated result

(c)

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

5

10

15

20

25

p

Monitored result Simulated result

(d)

Figure 12 Continued

0 700 1400 2100 2800 35000

5

10

15

20

25

30

35

k

Cum

ulat

ive v

alue

(mm

) mean (N1us) = 2820 mm

(a)

5 10 15 20 25 300

02

04

06

08

1

Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored result Simulated result

(b)

Figure 11 e simulated N1us and its CP (a) e simulated result (b) A comparison between the simulated and monitored results

10 Shock and Vibration

distribution function of Mius and Mids after least-square fitting

After F(Mius) and F(Mius) are determined [Mius] and[Mids] are substituted in Equations (6a) and (6b) to obtainthe value of Pf Furthermore the reliability level β can becalculated using Pf as follows

Pf 1113946+infin

β

12π

radic eminusx22

dx (7)

If the calculated reliability level β is larger than the targetreliability level [β] (ie βgt [β]) it can be inferred that thebearing is still in a good service state otherwise (ie βle [β])it can be inferred that the bearing is in a poor service stateand should be replaced in time According to the Eurocode[21] the target reliability level [β] for newly built bridgeswith a 100-year design lifetime is 38 which corresponds toa failure probability Pf 72 times 10minus5 However theDashengguan Railway Bridge has been in operation for 8years and it is not a newly built bridge From the viewpointof bridge reliability a reduction in the uncertainties of theload and resistance parameters decreases the failure prob-ability of existing bridge structures indicating the possibilityof adopting a lower reliability level for the evaluation ofexisting bridges rather than the reliability level that shouldbe used for newly built bridges [22] According to Kotesrsquoresearch [22] the target reliability level [β] with a 90-yearremaining lifetime is 3773 so the target reliability level [β]

of the Dashengguan Railway Bridge with a 92-yearremaining lifetime is 3778 using the linear interpolationcalculation namely [β] 3778

42 Evaluation Results Figure 8 reveals that M3d has thefastest growth so M3d is taken as an example to calculate itsfailure probability and evaluate the wear condition of B3d inthe service lifetime Before evaluation three parameters

should be determined namely J [M3ds] and W For theparameter J the designed service life of the NanjingDashengguan Yangtze River Bridge is 100 years and thenumber of passed trains in 2017 was 84343 so the value of J

is 8434300 for the parameter [M3ds] the China codespecifies that the linear wear rate of PTFE in a spherical steelbearing should be less than 15 μmkm and that the thicknessof PTFE should be more than 7mm and less than 8mm(ie only a 1mm wear thickness is allowed for an 8mmthickness of PTFE) [23] so the value of [M3ds] is 667 km forthe spherical steel bearing for the parameter W a larger W

can provide better evaluation results and the value of W is1000 for evaluation in this research

After the three parameters are determined the bearingwear life is evaluated by the following seven steps

(1) e values of N3d in 100-year service life are sim-ulated through Monte Carlo simulation usingEquation (4b) and 8434300 values of N3ds areobtained

(2) All the 8434300 values of N3ds are summed to obtainthe M3ds of B3d in 100-year service life usingEquation (6d)

(3) Steps (1) and (2) are repeated 1000 times to obtain1000 values of M3ds as shown in Figure 13(a) e1000 values of M3ds show good uniform and randomcharacteristics and these values can be used to an-alyze the cumulative probability characteristics

(4) e cumulative probability of M3ds is calculated andfitted by the ND WD and GEVD with the fittedresults shown in Figure 13(b) Both the ND andGEVD can well describe the cumulative probabilitycharacteristics of M3ds and GEVD is chosen for thetrue cumulative distribution function of M3ds

(5) e failure probability Pf of CBTover the upper limit[M3ds] for bearing B3d is about 10minus8 calculated by

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

5

10

15

20

25

p

Monitored result Simulated result

(e)

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

4

8

12

p

Monitored result Simulated result

(f )

Figure 12e simulated Mius and Mids (a) M1us and M1u (b) M1ds and M1d (c) M2us and M2u (d) M2ds and M2d (e) M3us and M3u(f ) M3ds and M3d

Shock and Vibration 11

Equation (6b) which means that M3ds has littleprobability of exceeding [M3ds] because the simu-lated 1000 values of M3ds are within a small interval[5874 km 5931 km]

(6) e reliability level β is 5612 calculated by Equation(7) which is larger than the target reliability level3778 indicating that bearing B3d does not reach thewear limit in the service life

(7) Because the linear growth rate of CBT for B3d isfaster than those of the other spherical steel bearingsit can be inferred that the other spherical steelbearings do not reach the wear limit as well in theservice life

5 Conclusions

Based on the monitored dynamic displacement data of steeltruss arch bridge bearings the CDD under the action ofa single train and the CBT under the continual actions ofmany passed trains are investigated Furthermore theprobability statistics and the Monte Carlo sampling simu-lationmethod for CDD are studied and the safety evaluationmethod for the bearing wear life is proposed using a re-liability index regarding the failure probability of monitoredCBT over the wear limit in the service lifetime e mainconclusions are drawn as follows

(1) rough monitored data analysis the monitoredlongitudinal displacement data mainly consist oftemperature-induced static displacements and train-induced dynamic displacements e wavelet packetdecomposition method can effectively extract thesmall-amplitude dynamic displacements caused bytrain loads from the monitored data All the CDDscontain uniform and random characteristics and allthe CBTs have an obvious linear correlation with the

number of passed trains but their linear growth ratesare different

(2) rough probability statistics analysis GEVD canwell describe the cumulative probability character-istics of CDDeMonte Carlo sampling simulationwas performed using GEVD to simulate CDD andCBT and the results show that the simulated cu-mulative probabilities of CDD have agreement withthemonitored ones of CDD and the simulated lineargrowth rates of CBT have agreement with themonitored ones of CBT

(3) rough safety evaluation analysis the safety eval-uation method for the bearing wear life is proposedusing a reliability index and it is finally judgedwhether the CBTs can exceed the wear limit in theservice lifetime e evaluation results show that allthe spherical steel bearings do not exceed the wearlimit in the service life

Notation

Biu ith spherical steel bearing at theupstream

Bid ith spherical steel bearing at thedownstream

CBT cumulative bearing travelCDD cumulative dynamic displacementCP cumulative probabilityDiu monitored longitudinal

displacement data at the ithspherical steel bearing Biu

Did monitored longitudinaldisplacement data at the ithspherical steel bearing Bid

1113957D(k) dynamic displacement caused by thekth passed train

0 200 400 600 800 1000585

5875

59

5925

595

The number of simulations

M3

ds (k

m)

(a)

M3ds (km)587 588 589 59 591 592 593 5940

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y

Cumulative probabilityGEVD

NDWD

(b)

Figure 13 e 1000 values of M3ds and their cumulative probabilities (a)1000M3ds (b) Cumulative probability of M3ds and its fittedresults

12 Shock and Vibration

1113957Dj(k) jth value in 1113957D(k)

F([Mius]) cumulative distribution functionwith assigned value [Mius]

F([Mids]) cumulative distribution functionwith assigned value [Mids]

GEVD general extreme value distributionG(Niu) cumulative distribution function of

NiuG(Nid) cumulative distribution function of

NidG(Nius) cumulative distribution function of

NiusG(Nids) cumulative distribution function of

NidsLVDT linear variable differential

transformermean(Niu) average value of 3495 CDDs for each

Niumean(Nid) average value of 3495 CDDs for each

NidM(p) CBT caused by p passed trainsMiu CBT of BiuMid CBT of Bid[Mius] wear limit of Biu[Mids] wear limit of BidN(k) cumulative travel of bearing caused

by the kth passed trainNiu set of N(1) N(2) N(3495)

for the bearings Biu

Nid set of N(1) N(2) N(3495)

for the bearings BidNius simulated CDDs of NiuNids simulated CDDs of NidNUM(Niu) total number of NiuNUM(Nid) total number of NidNUM(Niu ltNiu(j)) number of CDDs in Niu that are

smaller than Niu(j)

NUM(Nid ltNid(j)) number of CDDs in Nid that aresmaller than Nid(j)

ND normal distributionPTFE poly-tetra-fluoro-ethyleneP(Niu(j)) CP of the jth value Niu(j) in NiuP(Nid(j)) CP of the jth value Nid(j) in NidPf failure probabilityR value of cumulative probabilityWD Weibull distributionxjm wavelet packet coefficient in themth

frequency band of the jth scale[β] target reliability level

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors thank the Natural Science Foundation ofJiangsu Province of China (BK20180652) and the ChinaPostdoctoral Science Foundation (2017M621865)

References

[1] J W Zhan H Xia N Zhang and Y Lu ldquoA diagnosis methodfor bridge rubber support disease based on impact responsesrdquoJournal of Vibration and Shock vol 32 no 8 pp 153ndash1572013 in Chinese

[2] B Wei T H Yang L Z Jiang and X H He ldquoEffects offriction-based fixed bearings on the seismic vulnerability ofa high-speed railway continuous bridgerdquo Advances inStructural Engineering vol 21 no 5 pp 643ndash657 2018

[3] J Oh C Jang and J H Kim ldquoA study on the characteristics ofbridge bearings behavior by finite element analysis and modeltestrdquo Journal of Vibroengineering vol 17 no 5 pp 2559ndash2571 2015

[4] J J Huang Q Su L C Zhang Y J Li and B Liu ldquoRapidtreatment technique of diseases of the rocking axle bearing ofrailway simply supported beam bridgerdquo Applied Mechanicsand Materials vol 351-352 pp 1440ndash1444 2013

[5] A S Deshpande and J M C Kishen ldquoFatigue crack prop-agation in rocker and roller-rocker bearings of railway steelbridgesrdquo Engineering Fracture Mechanics vol 77 no 9pp 1454ndash1466 2010

[6] Q J Shi ldquoDefect analysis and reconstruction method forexisting railway bridge bearingrdquo Railway Engineering vol 57no 10 pp 12ndash14 2017

[7] S Y Yan ldquoAn analysis of causes for defects of roller bearingand treatment measuresrdquo Railway Standard Design vol 2002no 6 pp 30ndash32 2002 in Chinese

[8] H Wang Study on Damage Identification Method for RailwayBridge Rubber Bearings Based on Dynamic Responses BeijingJiaotong University Beijing China 2014 in Chinese

[9] Z Qiao Study on the Bridge Bearing Disease IdentificationTechnique Based on the Natural Frequency Change Rate EastChina Jiaotong University Nanchang China 2014 inChinese

[10] S L Chen Y Q Liu and Y B Zhang ldquoEffects of bearingdamage on vibration characteristics of a large span steel trussbridgerdquo Journal of Vibration and Shock vol 36 no 13pp 195ndash200 2017 in Chinese

[11] F Q Hu J Chen Y B Xu and X M Feng ldquoAnalysis ofmechanical properties of continuous T-beam bridge withdiseased bearingsrdquo Journal of Nanchang University Engi-neering and Technology vol 37 no 2 pp 123ndash127 2015 inChinese

[12] H Zhou K Liu G Shi Y Q Wang Y J Shi andG De Roeck ldquoFatigue assessment of a composite railwaybridge for high speed trains Part I modeling and fatiguecritical detailsrdquo Journal of Constructional Steel Researchvol 82 pp 234ndash245 2013

[13] T Guo J Liu and L Y Huang ldquoInvestigation and control ofexcessive cumulative girder movements of long-span steelsuspension bridgesrdquo Engineering Structures vol 125pp 217ndash226 2016

[14] 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 04014099 2015

Shock and Vibration 13

[15] Y L Ding and G X Wang ldquoEvaluation of dynamic loadfactors for a high-speed railway truss arch bridgerdquo Shock andVibration vol 2016 Article ID 5310769 15 pages 2016

[16] G X Wang Y L Ding Y S Song L Y Wu Q Yue andG H Mao ldquoDetection and location of the degraded bearingsbased on monitoring the longitudinal expansion performanceof the main girder of the Dashengguan Yangtze BridgerdquoJournal of Performance of Constructed Facilities vol 30 no 4article 04015074 2016

