Research Article A Damage Detection Algorithm Utilizing ...

Research ArticleA Damage Detection Algorithm Utilizing Dynamic Displacementof Bridge under Moving Vehicle

Zhen Sun12 Tomonori Nagayama1 Di Su1 and Yozo Fujino3

1Department of Civil Engineering The University of Tokyo 7-3-1 Hongo Bunkyo-ku Tokyo 113-8656 Japan2Jiangsu Transportation Institute 2200 Chengxin Street Nanjing 211112 China3Institute of Advanced Sciences Yokohama National University 79-1 Tokiwadai Hodogaya-ku Yokohama 240-8501 Japan

Correspondence should be addressed to Zhen Sun sunzhen08gmailcom

Received 30 July 2015 Revised 21 September 2015 Accepted 4 October 2015

Academic Editor Matteo Aureli

Copyright copy 2016 Zhen Sun et alThis is an open access article distributed under theCreativeCommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A damage detection method is proposed which utilizes dynamic displacement of bridge structures under moving vehicle Theproblem is first elaborated with closed-form solution of dynamic displacement which is decomposed into quasi-static componentand dynamic component Dynamic curvature is defined as second derivative of the dynamic displacement for detecting damagelocation and estimating damage extent Damage is modeled by local reduction of stiffness in this paper Numerical study wasconducted on a simply supported beam to verify the proposed method Vehicle model is analyzed with Newmarkrsquos method usingMatlab to obtain the contact force acting on the bridge Beammodel is established in commercial finite element software ABAQUSThe effects of road surface roughness and vehicle-bridge interaction are both considered in the analysis In order to identify damagelocation and extent dynamic curvature was calculated with midspan displacement Parametric study on measurement noise leveldamage location damage extent andmultiple damage cases is performed and the analysis results show both reliability and efficacyof this method in damage detection of bridge structures At last conclusions are drawn for its application to bridges in engineeringpractice

1 Introduction

During the service life of a bridge factors such as materialaging and harsh environmental conditions as well as extremeevents such as natural disasters may cause structural damageto varying degrees [1] In order to avoid severe accidentof bridge collapse detection of any potential damage isimportant for serviceability and safety of bridges

In recent years vibration basedmethods have beenwidelystudied for bridge assessment Detailed review of the damagedetection methods can be found in [2 3] Most of thesemethods utilize mode frequency or mode shapes as the iden-tification parameter However there are several difficultiesfor these frequency domain methods (1) baseline of previoustest data of undamaged structure is needed which is oftenunavailable (2) frequency as a globalmodal parameter is notsensitive to local damage (3) frequency is often affected by

environmental conditions such as temperature or humidityThese difficulties have prevented practical use of vibrationbased methods

Besides the frequency domain methods some effortsare devoted to investigating the problem with time domaindamage detection methods In the time domain methods amoving vehicle is often used as external excitation to obtainthe dynamic response of the bridge The basic concept is thatthemoving vehiclewill pass every section of the bridge whichincludes the damage part When the vehicle is right overthe damage range singularity will be caused in the responsecompared with other parts Through data-processing tech-niques to find singularity in the signal damage location couldbe detected Huang [4] applied Hilbert-Huang Transform(HHT) to a model of simply supported damaged bridgeunder loading of a moving truck The undamaged modelis taken as baseline and variation of local frequency and

Hindawi Publishing CorporationShock and VibrationVolume 2016 Article ID 8454567 9 pageshttpdxdoiorg10115520168454567

2 Shock and Vibration

deformation is employed to detect damage Some otherresearchers used wavelet analysis to detect singularity inbridge response signal hence detecting damage in bridgesZhu and Law [5] conducted wavelet analysis on bridgeresponse under amoving loadWavelet coefficient of dynamicdisplacement is obtained to detect damage in a bridge modelNguyen and Tran [6] conducted similar study but insteadof using bridge response they used vehicle response forwavelet analysis which makes it unnecessary for sensorinstallation on a bridge Hester and Gonzalez [7] performedwavelet transform on response of a damaged bridge undermoving load and investigated feasibility to detect damagelocation They concluded that multiple measurement pointswere required to achieve higher accuracy In the wavelet-based damage detection techniques there is a major inherentdifficulty In order to localize damage narrow window ofbridge response that is short time series is needed Howeverfrequency resolution is impaired when short time series arechosen to calculate the frequency Due to this contradictioneffective detection of damage is difficult

In this paper a curvature based damage detectionmethodis proposed which uses dynamic displacement of a bridgesubject to loading of a moving vehicle It does not requirebaseline of undamaged state of bridge and both damagelocation and damage extent can be detected Firstly the phys-ical mechanism of dynamic displacement of damaged bridgeunder loading of a moving vehicle is elaborated Dynamicdisplacement is decomposed into quasi-static componentand dynamic component and curvature is used to detectdamage location and estimate damage extent Secondly theproposedmethod is verifiedwith a finite elementmodel usingbeam elements in which vehicle model and beam model areestablished in Matlab and ABAQUS respectively Vehicle-bridge interaction and roughness are both considered anddynamic displacement is used to calculate curvature anddetect damage The effect of parameters of measurementnoise level damage location damage extent and multipledamage cases is studied which illustrates reliability andefficacy of the proposed method Finally conclusions aremade for application of this method in engineering practice

2 Damage Detection withDynamic Displacement

The dynamic problem of vehicle-bridge system has been asubject of interest for a long time In the earlier times mostof the studies are aimed at predicting dynamic response ofbridges and providing a dynamic amplification factor forengineering design [8] Detailed review of the early analyticaland numerical studies could be found in previous literaturesuch as [9 10] In the following section physical backgroundis elaborated and it is shown that dynamic curvature is a goodindicator for damage detection