[17] G X Wang and Y L Ding ldquoResearch on monitoring tem-perature difference from cross sections of steel truss archgirder of Dashengguan Yangtze Bridgerdquo International Journalof Steel Structures vol 15 no 3 pp 647ndash660 2015

[18] M Y Gokhale and K D Kaur ldquoTime domain signal analysisusing wavelet packet decomposition approachrdquo InternationalJournal of communications Networks and System Sciencesvol 3 no 3 pp 321ndash329 2010

[19] T Figlus S Liscak A Wilk and B Aazarz ldquoConditionmonitoring of engine timing system by using wavelet packetdecomposition of a acoustic signalrdquo Journal of MechanicalScience and Technology vol 28 no 5 pp 1663ndash1671 2014

[20] R Wald T M Khoshgoftaar and J C Sloan ldquoFeature se-lection for optimization of wavelet packet decomposition inreliability analysis of systemsrdquo International Journal on Ar-tificial Intelligence Tools vol 22 no 5 article 1360011 2013

[21] EN 1991-1-5 Eurocode 1 Actions on StructuresmdashPart 1ndash5General Actionsmdashampermal Actions European Committee forStandardization (CEN) Brussels Belgium Europe 2004

[22] P Kotes and J Vican ldquoReliability levels for existing bridgesevaluation according to eurocodesrdquo Procedia Engineeringvol 40 pp 211ndash216 2012

[23] TBT 3320-2013 Spherical Bearings for Railway BridgesChina Railway Publishing House Beijing China 2013 inChinese

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Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

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Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

5 10 15 20 25 300

02

04

06

08

1

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

WD

ND

GEVD

GEVDND

WDMonitored curve

(a)

10 20 30 40 500

02

04

06

08

1

r = 04765a = 10599b = 25108

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored curve Fitted curve

(b)

5 10 15 20 25 300

02

04

06

08

1

r = 01926a = 12320b = 28604

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored curveFitted curve

(c)

2 4 6 8 10 12 140

02

04

06

08

1

r = 00493a = 08440b = 24302

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored curveFitted curve

(d)

5 10 15 20 250

02

04

06

08

1

r = 00830a = 18184b = 53395

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored curveFitted curve

(e)

5 10 15 20 250

02

04

06

08

1

r = 01834a = 19436b = 54517

0Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored curveFitted curve

(f )

Figure 10 GEVDs of N1u minusN6u and N1u minusN6u (a) N1u (b) N1d (c) N2u (d) N2d (e) N3u (f ) N3d

8 Shock and Vibration

uniformly and randomly sampled from the interval (0 1)and then the demanded number of CDDs can be simulatedby substituting the K values into the cumulative probabilityR in the following equations

R G Nius1113872 1113873 (4a)

R G Nids1113872 1113873 (4b)

G Nius1113872 1113873 exp minus 1 + rNius minus b

a1113888 11138891113890 1113891

minus1r⎡⎣ ⎤⎦ (4c)

G Nids1113872 1113873 exp minus 1 + rNids minus b

a1113888 11138891113890 1113891

minus1r⎡⎣ ⎤⎦ (4d)

where Nius and Nids denote the simulated CDDs of Niu andNid respectively e values of r a and b have been cal-culated in the Section 32 However Niu and Nid areembedded in G(Niu) and G(Nids) so they cannot be di-rectly solved from Equations (4a) and (4b) Considering thatG(Nius) and G(Nids) are monotonically increasing func-tions Nius and Nids can be determined by the Newtoniteration method as follows

Nn+1ius N

nius minus

G Nnius1113872 1113873minusR

Gprime Nnius1113872 1113873

(5a)

Nn+1ids N

nids minus

G Nnids1113872 1113873minusR

Gprime Nnids1113872 1113873

(5b)

where Nn+1ius and Nn

ius denote the (n + 1)th and nth iterationresults of Nius respectively Nn+1

ids andNnids denote the (n + 1)

th and nth iteration results of Nids respectively Gprime(Nnius)

and Gprime(Nnids) denote the first-order derivatives of GEVD

(ie the probability density functions of GEVD) e iter-ation will terminate when the absolute difference betweentwo adjacent iterations is less than 10minus2 e last iterationresult is treated as the values of Nius and Nids

33 ampe Sampling Simulation Result Based on the samplingsimulation method above the simulation results of 3495N1us are shown in Figure 11(a) e mean value of thesimulated result is 2820mm which is very close to the meanvalue of Niu ie 2904mm Furthermore the CP of thesimulated result is calculated and then compared with themonitored CP as shown in Figure 11(b) It can be seen thatthe simulated CP coincides with the monitored CP whichverifies the accuracy of the sampling simulation method

Furthermore the simulated 3495 values of Nius andNidsare accumulated using Equation (1) to obtain the simulatedCBT (ie Mius and Mids) as shown in Figure 12 It can beseen that the simulated Mius and Mids present good lineargrowth characteristics with the number of passed trains Toverify the accuracy of the simulated linear growth ratesa comparison between the monitored and simulated lineargrowth rates was conducted as shown in Figure 12 It can beseen that the simulated linear growth rates have agreement

with the monitored ones which further verifies the accuracyof the sampling simulation method

4 Safety Evaluation Analysis

e CBT relates to the bearing wear condition If the CBTsexceed the wear limit in the service lifetime the sphericalsteel bearing is totally damaged and cannot be used forlongitudinal thermal expansion any longer erefore it isnecessary to analyze whether the CBTs at bridge site canexceed the wear limit in the service lifetime

41 EvaluationMethod e failure probabilities of sphericalsteel bearings in the service lifetime (ie the probabilities ofCBTover the wear limit in the service lifetime) are calculatedas follows

Pf 1minusF Mius1113960 11139611113872 1113873 P Mius gt Mius1113960 11139611113872 1113873 (6a)

or

Pf 1minusF Mids1113960 11139611113872 1113873 P Mids gt Mids1113960 11139611113872 1113873 (6b)

Mius 1113944

J

j1Nius(j) (6c)

Mids 1113944

J

j1Nids(j) (6d)

where Pf denotes the failure probability [Mius] and [Mids]

denote the wear limits of Biu and Bid respectivelyF([Mius]) and F([Mids]) denote the cumulative distribu-tion functions with assigned values [Mius] and [Mids]respectively namely F(Mius [Mius]) and F(Mids

[Mids]) Nius(j) and Nids(j) denote the jth values of Niusand Nids in the service lifetime respectively which can beobtained through sampling simulation as introduced in theSection 32 and J denotes the total number of passed trainsin the service lifetime

F(Mius) and F(Mius) are unknown which should befirst determined by the following two steps

(1) By referring to the sampling simulation method inthe Section 32 the values of Nius or Nids in theservice life are simulated through Monte Carlosampling en the values of Nius or Nids aresummed using Equation (6c) or (6d) respectivelyto obtain one value of Mius or Mids is step isrepeated W times to obtain W values of Mius orMids

(2) e cumulative probability characteristics for the W

values of Mius or Mids are fitted using cumulativedistribution function e true cumulative distri-bution function of Mius and Mids is unknown sothree possible cumulative distribution functions(namely ND WD and GEVD) are considered andthe cumulative distribution function with the bestfitting effect is selected as the true cumulative

Shock and Vibration 9

0 1000 2000 3000 40000

4

8

12

p

Cum

ulat

ive t

rave

l (m

m times

103 )

Monitored result Simulated result

(a)

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

5

10

15

p

Monitored result Simulated result

(b)

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

5

10

15

p

Monitored result Simulated result

(c)

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

5

10

15

20

25

p

Monitored result Simulated result

(d)

Figure 12 Continued

0 700 1400 2100 2800 35000

5

10

15

20

25

30

35

k

Cum

ulat

ive v

alue

(mm

) mean (N1us) = 2820 mm

(a)

5 10 15 20 25 300

02

04

06

08

1

Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored result Simulated result

(b)

Figure 11 e simulated N1us and its CP (a) e simulated result (b) A comparison between the simulated and monitored results

10 Shock and Vibration

distribution function of Mius and Mids after least-square fitting

After F(Mius) and F(Mius) are determined [Mius] and[Mids] are substituted in Equations (6a) and (6b) to obtainthe value of Pf Furthermore the reliability level β can becalculated using Pf as follows

Pf 1113946+infin

β

12π

radic eminusx22

dx (7)

If the calculated reliability level β is larger than the targetreliability level [β] (ie βgt [β]) it can be inferred that thebearing is still in a good service state otherwise (ie βle [β])it can be inferred that the bearing is in a poor service stateand should be replaced in time According to the Eurocode[21] the target reliability level [β] for newly built bridgeswith a 100-year design lifetime is 38 which corresponds toa failure probability Pf 72 times 10minus5 However theDashengguan Railway Bridge has been in operation for 8years and it is not a newly built bridge From the viewpointof bridge reliability a reduction in the uncertainties of theload and resistance parameters decreases the failure prob-ability of existing bridge structures indicating the possibilityof adopting a lower reliability level for the evaluation ofexisting bridges rather than the reliability level that shouldbe used for newly built bridges [22] According to Kotesrsquoresearch [22] the target reliability level [β] with a 90-yearremaining lifetime is 3773 so the target reliability level [β]

of the Dashengguan Railway Bridge with a 92-yearremaining lifetime is 3778 using the linear interpolationcalculation namely [β] 3778

42 Evaluation Results Figure 8 reveals that M3d has thefastest growth so M3d is taken as an example to calculate itsfailure probability and evaluate the wear condition of B3d inthe service lifetime Before evaluation three parameters

should be determined namely J [M3ds] and W For theparameter J the designed service life of the NanjingDashengguan Yangtze River Bridge is 100 years and thenumber of passed trains in 2017 was 84343 so the value of J

is 8434300 for the parameter [M3ds] the China codespecifies that the linear wear rate of PTFE in a spherical steelbearing should be less than 15 μmkm and that the thicknessof PTFE should be more than 7mm and less than 8mm(ie only a 1mm wear thickness is allowed for an 8mmthickness of PTFE) [23] so the value of [M3ds] is 667 km forthe spherical steel bearing for the parameter W a larger W

can provide better evaluation results and the value of W is1000 for evaluation in this research

After the three parameters are determined the bearingwear life is evaluated by the following seven steps

(1) e values of N3d in 100-year service life are sim-ulated through Monte Carlo simulation usingEquation (4b) and 8434300 values of N3ds areobtained

(2) All the 8434300 values of N3ds are summed to obtainthe M3ds of B3d in 100-year service life usingEquation (6d)

(3) Steps (1) and (2) are repeated 1000 times to obtain1000 values of M3ds as shown in Figure 13(a) e1000 values of M3ds show good uniform and randomcharacteristics and these values can be used to an-alyze the cumulative probability characteristics

(4) e cumulative probability of M3ds is calculated andfitted by the ND WD and GEVD with the fittedresults shown in Figure 13(b) Both the ND andGEVD can well describe the cumulative probabilitycharacteristics of M3ds and GEVD is chosen for thetrue cumulative distribution function of M3ds

(5) e failure probability Pf of CBTover the upper limit[M3ds] for bearing B3d is about 10minus8 calculated by

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

5

10

15

20

25

p

Monitored result Simulated result

(e)

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

4

8

12

p

Monitored result Simulated result

(f )

Figure 12e simulated Mius and Mids (a) M1us and M1u (b) M1ds and M1d (c) M2us and M2u (d) M2ds and M2d (e) M3us and M3u(f ) M3ds and M3d

Shock and Vibration 11

Equation (6b) which means that M3ds has littleprobability of exceeding [M3ds] because the simu-lated 1000 values of M3ds are within a small interval[5874 km 5931 km]

(6) e reliability level β is 5612 calculated by Equation(7) which is larger than the target reliability level3778 indicating that bearing B3d does not reach thewear limit in the service life

(7) Because the linear growth rate of CBT for B3d isfaster than those of the other spherical steel bearingsit can be inferred that the other spherical steelbearings do not reach the wear limit as well in theservice life

5 Conclusions

Based on the monitored dynamic displacement data of steeltruss arch bridge bearings the CDD under the action ofa single train and the CBT under the continual actions ofmany passed trains are investigated Furthermore theprobability statistics and the Monte Carlo sampling simu-lationmethod for CDD are studied and the safety evaluationmethod for the bearing wear life is proposed using a re-liability index regarding the failure probability of monitoredCBT over the wear limit in the service lifetime e mainconclusions are drawn as follows