21 Analysis on Dynamic Displacement under Moving Vehi-cle In simplified case of an undamaged uniform simplysupported beam vehicle is modeled as a moving load with

Pv

x

L

Figure 1 Schematic figure of simply supported beam undermovingload

constant speed v as illustrated in Figure 1 The closed-form solution was deduced by Weaver et al [11] and thedisplacement of the bridge is expressed as follows

119910 (119909 119905) = minus21198751198713

1198981205872

infin

sum

119894=1

sin (119894120587119909119871)1198942 (119894212058721198862 minus V21198712)

sin(119894120587V119905119871)

+21198751198714V

1198981205873119886

infin

sum

119894=1

sin (119894120587119909119871)1198943 (119894212058721198862 minus

21198712)

sin120596119899119905

(1)

where 119886 = radic119864119868119898120596119899= (119894212058721198712)radic119864119868119898 119871 is the length

of the bridge 119875 is the amplitude of the moving load119898 is themass per unit length119864119868 is the flexural rigidity of the bridge Vis the velocity of themoving force120596

119899is the natural frequency

of the bridgeIt is noticed that the first part is related to the vehicle

speed which is referred to as the quasi-static component thesecond part contains bridge natural frequency and is referredto as dynamic component which comes from the vibrationof the bridge Figure 2 illustrates the dynamic displacement atmidspan which is composed of these two parts quasi-staticcomponent and dynamic component The proposed methodconcentrates on calculating curvature from the quasi-staticcomponent of the dynamic displacement and an innovativetechnique is used in curvature calculation to cancel out thedynamic displacement

22 Damage Detection with Curvature The relation betweenbending moment and curvature of a beam is expressed in thefollowing equation [12]

119872(119909) = minus119864119868 (119909)1198892119908 (119909)

1198891199092 (2)

where 119864 denotes Youngrsquos modulus of the material 119868 denotesthe second moment of area of beam cross section 119908(119909)denotes deflection of the beam 119909 denotes the beam coordi-nate and119872(119909) denotes the bending moment

Curvature can be calculated from the second derivative ofdeflection measurements which is expressed as

120581 (119909) =1198892119908 (119909)

1198891199092= minus

119872(119909)

119864119868 (119909) (3)

If there is damage (stiffness loss) in some location of thebeam abrupt change is expected at the corresponding loca-tion on the curvature plot

Shock and Vibration 3

= +

1086420

Time (s)

1086420

Time (s)

2 4 6 80 10

Time (s)

Qua

si-st

atic

com

pone

nt (m

m)

minus15

minus1

minus05

0

05

Defl

ectio

n (m

m)

minus15

minus1

minus05

0

05

minus004

minus002

0

002

004

Dyn

amic

com

pone

nt (m

m)

Figure 2 Quasi-static and dynamic components in deflection signal

In this study damage extent is defined as 120572 whichindicates loss in stiffness 119864119868

120572 =119864119868119894minus 119864119868119889

119864119868119894

(4)

where the subscripts 119894 and 119889 denote ldquointactrdquo and ldquodamagedrdquorespectively

The stiffness in intact and damaged states is definedrespectively as

119864119868119894= minus

119872(119909)

120581119894(119909)

(5)

119864119868119889= minus

119872(119909)

120581119889(119909) (6)

Defining Δ120581 = 120581119889minus 120581119894and substituting (5) and (6) into (4)

yields

Δ120581

120581119894

=120572

1 minus 120572 (7)

Hence damage extent can be calculated as

120572 =Δ120581120581119894

Δ120581120581119894+ 1 (8)

Curvature can be approximated with second derivative ofdisplacement measurements

120581 (119909) = 11990810158401015840(119909) =

119908 (119909 + Δ119909) minus 2119908 (119909) + 119908 (119909 minus Δ119909)

Δ1199092 (9)

where 119908(119909) denotes displacement Illustration of the pro-posed method is shown in Figure 3 In order to obtain densedeflection measurement Maxwell-Betti Reciprocal Theoremis utilized and it is explained as follows Consider a beamin two states under different loads Maxwell-Betti ReciprocalTheorem states that the work of the forces of the first stateon the displacements of the second state is equal to the workof forces of the second state on the displacements of the firststate Based on this theorem the deflection of a beam atpoint A under a load at point B is the same as the deflectionat point B under the same load at point A By extendingthis equivalence relationship to a moving load the static


Discontinuity

y( l2 t)

Damage

Displacement transducer

Time (s)

minus15

minus1

minus05

0

Defl

ectio

n (m

m)

Figure 3 Illustration of damage identification from dynamicdisplacement

deflection w(x) of a beam under a load at point A P(A)is obtained with deflection measurement at point A w(A)under a moving load P(x) The load is assumed to moveslowly so that the dynamic effect is negligible In applicationto engineering practice the vehicle load is assumed to bepoint load and the bridge is modeled as a beam

For dynamic displacement of a damaged bridge undervehicle loading it contains three parts quasi-static compo-nent dynamic component and damage-induced componentTherefore (9) can be expressed as

120581 (119909) =119908static (119909 minus Δ119909) minus 2119908static (119909) + 119908static (119909 + Δ119909)

Δ1199092

+119908dynamic (119909 minus Δ119909) minus 2119908dynamic (119909) + 119908dynamic (119909 + Δ119909)