(1) rough monitored data analysis the monitoredlongitudinal displacement data mainly consist oftemperature-induced static displacements and train-induced dynamic displacements e wavelet packetdecomposition method can effectively extract thesmall-amplitude dynamic displacements caused bytrain loads from the monitored data All the CDDscontain uniform and random characteristics and allthe CBTs have an obvious linear correlation with the

number of passed trains but their linear growth ratesare different

(2) rough probability statistics analysis GEVD canwell describe the cumulative probability character-istics of CDDeMonte Carlo sampling simulationwas performed using GEVD to simulate CDD andCBT and the results show that the simulated cu-mulative probabilities of CDD have agreement withthemonitored ones of CDD and the simulated lineargrowth rates of CBT have agreement with themonitored ones of CBT

(3) rough safety evaluation analysis the safety eval-uation method for the bearing wear life is proposedusing a reliability index and it is finally judgedwhether the CBTs can exceed the wear limit in theservice lifetime e evaluation results show that allthe spherical steel bearings do not exceed the wearlimit in the service life

Notation

Biu ith spherical steel bearing at theupstream

Bid ith spherical steel bearing at thedownstream

CBT cumulative bearing travelCDD cumulative dynamic displacementCP cumulative probabilityDiu monitored longitudinal

displacement data at the ithspherical steel bearing Biu

Did monitored longitudinaldisplacement data at the ithspherical steel bearing Bid

1113957D(k) dynamic displacement caused by thekth passed train

0 200 400 600 800 1000585

5875

59

5925

595

The number of simulations

M3

ds (k

m)

(a)

M3ds (km)587 588 589 59 591 592 593 5940

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y

Cumulative probabilityGEVD

NDWD

(b)

Figure 13 e 1000 values of M3ds and their cumulative probabilities (a)1000M3ds (b) Cumulative probability of M3ds and its fittedresults

12 Shock and Vibration

1113957Dj(k) jth value in 1113957D(k)

F([Mius]) cumulative distribution functionwith assigned value [Mius]

F([Mids]) cumulative distribution functionwith assigned value [Mids]

GEVD general extreme value distributionG(Niu) cumulative distribution function of

NiuG(Nid) cumulative distribution function of

NidG(Nius) cumulative distribution function of

NiusG(Nids) cumulative distribution function of

NidsLVDT linear variable differential

transformermean(Niu) average value of 3495 CDDs for each

Niumean(Nid) average value of 3495 CDDs for each

NidM(p) CBT caused by p passed trainsMiu CBT of BiuMid CBT of Bid[Mius] wear limit of Biu[Mids] wear limit of BidN(k) cumulative travel of bearing caused

by the kth passed trainNiu set of N(1) N(2) N(3495)

for the bearings Biu

Nid set of N(1) N(2) N(3495)

for the bearings BidNius simulated CDDs of NiuNids simulated CDDs of NidNUM(Niu) total number of NiuNUM(Nid) total number of NidNUM(Niu ltNiu(j)) number of CDDs in Niu that are

smaller than Niu(j)

NUM(Nid ltNid(j)) number of CDDs in Nid that aresmaller than Nid(j)

ND normal distributionPTFE poly-tetra-fluoro-ethyleneP(Niu(j)) CP of the jth value Niu(j) in NiuP(Nid(j)) CP of the jth value Nid(j) in NidPf failure probabilityR value of cumulative probabilityWD Weibull distributionxjm wavelet packet coefficient in themth

frequency band of the jth scale[β] target reliability level

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors thank the Natural Science Foundation ofJiangsu Province of China (BK20180652) and the ChinaPostdoctoral Science Foundation (2017M621865)

References

[1] J W Zhan H Xia N Zhang and Y Lu ldquoA diagnosis methodfor bridge rubber support disease based on impact responsesrdquoJournal of Vibration and Shock vol 32 no 8 pp 153ndash1572013 in Chinese

[2] B Wei T H Yang L Z Jiang and X H He ldquoEffects offriction-based fixed bearings on the seismic vulnerability ofa high-speed railway continuous bridgerdquo Advances inStructural Engineering vol 21 no 5 pp 643ndash657 2018

[3] J Oh C Jang and J H Kim ldquoA study on the characteristics ofbridge bearings behavior by finite element analysis and modeltestrdquo Journal of Vibroengineering vol 17 no 5 pp 2559ndash2571 2015

[4] J J Huang Q Su L C Zhang Y J Li and B Liu ldquoRapidtreatment technique of diseases of the rocking axle bearing ofrailway simply supported beam bridgerdquo Applied Mechanicsand Materials vol 351-352 pp 1440ndash1444 2013

[5] A S Deshpande and J M C Kishen ldquoFatigue crack prop-agation in rocker and roller-rocker bearings of railway steelbridgesrdquo Engineering Fracture Mechanics vol 77 no 9pp 1454ndash1466 2010

[6] Q J Shi ldquoDefect analysis and reconstruction method forexisting railway bridge bearingrdquo Railway Engineering vol 57no 10 pp 12ndash14 2017

[7] S Y Yan ldquoAn analysis of causes for defects of roller bearingand treatment measuresrdquo Railway Standard Design vol 2002no 6 pp 30ndash32 2002 in Chinese

[8] H Wang Study on Damage Identification Method for RailwayBridge Rubber Bearings Based on Dynamic Responses BeijingJiaotong University Beijing China 2014 in Chinese

[9] Z Qiao Study on the Bridge Bearing Disease IdentificationTechnique Based on the Natural Frequency Change Rate EastChina Jiaotong University Nanchang China 2014 inChinese

[10] S L Chen Y Q Liu and Y B Zhang ldquoEffects of bearingdamage on vibration characteristics of a large span steel trussbridgerdquo Journal of Vibration and Shock vol 36 no 13pp 195ndash200 2017 in Chinese

[11] F Q Hu J Chen Y B Xu and X M Feng ldquoAnalysis ofmechanical properties of continuous T-beam bridge withdiseased bearingsrdquo Journal of Nanchang University Engi-neering and Technology vol 37 no 2 pp 123ndash127 2015 inChinese

[12] H Zhou K Liu G Shi Y Q Wang Y J Shi andG De Roeck ldquoFatigue assessment of a composite railwaybridge for high speed trains Part I modeling and fatiguecritical detailsrdquo Journal of Constructional Steel Researchvol 82 pp 234ndash245 2013

[13] T Guo J Liu and L Y Huang ldquoInvestigation and control ofexcessive cumulative girder movements of long-span steelsuspension bridgesrdquo Engineering Structures vol 125pp 217ndash226 2016

[14] 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 04014099 2015

Shock and Vibration 13

[15] Y L Ding and G X Wang ldquoEvaluation of dynamic loadfactors for a high-speed railway truss arch bridgerdquo Shock andVibration vol 2016 Article ID 5310769 15 pages 2016

[16] G X Wang Y L Ding Y S Song L Y Wu Q Yue andG H Mao ldquoDetection and location of the degraded bearingsbased on monitoring the longitudinal expansion performanceof the main girder of the Dashengguan Yangtze BridgerdquoJournal of Performance of Constructed Facilities vol 30 no 4article 04015074 2016

[17] G X Wang and Y L Ding ldquoResearch on monitoring tem-perature difference from cross sections of steel truss archgirder of Dashengguan Yangtze Bridgerdquo International Journalof Steel Structures vol 15 no 3 pp 647ndash660 2015

[18] M Y Gokhale and K D Kaur ldquoTime domain signal analysisusing wavelet packet decomposition approachrdquo InternationalJournal of communications Networks and System Sciencesvol 3 no 3 pp 321ndash329 2010

[19] T Figlus S Liscak A Wilk and B Aazarz ldquoConditionmonitoring of engine timing system by using wavelet packetdecomposition of a acoustic signalrdquo Journal of MechanicalScience and Technology vol 28 no 5 pp 1663ndash1671 2014

[20] R Wald T M Khoshgoftaar and J C Sloan ldquoFeature se-lection for optimization of wavelet packet decomposition inreliability analysis of systemsrdquo International Journal on Ar-tificial Intelligence Tools vol 22 no 5 article 1360011 2013

[21] EN 1991-1-5 Eurocode 1 Actions on StructuresmdashPart 1ndash5General Actionsmdashampermal Actions European Committee forStandardization (CEN) Brussels Belgium Europe 2004

[22] P Kotes and J Vican ldquoReliability levels for existing bridgesevaluation according to eurocodesrdquo Procedia Engineeringvol 40 pp 211ndash216 2012

[23] TBT 3320-2013 Spherical Bearings for Railway BridgesChina Railway Publishing House Beijing China 2013 inChinese

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uniformly and randomly sampled from the interval (0 1)and then the demanded number of CDDs can be simulatedby substituting the K values into the cumulative probabilityR in the following equations

R G Nius1113872 1113873 (4a)

R G Nids1113872 1113873 (4b)

G Nius1113872 1113873 exp minus 1 + rNius minus b

a1113888 11138891113890 1113891

minus1r⎡⎣ ⎤⎦ (4c)

G Nids1113872 1113873 exp minus 1 + rNids minus b

a1113888 11138891113890 1113891

minus1r⎡⎣ ⎤⎦ (4d)

where Nius and Nids denote the simulated CDDs of Niu andNid respectively e values of r a and b have been cal-culated in the Section 32 However Niu and Nid areembedded in G(Niu) and G(Nids) so they cannot be di-rectly solved from Equations (4a) and (4b) Considering thatG(Nius) and G(Nids) are monotonically increasing func-tions Nius and Nids can be determined by the Newtoniteration method as follows

Nn+1ius N

nius minus

G Nnius1113872 1113873minusR

Gprime Nnius1113872 1113873

(5a)

Nn+1ids N

nids minus

G Nnids1113872 1113873minusR

Gprime Nnids1113872 1113873

(5b)

where Nn+1ius and Nn

ius denote the (n + 1)th and nth iterationresults of Nius respectively Nn+1

ids andNnids denote the (n + 1)

th and nth iteration results of Nids respectively Gprime(Nnius)

and Gprime(Nnids) denote the first-order derivatives of GEVD

(ie the probability density functions of GEVD) e iter-ation will terminate when the absolute difference betweentwo adjacent iterations is less than 10minus2 e last iterationresult is treated as the values of Nius and Nids

33 ampe Sampling Simulation Result Based on the samplingsimulation method above the simulation results of 3495N1us are shown in Figure 11(a) e mean value of thesimulated result is 2820mm which is very close to the meanvalue of Niu ie 2904mm Furthermore the CP of thesimulated result is calculated and then compared with themonitored CP as shown in Figure 11(b) It can be seen thatthe simulated CP coincides with the monitored CP whichverifies the accuracy of the sampling simulation method

Furthermore the simulated 3495 values of Nius andNidsare accumulated using Equation (1) to obtain the simulatedCBT (ie Mius and Mids) as shown in Figure 12 It can beseen that the simulated Mius and Mids present good lineargrowth characteristics with the number of passed trains Toverify the accuracy of the simulated linear growth ratesa comparison between the monitored and simulated lineargrowth rates was conducted as shown in Figure 12 It can beseen that the simulated linear growth rates have agreement

with the monitored ones which further verifies the accuracyof the sampling simulation method

4 Safety Evaluation Analysis

e CBT relates to the bearing wear condition If the CBTsexceed the wear limit in the service lifetime the sphericalsteel bearing is totally damaged and cannot be used forlongitudinal thermal expansion any longer erefore it isnecessary to analyze whether the CBTs at bridge site canexceed the wear limit in the service lifetime

41 EvaluationMethod e failure probabilities of sphericalsteel bearings in the service lifetime (ie the probabilities ofCBTover the wear limit in the service lifetime) are calculatedas follows

Pf 1minusF Mius1113960 11139611113872 1113873 P Mius gt Mius1113960 11139611113872 1113873 (6a)

or

Pf 1minusF Mids1113960 11139611113872 1113873 P Mids gt Mids1113960 11139611113872 1113873 (6b)

Mius 1113944

J

j1Nius(j) (6c)

Mids 1113944

J

j1Nids(j) (6d)

where Pf denotes the failure probability [Mius] and [Mids]

denote the wear limits of Biu and Bid respectivelyF([Mius]) and F([Mids]) denote the cumulative distribu-tion functions with assigned values [Mius] and [Mids]respectively namely F(Mius [Mius]) and F(Mids

[Mids]) Nius(j) and Nids(j) denote the jth values of Niusand Nids in the service lifetime respectively which can beobtained through sampling simulation as introduced in theSection 32 and J denotes the total number of passed trainsin the service lifetime

F(Mius) and F(Mius) are unknown which should befirst determined by the following two steps

(1) By referring to the sampling simulation method inthe Section 32 the values of Nius or Nids in theservice life are simulated through Monte Carlosampling en the values of Nius or Nids aresummed using Equation (6c) or (6d) respectivelyto obtain one value of Mius or Mids is step isrepeated W times to obtain W values of Mius orMids

(2) e cumulative probability characteristics for the W

values of Mius or Mids are fitted using cumulativedistribution function e true cumulative distri-bution function of Mius and Mids is unknown sothree possible cumulative distribution functions(namely ND WD and GEVD) are considered andthe cumulative distribution function with the bestfitting effect is selected as the true cumulative

Shock and Vibration 9

0 1000 2000 3000 40000

4

8

12

p

Cum

ulat

ive t

rave

l (m

m times

103 )

Monitored result Simulated result

(a)

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

5

10

15

p

Monitored result Simulated result

(b)

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

5

10

15

p

Monitored result Simulated result

(c)

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

5

10

15

20

25

p

Monitored result Simulated result

(d)

Figure 12 Continued

0 700 1400 2100 2800 35000

5

10

15

20

25

30

35

k

Cum

ulat

ive v

alue

(mm

) mean (N1us) = 2820 mm

(a)

5 10 15 20 25 300

02

04

06

08

1

Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored result Simulated result

(b)

Figure 11 e simulated N1us and its CP (a) e simulated result (b) A comparison between the simulated and monitored results

10 Shock and Vibration

distribution function of Mius and Mids after least-square fitting

After F(Mius) and F(Mius) are determined [Mius] and[Mids] are substituted in Equations (6a) and (6b) to obtainthe value of Pf Furthermore the reliability level β can becalculated using Pf as follows

Pf 1113946+infin

β

12π

radic eminusx22

dx (7)

If the calculated reliability level β is larger than the targetreliability level [β] (ie βgt [β]) it can be inferred that thebearing is still in a good service state otherwise (ie βle [β])it can be inferred that the bearing is in a poor service stateand should be replaced in time According to the Eurocode[21] the target reliability level [β] for newly built bridgeswith a 100-year design lifetime is 38 which corresponds toa failure probability Pf 72 times 10minus5 However theDashengguan Railway Bridge has been in operation for 8years and it is not a newly built bridge From the viewpointof bridge reliability a reduction in the uncertainties of theload and resistance parameters decreases the failure prob-ability of existing bridge structures indicating the possibilityof adopting a lower reliability level for the evaluation ofexisting bridges rather than the reliability level that shouldbe used for newly built bridges [22] According to Kotesrsquoresearch [22] the target reliability level [β] with a 90-yearremaining lifetime is 3773 so the target reliability level [β]

of the Dashengguan Railway Bridge with a 92-yearremaining lifetime is 3778 using the linear interpolationcalculation namely [β] 3778

42 Evaluation Results Figure 8 reveals that M3d has thefastest growth so M3d is taken as an example to calculate itsfailure probability and evaluate the wear condition of B3d inthe service lifetime Before evaluation three parameters

should be determined namely J [M3ds] and W For theparameter J the designed service life of the NanjingDashengguan Yangtze River Bridge is 100 years and thenumber of passed trains in 2017 was 84343 so the value of J

is 8434300 for the parameter [M3ds] the China codespecifies that the linear wear rate of PTFE in a spherical steelbearing should be less than 15 μmkm and that the thicknessof PTFE should be more than 7mm and less than 8mm(ie only a 1mm wear thickness is allowed for an 8mmthickness of PTFE) [23] so the value of [M3ds] is 667 km forthe spherical steel bearing for the parameter W a larger W

can provide better evaluation results and the value of W is1000 for evaluation in this research

After the three parameters are determined the bearingwear life is evaluated by the following seven steps

(1) e values of N3d in 100-year service life are sim-ulated through Monte Carlo simulation usingEquation (4b) and 8434300 values of N3ds areobtained

(2) All the 8434300 values of N3ds are summed to obtainthe M3ds of B3d in 100-year service life usingEquation (6d)

(3) Steps (1) and (2) are repeated 1000 times to obtain1000 values of M3ds as shown in Figure 13(a) e1000 values of M3ds show good uniform and randomcharacteristics and these values can be used to an-alyze the cumulative probability characteristics

(4) e cumulative probability of M3ds is calculated andfitted by the ND WD and GEVD with the fittedresults shown in Figure 13(b) Both the ND andGEVD can well describe the cumulative probabilitycharacteristics of M3ds and GEVD is chosen for thetrue cumulative distribution function of M3ds

(5) e failure probability Pf of CBTover the upper limit[M3ds] for bearing B3d is about 10minus8 calculated by

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

5

10

15

20

25

p

Monitored result Simulated result

(e)

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

4

8

12

p

Monitored result Simulated result

(f )

Figure 12e simulated Mius and Mids (a) M1us and M1u (b) M1ds and M1d (c) M2us and M2u (d) M2ds and M2d (e) M3us and M3u(f ) M3ds and M3d

Shock and Vibration 11

Equation (6b) which means that M3ds has littleprobability of exceeding [M3ds] because the simu-lated 1000 values of M3ds are within a small interval[5874 km 5931 km]

(6) e reliability level β is 5612 calculated by Equation(7) which is larger than the target reliability level3778 indicating that bearing B3d does not reach thewear limit in the service life

(7) Because the linear growth rate of CBT for B3d isfaster than those of the other spherical steel bearingsit can be inferred that the other spherical steelbearings do not reach the wear limit as well in theservice life

5 Conclusions

Based on the monitored dynamic displacement data of steeltruss arch bridge bearings the CDD under the action ofa single train and the CBT under the continual actions ofmany passed trains are investigated Furthermore theprobability statistics and the Monte Carlo sampling simu-lationmethod for CDD are studied and the safety evaluationmethod for the bearing wear life is proposed using a re-liability index regarding the failure probability of monitoredCBT over the wear limit in the service lifetime e mainconclusions are drawn as follows

(1) rough monitored data analysis the monitoredlongitudinal displacement data mainly consist oftemperature-induced static displacements and train-induced dynamic displacements e wavelet packetdecomposition method can effectively extract thesmall-amplitude dynamic displacements caused bytrain loads from the monitored data All the CDDscontain uniform and random characteristics and allthe CBTs have an obvious linear correlation with the

number of passed trains but their linear growth ratesare different

(2) rough probability statistics analysis GEVD canwell describe the cumulative probability character-istics of CDDeMonte Carlo sampling simulationwas performed using GEVD to simulate CDD andCBT and the results show that the simulated cu-mulative probabilities of CDD have agreement withthemonitored ones of CDD and the simulated lineargrowth rates of CBT have agreement with themonitored ones of CBT

(3) rough safety evaluation analysis the safety eval-uation method for the bearing wear life is proposedusing a reliability index and it is finally judgedwhether the CBTs can exceed the wear limit in theservice lifetime e evaluation results show that allthe spherical steel bearings do not exceed the wearlimit in the service life

Notation

Biu ith spherical steel bearing at theupstream

Bid ith spherical steel bearing at thedownstream

CBT cumulative bearing travelCDD cumulative dynamic displacementCP cumulative probabilityDiu monitored longitudinal

displacement data at the ithspherical steel bearing Biu

Did monitored longitudinaldisplacement data at the ithspherical steel bearing Bid

1113957D(k) dynamic displacement caused by thekth passed train

0 200 400 600 800 1000585

5875

59

5925

595

The number of simulations

M3

ds (k

m)

(a)

M3ds (km)587 588 589 59 591 592 593 5940

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y

Cumulative probabilityGEVD

NDWD

(b)

Figure 13 e 1000 values of M3ds and their cumulative probabilities (a)1000M3ds (b) Cumulative probability of M3ds and its fittedresults

12 Shock and Vibration

1113957Dj(k) jth value in 1113957D(k)

F([Mius]) cumulative distribution functionwith assigned value [Mius]

F([Mids]) cumulative distribution functionwith assigned value [Mids]

GEVD general extreme value distributionG(Niu) cumulative distribution function of

NiuG(Nid) cumulative distribution function of

NidG(Nius) cumulative distribution function of

NiusG(Nids) cumulative distribution function of

NidsLVDT linear variable differential

transformermean(Niu) average value of 3495 CDDs for each

Niumean(Nid) average value of 3495 CDDs for each

NidM(p) CBT caused by p passed trainsMiu CBT of BiuMid CBT of Bid[Mius] wear limit of Biu[Mids] wear limit of BidN(k) cumulative travel of bearing caused

by the kth passed trainNiu set of N(1) N(2) N(3495)

for the bearings Biu

Nid set of N(1) N(2) N(3495)

for the bearings BidNius simulated CDDs of NiuNids simulated CDDs of NidNUM(Niu) total number of NiuNUM(Nid) total number of NidNUM(Niu ltNiu(j)) number of CDDs in Niu that are

smaller than Niu(j)

NUM(Nid ltNid(j)) number of CDDs in Nid that aresmaller than Nid(j)

ND normal distributionPTFE poly-tetra-fluoro-ethyleneP(Niu(j)) CP of the jth value Niu(j) in NiuP(Nid(j)) CP of the jth value Nid(j) in NidPf failure probabilityR value of cumulative probabilityWD Weibull distributionxjm wavelet packet coefficient in themth

frequency band of the jth scale[β] target reliability level

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors thank the Natural Science Foundation ofJiangsu Province of China (BK20180652) and the ChinaPostdoctoral Science Foundation (2017M621865)

References

[1] J W Zhan H Xia N Zhang and Y Lu ldquoA diagnosis methodfor bridge rubber support disease based on impact responsesrdquoJournal of Vibration and Shock vol 32 no 8 pp 153ndash1572013 in Chinese

[2] B Wei T H Yang L Z Jiang and X H He ldquoEffects offriction-based fixed bearings on the seismic vulnerability ofa high-speed railway continuous bridgerdquo Advances inStructural Engineering vol 21 no 5 pp 643ndash657 2018

[3] J Oh C Jang and J H Kim ldquoA study on the characteristics ofbridge bearings behavior by finite element analysis and modeltestrdquo Journal of Vibroengineering vol 17 no 5 pp 2559ndash2571 2015

[4] J J Huang Q Su L C Zhang Y J Li and B Liu ldquoRapidtreatment technique of diseases of the rocking axle bearing ofrailway simply supported beam bridgerdquo Applied Mechanicsand Materials vol 351-352 pp 1440ndash1444 2013

[5] A S Deshpande and J M C Kishen ldquoFatigue crack prop-agation in rocker and roller-rocker bearings of railway steelbridgesrdquo Engineering Fracture Mechanics vol 77 no 9pp 1454ndash1466 2010

[6] Q J Shi ldquoDefect analysis and reconstruction method forexisting railway bridge bearingrdquo Railway Engineering vol 57no 10 pp 12ndash14 2017

[7] S Y Yan ldquoAn analysis of causes for defects of roller bearingand treatment measuresrdquo Railway Standard Design vol 2002no 6 pp 30ndash32 2002 in Chinese

[8] H Wang Study on Damage Identification Method for RailwayBridge Rubber Bearings Based on Dynamic Responses BeijingJiaotong University Beijing China 2014 in Chinese

[9] Z Qiao Study on the Bridge Bearing Disease IdentificationTechnique Based on the Natural Frequency Change Rate EastChina Jiaotong University Nanchang China 2014 inChinese