Δ1199092

+119908damage (119909 minus Δ119909) minus 2119908damage (119909) + 119908damage (119909 + Δ119909)

Δ1199092

(10)

Noticing that dynamic component is a periodic function(Figure 2) in curvature calculation formula calculationinterval Δ119909 can be chosen as the length of the natural period

Δ119909 = 119879119899times 119891119904 (11)

where 119879119899denotes fundamental natural period of the bridge

119891119904denotes sampling frequency of data acquisition In this

way the three terms in the numerator of dynamic componentare canceled out while contribution from damage is still keptin the calculated curvature For a beam the fundamentalnatural period is integer times of higher bending modeperiods Therefore higher bending mode components arealso canceled out by the option of Δ119909

In this analysis 119908(119909) is dynamic displacement insteadof static deflection and Δ119909 is chosen as the length offundamental natural period 120581(119909) is defined as dynamiccurvature to distinguish with the curvature calculated withstatic deflection

In this proposed method as shown in Figure 3 adisplacement transducer is installed at one position while a

M

K C

Figure 4 Quarter-car model

vehicle passes through a bridge If there is discontinuity inrecorded dynamic displacement signal it will correspond todamage on the bridge Dynamic curvature will highlight thedamage location and damage extent can also be calculatedfrom curvature values

3 Investigation with a Finite Element Model

In order to verify the proposed damage detection methodinvestigation is conducted on a vehicle-bridge model usingMatlab and finite element software ABAQUS In the analysisthe vehicle model is established usingMatlab and the contactforce at the wheel is calculated with Newmarkrsquos method Thebridge is modeled as a simply supported beam in ABAQUSand contact force is imported fromMatlab calculation resultThe analysis scheme can be used to compute the bridgebehavior with good accuracy and high efficiency

31 Numerical Example

311 Vehicle Model Equation of motion for the vehicle isobtained using the principle of virtual work The vehicleloading is simulated with a quarter-car model shown inFigure 4 composed of a spring with stiffness coefficient of8times 10

3Nm and a damper with damping coefficient of 144times103Nsdotsm The mass of vehicle body is 100 kg Considering

the mass of tire is much smaller than the vehicle body it isneglected in the calculation

The equation of motion for the vehicle is

119872VV + 119862VV + 119870V119880V = 119891V (12)

where 119872V 119862V 119870V denote mass damping coefficient andstiffness of the vehicle respectively V V 119880V denote accel-eration velocity and displacement of vehicle motion respec-tively 119891V denotes the vertical force on the vehicle whichis contributed from road roughness function and bridgedisplacement at the contact points Constant vehicle speedand straight vehicle path are assumed in this study Thedynamic equation is calculated in Matlab with Newmarkrsquosmethod and the contact force on the bridge is calculated asfollows

119891 = 119891V +119872V119892 minus 119862VV minus 119870V119880V (13)

312 Bridge Model Steel beam model is established withthree-dimensional beam element B31 in commercial finite


Table 1 Properties of the beam model

Density 7800 kgm3

Youngrsquos modulus 21 times 1011 Nm2

Poissonrsquos ratio 03Damage location 11m from the left support

DamageDisplacement measurement

15m

11m

Figure 5 Layout of bridge model

element software ABAQUS shown in Figure 5 It is 15m longand 1m wide with a thickness of 006m and the boundarycondition is specified as simply supported The beam ismodeled totally with 200 beam elements

The bridge specifications are provided in Table 1 Damageis simulated by reduction of cross section which is 11m awayfrom the left end with a length of 075m and the thicknessreduced to 005m Damage extent is 42 stiffness loss Inanalysis vehicle passes at a constant speed of 1ms so totallyit takes 15 s to pass the bridge Rayleigh damping is definedfor the model

Road surface is continuously distributed in a randomfashion which affects dynamic behavior of both vehicle andbridge It is usually assumed to be a stationary Gaussianrandom process [13 14] and is generated from power spectraldensity functions Consider

119903 (119909) =

119873

sum

119896=1

radic2119866119889(119899119896) Δ119899 cos (2120587119899

119896119909 minus 120579119896) (14)

where 119866119889(119899119896) = 119886(2120587119899

119896)2 is power spectral density function

and 119886 denotes road condition and is chosen as 2 times 10minus6 Δ119899 isthe frequency interval Δ119899 = (119899max minus 119899min)119873 119899max and 119899minare the upper and lower cut-off frequencies respectively herethey are chosen as 5Hz and 01Hz 119899

119896is the wave number 120579

119896

is the random phase angle uniformly distributed from 0 to2120587

When calculating the dynamic behavior of the vehiclethe space function is transformed to time function accordingto the vehicle speed The generated road profile is shownin Figure 6 with the largest amplitude of roughness below2mm

313 Calculation Algorithm In this analysis interactionbetween the bridge and the vehicle is realized through contactforce At the contact points between the vehicle wheel andthe bridge displacement of the vehicle and the bridge shouldsatisfy compatibility at the contact points where the samedeformation of both the vehicle and the bridge should bemaintained Convergence should be satisfied at every contactpoint

minus2

minus15

minus1

minus05

0

05

1

15

2

Road

roug

hnes

s (m

m)

3 6 9 12 150Bridge (m)