[10] S L Chen Y Q Liu and Y B Zhang ldquoEffects of bearingdamage on vibration characteristics of a large span steel trussbridgerdquo Journal of Vibration and Shock vol 36 no 13pp 195ndash200 2017 in Chinese

[11] F Q Hu J Chen Y B Xu and X M Feng ldquoAnalysis ofmechanical properties of continuous T-beam bridge withdiseased bearingsrdquo Journal of Nanchang University Engi-neering and Technology vol 37 no 2 pp 123ndash127 2015 inChinese

[12] H Zhou K Liu G Shi Y Q Wang Y J Shi andG De Roeck ldquoFatigue assessment of a composite railwaybridge for high speed trains Part I modeling and fatiguecritical detailsrdquo Journal of Constructional Steel Researchvol 82 pp 234ndash245 2013

[13] T Guo J Liu and L Y Huang ldquoInvestigation and control ofexcessive cumulative girder movements of long-span steelsuspension bridgesrdquo Engineering Structures vol 125pp 217ndash226 2016

[14] 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 04014099 2015

Shock and Vibration 13

[15] Y L Ding and G X Wang ldquoEvaluation of dynamic loadfactors for a high-speed railway truss arch bridgerdquo Shock andVibration vol 2016 Article ID 5310769 15 pages 2016

[16] G X Wang Y L Ding Y S Song L Y Wu Q Yue andG H Mao ldquoDetection and location of the degraded bearingsbased on monitoring the longitudinal expansion performanceof the main girder of the Dashengguan Yangtze BridgerdquoJournal of Performance of Constructed Facilities vol 30 no 4article 04015074 2016

[17] G X Wang and Y L Ding ldquoResearch on monitoring tem-perature difference from cross sections of steel truss archgirder of Dashengguan Yangtze Bridgerdquo International Journalof Steel Structures vol 15 no 3 pp 647ndash660 2015

[18] M Y Gokhale and K D Kaur ldquoTime domain signal analysisusing wavelet packet decomposition approachrdquo InternationalJournal of communications Networks and System Sciencesvol 3 no 3 pp 321ndash329 2010

[19] T Figlus S Liscak A Wilk and B Aazarz ldquoConditionmonitoring of engine timing system by using wavelet packetdecomposition of a acoustic signalrdquo Journal of MechanicalScience and Technology vol 28 no 5 pp 1663ndash1671 2014

[20] R Wald T M Khoshgoftaar and J C Sloan ldquoFeature se-lection for optimization of wavelet packet decomposition inreliability analysis of systemsrdquo International Journal on Ar-tificial Intelligence Tools vol 22 no 5 article 1360011 2013

[21] EN 1991-1-5 Eurocode 1 Actions on StructuresmdashPart 1ndash5General Actionsmdashampermal Actions European Committee forStandardization (CEN) Brussels Belgium Europe 2004

[22] P Kotes and J Vican ldquoReliability levels for existing bridgesevaluation according to eurocodesrdquo Procedia Engineeringvol 40 pp 211ndash216 2012

[23] TBT 3320-2013 Spherical Bearings for Railway BridgesChina Railway Publishing House Beijing China 2013 inChinese

14 Shock and Vibration

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

0 1000 2000 3000 40000

4

8

12

p

Cum

ulat

ive t

rave

l (m

m times

103 )

Monitored result Simulated result

(a)

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

5

10

15

p

Monitored result Simulated result

(b)

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

5

10

15

p

Monitored result Simulated result

(c)

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

5

10

15

20

25

p

Monitored result Simulated result

(d)

Figure 12 Continued

0 700 1400 2100 2800 35000

5

10

15

20

25

30

35

k

Cum

ulat

ive v

alue

(mm

) mean (N1us) = 2820 mm

(a)

5 10 15 20 25 300

02

04

06

08

1

Cumulative value (mm)

Cum

ulat

ive p

roba

bilit

y

Monitored result Simulated result

(b)

Figure 11 e simulated N1us and its CP (a) e simulated result (b) A comparison between the simulated and monitored results

10 Shock and Vibration

distribution function of Mius and Mids after least-square fitting

After F(Mius) and F(Mius) are determined [Mius] and[Mids] are substituted in Equations (6a) and (6b) to obtainthe value of Pf Furthermore the reliability level β can becalculated using Pf as follows

Pf 1113946+infin

β

12π

radic eminusx22

dx (7)

If the calculated reliability level β is larger than the targetreliability level [β] (ie βgt [β]) it can be inferred that thebearing is still in a good service state otherwise (ie βle [β])it can be inferred that the bearing is in a poor service stateand should be replaced in time According to the Eurocode[21] the target reliability level [β] for newly built bridgeswith a 100-year design lifetime is 38 which corresponds toa failure probability Pf 72 times 10minus5 However theDashengguan Railway Bridge has been in operation for 8years and it is not a newly built bridge From the viewpointof bridge reliability a reduction in the uncertainties of theload and resistance parameters decreases the failure prob-ability of existing bridge structures indicating the possibilityof adopting a lower reliability level for the evaluation ofexisting bridges rather than the reliability level that shouldbe used for newly built bridges [22] According to Kotesrsquoresearch [22] the target reliability level [β] with a 90-yearremaining lifetime is 3773 so the target reliability level [β]

of the Dashengguan Railway Bridge with a 92-yearremaining lifetime is 3778 using the linear interpolationcalculation namely [β] 3778

42 Evaluation Results Figure 8 reveals that M3d has thefastest growth so M3d is taken as an example to calculate itsfailure probability and evaluate the wear condition of B3d inthe service lifetime Before evaluation three parameters

should be determined namely J [M3ds] and W For theparameter J the designed service life of the NanjingDashengguan Yangtze River Bridge is 100 years and thenumber of passed trains in 2017 was 84343 so the value of J

is 8434300 for the parameter [M3ds] the China codespecifies that the linear wear rate of PTFE in a spherical steelbearing should be less than 15 μmkm and that the thicknessof PTFE should be more than 7mm and less than 8mm(ie only a 1mm wear thickness is allowed for an 8mmthickness of PTFE) [23] so the value of [M3ds] is 667 km forthe spherical steel bearing for the parameter W a larger W

can provide better evaluation results and the value of W is1000 for evaluation in this research

After the three parameters are determined the bearingwear life is evaluated by the following seven steps

(1) e values of N3d in 100-year service life are sim-ulated through Monte Carlo simulation usingEquation (4b) and 8434300 values of N3ds areobtained

(2) All the 8434300 values of N3ds are summed to obtainthe M3ds of B3d in 100-year service life usingEquation (6d)

(3) Steps (1) and (2) are repeated 1000 times to obtain1000 values of M3ds as shown in Figure 13(a) e1000 values of M3ds show good uniform and randomcharacteristics and these values can be used to an-alyze the cumulative probability characteristics

(4) e cumulative probability of M3ds is calculated andfitted by the ND WD and GEVD with the fittedresults shown in Figure 13(b) Both the ND andGEVD can well describe the cumulative probabilitycharacteristics of M3ds and GEVD is chosen for thetrue cumulative distribution function of M3ds

(5) e failure probability Pf of CBTover the upper limit[M3ds] for bearing B3d is about 10minus8 calculated by

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

5

10

15

20

25

p

Monitored result Simulated result

(e)

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

4

8

12

p

Monitored result Simulated result

(f )

Figure 12e simulated Mius and Mids (a) M1us and M1u (b) M1ds and M1d (c) M2us and M2u (d) M2ds and M2d (e) M3us and M3u(f ) M3ds and M3d

Shock and Vibration 11

Equation (6b) which means that M3ds has littleprobability of exceeding [M3ds] because the simu-lated 1000 values of M3ds are within a small interval[5874 km 5931 km]

(6) e reliability level β is 5612 calculated by Equation(7) which is larger than the target reliability level3778 indicating that bearing B3d does not reach thewear limit in the service life

(7) Because the linear growth rate of CBT for B3d isfaster than those of the other spherical steel bearingsit can be inferred that the other spherical steelbearings do not reach the wear limit as well in theservice life

5 Conclusions

Based on the monitored dynamic displacement data of steeltruss arch bridge bearings the CDD under the action ofa single train and the CBT under the continual actions ofmany passed trains are investigated Furthermore theprobability statistics and the Monte Carlo sampling simu-lationmethod for CDD are studied and the safety evaluationmethod for the bearing wear life is proposed using a re-liability index regarding the failure probability of monitoredCBT over the wear limit in the service lifetime e mainconclusions are drawn as follows

(1) rough monitored data analysis the monitoredlongitudinal displacement data mainly consist oftemperature-induced static displacements and train-induced dynamic displacements e wavelet packetdecomposition method can effectively extract thesmall-amplitude dynamic displacements caused bytrain loads from the monitored data All the CDDscontain uniform and random characteristics and allthe CBTs have an obvious linear correlation with the

number of passed trains but their linear growth ratesare different

(2) rough probability statistics analysis GEVD canwell describe the cumulative probability character-istics of CDDeMonte Carlo sampling simulationwas performed using GEVD to simulate CDD andCBT and the results show that the simulated cu-mulative probabilities of CDD have agreement withthemonitored ones of CDD and the simulated lineargrowth rates of CBT have agreement with themonitored ones of CBT

(3) rough safety evaluation analysis the safety eval-uation method for the bearing wear life is proposedusing a reliability index and it is finally judgedwhether the CBTs can exceed the wear limit in theservice lifetime e evaluation results show that allthe spherical steel bearings do not exceed the wearlimit in the service life

Notation

Biu ith spherical steel bearing at theupstream

Bid ith spherical steel bearing at thedownstream

CBT cumulative bearing travelCDD cumulative dynamic displacementCP cumulative probabilityDiu monitored longitudinal

displacement data at the ithspherical steel bearing Biu

Did monitored longitudinaldisplacement data at the ithspherical steel bearing Bid

1113957D(k) dynamic displacement caused by thekth passed train

0 200 400 600 800 1000585

5875

59

5925

595

The number of simulations

M3

ds (k

m)

(a)

M3ds (km)587 588 589 59 591 592 593 5940

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y

Cumulative probabilityGEVD

NDWD

(b)

Figure 13 e 1000 values of M3ds and their cumulative probabilities (a)1000M3ds (b) Cumulative probability of M3ds and its fittedresults

12 Shock and Vibration

1113957Dj(k) jth value in 1113957D(k)

F([Mius]) cumulative distribution functionwith assigned value [Mius]

F([Mids]) cumulative distribution functionwith assigned value [Mids]

GEVD general extreme value distributionG(Niu) cumulative distribution function of

NiuG(Nid) cumulative distribution function of

NidG(Nius) cumulative distribution function of

NiusG(Nids) cumulative distribution function of

NidsLVDT linear variable differential

transformermean(Niu) average value of 3495 CDDs for each

Niumean(Nid) average value of 3495 CDDs for each

NidM(p) CBT caused by p passed trainsMiu CBT of BiuMid CBT of Bid[Mius] wear limit of Biu[Mids] wear limit of BidN(k) cumulative travel of bearing caused

by the kth passed trainNiu set of N(1) N(2) N(3495)

for the bearings Biu

Nid set of N(1) N(2) N(3495)

for the bearings BidNius simulated CDDs of NiuNids simulated CDDs of NidNUM(Niu) total number of NiuNUM(Nid) total number of NidNUM(Niu ltNiu(j)) number of CDDs in Niu that are

smaller than Niu(j)

NUM(Nid ltNid(j)) number of CDDs in Nid that aresmaller than Nid(j)

ND normal distributionPTFE poly-tetra-fluoro-ethyleneP(Niu(j)) CP of the jth value Niu(j) in NiuP(Nid(j)) CP of the jth value Nid(j) in NidPf failure probabilityR value of cumulative probabilityWD Weibull distributionxjm wavelet packet coefficient in themth

frequency band of the jth scale[β] target reliability level

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors thank the Natural Science Foundation ofJiangsu Province of China (BK20180652) and the ChinaPostdoctoral Science Foundation (2017M621865)

References

[1] J W Zhan H Xia N Zhang and Y Lu ldquoA diagnosis methodfor bridge rubber support disease based on impact responsesrdquoJournal of Vibration and Shock vol 32 no 8 pp 153ndash1572013 in Chinese

[2] B Wei T H Yang L Z Jiang and X H He ldquoEffects offriction-based fixed bearings on the seismic vulnerability ofa high-speed railway continuous bridgerdquo Advances inStructural Engineering vol 21 no 5 pp 643ndash657 2018

[3] J Oh C Jang and J H Kim ldquoA study on the characteristics ofbridge bearings behavior by finite element analysis and modeltestrdquo Journal of Vibroengineering vol 17 no 5 pp 2559ndash2571 2015

[4] J J Huang Q Su L C Zhang Y J Li and B Liu ldquoRapidtreatment technique of diseases of the rocking axle bearing ofrailway simply supported beam bridgerdquo Applied Mechanicsand Materials vol 351-352 pp 1440ndash1444 2013

[5] A S Deshpande and J M C Kishen ldquoFatigue crack prop-agation in rocker and roller-rocker bearings of railway steelbridgesrdquo Engineering Fracture Mechanics vol 77 no 9pp 1454ndash1466 2010

[6] Q J Shi ldquoDefect analysis and reconstruction method forexisting railway bridge bearingrdquo Railway Engineering vol 57no 10 pp 12ndash14 2017

[7] S Y Yan ldquoAn analysis of causes for defects of roller bearingand treatment measuresrdquo Railway Standard Design vol 2002no 6 pp 30ndash32 2002 in Chinese

[8] H Wang Study on Damage Identification Method for RailwayBridge Rubber Bearings Based on Dynamic Responses BeijingJiaotong University Beijing China 2014 in Chinese

[9] Z Qiao Study on the Bridge Bearing Disease IdentificationTechnique Based on the Natural Frequency Change Rate EastChina Jiaotong University Nanchang China 2014 inChinese

[10] S L Chen Y Q Liu and Y B Zhang ldquoEffects of bearingdamage on vibration characteristics of a large span steel trussbridgerdquo Journal of Vibration and Shock vol 36 no 13pp 195ndash200 2017 in Chinese

[11] F Q Hu J Chen Y B Xu and X M Feng ldquoAnalysis ofmechanical properties of continuous T-beam bridge withdiseased bearingsrdquo Journal of Nanchang University Engi-neering and Technology vol 37 no 2 pp 123ndash127 2015 inChinese

[12] H Zhou K Liu G Shi Y Q Wang Y J Shi andG De Roeck ldquoFatigue assessment of a composite railwaybridge for high speed trains Part I modeling and fatiguecritical detailsrdquo Journal of Constructional Steel Researchvol 82 pp 234ndash245 2013

[13] T Guo J Liu and L Y Huang ldquoInvestigation and control ofexcessive cumulative girder movements of long-span steelsuspension bridgesrdquo Engineering Structures vol 125pp 217ndash226 2016

[14] 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 04014099 2015

Shock and Vibration 13

[15] Y L Ding and G X Wang ldquoEvaluation of dynamic loadfactors for a high-speed railway truss arch bridgerdquo Shock andVibration vol 2016 Article ID 5310769 15 pages 2016

[16] G X Wang Y L Ding Y S Song L Y Wu Q Yue andG H Mao ldquoDetection and location of the degraded bearingsbased on monitoring the longitudinal expansion performanceof the main girder of the Dashengguan Yangtze BridgerdquoJournal of Performance of Constructed Facilities vol 30 no 4article 04015074 2016

[17] G X Wang and Y L Ding ldquoResearch on monitoring tem-perature difference from cross sections of steel truss archgirder of Dashengguan Yangtze Bridgerdquo International Journalof Steel Structures vol 15 no 3 pp 647ndash660 2015

[18] M Y Gokhale and K D Kaur ldquoTime domain signal analysisusing wavelet packet decomposition approachrdquo InternationalJournal of communications Networks and System Sciencesvol 3 no 3 pp 321ndash329 2010

[19] T Figlus S Liscak A Wilk and B Aazarz ldquoConditionmonitoring of engine timing system by using wavelet packetdecomposition of a acoustic signalrdquo Journal of MechanicalScience and Technology vol 28 no 5 pp 1663ndash1671 2014

[20] R Wald T M Khoshgoftaar and J C Sloan ldquoFeature se-lection for optimization of wavelet packet decomposition inreliability analysis of systemsrdquo International Journal on Ar-tificial Intelligence Tools vol 22 no 5 article 1360011 2013

[21] EN 1991-1-5 Eurocode 1 Actions on StructuresmdashPart 1ndash5General Actionsmdashampermal Actions European Committee forStandardization (CEN) Brussels Belgium Europe 2004

[22] P Kotes and J Vican ldquoReliability levels for existing bridgesevaluation according to eurocodesrdquo Procedia Engineeringvol 40 pp 211ndash216 2012

[23] TBT 3320-2013 Spherical Bearings for Railway BridgesChina Railway Publishing House Beijing China 2013 inChinese

14 Shock and Vibration

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

distribution function of Mius and Mids after least-square fitting

After F(Mius) and F(Mius) are determined [Mius] and[Mids] are substituted in Equations (6a) and (6b) to obtainthe value of Pf Furthermore the reliability level β can becalculated using Pf as follows

Pf 1113946+infin

β

12π

radic eminusx22

dx (7)

If the calculated reliability level β is larger than the targetreliability level [β] (ie βgt [β]) it can be inferred that thebearing is still in a good service state otherwise (ie βle [β])it can be inferred that the bearing is in a poor service stateand should be replaced in time According to the Eurocode[21] the target reliability level [β] for newly built bridgeswith a 100-year design lifetime is 38 which corresponds toa failure probability Pf 72 times 10minus5 However theDashengguan Railway Bridge has been in operation for 8years and it is not a newly built bridge From the viewpointof bridge reliability a reduction in the uncertainties of theload and resistance parameters decreases the failure prob-ability of existing bridge structures indicating the possibilityof adopting a lower reliability level for the evaluation ofexisting bridges rather than the reliability level that shouldbe used for newly built bridges [22] According to Kotesrsquoresearch [22] the target reliability level [β] with a 90-yearremaining lifetime is 3773 so the target reliability level [β]

of the Dashengguan Railway Bridge with a 92-yearremaining lifetime is 3778 using the linear interpolationcalculation namely [β] 3778

42 Evaluation Results Figure 8 reveals that M3d has thefastest growth so M3d is taken as an example to calculate itsfailure probability and evaluate the wear condition of B3d inthe service lifetime Before evaluation three parameters

should be determined namely J [M3ds] and W For theparameter J the designed service life of the NanjingDashengguan Yangtze River Bridge is 100 years and thenumber of passed trains in 2017 was 84343 so the value of J

is 8434300 for the parameter [M3ds] the China codespecifies that the linear wear rate of PTFE in a spherical steelbearing should be less than 15 μmkm and that the thicknessof PTFE should be more than 7mm and less than 8mm(ie only a 1mm wear thickness is allowed for an 8mmthickness of PTFE) [23] so the value of [M3ds] is 667 km forthe spherical steel bearing for the parameter W a larger W

can provide better evaluation results and the value of W is1000 for evaluation in this research

After the three parameters are determined the bearingwear life is evaluated by the following seven steps

(1) e values of N3d in 100-year service life are sim-ulated through Monte Carlo simulation usingEquation (4b) and 8434300 values of N3ds areobtained

(2) All the 8434300 values of N3ds are summed to obtainthe M3ds of B3d in 100-year service life usingEquation (6d)

(3) Steps (1) and (2) are repeated 1000 times to obtain1000 values of M3ds as shown in Figure 13(a) e1000 values of M3ds show good uniform and randomcharacteristics and these values can be used to an-alyze the cumulative probability characteristics

(4) e cumulative probability of M3ds is calculated andfitted by the ND WD and GEVD with the fittedresults shown in Figure 13(b) Both the ND andGEVD can well describe the cumulative probabilitycharacteristics of M3ds and GEVD is chosen for thetrue cumulative distribution function of M3ds

(5) e failure probability Pf of CBTover the upper limit[M3ds] for bearing B3d is about 10minus8 calculated by

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

5

10

15

20

25

p

Monitored result Simulated result

(e)

Cum

ulat

ive t

rave

l (m

m times

103 )

0 1000 2000 3000 40000

4

8

12

p

Monitored result Simulated result

(f )

Figure 12e simulated Mius and Mids (a) M1us and M1u (b) M1ds and M1d (c) M2us and M2u (d) M2ds and M2d (e) M3us and M3u(f ) M3ds and M3d

Shock and Vibration 11

Equation (6b) which means that M3ds has littleprobability of exceeding [M3ds] because the simu-lated 1000 values of M3ds are within a small interval[5874 km 5931 km]

(6) e reliability level β is 5612 calculated by Equation(7) which is larger than the target reliability level3778 indicating that bearing B3d does not reach thewear limit in the service life

(7) Because the linear growth rate of CBT for B3d isfaster than those of the other spherical steel bearingsit can be inferred that the other spherical steelbearings do not reach the wear limit as well in theservice life

5 Conclusions

Based on the monitored dynamic displacement data of steeltruss arch bridge bearings the CDD under the action ofa single train and the CBT under the continual actions ofmany passed trains are investigated Furthermore theprobability statistics and the Monte Carlo sampling simu-lationmethod for CDD are studied and the safety evaluationmethod for the bearing wear life is proposed using a re-liability index regarding the failure probability of monitoredCBT over the wear limit in the service lifetime e mainconclusions are drawn as follows

(1) rough monitored data analysis the monitoredlongitudinal displacement data mainly consist oftemperature-induced static displacements and train-induced dynamic displacements e wavelet packetdecomposition method can effectively extract thesmall-amplitude dynamic displacements caused bytrain loads from the monitored data All the CDDscontain uniform and random characteristics and allthe CBTs have an obvious linear correlation with the

number of passed trains but their linear growth ratesare different

(2) rough probability statistics analysis GEVD canwell describe the cumulative probability character-istics of CDDeMonte Carlo sampling simulationwas performed using GEVD to simulate CDD andCBT and the results show that the simulated cu-mulative probabilities of CDD have agreement withthemonitored ones of CDD and the simulated lineargrowth rates of CBT have agreement with themonitored ones of CBT

(3) rough safety evaluation analysis the safety eval-uation method for the bearing wear life is proposedusing a reliability index and it is finally judgedwhether the CBTs can exceed the wear limit in theservice lifetime e evaluation results show that allthe spherical steel bearings do not exceed the wearlimit in the service life

Notation

Biu ith spherical steel bearing at theupstream

Bid ith spherical steel bearing at thedownstream

CBT cumulative bearing travelCDD cumulative dynamic displacementCP cumulative probabilityDiu monitored longitudinal

displacement data at the ithspherical steel bearing Biu

Did monitored longitudinaldisplacement data at the ithspherical steel bearing Bid

1113957D(k) dynamic displacement caused by thekth passed train

0 200 400 600 800 1000585

5875

59

5925

595

The number of simulations

M3

ds (k

m)

(a)

M3ds (km)587 588 589 59 591 592 593 5940

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y

Cumulative probabilityGEVD

NDWD

(b)

Figure 13 e 1000 values of M3ds and their cumulative probabilities (a)1000M3ds (b) Cumulative probability of M3ds and its fittedresults

12 Shock and Vibration

1113957Dj(k) jth value in 1113957D(k)

F([Mius]) cumulative distribution functionwith assigned value [Mius]

F([Mids]) cumulative distribution functionwith assigned value [Mids]

GEVD general extreme value distributionG(Niu) cumulative distribution function of

NiuG(Nid) cumulative distribution function of

NidG(Nius) cumulative distribution function of

NiusG(Nids) cumulative distribution function of

NidsLVDT linear variable differential

transformermean(Niu) average value of 3495 CDDs for each

Niumean(Nid) average value of 3495 CDDs for each

NidM(p) CBT caused by p passed trainsMiu CBT of BiuMid CBT of Bid[Mius] wear limit of Biu[Mids] wear limit of BidN(k) cumulative travel of bearing caused

by the kth passed trainNiu set of N(1) N(2) N(3495)

for the bearings Biu

Nid set of N(1) N(2) N(3495)

for the bearings BidNius simulated CDDs of NiuNids simulated CDDs of NidNUM(Niu) total number of NiuNUM(Nid) total number of NidNUM(Niu ltNiu(j)) number of CDDs in Niu that are

smaller than Niu(j)