Figure 6 Bridge road surface profile

There has been step by step algorithm for the computation[8] in which the vehicle moves through the bridge andthe contact force is calculated from the vehicle behavior ateach contact point Then the contact force is input to thebridge which causes bridge response and the response is fedback to the vehicle model After that the vehicle moves tothe next point and the process continues until the vehiclepasses the bridge In addition to step by step algorithm thereis also iterative procedure which treats the contact force astime history [15] In this analysis computation procedure isconducted in a global manner presented in Figure 7 andinteraction between the vehicle and the bridge is realizedthrough the contact force time history First road profile isused as the initial input for the vehicle and the contact forcehistory is calculated for each contact point The contact forcehistory is then applied to the bridge model in ABAQUSwhich leads to bridge response Then the bridge responseis incorporated in the vehicle dynamic equation as inputtogether with the road profile and a new contact force timehistory is generated This procedure is repeated until userdefined convergence criteria are satisfied Particularly theconvergence criterion is set that the maximum deflection dif-ference between successive iterations is smaller than 001 ofthe dynamic displacement at midspan Once the convergencecriterion is satisfied displacement response of the bridge isextracted for damage detection

314 Analysis Result Analysis is conducted until conver-gence is satisfied The difference of dynamic displacementat the midspan point is shown in Figure 8 It indicates thatafter three iterations the maximum difference between twoiterations is 5 times 10minus7m around 00025 of the midspandeflection of the bridge It satisfies the convergence criterionthat the displacement difference is smaller than 001 of themidspan deflection

Dynamic displacement at the midspan point is extractedfor damage detection given in Figure 9 The natural fre-quencies of the bridge were also extracted which shows that


Nodes at different times

Vehicle mass Road profile Bridge damping and stiffness displacement

StartCalculate contact force

Calculate bridge response using FEM

Check convergenceNo

Yes

Result

V = r(x)

MU + CU + KU = CV + KV

f = Mg minus C(U minus V) minus K(U minus V)

V = r(x) + Ub(x t) Δ(Ub(L2 t)) le 001 times Ub(L2 td2)

M C K r(x) Ub(x t)

Figure 7 Calculation algorithm for vehicle-bridge model

3 6 9 12 150Time (s)

minus01

minus005

0

005

01

015

Defl

ectio

n (m

m)

1st iteration2nd iteration

3rd iteration4th iteration

Figure 8 Convergence of displacement at contact points

the fundamental natural frequency is 060765Hz Hence thefundamental natural period is obtained as 1648 s In dataprocessing the second derivatives of the dynamic displace-ment data are taken at an interval of the natural period of thebridgeThe vehicle passes the bridge in 15 s and displacementis recorded Therefore 119891

119904is calculated as 134Hz For the

calculation of dynamic curvature Δ119909 is chosen as Δ119909 = 119879119899times

119891119904 Δ119909 is then rounded to an integer value which is 22

3 6 9 12 150Time (s)

minus20

minus15

minus10

minus5

0

5

Defl

ectio

n (m

m)

Figure 9 Dynamic displacement at the center of the bridge span

The dynamic curvature is illustrated in Figure 10 Whenthe vehicle-bridge interaction is not considered the dynamiccurvature is also presented for comparison given by the reddashed line In the curvature figure there are no calculatedcurvature values at both ends of the bridge span because thesecond derivative was taken at an interval Δ119909 It can be indi-cated from the figure that when interaction is not consideredcurvature plot is smoother and peak caused by damage ismore apparent In contrast there aremore fluctuations on thecurvature result which incorporated interaction between thevehicle and the bridge But even considering the interactioneffect the damage location is clearly identified


times10minus4

Without interactionWith interaction

3 6 9 12 150Bridge (m)

1

2

3

4

5

6

7

8

9

10

Curv

atur

e (1

m)

Figure 10 Curvature of bridge at midpoint of the span

Apparent peak at the location of 11m on the dynamiccurvature plot indicates the damage location Using thedamage extent calculation equation (9) the damage extent isobtained as 3998 Compared with the actual damage extentof 42 the difference is around 2 It demonstrates thateven with the effects from road roughness and vehicle-bridgeinteraction the damage detectionmethod can detect damagelocation and identify damage extent with good accuracy

32 Effect of Measurement Noise In order to investigate theeffect of measurement noise on efficacy of the proposedmethod random noise is generated and added to the calcu-lated displacement response of the bridge The displacementwith measurement noise is calculated as [5 6]

119908 = 119908119888+ 119864119901119873120590 (119908

119888) (15)

where 119908 is the polluted bridge displacement with mea-surement noise 119864

119901is the noise level and 119873 is a standard

normal distribution with zero mean value and unit standarddeviation 119908

119888is the calculated displacement at midpoint of

bridge span and 120590(119908119888) is its standard deviation

Three different noise levels of 01 03 and 05 areinvestigated and curvature result is shown in Figure 11 It isshown that damage location can be successfully determinedfor all three cases However damage extent is less accuratewhich is identified as 313 301 and 282 respectivelyThis is due to the fact that curvature values are distortedby measurement noise The proposed method can bothsuccessfully locate the damage and provide a quantitativeestimation of the damage extent

33 Effect of Damage Location and Extent In order toinvestigate the effect of damage location and extent differ-ent damage locations and extents are analyzed Damage is

times10minus4

01 noise03 noise05 noise

3 6 9 12 150Bridge (m)

1

2

3

4

5

6

7

8

9

10

Curv

atur

e (1

m)

Figure 11 Curvature at midspan with different noise level

Damage at 12 mDamage at 5 mDamage at 3 m

times10minus4

2

3

4

5

6

7

8

9

10

Curv

atur

e (1

m)

3 6 9 12 150Bridge (m)

Figure 12 Curvature at midspan with different damage location

assumed at location of 3m (L5) 5m (L3) and 12m (4L5)respectively and damage length is 075m with thicknessreduced to 005m Noise level is assumed as 01 and it isadded to calculated displacement in the same way as in thelast subsectionAll other conditions are defined the same as inSection 31 The curvature results are illustrated in Figure 12It is shown that damage can be detected reliably regardless ofits location Damage extent is identified as 368 329 and339 respectively


Damage intensity 20Damage intensity 40Damage intensity 60

3 6 9 12 150Bridge (m)

1

2

3

4

5

6

7

8

9

10

11

12

Curv

atur

e (1

m)

times10minus4

Figure 13 Curvature at midspan with different damage extent

Effect of damage extent is also studied by assumingdifferent damage extent of 20 40 and 60 Noise levelis assumed as 01 and all other conditions are defined thesame as in Section 31 The curvature results are plotted inFigure 13 which shows that the peak caused by damageincreases as damage extent increases from 20 to 60Damage extent is identified as 179 322 and 496 Thediscrepancy can be attributed to the road surface roughnessand measurement noise

34 Effect of Multi-Damage Case In this subsection thebeam with multiple damage is studied The bridge model isthe same as in Section 31 except that multiple damage casesare artificially created Totally damage is generated at threelocations which are 5m (13L) 10m (23L) and 12m (45L)of the bridgeThe damage extent is 42 for all cases and 01noise level is added to calculated displacement

Figure 14 shows results of the curvature at midspan ofthe bridge It is observed that all three damage cases canbe detected at corresponding locations Damage extent isalso identified using (9) which is 337 348 and 323respectively

A finite element bridge model is used to investigatereliability and efficacy of the proposed method Effects ofmeasurement noise damage location and extent and amulti-damage case are studied which show that the proposedmethod can be used to detect damage location and estimatedamage extent successfully It is noted that the purpose of thispaper is to verify the proposed damage detection algorithmnumerically and a simple supported beammodel is adoptedFurther application to real bridges should take into accounthigh local modes and torsion modes

3 6 9 12 150Bridge (m)

1

2

3

4

5

6

7

8

9

10

Curv

atur

e (1

m)

times10minus4

Figure 14 Curvature at midspan with multiple damage cases

4 Conclusions

Safety of bridges during operation is of much concern forthe society which requiresmore reliable and efficient damagedetectionmethods In this paper a damage detectionmethodis proposed based on dynamic displacement of bridge underthe loading of a moving vehicle Through this study severalconclusions are drawn as follows

(1) Closed-form solution of the vehicle-bridge problemshed light on decomposing dynamic displacementinto quasi-static component and dynamic compo-nent and curvature of the quasi-static componentcan be used to detect damage location and estimatedamage extent

(2) In curvature calculation using dynamic displace-ment the calculation interval is chosen as naturalperiod of the beam by which dynamic componentcan be canceled out while damage is detected withquasi-static component

(3) Numerical simulation of a beam model is investi-gated in which road surface roughness and vehicle-bridge interaction are considered Parametric studyon measurement noise damage location damageextent and multiple damage cases is performedwhich verified the reliability and efficacy of thismethod

In light of damage detection of bridges this method utilizesdynamic displacement of bridges under moving vehicleloading which can be operated with low cost and simpleimplementation Moreover baseline from undamaged stateis not required and both damage location and damage extentcan be assessed In field test measurements of real bridgesthere could be higher local modes or torsional modes whichshould be considered in application of the proposed methodFurther investigations will be conducted to study these issuesin future


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to express gratitude to ProfessorDionysius fromYokohamaNationalUniversity for his adviceson this research The first author is also grateful for thescholarship from MEXT (Ministry of Education CultureSports Science and Technology Japan) during his PhDstudy

References

[1] Y Fujino and D M Siringoringo ldquoBridge monitoring inJapan the needs and strategiesrdquo Structure and InfrastructureEngineering vol 7 no 7-8 pp 597ndash611 2011

[2] O S Salawu ldquoDetection of structural damage through changesin frequency a reviewrdquo Engineering Structures vol 19 no 9 pp718ndash723 1997

[3] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoTheShock and Vibration Digest vol 30 pp 91ndash105 1998

[4] K Huang US 6192758 B1 United States 2001[5] X Q Zhu and S S Law ldquoWavelet-based crack identification

of bridge beam from operational deflection time historyrdquoInternational Journal of Solids and Structures vol 43 no 7-8pp 2299ndash2317 2006

[6] K V Nguyen and H T Tran ldquoMulti-cracks detection of abeam-like structure based on the on-vehicle vibration signaland wavelet analysisrdquo Journal of Sound and Vibration vol 329no 21 pp 4455ndash4465 2010

[7] D Hester and A Gonzalez ldquoA wavelet-based damage detectionalgorithm based on bridge acceleration response to a vehiclerdquoMechanical Systems and Signal Processing vol 28 pp 145ndash1662012

[8] J-C Wyss D Su and Y Fujino ldquoPrediction of vehicle-inducedlocal responses and application to a skewed girder bridgerdquoEngineering Structures vol 33 no 4 pp 1088ndash1097 2011

[9] S P Timoshenko History of Strength of Materials D VanNostrand Company New York NY USA 1953

[10] L FrybaVibration of Solids and Structures UnderMoving LoadsAcademia Prague Prague Czech Republic 1999

[11] J R W Weaver S P Timoshenko and D H Young VibrationProblems in Engineering John Wiley amp Sons New York NYUSA 1990

[12] S P Timoshenko Strength of Materials D Van NostrandCompany New York NY USA 1955

[13] C J Dodds and J D Robson ldquoThe description of road surfaceroughnessrdquo Journal of Sound and Vibration vol 31 no 2 pp175ndash183 1973

[14] Y-B Yang and B-H Lin ldquoVehicle-bridge interaction analysisby dynamic condensation methodrdquo Journal of Structural Engi-neering vol 121 no 11 pp 1636ndash1643 1995

[15] M David Finite Element Analysis Sciyo 2010

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deformation is employed to detect damage Some otherresearchers used wavelet analysis to detect singularity inbridge response signal hence detecting damage in bridgesZhu and Law [5] conducted wavelet analysis on bridgeresponse under amoving loadWavelet coefficient of dynamicdisplacement is obtained to detect damage in a bridge modelNguyen and Tran [6] conducted similar study but insteadof using bridge response they used vehicle response forwavelet analysis which makes it unnecessary for sensorinstallation on a bridge Hester and Gonzalez [7] performedwavelet transform on response of a damaged bridge undermoving load and investigated feasibility to detect damagelocation They concluded that multiple measurement pointswere required to achieve higher accuracy In the wavelet-based damage detection techniques there is a major inherentdifficulty In order to localize damage narrow window ofbridge response that is short time series is needed Howeverfrequency resolution is impaired when short time series arechosen to calculate the frequency Due to this contradictioneffective detection of damage is difficult

In this paper a curvature based damage detectionmethodis proposed which uses dynamic displacement of a bridgesubject to loading of a moving vehicle It does not requirebaseline of undamaged state of bridge and both damagelocation and damage extent can be detected Firstly the phys-ical mechanism of dynamic displacement of damaged bridgeunder loading of a moving vehicle is elaborated Dynamicdisplacement is decomposed into quasi-static componentand dynamic component and curvature is used to detectdamage location and estimate damage extent Secondly theproposedmethod is verifiedwith a finite elementmodel usingbeam elements in which vehicle model and beam model areestablished in Matlab and ABAQUS respectively Vehicle-bridge interaction and roughness are both considered anddynamic displacement is used to calculate curvature anddetect damage The effect of parameters of measurementnoise level damage location damage extent and multipledamage cases is studied which illustrates reliability andefficacy of the proposed method Finally conclusions aremade for application of this method in engineering practice

2 Damage Detection withDynamic Displacement

The dynamic problem of vehicle-bridge system has been asubject of interest for a long time In the earlier times mostof the studies are aimed at predicting dynamic response ofbridges and providing a dynamic amplification factor forengineering design [8] Detailed review of the early analyticaland numerical studies could be found in previous literaturesuch as [9 10] In the following section physical backgroundis elaborated and it is shown that dynamic curvature is a goodindicator for damage detection

21 Analysis on Dynamic Displacement under Moving Vehi-cle In simplified case of an undamaged uniform simplysupported beam vehicle is modeled as a moving load with

Pv

x

L

Figure 1 Schematic figure of simply supported beam undermovingload

constant speed v as illustrated in Figure 1 The closed-form solution was deduced by Weaver et al [11] and thedisplacement of the bridge is expressed as follows

119910 (119909 119905) = minus21198751198713

1198981205872

infin

sum

119894=1

sin (119894120587119909119871)1198942 (119894212058721198862 minus V21198712)

sin(119894120587V119905119871)

+21198751198714V

1198981205873119886

infin

sum

119894=1

sin (119894120587119909119871)1198943 (119894212058721198862 minus

21198712)

sin120596119899119905

(1)

where 119886 = radic119864119868119898120596119899= (119894212058721198712)radic119864119868119898 119871 is the length

of the bridge 119875 is the amplitude of the moving load119898 is themass per unit length119864119868 is the flexural rigidity of the bridge Vis the velocity of themoving force120596

119899is the natural frequency

of the bridgeIt is noticed that the first part is related to the vehicle

speed which is referred to as the quasi-static component thesecond part contains bridge natural frequency and is referredto as dynamic component which comes from the vibrationof the bridge Figure 2 illustrates the dynamic displacement atmidspan which is composed of these two parts quasi-staticcomponent and dynamic component The proposed methodconcentrates on calculating curvature from the quasi-staticcomponent of the dynamic displacement and an innovativetechnique is used in curvature calculation to cancel out thedynamic displacement

22 Damage Detection with Curvature The relation betweenbending moment and curvature of a beam is expressed in thefollowing equation [12]

119872(119909) = minus119864119868 (119909)1198892119908 (119909)

1198891199092 (2)

where 119864 denotes Youngrsquos modulus of the material 119868 denotesthe second moment of area of beam cross section 119908(119909)denotes deflection of the beam 119909 denotes the beam coordi-nate and119872(119909) denotes the bending moment

Curvature can be calculated from the second derivative ofdeflection measurements which is expressed as

120581 (119909) =1198892119908 (119909)

1198891199092= minus

119872(119909)

119864119868 (119909) (3)

If there is damage (stiffness loss) in some location of thebeam abrupt change is expected at the corresponding loca-tion on the curvature plot


= +

1086420

Time (s)

1086420

Time (s)

2 4 6 80 10

Time (s)

Qua

si-st

atic

com

pone

nt (m

m)

minus15

minus1

minus05

0

05

Defl

ectio

n (m

m)

minus15

minus1

minus05

0

05

minus004

minus002

0

002

004

Dyn

amic

com

pone

nt (m

m)



120572 =119864119868119894minus 119864119868119889

119864119868119894

(4)



119864119868119894= minus

119872(119909)

120581119894(119909)

(5)

119864119868119889= minus

119872(119909)

120581119889(119909) (6)


yields

Δ120581

120581119894

=120572

1 minus 120572 (7)


120572 =Δ120581120581119894

Δ120581120581119894+ 1 (8)


120581 (119909) = 11990810158401015840(119909) =

119908 (119909 + Δ119909) minus 2119908 (119909) + 119908 (119909 minus Δ119909)

Δ1199092 (9)



Discontinuity

y( l2 t)

Damage


Time (s)

minus15

minus1

minus05

0

Defl

ectio

n (m

m)





Δ1199092


Δ1199092


Δ1199092

(10)


Δ119909 = 119879119899times 119891119904 (11)






M

K C










119872VV + 119862VV + 119870V119880V = 119891V (12)






Density 7800 kgm3




15m

11m





119903 (119909) =

119873

sum

119896=1

radic2119866119889(119899119896) Δ119899 cos (2120587119899

119896119909 minus 120579119896) (14)

where 119866119889(119899119896) = 119886(2120587119899




119896




minus2

minus15

minus1

minus05

0

05

1

15

2

Road

roug

hnes

s (m

m)

3 6 9 12 150Bridge (m)










Check convergenceNo

Yes

Result

V = r(x)




M C K r(x) Ub(x t)


3 6 9 12 150Time (s)

minus01

minus005

0

005

01

015

Defl

ectio

n (m

m)








3 6 9 12 150Time (s)

minus20

minus15

minus10

minus5

0

5

Defl

ectio

n (m

m)




times10minus4


3 6 9 12 150Bridge (m)

1

2

3

4

5

6

7

8

9

10

Curv

atur

e (1

m)




119908 = 119908119888+ 119864119901119873120590 (119908

119888) (15)








times10minus4


3 6 9 12 150Bridge (m)

1

2

3

4

5

6

7

8

9

10

Curv

atur

e (1

m)



times10minus4

2

3

4

5

6

7

8

9

10

Curv

atur

e (1

m)

3 6 9 12 150Bridge (m)





3 6 9 12 150Bridge (m)

1

2

3

4

5

6

7

8

9

10

11

12

Curv

atur

e (1

m)

times10minus4






3 6 9 12 150Bridge (m)

1

2

3

4

5

6

7

8

9

10

Curv

atur

e (1

m)

times10minus4


4 Conclusions









Acknowledgments


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










= +

1086420

Time (s)

1086420

Time (s)

2 4 6 80 10

Time (s)

Qua

si-st

atic

com

pone

nt (m

m)

minus15

minus1

minus05

0

05

Defl

ectio

n (m

m)

minus15

minus1

minus05

0

05

minus004

minus002

0

002

004

Dyn

amic

com

pone

nt (m

m)



120572 =119864119868119894minus 119864119868119889

119864119868119894

(4)



119864119868119894= minus

119872(119909)

120581119894(119909)

(5)

119864119868119889= minus

119872(119909)

120581119889(119909) (6)


yields

Δ120581

120581119894

=120572

1 minus 120572 (7)


120572 =Δ120581120581119894

Δ120581120581119894+ 1 (8)


120581 (119909) = 11990810158401015840(119909) =

119908 (119909 + Δ119909) minus 2119908 (119909) + 119908 (119909 minus Δ119909)

Δ1199092 (9)



Discontinuity

y( l2 t)

Damage


Time (s)

minus15

minus1

minus05

0

Defl

ectio

n (m

m)





Δ1199092


Δ1199092


Δ1199092

(10)


Δ119909 = 119879119899times 119891119904 (11)






M

K C










119872VV + 119862VV + 119870V119880V = 119891V (12)






Density 7800 kgm3




15m

11m





119903 (119909) =

119873

sum

119896=1

radic2119866119889(119899119896) Δ119899 cos (2120587119899

119896119909 minus 120579119896) (14)

where 119866119889(119899119896) = 119886(2120587119899




119896




minus2

minus15

minus1

minus05

0

05

1

15

2

Road

roug

hnes

s (m

m)

3 6 9 12 150Bridge (m)










Check convergenceNo

Yes

Result

V = r(x)




M C K r(x) Ub(x t)


3 6 9 12 150Time (s)

minus01

minus005

0

005

01

015

Defl

ectio

n (m

m)








3 6 9 12 150Time (s)

minus20

minus15

minus10

minus5

0

5

Defl

ectio

n (m

m)




times10minus4


3 6 9 12 150Bridge (m)

1

2

3

4

5

6

7

8

9

10

Curv

atur

e (1

m)




119908 = 119908119888+ 119864119901119873120590 (119908

119888) (15)








times10minus4


3 6 9 12 150Bridge (m)

1

2

3

4

5

6

7

8

9

10

Curv

atur

e (1

m)



times10minus4

2

3

4

5

6

7

8

9

10

Curv

atur

e (1

m)

3 6 9 12 150Bridge (m)





3 6 9 12 150Bridge (m)

1

2

3

4

5

6

7

8

9

10

11

12

Curv

atur

e (1

m)

times10minus4






3 6 9 12 150Bridge (m)

1

2

3

4

5

6

7

8

9

10

Curv

atur

e (1

m)

times10minus4


4 Conclusions









Acknowledgments


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Discontinuity

y( l2 t)

Damage


Time (s)

minus15

minus1

minus05

0

Defl

ectio

n (m

m)





Δ1199092


Δ1199092


Δ1199092

(10)


Δ119909 = 119879119899times 119891119904 (11)






M

K C










119872VV + 119862VV + 119870V119880V = 119891V (12)






Density 7800 kgm3




15m

11m





119903 (119909) =

119873

sum

119896=1

radic2119866119889(119899119896) Δ119899 cos (2120587119899

119896119909 minus 120579119896) (14)

where 119866119889(119899119896) = 119886(2120587119899




119896




minus2

minus15

minus1

minus05

0

05

1

15

2

Road

roug

hnes

s (m

m)

3 6 9 12 150Bridge (m)










Check convergenceNo

Yes

Result

V = r(x)




M C K r(x) Ub(x t)


3 6 9 12 150Time (s)

minus01

minus005

0

005

01

015

Defl

ectio

n (m

m)








3 6 9 12 150Time (s)

minus20

minus15

minus10

minus5

0

5

Defl

ectio

n (m

m)




times10minus4


3 6 9 12 150Bridge (m)

1

2

3

4

5

6

7

8

9

10

Curv

atur

e (1

m)




119908 = 119908119888+ 119864119901119873120590 (119908

119888) (15)








times10minus4


3 6 9 12 150Bridge (m)

1

2

3

4

5

6

7

8

9

10

Curv

atur

e (1

m)



times10minus4

2

3

4

5

6

7

8

9

10

Curv

atur

e (1

m)

3 6 9 12 150Bridge (m)





3 6 9 12 150Bridge (m)

1

2

3

4

5

6

7

8

9

10

11

12

Curv

atur

e (1

m)

times10minus4






3 6 9 12 150Bridge (m)

1

2

3

4

5

6

7

8

9

10

Curv

atur

e (1

m)

times10minus4


4 Conclusions









Acknowledgments


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Density 7800 kgm3




15m

11m





119903 (119909) =

119873

sum

119896=1

radic2119866119889(119899119896) Δ119899 cos (2120587119899

119896119909 minus 120579119896) (14)

where 119866119889(119899119896) = 119886(2120587119899




119896




minus2

minus15

minus1

minus05

0

05

1

15

2

Road

roug

hnes

s (m

m)

3 6 9 12 150Bridge (m)










Check convergenceNo

Yes

Result

V = r(x)




M C K r(x) Ub(x t)


3 6 9 12 150Time (s)

minus01

minus005

0

005

01

015

Defl

ectio

n (m

m)








3 6 9 12 150Time (s)

minus20

minus15

minus10

minus5

0

5

Defl

ectio

n (m

m)




times10minus4


3 6 9 12 150Bridge (m)

1

2

3

4

5

6

7

8

9

10

Curv

atur

e (1

m)




119908 = 119908119888+ 119864119901119873120590 (119908

119888) (15)








times10minus4


3 6 9 12 150Bridge (m)

1

2

3

4

5

6

7

8

9

10

Curv

atur

e (1

m)



times10minus4

2

3

4

5

6

7

8

9

10

Curv

atur

e (1

m)

3 6 9 12 150Bridge (m)





3 6 9 12 150Bridge (m)

1

2

3

4

5

6

7

8

9

10

11

12

Curv

atur

e (1

m)

times10minus4






3 6 9 12 150Bridge (m)

1

2

3

4

5

6

7

8

9

10

Curv

atur

e (1

m)

times10minus4


4 Conclusions









Acknowledgments


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














Check convergenceNo

Yes

Result

V = r(x)




M C K r(x) Ub(x t)


3 6 9 12 150Time (s)

minus01

minus005

0

005

01

015

Defl

ectio

n (m

m)








3 6 9 12 150Time (s)

minus20

minus15

minus10

minus5

0

5

Defl

ectio

n (m

m)




times10minus4


3 6 9 12 150Bridge (m)

1

2

3

4

5

6

7

8

9

10

Curv

atur

e (1

m)




119908 = 119908119888+ 119864119901119873120590 (119908

119888) (15)








times10minus4


3 6 9 12 150Bridge (m)

1

2

3

4

5

6

7

8

9

10

Curv

atur

e (1

m)



times10minus4

2

3

4

5

6

7

8

9

10

Curv

atur

e (1

m)

3 6 9 12 150Bridge (m)





3 6 9 12 150Bridge (m)

1

2

3

4

5

6

7

8

9

10

11

12

Curv

atur

e (1

m)

times10minus4






3 6 9 12 150Bridge (m)

1

2

3

4

5

6

7

8

9

10

Curv

atur

e (1

m)

times10minus4


4 Conclusions









Acknowledgments


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










times10minus4


3 6 9 12 150Bridge (m)

1

2

3

4

5

6

7

8

9

10

Curv

atur

e (1

m)




119908 = 119908119888+ 119864119901119873120590 (119908

119888) (15)








times10minus4


3 6 9 12 150Bridge (m)

1

2

3

4

5

6

7

8

9

10

Curv

atur

e (1

m)



times10minus4

2

3

4

5

6

7

8

9

10

Curv

atur

e (1

m)

3 6 9 12 150Bridge (m)





3 6 9 12 150Bridge (m)

1

2

3

4

5

6

7

8

9

10

11

12

Curv

atur

e (1

m)

times10minus4






3 6 9 12 150Bridge (m)

1

2

3

4

5

6

7

8

9

10

Curv

atur

e (1

m)

times10minus4


4 Conclusions









Acknowledgments


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











3 6 9 12 150Bridge (m)

1

2

3

4

5

6

7

8

9

10

11

12

Curv

atur

e (1

m)

times10minus4






3 6 9 12 150Bridge (m)

1

2

3

4

5

6

7

8

9

10

Curv

atur

e (1

m)

times10minus4


4 Conclusions









Acknowledgments


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Acknowledgments


References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article A Damage Detection Algorithm Utilizing ...

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