NUM(Nid ltNid(j)) number of CDDs in Nid that aresmaller than Nid(j)

ND normal distributionPTFE poly-tetra-fluoro-ethyleneP(Niu(j)) CP of the jth value Niu(j) in NiuP(Nid(j)) CP of the jth value Nid(j) in NidPf failure probabilityR value of cumulative probabilityWD Weibull distributionxjm wavelet packet coefficient in themth

frequency band of the jth scale[β] target reliability level

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors thank the Natural Science Foundation ofJiangsu Province of China (BK20180652) and the ChinaPostdoctoral Science Foundation (2017M621865)

References

[1] J W Zhan H Xia N Zhang and Y Lu ldquoA diagnosis methodfor bridge rubber support disease based on impact responsesrdquoJournal of Vibration and Shock vol 32 no 8 pp 153ndash1572013 in Chinese

[2] B Wei T H Yang L Z Jiang and X H He ldquoEffects offriction-based fixed bearings on the seismic vulnerability ofa high-speed railway continuous bridgerdquo Advances inStructural Engineering vol 21 no 5 pp 643ndash657 2018

[3] J Oh C Jang and J H Kim ldquoA study on the characteristics ofbridge bearings behavior by finite element analysis and modeltestrdquo Journal of Vibroengineering vol 17 no 5 pp 2559ndash2571 2015

[4] J J Huang Q Su L C Zhang Y J Li and B Liu ldquoRapidtreatment technique of diseases of the rocking axle bearing ofrailway simply supported beam bridgerdquo Applied Mechanicsand Materials vol 351-352 pp 1440ndash1444 2013

[5] A S Deshpande and J M C Kishen ldquoFatigue crack prop-agation in rocker and roller-rocker bearings of railway steelbridgesrdquo Engineering Fracture Mechanics vol 77 no 9pp 1454ndash1466 2010

[6] Q J Shi ldquoDefect analysis and reconstruction method forexisting railway bridge bearingrdquo Railway Engineering vol 57no 10 pp 12ndash14 2017

[7] S Y Yan ldquoAn analysis of causes for defects of roller bearingand treatment measuresrdquo Railway Standard Design vol 2002no 6 pp 30ndash32 2002 in Chinese

[8] H Wang Study on Damage Identification Method for RailwayBridge Rubber Bearings Based on Dynamic Responses BeijingJiaotong University Beijing China 2014 in Chinese

[9] Z Qiao Study on the Bridge Bearing Disease IdentificationTechnique Based on the Natural Frequency Change Rate EastChina Jiaotong University Nanchang China 2014 inChinese

[10] S L Chen Y Q Liu and Y B Zhang ldquoEffects of bearingdamage on vibration characteristics of a large span steel trussbridgerdquo Journal of Vibration and Shock vol 36 no 13pp 195ndash200 2017 in Chinese

[11] F Q Hu J Chen Y B Xu and X M Feng ldquoAnalysis ofmechanical properties of continuous T-beam bridge withdiseased bearingsrdquo Journal of Nanchang University Engi-neering and Technology vol 37 no 2 pp 123ndash127 2015 inChinese

[12] H Zhou K Liu G Shi Y Q Wang Y J Shi andG De Roeck ldquoFatigue assessment of a composite railwaybridge for high speed trains Part I modeling and fatiguecritical detailsrdquo Journal of Constructional Steel Researchvol 82 pp 234ndash245 2013

[13] T Guo J Liu and L Y Huang ldquoInvestigation and control ofexcessive cumulative girder movements of long-span steelsuspension bridgesrdquo Engineering Structures vol 125pp 217ndash226 2016

[14] 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 04014099 2015

Shock and Vibration 13

[15] Y L Ding and G X Wang ldquoEvaluation of dynamic loadfactors for a high-speed railway truss arch bridgerdquo Shock andVibration vol 2016 Article ID 5310769 15 pages 2016

[16] G X Wang Y L Ding Y S Song L Y Wu Q Yue andG H Mao ldquoDetection and location of the degraded bearingsbased on monitoring the longitudinal expansion performanceof the main girder of the Dashengguan Yangtze BridgerdquoJournal of Performance of Constructed Facilities vol 30 no 4article 04015074 2016

[17] G X Wang and Y L Ding ldquoResearch on monitoring tem-perature difference from cross sections of steel truss archgirder of Dashengguan Yangtze Bridgerdquo International Journalof Steel Structures vol 15 no 3 pp 647ndash660 2015

[18] M Y Gokhale and K D Kaur ldquoTime domain signal analysisusing wavelet packet decomposition approachrdquo InternationalJournal of communications Networks and System Sciencesvol 3 no 3 pp 321ndash329 2010

[19] T Figlus S Liscak A Wilk and B Aazarz ldquoConditionmonitoring of engine timing system by using wavelet packetdecomposition of a acoustic signalrdquo Journal of MechanicalScience and Technology vol 28 no 5 pp 1663ndash1671 2014

[20] R Wald T M Khoshgoftaar and J C Sloan ldquoFeature se-lection for optimization of wavelet packet decomposition inreliability analysis of systemsrdquo International Journal on Ar-tificial Intelligence Tools vol 22 no 5 article 1360011 2013

[21] EN 1991-1-5 Eurocode 1 Actions on StructuresmdashPart 1ndash5General Actionsmdashampermal Actions European Committee forStandardization (CEN) Brussels Belgium Europe 2004

[22] P Kotes and J Vican ldquoReliability levels for existing bridgesevaluation according to eurocodesrdquo Procedia Engineeringvol 40 pp 211ndash216 2012

[23] TBT 3320-2013 Spherical Bearings for Railway BridgesChina Railway Publishing House Beijing China 2013 inChinese

14 Shock and Vibration

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Equation (6b) which means that M3ds has littleprobability of exceeding [M3ds] because the simu-lated 1000 values of M3ds are within a small interval[5874 km 5931 km]

(6) e reliability level β is 5612 calculated by Equation(7) which is larger than the target reliability level3778 indicating that bearing B3d does not reach thewear limit in the service life

(7) Because the linear growth rate of CBT for B3d isfaster than those of the other spherical steel bearingsit can be inferred that the other spherical steelbearings do not reach the wear limit as well in theservice life

5 Conclusions

Based on the monitored dynamic displacement data of steeltruss arch bridge bearings the CDD under the action ofa single train and the CBT under the continual actions ofmany passed trains are investigated Furthermore theprobability statistics and the Monte Carlo sampling simu-lationmethod for CDD are studied and the safety evaluationmethod for the bearing wear life is proposed using a re-liability index regarding the failure probability of monitoredCBT over the wear limit in the service lifetime e mainconclusions are drawn as follows

(1) rough monitored data analysis the monitoredlongitudinal displacement data mainly consist oftemperature-induced static displacements and train-induced dynamic displacements e wavelet packetdecomposition method can effectively extract thesmall-amplitude dynamic displacements caused bytrain loads from the monitored data All the CDDscontain uniform and random characteristics and allthe CBTs have an obvious linear correlation with the

number of passed trains but their linear growth ratesare different

(2) rough probability statistics analysis GEVD canwell describe the cumulative probability character-istics of CDDeMonte Carlo sampling simulationwas performed using GEVD to simulate CDD andCBT and the results show that the simulated cu-mulative probabilities of CDD have agreement withthemonitored ones of CDD and the simulated lineargrowth rates of CBT have agreement with themonitored ones of CBT

(3) rough safety evaluation analysis the safety eval-uation method for the bearing wear life is proposedusing a reliability index and it is finally judgedwhether the CBTs can exceed the wear limit in theservice lifetime e evaluation results show that allthe spherical steel bearings do not exceed the wearlimit in the service life

Notation

Biu ith spherical steel bearing at theupstream

Bid ith spherical steel bearing at thedownstream

CBT cumulative bearing travelCDD cumulative dynamic displacementCP cumulative probabilityDiu monitored longitudinal

displacement data at the ithspherical steel bearing Biu

Did monitored longitudinaldisplacement data at the ithspherical steel bearing Bid

1113957D(k) dynamic displacement caused by thekth passed train

0 200 400 600 800 1000585

5875

59

5925

595

The number of simulations

M3

ds (k

m)

(a)

M3ds (km)587 588 589 59 591 592 593 5940

02

04

06

08

1

Cum

ulat

ive p

roba

bilit

y

Cumulative probabilityGEVD

NDWD

(b)

Figure 13 e 1000 values of M3ds and their cumulative probabilities (a)1000M3ds (b) Cumulative probability of M3ds and its fittedresults

12 Shock and Vibration

1113957Dj(k) jth value in 1113957D(k)

F([Mius]) cumulative distribution functionwith assigned value [Mius]

F([Mids]) cumulative distribution functionwith assigned value [Mids]

GEVD general extreme value distributionG(Niu) cumulative distribution function of

NiuG(Nid) cumulative distribution function of

NidG(Nius) cumulative distribution function of

NiusG(Nids) cumulative distribution function of

NidsLVDT linear variable differential

transformermean(Niu) average value of 3495 CDDs for each

Niumean(Nid) average value of 3495 CDDs for each

NidM(p) CBT caused by p passed trainsMiu CBT of BiuMid CBT of Bid[Mius] wear limit of Biu[Mids] wear limit of BidN(k) cumulative travel of bearing caused

by the kth passed trainNiu set of N(1) N(2) N(3495)

for the bearings Biu

Nid set of N(1) N(2) N(3495)

for the bearings BidNius simulated CDDs of NiuNids simulated CDDs of NidNUM(Niu) total number of NiuNUM(Nid) total number of NidNUM(Niu ltNiu(j)) number of CDDs in Niu that are

smaller than Niu(j)

NUM(Nid ltNid(j)) number of CDDs in Nid that aresmaller than Nid(j)

ND normal distributionPTFE poly-tetra-fluoro-ethyleneP(Niu(j)) CP of the jth value Niu(j) in NiuP(Nid(j)) CP of the jth value Nid(j) in NidPf failure probabilityR value of cumulative probabilityWD Weibull distributionxjm wavelet packet coefficient in themth

frequency band of the jth scale[β] target reliability level

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors thank the Natural Science Foundation ofJiangsu Province of China (BK20180652) and the ChinaPostdoctoral Science Foundation (2017M621865)

References

[1] J W Zhan H Xia N Zhang and Y Lu ldquoA diagnosis methodfor bridge rubber support disease based on impact responsesrdquoJournal of Vibration and Shock vol 32 no 8 pp 153ndash1572013 in Chinese

[2] B Wei T H Yang L Z Jiang and X H He ldquoEffects offriction-based fixed bearings on the seismic vulnerability ofa high-speed railway continuous bridgerdquo Advances inStructural Engineering vol 21 no 5 pp 643ndash657 2018

[3] J Oh C Jang and J H Kim ldquoA study on the characteristics ofbridge bearings behavior by finite element analysis and modeltestrdquo Journal of Vibroengineering vol 17 no 5 pp 2559ndash2571 2015

[4] J J Huang Q Su L C Zhang Y J Li and B Liu ldquoRapidtreatment technique of diseases of the rocking axle bearing ofrailway simply supported beam bridgerdquo Applied Mechanicsand Materials vol 351-352 pp 1440ndash1444 2013

[5] A S Deshpande and J M C Kishen ldquoFatigue crack prop-agation in rocker and roller-rocker bearings of railway steelbridgesrdquo Engineering Fracture Mechanics vol 77 no 9pp 1454ndash1466 2010

[6] Q J Shi ldquoDefect analysis and reconstruction method forexisting railway bridge bearingrdquo Railway Engineering vol 57no 10 pp 12ndash14 2017

[7] S Y Yan ldquoAn analysis of causes for defects of roller bearingand treatment measuresrdquo Railway Standard Design vol 2002no 6 pp 30ndash32 2002 in Chinese

[8] H Wang Study on Damage Identification Method for RailwayBridge Rubber Bearings Based on Dynamic Responses BeijingJiaotong University Beijing China 2014 in Chinese

[9] Z Qiao Study on the Bridge Bearing Disease IdentificationTechnique Based on the Natural Frequency Change Rate EastChina Jiaotong University Nanchang China 2014 inChinese

[10] S L Chen Y Q Liu and Y B Zhang ldquoEffects of bearingdamage on vibration characteristics of a large span steel trussbridgerdquo Journal of Vibration and Shock vol 36 no 13pp 195ndash200 2017 in Chinese

[11] F Q Hu J Chen Y B Xu and X M Feng ldquoAnalysis ofmechanical properties of continuous T-beam bridge withdiseased bearingsrdquo Journal of Nanchang University Engi-neering and Technology vol 37 no 2 pp 123ndash127 2015 inChinese

[12] H Zhou K Liu G Shi Y Q Wang Y J Shi andG De Roeck ldquoFatigue assessment of a composite railwaybridge for high speed trains Part I modeling and fatiguecritical detailsrdquo Journal of Constructional Steel Researchvol 82 pp 234ndash245 2013

[13] T Guo J Liu and L Y Huang ldquoInvestigation and control ofexcessive cumulative girder movements of long-span steelsuspension bridgesrdquo Engineering Structures vol 125pp 217ndash226 2016

[14] 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 04014099 2015

Shock and Vibration 13

[15] Y L Ding and G X Wang ldquoEvaluation of dynamic loadfactors for a high-speed railway truss arch bridgerdquo Shock andVibration vol 2016 Article ID 5310769 15 pages 2016

[16] G X Wang Y L Ding Y S Song L Y Wu Q Yue andG H Mao ldquoDetection and location of the degraded bearingsbased on monitoring the longitudinal expansion performanceof the main girder of the Dashengguan Yangtze BridgerdquoJournal of Performance of Constructed Facilities vol 30 no 4article 04015074 2016

[17] G X Wang and Y L Ding ldquoResearch on monitoring tem-perature difference from cross sections of steel truss archgirder of Dashengguan Yangtze Bridgerdquo International Journalof Steel Structures vol 15 no 3 pp 647ndash660 2015

[18] M Y Gokhale and K D Kaur ldquoTime domain signal analysisusing wavelet packet decomposition approachrdquo InternationalJournal of communications Networks and System Sciencesvol 3 no 3 pp 321ndash329 2010

[19] T Figlus S Liscak A Wilk and B Aazarz ldquoConditionmonitoring of engine timing system by using wavelet packetdecomposition of a acoustic signalrdquo Journal of MechanicalScience and Technology vol 28 no 5 pp 1663ndash1671 2014

[20] R Wald T M Khoshgoftaar and J C Sloan ldquoFeature se-lection for optimization of wavelet packet decomposition inreliability analysis of systemsrdquo International Journal on Ar-tificial Intelligence Tools vol 22 no 5 article 1360011 2013

[21] EN 1991-1-5 Eurocode 1 Actions on StructuresmdashPart 1ndash5General Actionsmdashampermal Actions European Committee forStandardization (CEN) Brussels Belgium Europe 2004

[22] P Kotes and J Vican ldquoReliability levels for existing bridgesevaluation according to eurocodesrdquo Procedia Engineeringvol 40 pp 211ndash216 2012

[23] TBT 3320-2013 Spherical Bearings for Railway BridgesChina Railway Publishing House Beijing China 2013 inChinese

14 Shock and Vibration

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

1113957Dj(k) jth value in 1113957D(k)

F([Mius]) cumulative distribution functionwith assigned value [Mius]

F([Mids]) cumulative distribution functionwith assigned value [Mids]

GEVD general extreme value distributionG(Niu) cumulative distribution function of

NiuG(Nid) cumulative distribution function of

NidG(Nius) cumulative distribution function of

NiusG(Nids) cumulative distribution function of

NidsLVDT linear variable differential

transformermean(Niu) average value of 3495 CDDs for each

Niumean(Nid) average value of 3495 CDDs for each

NidM(p) CBT caused by p passed trainsMiu CBT of BiuMid CBT of Bid[Mius] wear limit of Biu[Mids] wear limit of BidN(k) cumulative travel of bearing caused

by the kth passed trainNiu set of N(1) N(2) N(3495)

for the bearings Biu

Nid set of N(1) N(2) N(3495)

for the bearings BidNius simulated CDDs of NiuNids simulated CDDs of NidNUM(Niu) total number of NiuNUM(Nid) total number of NidNUM(Niu ltNiu(j)) number of CDDs in Niu that are

smaller than Niu(j)

NUM(Nid ltNid(j)) number of CDDs in Nid that aresmaller than Nid(j)

ND normal distributionPTFE poly-tetra-fluoro-ethyleneP(Niu(j)) CP of the jth value Niu(j) in NiuP(Nid(j)) CP of the jth value Nid(j) in NidPf failure probabilityR value of cumulative probabilityWD Weibull distributionxjm wavelet packet coefficient in themth

frequency band of the jth scale[β] target reliability level

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

e authors thank the Natural Science Foundation ofJiangsu Province of China (BK20180652) and the ChinaPostdoctoral Science Foundation (2017M621865)

References

[1] J W Zhan H Xia N Zhang and Y Lu ldquoA diagnosis methodfor bridge rubber support disease based on impact responsesrdquoJournal of Vibration and Shock vol 32 no 8 pp 153ndash1572013 in Chinese

[2] B Wei T H Yang L Z Jiang and X H He ldquoEffects offriction-based fixed bearings on the seismic vulnerability ofa high-speed railway continuous bridgerdquo Advances inStructural Engineering vol 21 no 5 pp 643ndash657 2018

[3] J Oh C Jang and J H Kim ldquoA study on the characteristics ofbridge bearings behavior by finite element analysis and modeltestrdquo Journal of Vibroengineering vol 17 no 5 pp 2559ndash2571 2015

[4] J J Huang Q Su L C Zhang Y J Li and B Liu ldquoRapidtreatment technique of diseases of the rocking axle bearing ofrailway simply supported beam bridgerdquo Applied Mechanicsand Materials vol 351-352 pp 1440ndash1444 2013

[5] A S Deshpande and J M C Kishen ldquoFatigue crack prop-agation in rocker and roller-rocker bearings of railway steelbridgesrdquo Engineering Fracture Mechanics vol 77 no 9pp 1454ndash1466 2010

[6] Q J Shi ldquoDefect analysis and reconstruction method forexisting railway bridge bearingrdquo Railway Engineering vol 57no 10 pp 12ndash14 2017

[7] S Y Yan ldquoAn analysis of causes for defects of roller bearingand treatment measuresrdquo Railway Standard Design vol 2002no 6 pp 30ndash32 2002 in Chinese

[8] H Wang Study on Damage Identification Method for RailwayBridge Rubber Bearings Based on Dynamic Responses BeijingJiaotong University Beijing China 2014 in Chinese

[9] Z Qiao Study on the Bridge Bearing Disease IdentificationTechnique Based on the Natural Frequency Change Rate EastChina Jiaotong University Nanchang China 2014 inChinese

[10] S L Chen Y Q Liu and Y B Zhang ldquoEffects of bearingdamage on vibration characteristics of a large span steel trussbridgerdquo Journal of Vibration and Shock vol 36 no 13pp 195ndash200 2017 in Chinese

[11] F Q Hu J Chen Y B Xu and X M Feng ldquoAnalysis ofmechanical properties of continuous T-beam bridge withdiseased bearingsrdquo Journal of Nanchang University Engi-neering and Technology vol 37 no 2 pp 123ndash127 2015 inChinese

[12] H Zhou K Liu G Shi Y Q Wang Y J Shi andG De Roeck ldquoFatigue assessment of a composite railwaybridge for high speed trains Part I modeling and fatiguecritical detailsrdquo Journal of Constructional Steel Researchvol 82 pp 234ndash245 2013

[13] T Guo J Liu and L Y Huang ldquoInvestigation and control ofexcessive cumulative girder movements of long-span steelsuspension bridgesrdquo Engineering Structures vol 125pp 217ndash226 2016

[14] 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 04014099 2015

Shock and Vibration 13

[15] Y L Ding and G X Wang ldquoEvaluation of dynamic loadfactors for a high-speed railway truss arch bridgerdquo Shock andVibration vol 2016 Article ID 5310769 15 pages 2016

[16] G X Wang Y L Ding Y S Song L Y Wu Q Yue andG H Mao ldquoDetection and location of the degraded bearingsbased on monitoring the longitudinal expansion performanceof the main girder of the Dashengguan Yangtze BridgerdquoJournal of Performance of Constructed Facilities vol 30 no 4article 04015074 2016

[17] G X Wang and Y L Ding ldquoResearch on monitoring tem-perature difference from cross sections of steel truss archgirder of Dashengguan Yangtze Bridgerdquo International Journalof Steel Structures vol 15 no 3 pp 647ndash660 2015

[18] M Y Gokhale and K D Kaur ldquoTime domain signal analysisusing wavelet packet decomposition approachrdquo InternationalJournal of communications Networks and System Sciencesvol 3 no 3 pp 321ndash329 2010

[19] T Figlus S Liscak A Wilk and B Aazarz ldquoConditionmonitoring of engine timing system by using wavelet packetdecomposition of a acoustic signalrdquo Journal of MechanicalScience and Technology vol 28 no 5 pp 1663ndash1671 2014

[20] R Wald T M Khoshgoftaar and J C Sloan ldquoFeature se-lection for optimization of wavelet packet decomposition inreliability analysis of systemsrdquo International Journal on Ar-tificial Intelligence Tools vol 22 no 5 article 1360011 2013

[21] EN 1991-1-5 Eurocode 1 Actions on StructuresmdashPart 1ndash5General Actionsmdashampermal Actions European Committee forStandardization (CEN) Brussels Belgium Europe 2004

[22] P Kotes and J Vican ldquoReliability levels for existing bridgesevaluation according to eurocodesrdquo Procedia Engineeringvol 40 pp 211ndash216 2012

[23] TBT 3320-2013 Spherical Bearings for Railway BridgesChina Railway Publishing House Beijing China 2013 inChinese

14 Shock and Vibration

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

[15] Y L Ding and G X Wang ldquoEvaluation of dynamic loadfactors for a high-speed railway truss arch bridgerdquo Shock andVibration vol 2016 Article ID 5310769 15 pages 2016

[16] G X Wang Y L Ding Y S Song L Y Wu Q Yue andG H Mao ldquoDetection and location of the degraded bearingsbased on monitoring the longitudinal expansion performanceof the main girder of the Dashengguan Yangtze BridgerdquoJournal of Performance of Constructed Facilities vol 30 no 4article 04015074 2016

[17] G X Wang and Y L Ding ldquoResearch on monitoring tem-perature difference from cross sections of steel truss archgirder of Dashengguan Yangtze Bridgerdquo International Journalof Steel Structures vol 15 no 3 pp 647ndash660 2015

[18] M Y Gokhale and K D Kaur ldquoTime domain signal analysisusing wavelet packet decomposition approachrdquo InternationalJournal of communications Networks and System Sciencesvol 3 no 3 pp 321ndash329 2010

[19] T Figlus S Liscak A Wilk and B Aazarz ldquoConditionmonitoring of engine timing system by using wavelet packetdecomposition of a acoustic signalrdquo Journal of MechanicalScience and Technology vol 28 no 5 pp 1663ndash1671 2014

[20] R Wald T M Khoshgoftaar and J C Sloan ldquoFeature se-lection for optimization of wavelet packet decomposition inreliability analysis of systemsrdquo International Journal on Ar-tificial Intelligence Tools vol 22 no 5 article 1360011 2013

[21] EN 1991-1-5 Eurocode 1 Actions on StructuresmdashPart 1ndash5General Actionsmdashampermal Actions European Committee forStandardization (CEN) Brussels Belgium Europe 2004

[22] P Kotes and J Vican ldquoReliability levels for existing bridgesevaluation according to eurocodesrdquo Procedia Engineeringvol 40 pp 211ndash216 2012

[23] TBT 3320-2013 Spherical Bearings for Railway BridgesChina Railway Publishing House Beijing China 2013 inChinese

14 Shock and Vibration

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom