Vibration-BasedDamageDetectionofaSteel ...downloads.hindawi.com/journals/ace/2020/8889277.pdfdeﬁnedamageprobabilityfunctions[2].Wheninvisible damageoccursonacomplicatedstructuresuchasacom-positebridgedeckwithshearconnectors,theVBDDmethod

Research ArticleVibration-BasedDamageDetectionof a Steel-ConcreteCompositeSlab Using Non-Model-Based and Model-Based Methods

Liang Fang 123 Yun Zhou 12 Yunzhong Jiang1 Yilin Pei1 and Weijian Yi12

1College of Civil Engineering Hunan University Changsha 410082 China2Hunan Provincial Key Lab on Damage Diagnosis for Engineering Structures Hunan University Changsha 410082 China3College of Water Resource and Civil Engineering Hunan Agricultural University Changsha 410128 China

Correspondence should be addressed to Yun Zhou zhouyun05hnueducn

Received 8 March 2020 Revised 9 August 2020 Accepted 24 August 2020 Published 11 September 2020

Academic Editor Paolo Castaldo

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

is paper presents vibration-based damage detection (VBDD) for testing a steel-concrete composite bridge deck in a laboratoryusing bothmodel-based and non-model-based methods Damage that appears on a composite bridge deck may occur either in theservice condition or in the loading condition To verify the efficiency of the dynamic test methods for assessing different damagescenarios two defect cases were designed in the service condition by removing the connection bolts along half of a steel girder andreplacing the boundary conditions while three damage cases were introduced in the loading condition by increasing the appliedload A static test and a multiple reference impact test (MRIT) were conducted in each case to obtain the corresponding deflectionandmodal data For the non-model-basedmethod modal flexibility andmodal flexibility displacement (MFD) were used to detectthe location and extent of the damage e test results showed that the appearance and location of the damage in defect cases andloading conditions can be successfully identified by the MFD values A finite element (FE) model was rationally selected torepresent the dynamic characteristics of the physical model while four highly sensitive physical parameters were rationallyselected using sensitivity analysis e model updating technique was used to assess the condition of the whole deck in the servicecondition including the boundary conditions connectors and slab Using damage functions Strand7 software was used toconduct FE analysis coupled with theMATLAB application programming interface to updatemultiple physical parameters Of thethree different FE models used to simulate the behavior of the composite slab the calculated MFD of the shell-solid FE model wasalmost identical to the test results indicating that the performance of the tested composite structure could be accurately predictedby this type of FE model

1 Introduction

Steel-concrete composite bridges are used extensively inhighway networks In this type of bridge system reinforcedconcrete (RC) decks are connected to steel girders to obtainthe merits of both materials and to increase the rigidity ofboth the girders and the slabs Shear connectors (headedstuds) provide integrity to the composite bridge by in-creasing the composite rigidity for uniform actions underlive loads and resisting horizontal shear at the girder-deckinterface However deterioration of the connectors overseveral years will decrease the composite action and cracksthat occur under the service load due to insufficient strengthor fatigue may result in increased damage plus settlement

caused by increased traffic weights can induce or change theintrinsic forces in the bridge deck Since all defects anddamage can influence the load-carrying capacity of thebridge early detection of structural damage in a bridge istherefore crucial to avoid life and economic losses due tocatastrophic failures To detect structural damage tradi-tional nondestructive techniques (NDT) such as ultrasonicmethods magnetic fieldmethods radiographmethods eddycurrent methods and thermal field methods have beendeveloped to identify local damage [1] However a new rapidnondestructive assessment technique needs to be developedfor practical applications Among the various methods thevibration-based damage detection (VBDD) is preferred as ituses changes in modal curvature and natural frequencies to

HindawiAdvances in Civil EngineeringVolume 2020 Article ID 8889277 20 pageshttpsdoiorg10115520208889277

define damage probability functions [2] When invisibledamage occurs on a complicated structure such as a com-posite bridge deck with shear connectors the VBDDmethodmay be particularly well suited as a global testing method todetermine the structural condition of the bridge Because ofthe easy acquisition of vibration characteristics and globalinformation on structural conditions [3] the VBDDmethodcan assess the condition of an entire structural componentwhen some portions of the structure remain inaccessible toinspection by common NDE methods In addition whenvisible damage appears on a composite deck the damagedetected by the vibration-based method can be investigatedusing the provided dynamic information

e VBDD method detects changes in the dynamiccharacteristics (eg natural frequencies modal shapes anddamping ratios) which can be regarded as a global responsesignature resulting from changes in the stiffness mass andboundary conditions It is a great challenge to develop robustalgorithms with the ability to detect locate and quantifydamage according to the measured dynamic responses of thestructure Some complex situations and conditions such asboundary conditions bond-slip occurrences and sub-structure damage would be reflected in the vibration sig-nature e advantage of the VBDD method is thecontinuous monitoring of the structural conditions andidentifying the earliest occurrence of possible damage eVBDD method comprises the following four levels thatdetermine changes in dynamic structural characteristics [4](1) qualitatively indicating damage occurrences (2) local-izing spatial information (3) estimating the extent ofdamage and (4) predicting the actual safety of the structureModal parameters such as natural frequencies and modeshapes are frequently used as damage-sensitive features toidentify damage [5 6] Many applications of the VBDDmethod have been developed in the past two decades aspresented by Doebling et al [7] and Sohn et al [8] Moughtyand Casas [9] provided an in-depth review of the devel-opment of modal-based damage-sensitive features (DSFs) inbridges and a synopsis of the challenges involved eyaddressed the challenges by drawing upon advanced tech-niques outlined in recent literature Yang et al [10] proposedthe extraction of bridge frequencies from the dynamic re-sponse of a moving test vehicle ey also verified thetechnique by a field test and presented a state-of-the-artreview of the related research works conducted worldwidee VBDD method can be classified into the non-model-based method and the model-based approach e biggestdifference in these two methods lies in whether a calibratedfinite element (FE) has been used in the data analysis or note non-model-basedmethod utilizes dynamic properties inmeasurements to directly deduce the physical characteris-tics which is also known as a damage fingerprint methode model-based method uses dynamic properties to cali-brate the initial FE model and then deduce the damagedphysical parameters by comparing updated and baselinecases Although the non-model-basedmethod is much easierto apply since no time and effort need be spent on modelingand model calibration it is harder to detect damage in somestructures than with the model-based method

For a steel-concrete composite deck damage that occursin the structure such as the loosening of shear connectionschanges in the boundary conditions and the development ofcracks caused by loading changes the structural vibrationcharacteristics Several studies have attempted to detectdamage in the connectors between steel-concrete compositegirders both in the laboratory and on actual projectsHowever limited studies have focused on damage detectionapproaches for steel-concrete composite structures and verylittle research has used the VBDD method to provide asatisfactory assessment of different damage cases for com-posite structures Xia et al [11] made the first attempt atdetecting possible damage of shear connectors in slab-girderbridges through vibration methods A 1 3 scale bridgemodel was constructed in a laboratory to test the suitabilityand efficiency of the VBDDmethods Since it was found thatthe local approach is more sensitive to identifying the localdamage of shear connectors than global methods the localapproach was developed and applied to assess the conditionof the shear connectors in a full-size slab-girder bridge [12]It was concluded that the newly developed method is a morepractical approach for detection of damaged connectorsthan the model updating technique Dilena andMorassi [13]presented a EulerndashBernoulli model of a composite beam thataccurately described the measured dynamic response ofcomposite beams with either severe or intermediate levels ofdamage A diagnostic technique based on frequency mea-surements was then applied to the suggested model whichprovided positive results Xu and Jiang [14] used a number ofpiezoelectric zirconate titanate (PZT) patches on the upperflange of a steel girder and concrete slab and the corre-sponding electromechanical impedance was measured usingan impedance analyzer before and after loosening theconnection bolts Allahyari et al [15] investigated the dy-namic characteristics of unfilled steel-concrete decks in-cluding normal-weight high-strength concrete (HC) andlightweight high-strength concrete (LHC) It was concludedthat the decks fabricated with plain concrete (DPC) and thedecks fabricated with LHC (DLHC) had approximatelysimilar serviceability whereas DLHC can be more applicablethan DPC due to lower weight Zhang et al [16] presented anon-model-based damage detection approach for bridgestructures based on the phase trajectory change of multitypevibration measurements e experimental study demon-strated that the proposed approach can be used to identifythe shear connection failure in a composite bridge modelsubjected to moving loads Wroblewski et al [17] demon-strated a method to show how changes in energy transferratios (ETRs) could be used for damage detection and lo-calization in steel-concrete composite beams

e dynamic computer simulation technique was usedby Shih et al [18] to identify damage to slab-on-girderbridges Based on that technique they used modal flexibilitychange and modal strain energy change for damage local-ization of shear connectors Results showed that theirmethod was effective in damage assessment for slab-on-girder bridge superstructure e autoregressive with ex-ogenous input (ARX) models and the sensor clusteringdamage identification technique [19] were then adopted to

2 Advances in Civil Engineering

identify the damage of a full-scale five-girder bridge Zhanget al [20] used quantum-inspired genetic algorithms(QIGAs) to improve computational efficiency by trans-forming the scaling factor sign determination problem to anoptimization problem And an experimental example of asteel-concrete composite slab and numerical example of athree-span continuous rigid-frame bridge were studied toverify the effectiveness of the proposed method Tan et al[21] used the vibration characteristics and artificial neuralnetwork (ANN) to detect damage in composite slab-on-girder bridge structure Wang et al [22] proposed a sparseBayesianmodel for structural damage detection based on thevariational Bayesian inference (VBI) and delayed rejectionadaptive Metropolis (DRAM) algorithms and a laboratory-tested frame was utilized to verify its effectiveness

e developed signal processing technique has re-cently been proposed for damage identification of com-posite structures Liu and De Roeck [23] proposed adamage indicator for identifying stud damage based onthe local modal curvature and the wavelet transformmodulus maxima e efficiency of the damage indicatorwas investigated using numerical simulations that in-troduced varying levels of damage to the stud by de-creasing the spring stiffness Ren et al [24] simulateddifferent damage scenarios by removing some shearconnectors in a bridge model and a signal-based damagedetection method in which the damage feature wascharacterized by the wavelet packet energy changes wasused to identify damage in the shear connectors Li et al[25] proposed a dynamic damage detection approachbased on the wavelet packet energy of cross-correlationfunctions from ambient vibration measurements toidentify the damage of shear connectors in slab-on-girderbridges Bao et al [26] presented an improved Hil-bertndashHuang transform (HHT) algorithm for identifyingtime-varying systems and analyzing nonlinear structuralresponses with closely spaced modes e robustness andeffectiveness of this algorithm were verified using bothnumerical simulations and laboratory measurements ofvibration data on a scaled concrete-steel composite beammodel

To overcome the limitation of visual-based and vibra-tion-based approaches to access the shear connection incomposite bridges an innovative relative displacementsensor has been developed to directly measure the relativedisplacement between the slab and girder in compositebridges [27] Continuous wavelet transform (CWT) andHHT [28] were used to analyze the measured dynamicresponses and to identify the damage of shear connectors ina composite bridge model under moving loads e resultsdemonstrate that relative displacement is a better responsequantity for structural health monitoring of compositebridges Moreover Dackermann et al [29] presented adynamic-based method for evaluating the connection sys-tems of timber composite structures e proposed dynamicmethod provided an alternative to traditional static loadtesting and used vibration measurements to derive a loss ofcomposite action index

Detecting minor structural damage at an early stage toprevent further structural degradation and potential pro-gressive failure was of great importance for bridge conditionassessment However the detection of damage in thesestructures is difficult because the baseline data of thesestructures such as their design details and an accurate FEmodel are usually not available To overcome that in thisstudy two kinds of VBDD methods namely model-basedand non-model-based damage detectionmethods were usedfor damage detectione efficiency and reliability of the twodetection methods were also described To the best of theauthorsrsquo knowledge it was the first attempt to use bothVBDD methods to detect the occurrence location andextent of the damage in a steel-concrete composite slab eeffectiveness of both methods was demonstrated in pre-dicting damage of a composite bridge to enable early ret-rofitting and prevent bridge failure Moreover two differentdamage simulation strategies were carried out in the damagedetection experiment which enhanced the practicability ofthe experimental study e aim of this study was thereforeto assess the feasibility of applying VBDD to identify thedifferent damage cases of a steel-concrete compositestructure To achieve this aim a 1 3 scaled steel-concretecomposite bridge model was built in the laboratory to in-vestigate the possibility of using VBDD Although the ex-periment was conducted in the laboratory it should beemphasized that the proposed techniques can be applied toall types of short bridges in the field e focus was ondetecting small levels and different types of damage using aglobal vibration testing method combined with the elaborateproposed techniques Five different cases were designed tosimulate damage in the service state and in the loading stateand static and dynamic experiments were performed underdifferent damage conditions e removable bolts andsupports were specially designed as the controllable damageand different levels of loading were carried out to obtain theactual damage incrementally Owning to using these damagesimulation strategies the interfacial bond-slip damage be-tween the concrete and the steel girder was simulated ef-fectively which was rarely considered in previous studiesDamage of the concrete deck was evaluated using twodifferent strategies e first strategy was to assess thecondition using the damage index method which does notrely on an FE model so the modal flexibility was directlyobtained through a multiple reference impact test (MRIT)e second strategy was to detect the damage using the FEmodel In this case the calculation and analysis were per-formed using the Strand7 FE analysis package coupled withthe MATLAB application programming interface (API)after themodel was updated with the data from the static anddynamic experiments on the slab It was the first attempt touse both model-based and non-model-based damage de-tection methods to detect the occurrence location andextent of the damage in the steel-concrete composite slabe effectiveness of both methods was demonstrated andthe findings of this research will be useful in predictingdamage of composite bridges to enable early retrofitting andprevent bridge failure

Advances in Civil Engineering 3

2 Test Specimen and Defect Case Design

21 Test Specimen A 1 3 scaled steel-concrete compositedeck model was designed and constructed to represent half ofthe span of Pennsauken Creek Bridge in New Jersey USAe specimen was constructed in Hunan Provincial Key Labon Damage Diagnosis for Engineering Structures at HunanUniversity e steel-concrete composite deck had dimen-sions of 400mtimes 205m to fit within the available laboratoryspace A 60mm C30 concrete slab was connected to threeQ235 I-girders by shear connectors e cubic compressivestrength of the concrete was 5368MPa which was used toestimate the elastic modulus of 351times 104Nmm2 based onthe Chinese Code for the design of concrete structures(GB50010-2010) [30] e HPB 300 Φ 6mm rebars in theconcrete slab were arranged in double layers e yieldstrength of the rebar was 30784MPa and its ultimatestrength was 42901MPa according to the reinforcementtensile test ree steel I-girders were connected at the middleand at both ends by steel channel diaphragms e specimenwas simply supported on adjustable bearings which were 3fixed supports and 3 rolling supports on each side especimen cross section and the dynamic test instrumentlayout are shown in Figure 1 Two types of connections weremade on this specimen headed studs and removable boltse removable bolts were designed to simulate damage and tothen be reset to an undamaged state e 13mm diameterheaded studs were welded onto the upper flange of the middlegirder while the 12mm removable bolts connected theconcrete slab to girder 1 and girder 3 by drilling holes in theupper flange of the steel girder After the concrete was pouredthe bolt sleeves were permanently fixed e connection wasthen completed by threading the bolt through the bolt sleevee tensile strength of the bolt was 800MPa with a yield ratioof 08 To design the removable bolts in an undamaged statethe bolts were tightly screwed into the bolt sleeves in adamaged state the bolts were simply unscrewed from the boltsleeve and completely removed e distribution of bolts andconnectors is presented in Figure 2 and the detailed designinformation can be found in Jiang [31]

22 Defect Cases in the Service Condition (Case 1 and Case 2)Considering the appearance of corrosion and fatigue underunexpected overloading two cases were designed to simu-late different damage scenarios of a composite deck in theservice condition e possible defects in the service con-dition were simulated as shown in Figure 3 In Case 1 thesteel support (E 2times105MPa) at point 1 was replaced by apolyurethane panel rubber support (E 60MPa) to simulatea boundary condition deficiency as shown in Figure 3(a) InCase 2 the bolts between points 5 and 9 on girder 1 wereloosened to simulate interfacial bond-slip damage betweenthe concrete and the girder as shown in Figure 3(b) edesigned defect cases had no material damage on theconcrete slab or on the steel girders e specimen in theintact stage corresponding to the two defect cases was de-fined as reference case I (Ref Case I)

23 Damage Cases in the Loading Conditions (Cases 3 4 and5) e static load tests performed on the composite deckcan be classified into two different procedures the weightloading procedure in the undamaged state and the hydraulicloading procedure in the damaged state e vertical de-flections were measured by 15 displacement sensors alongthree longitudinal girders (denoted as D01-15) Fifteenstrain gauges were used to measure the concrete strain(denoted as C01-15) which were attached at the 14 38 and12 locations of the slab Twenty-four strain gauges wereinstalled on the steel girder four strain gauges were installedat the 14 and 12 span locations of each steel girder whilethree strain gauges were installed on the web of the girderand one was installed at the bottom of the flange (denoted asST01-24) e static stain data were recorded by TDS-530data loggere instrumentation layout is shown in Figure 4

In the first loading procedure the weights were stackedon different instrumentation points except for the boundarysupports to measure the deflection of the composite deck inthe undamaged state this state was defined as reference caseII (Ref Case II) as shown in Figure 5(a) and the static anddynamic data were used as the baseline for further analysisIn the second loading procedure a multilevel hydraulicloading test was conducted on the composite deck with threeloading-unloading cycles that increased until the deck lostcarrying capacity as shown in Figure 5(b) ree damagecases (Cases 3 4 and 5) were defined according to theseverity of the damage on the model under cyclic loadinge typical load-displacement and load-strain curves areshown in Figure 6 e three damage cases were classified asfollows

(1) Case 3 when the load was 10 kN the strain increasedlinearly as the residual displacement remained at95mm e structure then remained in the elasticstage as it was unloaded and visible cracks began toappear on the concrete deck near girder 1

(2) Case 4 when the load was continually increased to17813 kN the steel strain showed obvious plasticdeformation especially along girder 1 and theconcrete strain gauge attached to the bottom surfaceof the concrete presented an obvious bilinear load-strain relationship until the strain gauge failed estiffness of the structure obviously decreased with theappearance of new cracks around the three steelgirders and the residual displacement remained at173mm e load was increased until a moderatenumber of cracks appeared and the damage accu-mulated and extended from the middle of the slab tothe two sides of the supports

(3) Case 5 when the load was 258 kN the displacementincreased rapidly until the bolts on girder 1 werebroken by the interfacial shear force e steel strainalmost reached its ultimate strain and the residualdisplacement remained at 925mm In this stage anobvious flexural deformation appeared and thecracking range increased before the steel girderbegan to yielde crack width continued to increase


and further penetrate the entire concrete deck Fi-nally the bolt at the critical location was broken bythe interfacial shear force which was followed byspalling of the concrete e typical crack patterns ofthe three damage cases are shown in Figure 7 Be-cause the cross section of composite deck wasasymmetric the crack patterns displayed eccentriccharacteristics

3 Dynamic Test and ExperimentalModal Analysis

31 Multiple Reference Impact Test To determine the dy-namic characteristics of the composite deck a MRIT wasused to generate modal information and further detectpossible defects or damage in different cases In the test aPCB-086D20 sledge hammer (with a sensitivity coefficient of023mvN and a frequency range of 0ndash6000Hz) was used togenerate an impulse signal and the vibration responses werecollected by ICP-type KD1010L accelerometers (with asensitivity of 10 mvmsminus2 and a frequency range of05ndash7000Hz) Both of the signals were collected using SignalCalc DP730 data acquisition e sampling frequency wasset to 4096Hz with a duration of 8 s To mitigate the ambientnoise in the dynamic test each point was impacted 5 times to

generate average values MRIT was performed in the un-damaged state to obtain the modal data for the referencecasese dynamic test was then conducted for each damagecase to generate the modal information In Case 1 and Case2 only girder 1 and girder 2 were tested

32Modal Parameter Identification In MRIT the measureddynamic signals of the force and response in the time do-main were processed by fast Fourier transform (FFT) andthen analyzed using the autopower spectrum and the cross-power spectrum in the frequency domainH1 estimation wasthen used to analyze the frequency response function (FRF)thereafter the modal poles were extracted from the peakvalue of the FRF curves using the complex mode indicatorfunction (CMIF) methode peak-picking extraction of theeigenvalue from the singular value figure in Ref Case II isshown in Figure 8 and the first 9 identified mode shapes areshown in Figure 9

e modal parameters such as the frequencies anddamping ratios of defect Case 1 and Case 2 were comparedwith those of Ref Case I as shown in Table 1 e naturalfrequencies of the specimen in different cases generallydecreased when defect damage was introduced Comparedwith the higher modes the lower modes were more sensitiveto slab defects e first modal frequency in defect Cases 1

150 760 760 380

HM 148 times 100 times 6 times 9 80 times 43 times 5 times 8

60

30

(a)

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18

19 20 21 22 23 24 25 26 27

1900 1001900100

Girder 3

Girder 2

Girder 1HM 148 times 100 times 6 times 9 HM 148 times 100 times 6 times 9

HM 148 times 100 times 6 times 9 HM 148 times 100 times 6 times 9


760

150

760

380

Location of acceleration sensors

(b)

Figure 1 Steel-concrete composite slab model (a) cross section (b) instrumentation layout for the dynamic test (unit mm)

Headed stud

Coating oil

Bolt

Bolt

(a)

Concrete slab

Bolt

Bolt sleeveburied inthe slab

Steel grider

(b)

Figure 2 Design of composite slab connection (a) distribution of the headed shear stud and bolt connections (b) detailed configuration ofthe bolted connection along girder 1 and girder 3


and 2 decreased by 625 and 305 respectively emodal frequencies were more sensitive to changes in thebearing stiffness due to the introduced defects in the model

In Cases 3 4 and 5 the modal parameters of the deckidentified using the CMIF method are listed in Table 2 Ingeneral with increasing damage the natural frequenciesdecreased while the damping ratios increased in the samemode e frequencies and damping ratios of the lowermodes changed more obviously than those of the highermodes and the decreasing tendency of the modal fre-quencies and damping ratios followed the static damageseverity e first modal frequency in Cases 3 4 and 5

decreased by 083 1237 and 2431 respectively whichwere generally larger than the decreases due to the intro-duced defect

4 Damage Detection Using the Non-Model-Based Method

41 Modal Flexibility by MRIT e concept of structuralmodal flexibility was proposed by Clough and Penzien [32]as one of the common dynamic damage fingerprints [33]Modal flexibility which refers to a close approximation offlexibility extracted from modal test results has been shown

(a)

(b)

Figure 3 Two defect cases in the service condition (a) Case 1 (b) Case 2

ST05-0817-20

ST09-1221-24

ST01-0413-16

Girder 3

Girder 2

Girder 1C01

C02

C03

C04

C05

D02

D13

C06

C07

C08

C09

C10

D08

C11

C12

C13

C14

C15

D03

D14

D09

D04

D15

D10

D05

D16

D11

D01

D12

D07

1900 1900 1900 1900 1900100 100

380

380

380

380

380

150

Steel strain gauge location (ST01 ~ 24)

Displacement sensor location (D01 ~ 15)Concrete strain gauge location (C01 ~ 15)

Figure 4 Instrumentation layout of static test


to be an excellent measure of flexibility if a sufficient numberof modes are included [34] It has been devised as a reliableexperimental signature reflecting the existing condition ofstructures and can also serve as a stringent check to validatethe accuracy and completeness of modal analysis results[35] Structural modal flexibility and its changes are effectiveindices for evaluating structural performance and are moresensitive than frequency and mode shape rough dynamictesting using MRIT the modal flexibility can be directlydetermined from the impact and response data and thevariation of the modal flexibility can be used to detectdamage without relying on an FE model e two ap-proaches for extracting modal flexibility were (1) the ex-traction of mass normalized mode shapes and modalfrequencies and (2) the identification of a synthesized FRFmatrix [34] In the calculation strategy adopted by this studythe modal mass coefficient is directly extracted from the FRFmeasured by the MRIT e FRF of the experiment can bewritten in partial fraction form

Hpq(ω) 1113944n

r1

Apqr

jω minus λr( 1113857+

Alowastpqr

jω minus λlowastr( 11138571113890 1113891 (1)

whereHpq(ω) is the FRF at point pwhen the impacting takesplace at point q j represents an imaginary number ω is thefrequency λr represents the r

th order pole of the system andApqr is the rth order residue caught at point p when thehitting takes place at point q Apqr QArψprψqr where QAris the modal scale factor of the rth order mode ψpr andψqr represent the mode shape coefficient of the rth ordermode at point p and point q respectively and lowast representscomplex conjugates

Using the modal parameter estimation algorithm [35]the FRF between the degree of freedom (DOF) of p and q atω 0 was calculated and substituted into the previous ex-pression as follows

Hpq(ω) 1113944m

r1

ψprψqr

MAr minusλr( 1113857+

ψ lowastprψlowastqr

MlowastAr minusλlowastr( 1113857

1113890 1113891 (2)

whereHpq(ω) is the FRF at point p due to the hitting input atpoint q andMAr is the modal mass coefficient of the rth ordermode MAr 1QAr

e damage index method is a vibration-based methodused for identifying damage in a structure including thedamage severity After the modal flexibility matrix of thestructure is obtained it can be multiplied by the corre-sponding load force vector at each measurement point edifference between the modal flexibility displacements(MFDs) of different cases is considered to be a damageindex as shown in the following equation

MFD Dd minus Dr

11138681113868111386811138681113868111386811138681113868

Dr

times 100 (3)

where Dd is the MFD under the damage condition and Dr isthe MFD of the reference case

42 Defect Detection Two defect cases were analyzed usingthe CMIF method to select the peak values on the signalvalue curves the modal flexibility can be estimated usingequations (1) and (2) to predict the displacement value of thecomposite model under the applied weights e displace-ment values obtained from the dynamic test analysis andmeasured by the static load test in the reference state wereanalyzed Comparing the generated modal flexibility in RefCase I and the defect cases theMFD values of girders 1 and 2are shown in Figure 10 In Case 1 when the defect occurredat point 1 of the support the MFD values were sensitive tothe induced damage location and the maximum MFDoccurred at point 1 with a difference of 1196 In Case 2the displacement values of girder 1 changed significantlywhen the bolts were loosened from point 5 to point 9 alonghalf of girder 1 e MFD values at point 8 and point 9reached 221 and 898 respectively Comparing thedifferent defect cases in the service condition the appearanceand location of the damage due to changing the boundaryconditions and loosening the bolts were effectively identifiedby the MFD values is means that the non-model-basedmethod can accurately identify both the damage and itslocation

43 Loading Damage Detection In Cases 3 4 and 5 thedisplacements caused by the measured static test and thedynamic modal flexibility were compared for different cases

(a) (b)

Figure 5 Pictures of the static tests (a) weight loading test (b) hydraulic loading test


e difference in MFD values between the damage cases andRef Case II is shown in Figure 11 In Case 3 a 175 dif-ference in the MFD appeared at point 5 which is the exactposition where the loading cracks first appeared thereforethe MFD value changed more obviously on girders 1 and 2than on girder 3 In Case 4 the MFD values of girder 2 arelarger than those of girder 1 because the cracks extendedfrom girder 1 to girder 2 However the changes in the MFDvalue tended to be uniform at the other instrumentationpoints In Case 5 with increasing applied load the changesin the MFD value increased up to 928 At this stage the

visible damage on the specimen indicated ultimate failure ofthe structure As a method that relies on input and outputdynamic test data the modal flexibility method along withthe MFD value can effectively identify the location andextent of damage It can be used for structural performanceprediction and damage detection with reliable accuracy eprimary advantage of the proposed non-model-based ap-proach is that it does not need baseline data (recorded withthe structure in an undamaged state) or an accurate FEmodel is method was suitable for damage detection withlarge modeling errors

1st loading2nd loading3rd loading

Load

(kN

)

0

50

100

150

200

250

300

20 40 60 80 1000Displacement (mm)

(a)

ST04ST08ST12

ST16ST20ST24

Load

(kN

)

0

50

100

150

200

250

300

5000 10000 15000 20000 250000Strain (με)

(b)

C11C12C13

C14C15

Load

(kN

)

0

50

100

150

200

250

300

500 1000 1500 2000 25000Strain (με)

(c)

Figure 6 Measured displacement and strain on the structure (a) measured vertical displacement at the middle span of girder 2 (b)measured steel strain at the 14 and 12 span locations of girder 1 2 and 3 (c) measured concrete strain on the bottom surface at the 12 spanlocation


5 Damage Identification Using the Model-Based Method

51 FE Model Construction In addition to the non-model-based method an FE model can also be used to detectdamage via a model updating technique is model-basedmethod depends on rational modeling to fully consider thedetailed modeling configuration A precise model can beused to simulate the mechanical behavior of the element andthen relate it to the dynamic parameters In order to obtainan accurate and reliable finite element model an FE modelwas constructed using Strand7 software and computationalsimulation was conducted with different modeling strategiesfor the composite slab ree different modeling strategies

were used according to the test specimens shown in Fig-ure 12 a beam-shell element model a beam-solid elementmodel and a shell-solid element model In the FEmodel therebar was considered to be a uniformly distributed materialin the concrete and the integrity model was adopted for thedesign of the concrete slab

e interfacial adhesion of the two materials the shearstiffness of the connectors and the boundary conditionsshould all be considered in the model erefore definingthe link element between different components was veryimportant in the modeling process According to the dif-ferent objects in elements there were four kinds of linkelements in three models beam-support link element beam-slab link element slab-slab link element and connection

Girder 1

Girder 2

Girder 3

(a)

Girder 1

Girder 2

Girder 3

(b)

Girder 1

Girder 2

Girder 3

(c)

Figure 7 Crack patterns in different hydraulic loading cases (a) Case 3 (b) Case 4 (c) Case 5

100

10ndash2

10ndash4

10ndash6

10ndash80 50 100

Frequency (Hz)150 200

Log

mag

nitu

de (m

mN

)

Figure 8 Peak-picking extraction in the singular value in Ref Case II


500

0

ndash5001500

1000

Y X5000 0

10002000

3000

Z

(a)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(b)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(c)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(d)

15001000

Y X5000 0

10002000

3000

500

0

ndash500Z

(e)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(f)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(g)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(h)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z(i)

Figure 9 First 9 identified mode shapes in Ref CaseII (a) 1st mode (b) 2nd mode (c) 3rd mode (d) 4th mode (e) 5th mode (f ) 6th mode (g)7th mode (h) 8th mode (i) 9th mode

Table 1 Comparison of the identified modal information for Case 1 and Case 2

No Ref Case I Case 1 Case 2Freq (Hz) Damp () Freq (Hz) Damp () Diff () Freq (Hz) Damp () Diff ()

1 2001 247 1876 minus308 minus625 1940 271 minus3052 2532 152 2406 minus180 498 2493 202 minus1543 3174 172 3089 minus167 268 3137 154 minus1174 6940 680 6563 minus271 543 6950 803 0145 7454 411 mdash mdash mdash 7375 305 minus1066 8233 234 8568 minus243 407 8210 207 minus0287 11061 460 10340 minus376 652 mdash mdash mdash8 11350 319 11755 minus349 357 11143 368 minus1829 15439 275 14980 minus380 297 15013 369 minus276Note Diff (f1 minus fr1)fr1 times 100 f1 is the measured frequency of Case 1 or Case 2 and fr1 is the measured frequency of Ref Case I

Table 2 Comparison of the identified modal information for Cases 3 4 and 5

Mode Ref Case II Case 3 Case 4 Case 5Freq (Hz) Damp () Freq (Hz) Damp () Diff () Freq (Hz) Damp () Diff () Freq (Hz) Damp () Diff ()

1 2530 099 2509 147 minus083 2217 191 minus1237 1915 221 minus24312 3349 193 3332 103 minus051 3170 258 minus534 2846 199 minus15023 6776 175 6756 226 minus030 6524 231 minus372 6370 207 minus5994 7789 258 7631 265 minus203 7314 332 minus610 7063 277 minus9325 8471 143 8309 125 minus191 8002 154 minus554 7974 165 minus5876 9949 386 9742 424 minus208 9546 538 minus405 9717 283 minus2337 12537 312 12113 282 minus338 11956 310 minus463 11725 252 minus6488 13685 352 13291 303 minus288 13011 317 minus493 13685 296 0009 16526 333 16744 325 132 16274 287 minus152 16151 361 minus227Note Diff (f2 minus fr2)fr2 times 100 f2 is the frequency of Cases 3 4 or 5 and fr2 is the frequency of Ref Case II


element e first three link elements were simulated byspring-damping elements which have different axial andlateral stiffness e connection element was then modeledas an octahedron element which consists of two spring-damping elements and four rigid elements as shown inFigure 13 e spring-damping element in the connectorincluded only the axial stiffness of the spring And the in-tersections of rigid elements were located in the verticalalignment nodes of the concrete slab and steel girder evalues for the spring stiffness are shown in Table 3

e nominal physical properties of the concrete and thesteel were used to model the deck e uniaxial stress-strainrelationship of concrete prescribed in GB50010-2010 [30]was employed to simulate the constitutive relationship of theconcrete component (Figure 14(a)) e compressive

strength of the concrete was fc 5386MPa which resultedin an estimated elastic modulus of Ec 351times 104MPaPoissonrsquos ratio was 02 and the density was 2450 kgm3 Todescribe the stress-strain relationship of steel the trilinearisotropic hardening model was employed addressing thestrengthening stage after yielding and the descending stageafter fracture (Figure 14(b)) e elastic modulus of the steelgirder was Es 20times105MPa the density was 7850 kgm3and Poissonrsquos ratio was 03

e concrete damage plasticity (CDP) model is shownin Figure 15 In the CDP model tensile cracking andcompressive crushing of concrete are considered the maintwo failure mechanisms Based on the smeared crackingapproach the damage or stiffness degradation is assumedto be uniformly distributed e cracked concrete is

MFD

val

ue (

)

Case 1Case 2

0

20

40

60

80

100

120

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 181Points 1ndash18

Figure 10 Relative difference of MFD in Case 1 and Case 2

Diff

eren

ce v

alue

()

Case 3Case 4Case 5

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 271Points 1ndash27

0

20

40

60

80

100

Figure 11 Relative difference of MFD in Cases 3 4 and 5 (except for the support points)


regarded as an elastic orthotropic material with a reducedelastic modulus e stiffness degradation is assumed tooccur in the softening response in both compression andtension [36] In this study the corresponding parameterswere defined as follows the shape parameter wasKc 06667 the dilation angle was ψ 30∘ the plasticpotential eccentricity was ε 01 and the ratio of thebiaxial stress to uniaxial stress was σb0σc0 116according to the study by Genikomsou and Polak [37]

e beam-shell model is relatively simple the onlydifficulty lies in modeling the plate bracket due to thecomplex geometry of the height variations e beam-solidmodel can overcome the difficulties of plate bracket mod-eling but the interfacial bonding condition between the steel

girder and concrete deck cannot be rationally simulated inthe model To overcome the difficulties of both of thesemodeling strategies the shell-solid model reflects the realinterface geometry and boundary conditions To determinea suitable FE model all three models were used for dynamicand static modal analysis and the results were comparedwith the measured dynamic and static data After the statictests were performed the measured displacement and thedisplacement estimated by the FE model were compared asshown in Figure 16 Because the shell-solid model has thepreviously mentioned advantages its results are consistentwith the measured results in the tested model erefore theshell-solid model as shown in Figure 17 was selected for themodel updating analysis In total the shell-solid model

Support

Concrete plate(shell element)

Spring element

Steel beam(beam element)

Support plate(shell element)

(a)

Support

Concrete plate(brick element)

Spring element


(b)

Support


Spring element

Web plate(shell element)

Top flange(shell element)

Bottom flange(shell element)

(c)

Figure 12 Comparison of the detailed configurations of three FE models (a) beam-shell element (b) beam-solid element (c) shell-solidelement

X

Y

Z

Figure 13 Octahedron element

Table 3 Stiffness of the spring elements in the three models (unit times106 kNmm)Model Axial stiffness Lateral stiffness

Beam-shell element model

Beam-support 20 109

Beam-slab 102 05Slab-slab 0038 01

Connection 10 0

Beam-solid element modelBeam-support 20 109

Beam-slab 102 05Connection 10 0

Shell-solid element modelBeam-support 20 109



consisted of 1437 beam elements 2400 shell elements 9280brick elements and 264 link elements

52 Sensitivity Analysis In the damage identification tech-niques based on dynamic responses the physical parametersof a structure in a healthy condition can be calibrated by

minimizing residual errorsesemodels are then comparedwith any unknown (damaged) condition to obtain thecorresponding residual errors Any minor changes in thesecoefficients or residual errors are assumed to indicatedamage in the structure e physical parameters are esti-mated according to design drawings standards and theliterature Sensitivity analysis was used to analyze the

σ

εtrεcr

ftr

εcu ε0

fcr

05fcr

(a)

σ

ε0

σu

εuεy

σy

(b)

Figure 14 Stress-strain relationship of materials (a) concrete (b) steel reinforcement

ndashS2

ndashS3

ndashS1

(T M)

(C M)Kc = 23

Kc = 1

(a)

Uniaxialcompression

σ2

σ1

Biaxialcompression

Uniaxialtension

Biaxialtension

(b)

Figure 15 Illustration of concrete damage plasticity model (a) deviatoric plane (b) plane stress yield surface

Measured valueShell-solid model

Beam-solid modelBeam-shell model

ndash05ndash04ndash03ndash02ndash01

000102030405

Disp

lace

men

t (m

m)

2 3 4 5 6 7 8 9 1011121314151617181920212223242526271Points 1ndash27

Figure 16 Comparison of the displacement results from the static test and the FE model analysis


influence of different physical parameters on the objectivefunction In the nominal FE model five physical parameterswere selected and the data range was set as shown in Table 4e objective function was built using the dynamic mea-surement data as shown in equations (4) (5) and (6) emodes that have relatively large MAC values were used inthe following objective function analysis e sensitivityanalysis results based on modal data as shown in Figure 18reveal that four physical parameters were sensitive to theobjective function while the beam-slab axial spring stiffnesswas not

objModal(x) 1113936

ni1 fi(x) + 1 minus MACi( 1113857( 1113857

n (4)

MACi(x) ϕTAi(x)ϕEi(x)

11138681113868111386811138681113868

111386811138681113868111386811138682

ϕTAi(x)ϕAi(x)1113872 1113873 ϕT

EiϕEi1113872 1113873 (5)

fi(x) f

j

E minus fj

A(x)

fj

E

(6)

where E represents the measured data A denotes the ana-lytical data from the Strand7 FE model f is the frequency Φis the mode shape and x is the updated parameter such asthe elastic modulus

53 Model Updating for the Baseline FE Model Finite ele-ment (FE) models were constructed on the basis of idealizedengineering designs that may not truly represent all thephysical aspects of an actual structure and therefore need tobe updated to match the measured data [38] In manystudies the sensitivity-based model updating method de-scribed by Mottershead et al [39] has achieved great successe FEmodel updating method considered in this paper wasthe iterative method which involved the use of the sensi-tivity of the parameters to update the model

An FE model as illustrated in Figure 17 was built todetect the damage via a sensitivity-based model updatingtechnique Based on sensitivity analysis of the dynamic testresults the elastic modulus of concrete the density ofconcrete the axial stiffness of the bearing spring and theaxial stiffness of the connector spring were selected to updatethe model in the undamaged reference case to obtain abaseline model e Strand7 FE analysis package was cou-pled with the MATLAB API for automated multiparametermodel updating e iterative results for multiple parameteridentification based on the modal data are shown in

Figure 19 while the objective function which convergedafter 67 iterative cycles is shown in Figure 19(b) e initialvalues and updated values are shown in Table 5

e updated baseline model was then used for static anddynamic modal analysis e calculated static displace-ments under a uniformly distributed load before and aftermodel updating as well as the measured displacements areshown in Figure 20 e modal frequencies and modeshapes before and after model updating were comparedwith the dynamic test results shown in Table 6 Comparedwith the analysis results of the initial model the static anddynamic analysis results using the updated model im-proved significantly

54 Damage Detection in the Defect Cases After the baselinemodel was obtained the damage information was deducedby an additional model updating technique using the modaltest results In Case 1 the updating parameters were set asthe stiffness of the six supports at points 1 9 10 18 19and 27 In Case 2 all the bolts were divided into 6 regionsas shown in Figure 21 and the updating parameters were theaverage stiffness of the bolts in each region e initialnumbers of updating parameters in Case 1 and Case 2 weredefined as a normalized stiffness of 10 e stiffness deg-radation distributions of the supports and the bolts aftermodel updating are shown in Figure 22 Using the updatedmodels the correct locations of defective supports and boltswere identified In Case 1 the normalized stiffness of point 1decreased to 0131 due to the support changes In Case 2 thestiffness value in the loosened bolt region between points 5

Figure 17 Shell-solid model

Table 4 Updating parameters for sensitivity analysisUpdatingparameter

Lowerlimit

Upperlimit

Distributionpattern Nominal value

Elasticmodulus E 08E0 15E0 Linear E0 351times 105MPa

Density ρ 08ρ0 15ρ0 Linear ρ 2450 kgm3

Axialstiffness ofsupportspring K1

001K1 100K1 Logarithmic K1 200 kNmm

Axialstiffness ofbeam-slabspring K2


Axialstiffness ofconnectorspring K3



and 9 decreased to 0298 as shown in Figure 22 e modalparameters calculated using the updated physical parameterswere compared with the measured data in Table 7 eresults from the updated models in Case 1 and Case 2 matchthe results of the experiment models well

55 Damage Detection in the Loading Cases Instead ofadjusting the stiffness properties of all the elements sepa-rately the stiffness can be determined by damage functionswhich have to be multiplied by the appropriate factors isapproach decreases the number of unknowns and ensures aphysically significant solution [40] e distribution of theunknown physical properties can be estimated by combininga limited set of damage functions and the updating pa-rameters are multiplied by the damage functions before

combining them [33] e one-dimensional hierarchicalshape function can be defined by

Ne1 x1( 1113857

1 minus x1

2

Ne2 x1( 1113857

1 + x1

2

(7)

where minus1lex1 le 1 max|Ne2(x1)| 1

In the process of damage detection the element-leveldamage functions are constructed by mapping the standardshape functions onto ldquodamage elementsrdquo ese are definedas a series of neighboring elements that are connected bycommon nodes e correction parameter ae of each ele-ment is determined by the linear combination of the global

018

020

022

024

026

028

030O

bjec

tive f

unct

ion

09 10 11 12 13 14 1508Elastic modulus ratio (EE0)

(a)

018

020

022

024

026

028

030

Obj

ectiv

e fun

ctio

n

09 10 11 12 13 14 1508Density ratio (ρρ0)

(b)

Support spring log (K1K0)Beam-plate spring log (K2K0)Connector spring log (K3K0)

015

020

025

030

035

040

045

050

055

Obj

ectiv

e fun

ctio

n

ndash1 0 1 2ndash2Axial stiffness proportional coefficent (log(KiK0))

(c)

Figure 18 Physical parameter sensitivity analysis using dynamic data (a) elastic modulus of the concrete (b) density of the concrete (c)axial stiffness of the springs


damage functions Ni [40] as shown in the followingequation

ae

1113944ni

i1piNi xe

( 1113857 (8)

where ni is the number of damage functions Ni(x) pi is themultiplication coefficient and xe denotes the coordinates ofthe center point of element e e initial parameter in theundamaged state is characterized by Youngrsquos modulus of

E0 431times 104MPa and the elastic modulus is selected as theupdating parameter in equation (9) e connection pa-rameter ae lies in the range of 0ndash1

Ee

1 minus ae

( 1113857E0 (9)

Each steel girder under the composite deck was dividedinto 9 FE clusters as shown in Figure 23(a) and each girderconsists of 5 damage element regions as shown in

Elastic modulusDensity

Stiffness of supportsStiffness of connections

08

10

12

14

16

Itera

tive r

esul

ts

10 20 30 40 50 60 700Iterations (n)

(a)

016

018

020

022

024

026

028

030

Obj

ectiv

e fun

ctio

n

10 20 30 40 50 60 700Iterations (n)

(b)

Figure 19 Multiple parameter model updating based on dynamic data (a) variation of updating parameters (b) variation of objectivefunction

Table 5 Comparison of initial values and updated valuesData Elastic modulus (Nmm2) Concrete density (kgm3) Support stiffness (kNmm) Connector stiffness (kNmm)Initial value 15 09 11 11Updated value 123 115 105 096

1 3 5 7 9 10 12 14 16 1819 21 23 25 27Points 1ndash27(matching the displacement

sensor location D01-15)

Measured valueInitial modelUpdating model


000102030405

Disp

lace

men

t (m

m)

Figure 20 Displacement under a uniformly distributed load before and after model updating


Figure 23(b) thus 5 parameters need to be updated for eachgirder A triangular function was used as the shape functionin each damage region instead of modifying all 9 elements

separately the FE model was divided into five damage el-ements consisting of three or five neighboring beam ele-ments Each girder consisted of n connection parameters

Table 6 Comparison of modal data before and after model updating

Order Measured Initial model Updating modelFreq (Hz) Freq (Hz) Diff () MAC Freq (Hz) Diff () MAC

1 2478 2367 minus448 0966 2468 minus040 09772 3126 3130 013 0959 3079 minus151 09643 5677 6216 949 0503 5937 491 06125 8268 7620 minus784 0870 7855 minus499 09197 12001 11442 minus466 0755 12252 209 0788Note Diff (f3 minus fm)fm times 100 fm is the measured frequency of the composite deck and f3 is the frequency of the initial or updated model


5

3

11

10

19 20 21 22 23 24 25 26 27

11 12 13 14 15 16 17 18

2 3 4 5 6 7 8 9

6

4

2

Girder 3

Girder 2

Girder 1

Figure 21 Selection of the updating parameters in Case 2

Case 1

00

02

04

06

08

10

12

14

16

Stiff

ness

(nor

mal

ized

)

9 10 18 19 271Updating parameters (supports)

(a)

Case 2

00

02

04

06

08

10

12

14

16

Stiff

ness

(nor

mal

ized

)

2 3 4 5 61Updating parameters (bolts)

(b)

Figure 22 Updating results of the stiffness degradation (a) Case 1 (b) Case 2

Table 7 Comparison of the identified and tested modal parameters in Case 1 and Case 2Case 1 Case 2

Mode Exp Freq (Hz) Updated Freq (Hz) Diff () MAC Exp Freq (Hz) Updated Freq (Hz) Diff () MAC1st 1827 1604 1221 089 mdash mdash mdash mdash2nd 2391 2242 623 096 2470 2370 405 0973rd 3076 2961 374 086 3171 3113 183 0836th 8443 7347 minus1298 060 7393 7601 281 089NoteDiff (fu1 minus fe1)fe1 times 100 fe1 is the experimental frequency in Case 1 or Case 2 and fu1 is the frequency of the updating model in Case 1 or Case 2


which can be obtained by multiplying the shape functionmatrix by the damage coefficient vector as shown in thefollowing equations

a ntimes1 [N]ntimesniP nitimes1 (10)

a1

a2

⋮a9

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

N1 x1

1113872 1113873 N2 x1

1113872 1113873 N3 x1

1113872 1113873 N4 x1

1113872 1113873 N5 x1

1113872 1113873

N1 x2

1113872 1113873 N2 x2

1113872 1113873 N3 x2

1113872 1113873 N4 x2

1113872 1113873 N5 x2

1113872 1113873

⋮ ⋮ ⋮ ⋮ ⋮N1 x

91113872 1113873 N2 x

91113872 1113873 N3 x

91113872 1113873 N4 x

91113872 1113873 N5 x

91113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

P1

P2

P3

P4

P5

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎭

(11)

e damage value in each element cluster can beexpressed by a shape function after Pi has been identifiedthe stiffness reduction coefficient in each element can becalculated After assembly five global (linear and step-wise) damage functions Ni were obtained and N1ndashN5 areplotted in Figure 23(b) For damage detection the firstfour identified modes were used to update the model

With increasing structural damage in Cases 3 4 and5 the stiffness continuously decreased as shown in Fig-ure 21 and Table 8 Under the same damage case thestiffness of girder 1 decreased more than that of girder 2and girder 3 e updating results under the static loadingtest were consistent with the crack development (Fig-ure 7) When the structure was still in the linear elasticstate the maximum value of stiffness loss which was134 appeared in the 5th zone of girder 1 however thesame value became 414 in damage Case 5 as shown inFigure 24

6 Conclusions

is study proposed two VBDDmethods for structure damageidentification An experimental test was firstly conducted byutilizing the non-model-based VBDD method to detect thedamage of a scaled steel-concrete composite specimen withvarious damage settings ereafter based on FE model pa-rameters using Strand7 and MATLAB software the model-based VBDD method was utilized to locate and quantify thedamage of three calibrated FE modelse proposed procedureis a new application of the damage detection technique for steel-concrete structures Conclusions from the study are as follows

(1) e non-model-based VBDD method was utilizedfor damage identification of the tested steel-concrete

4000

19 20 21 22 23 24 25 26 27

10 11 12 13 14 15 16 17 18

1 2 3 4 5 6 7 8 9

530

760

760

(a)

1 2 3 4 5 6 7 8 9

1

05

N1 N3 N5 N7 N9

(b)

Figure 23 Element division and damage function (a) element division of the updating model (b) damage function of a girder

Table 8 Comparison of the identified and tested modal parameters for Cases 3 4 and 5Case 3 Case 4 Case 5

Mode Exp Freq(Hz)

UpdatedFreq (Hz)

Diff() MAC Exp Freq

(Hz)UpdatedFreq (Hz)

Diff() MAC Exp Freq


Diff() MAC

1st 2457 2434 090 096 2217 2185 144 097 1915 1879 187 0932nd 3109 3165 181 096 3090 3142 168 094 2846 2928 288 0913rd 5756 6036 486 063 5524 5978 821 059 5370 5893 974 0615th 8309 7811 599 091 8002 7643 449 089 7974 7586 487 0837th 12113 12209 079 079 11956 12143 156 078 11725 12085 307 075Note Diff (fu2 minus fe2)fe2 times 100 fe2 is the experimental frequency in Cases 3 4 or 5 and fu2 is the frequency of updating model in Cases 3 4 or 5

1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627

00

01

02

03

04

05

Stiff

ness

deg

ener

atio

n co

effic

ient

Points 1ndash27

Case 3Case 4Case 5

Figure 24 Stiffness degradation curve of the steel-concretecomposite deck


composite specimen where the largest MFD ob-tained by the MRITwas 1196 and 898 when thesupport and connected condition changed respec-tively e MFD also significantly varied underdifferent loading schemes which can be referred toas an index for damage assessment is method hasthe advantages of high accuracy and wide applica-bility for damage detection of steel-concrete com-posite structures

(2) e model-based VBDD method was used fordamage identification by conducting numericalanalysis ree different FE models the beam-shellmodel the beam-solid model and the shell-solidmodel were utilized to simulate the behavior of thecomposite slab e calculated MFD of the shell-solid FE model was almost identical to the test re-sults indicating that the performance of the testedcomposite structure could be accurately predicted bythis type of FE model

(3) Based on the proposed parameter updating programusing Strand7 and MATLAB software the estab-lished shell-solid model was calibrated with the se-lected highly sensitive physical parameters wherethe error between calculated and tested 1st naturalfrequency had been decreased from 448 to 040Furthermore the stiffness of the FE model signifi-cantly decreased from 866 (under crack stage) to586 (under failure stage) indicating that thedamage of the specimen could be effectively locatedand quantified with the proposed VBDD method

Data Availability

e datasets generatedanalyzed in the present study areavailable upon reasonable request from the correspondingauthor

Conflicts of Interest

e authors declare that there are no conflicts of interest

Acknowledgments

e authors sincerely appreciate the funding support pro-vided by the National Natural Science Foundation of China(NSFC) (no 51878264) the Science and TechnologyProgress and Innovation Project of the Department ofTransportation of Hunan Province (201912) and the KeyResearch and Development Program of Changsha City(kq1801010)

References

[1] A S Kobayashi Handbook on Experimental MechanicsPrentice-Hall Englewood CL USA 1987

[2] S Das P Saha and S K Patro ldquoVibration-based damagedetection techniques used for health monitoring of structuresa reviewrdquo Journal of Civil Structural Health Monitoring vol 6no 3 pp 477ndash507 2016

[3] M U Hanif Z Ibrahim K Ghaedi H Hashim andA Javanmardi ldquoDamage assessment of reinforced concretestructures using a model-based nonlinear approach-a com-prehensive reviewrdquo Construction and Building Materialsvol 192 pp 846ndash865 2018

[4] A Rytter Vibration Based Inspection of Civil EngineeringStructures PhD thesis Aalborg University Aalborg Den-mark 1993

[5] S W Doebling C R Farrar and M B Prime ldquoA summaryreview of vibration-based damage identification methodsrdquoBe Shock and Vibration Digest vol 30 no 2 pp 91ndash1051998

[6] J Kullaa ldquoDamage detection of the Z24 bridge using controlchartsrdquo Mechanical Systems and Signal Processing vol 17no 1 pp 163ndash170 2003

[7] S W Doebling C R Farrar M B Prime and D W ShevitzldquoDamage identification and health monitoring of structuraland mechanical systems from changes in their vibrationcharacteristics a literature reviewrdquo Technical Report LA-13070-MS Los Alamos National Laboratory Los AlamosNM USA 1996

[8] H Sohn C R Farrar F M Hemez et al ldquoA review ofstructural health monitoring literature from 1996-2001rdquoTechnical Report LA-13976-MS Los Alamos National Lab-oratory Los Alamos NM USA 2003

[9] J J Moughty and J R Casas ldquoA state-of-the-art review ofmodal-based damage detection in bridges developmentchallenges and solutionsrdquo Applied Sciences vol 7 no 7p 510 2017

[10] Y B Yang and J P Yang ldquoState-of-the-Art review on modalidentification and damage detection of bridges by moving testvehiclesrdquo International Journal of Structural Stability andDynamics vol 18 no 2 Article ID 1850025 2018

[11] Y Xia H Hao and A J Deeks ldquoDynamic assessment of shearconnectors in slab-girder bridgesrdquo Engineering Structuresvol 29 no 7 pp 1475ndash1486 2007

[12] Y Xia H Hao A J Deeks and X Zhu ldquoCondition as-sessment of shear connectors in slab-girder bridges via vi-bration measurementsrdquo Journal of Bridge Engineering vol 13no 1 pp 43ndash54 2008

[13] M Dilena and A Morassi ldquoVibrations of steel-concretecomposite beams with partially degraded connection andapplications to damage detectionrdquo Journal of Sound andVibration vol 320 no 1-2 pp 101ndash124 2009

[14] B Xu and F Jiang ldquoConcrete-steel composite girder boltloosening monitoring using electromechanical impedancemeasurementsrdquo Earth and Space pp 629ndash634 2012

[15] H Allahyari I Rahimi and A Allahyari ldquoExperimentalmeasurement of dynamic properties of composite slabs fromfrequency responserdquo Measurement vol 114 pp 150ndash1612018

[16] W Zhang J Li H Hao and H Ma ldquoDamage detection inbridge structures under moving loads with phase trajectorychange of multi-type vibration measurementsrdquo MechanicalSystems and Signal Processing vol 87 pp 410ndash425 2017

[17] T Wroblewski M Jarosinska M Abramowicz andS Berczynski ldquoExperimental validation of the use of energytransfer ratio (ETR) for damage diagnosis of steel-concretecomposite beamsrdquo Journal of Beoretical and Applied Me-chanics vol 55 no 1 pp 241ndash252 2017

[18] H W Shih T H T Chan and D P ambiratnamldquoStructural damage localization in slab-on-girder bridgesusing vibration characteristicsrdquo in Proceedings of the 3rd


World Congress on Engineering Asset Management And In-telligent Maintenance System Beijing China October 2008

[19] R V Farahani and D Penumadu ldquoDamage identification of afull-scale five-girder bridge using time-series analysis of vi-bration datardquo Engineering Structures vol 115 pp 129ndash1392016

[20] W Zhao S Guo Y Zhou and J Zhang ldquoA quantum-inspiredgenetic algorithm-based optimization method for mobileimpact test data integrationrdquo Computer-Aided Civil and In-frastructure Engineering vol 33 no 5 pp 411ndash422 2018

[21] Z X Tan D P ambiratnam T H T Chan et al ldquoDamagedetection in steel-concrete composite bridge using vibrationcharacteristics and artificial neural networkrdquo Structure AndInfrastructure Engineering vol 16 pp 1ndash15 2019

[22] X Wang R Hou Y Xia et al ldquoStructural damage detectionbased on variational bayesian inference and delayed rejectionadaptive metropolis algorithmrdquo Structural Health Monitor-ing pp 1ndash18 Article ID 147592172092125 2020

[23] K Liu and G De Roeck ldquoDamage detection of shear con-nectors in composite bridgesrdquo Structural Health MonitoringAn International Journal vol 8 no 5 pp 345ndash356 2009

[24] W-X Ren Z-S Sun Y Xia H Hao and A J DeeksldquoDamage identification of shear connectors with waveletpacket energy laboratory test studyrdquo Journal of StructuralEngineering vol 134 no 5 pp 832ndash841 2008

[25] J Li H Hao and H-P Zhu ldquoDynamic assessment of shearconnectors in composite bridges with ambient vibrationmeasurementsrdquo Advances in Structural Engineering vol 17no 5 pp 617ndash637 2014

[26] C Bao H Hao Z Li and X Zhu ldquoTime-varying systemidentification using a newly improved HHT algorithmrdquoComputers and Structures vol 87 no 23-24 pp 1611ndash16232009

[27] J Li H Hao K Fan and J Brownjohn ldquoDevelopment andapplication of a relative displacement sensor for structuralhealth monitoring of composite bridgesrdquo Structural Controland Health Monitoring and Health Monitoring vol 22 no 4pp 726ndash742 2015

[28] J Li and H Hao ldquoDamage detection of shear connectorsunder moving loads with relative displacement measure-mentsrdquoMechanical Systems and Signal Processing vol 60-61pp 124ndash150 2015

[29] U Dackermann J Li R Rijal and K Crews ldquoA dynamic-based method for the assessment of connection systems oftimber composite structuresrdquo Construction and BuildingMaterials vol 102 pp 999ndash1008 2016

[30] Ministry of Housing andUrban-Rural Development of ChinaCode For Design of Concrete Structures (GB 50010-2010)China Architecture and Building Press Beijing China 2010

[31] Y Jiang ldquoSub-structural modal flexibility integration theoryand experiment research for bridge deck identificationrdquopp 87ndash102 Hunan University Changsha China 2015Master esis

[32] R Clough and J Penzien Dynamics of Structure pp 175-176McGraw-Hill Book Co Berkeley CA USA 1975

[33] F N Catbas D L Brown and A E Aktan ldquoUse of modalflexibility for damage detection and condition assessmentcase studies and demonstrations on large structuresrdquo Journalof Structural Engineering vol 132 no 11 pp 1699ndash17122006

[34] Y Zhou J Prader J Weidner N Dubbs F Moon andA E Aktan ldquoStructural identification of a deterioratedreinforced concrete bridgerdquo Journal of Bridge Engineeringvol 17 no 5 pp 774ndash787 2012

[35] R J Allemang and D L Brown ldquoA unified matrix polynomialapproach to modal identificationrdquo Journal of Sound andVibration vol 211 no 3 pp 301ndash322 1998

[36] Y Zhou T Chen Y Pei et al ldquoStatic load test on progressivecollapse resistance of fully assembled precast concrete framestructurerdquo Engineering Structures vol 200 Article ID 1097192019

[37] A S Genikomsou andM A Polak ldquoFinite element analysis ofpunching shear of concrete slabs using damaged plasticitymodel in ABAQUSrdquo Engineering Structures vol 98 no 1pp 38ndash48 2015

[38] B Jaishi H-J Kim M K Kim W-X Ren and S-H LeeldquoFinite elementmodel updating of concrete-filled steel tubulararch bridge under operational condition using modal flexi-bilityrdquo Mechanical Systems and Signal Processing vol 21no 6 pp 2406ndash2426 2007

[39] J E Mottershead M Link and M I Friswell ldquoe sensitivitymethod in finite element model updating a tutorialrdquo Me-chanical Systems and Signal Processing vol 25 no 7pp 2275ndash2296 2011

[40] A Teughels J Maeck and G De Roeck ldquoDamage assessmentby FE model updating using damage functionsrdquo Computers ampStructures and Structures vol 80 no 25 pp 1869ndash1879 2002


define damage probability functions [2] When invisibledamage occurs on a complicated structure such as a com-posite bridge deck with shear connectors the VBDDmethodmay be particularly well suited as a global testing method todetermine the structural condition of the bridge Because ofthe easy acquisition of vibration characteristics and globalinformation on structural conditions [3] the VBDDmethodcan assess the condition of an entire structural componentwhen some portions of the structure remain inaccessible toinspection by common NDE methods In addition whenvisible damage appears on a composite deck the damagedetected by the vibration-based method can be investigatedusing the provided dynamic information

e VBDD method detects changes in the dynamiccharacteristics (eg natural frequencies modal shapes anddamping ratios) which can be regarded as a global responsesignature resulting from changes in the stiffness mass andboundary conditions It is a great challenge to develop robustalgorithms with the ability to detect locate and quantifydamage according to the measured dynamic responses of thestructure Some complex situations and conditions such asboundary conditions bond-slip occurrences and sub-structure damage would be reflected in the vibration sig-nature e advantage of the VBDD method is thecontinuous monitoring of the structural conditions andidentifying the earliest occurrence of possible damage eVBDD method comprises the following four levels thatdetermine changes in dynamic structural characteristics [4](1) qualitatively indicating damage occurrences (2) local-izing spatial information (3) estimating the extent ofdamage and (4) predicting the actual safety of the structureModal parameters such as natural frequencies and modeshapes are frequently used as damage-sensitive features toidentify damage [5 6] Many applications of the VBDDmethod have been developed in the past two decades aspresented by Doebling et al [7] and Sohn et al [8] Moughtyand Casas [9] provided an in-depth review of the devel-opment of modal-based damage-sensitive features (DSFs) inbridges and a synopsis of the challenges involved eyaddressed the challenges by drawing upon advanced tech-niques outlined in recent literature Yang et al [10] proposedthe extraction of bridge frequencies from the dynamic re-sponse of a moving test vehicle ey also verified thetechnique by a field test and presented a state-of-the-artreview of the related research works conducted worldwidee VBDD method can be classified into the non-model-based method and the model-based approach e biggestdifference in these two methods lies in whether a calibratedfinite element (FE) has been used in the data analysis or note non-model-basedmethod utilizes dynamic properties inmeasurements to directly deduce the physical characteris-tics which is also known as a damage fingerprint methode model-based method uses dynamic properties to cali-brate the initial FE model and then deduce the damagedphysical parameters by comparing updated and baselinecases Although the non-model-basedmethod is much easierto apply since no time and effort need be spent on modelingand model calibration it is harder to detect damage in somestructures than with the model-based method

For a steel-concrete composite deck damage that occursin the structure such as the loosening of shear connectionschanges in the boundary conditions and the development ofcracks caused by loading changes the structural vibrationcharacteristics Several studies have attempted to detectdamage in the connectors between steel-concrete compositegirders both in the laboratory and on actual projectsHowever limited studies have focused on damage detectionapproaches for steel-concrete composite structures and verylittle research has used the VBDD method to provide asatisfactory assessment of different damage cases for com-posite structures Xia et al [11] made the first attempt atdetecting possible damage of shear connectors in slab-girderbridges through vibration methods A 1 3 scale bridgemodel was constructed in a laboratory to test the suitabilityand efficiency of the VBDDmethods Since it was found thatthe local approach is more sensitive to identifying the localdamage of shear connectors than global methods the localapproach was developed and applied to assess the conditionof the shear connectors in a full-size slab-girder bridge [12]It was concluded that the newly developed method is a morepractical approach for detection of damaged connectorsthan the model updating technique Dilena andMorassi [13]presented a EulerndashBernoulli model of a composite beam thataccurately described the measured dynamic response ofcomposite beams with either severe or intermediate levels ofdamage A diagnostic technique based on frequency mea-surements was then applied to the suggested model whichprovided positive results Xu and Jiang [14] used a number ofpiezoelectric zirconate titanate (PZT) patches on the upperflange of a steel girder and concrete slab and the corre-sponding electromechanical impedance was measured usingan impedance analyzer before and after loosening theconnection bolts Allahyari et al [15] investigated the dy-namic characteristics of unfilled steel-concrete decks in-cluding normal-weight high-strength concrete (HC) andlightweight high-strength concrete (LHC) It was concludedthat the decks fabricated with plain concrete (DPC) and thedecks fabricated with LHC (DLHC) had approximatelysimilar serviceability whereas DLHC can be more applicablethan DPC due to lower weight Zhang et al [16] presented anon-model-based damage detection approach for bridgestructures based on the phase trajectory change of multitypevibration measurements e experimental study demon-strated that the proposed approach can be used to identifythe shear connection failure in a composite bridge modelsubjected to moving loads Wroblewski et al [17] demon-strated a method to show how changes in energy transferratios (ETRs) could be used for damage detection and lo-calization in steel-concrete composite beams

e dynamic computer simulation technique was usedby Shih et al [18] to identify damage to slab-on-girderbridges Based on that technique they used modal flexibilitychange and modal strain energy change for damage local-ization of shear connectors Results showed that theirmethod was effective in damage assessment for slab-on-girder bridge superstructure e autoregressive with ex-ogenous input (ARX) models and the sensor clusteringdamage identification technique [19] were then adopted to






















150 760 760 380


60

30

(a)

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18

19 20 21 22 23 24 25 26 27

1900 1001900100

Girder 3

Girder 2




760

150

760

380


(b)


Headed stud

Coating oil

Bolt

Bolt

(a)

Concrete slab

Bolt


Steel grider

(b)








(a)

(b)


ST05-0817-20

ST09-1221-24

ST01-0413-16

Girder 3

Girder 2

Girder 1C01

C02

C03

C04

C05

D02

D13

C06

C07

C08

C09

C10

D08

C11

C12

C13

C14

C15

D03

D14

D09

D04

D15

D10

D05

D16

D11

D01

D12

D07

1900 1900 1900 1900 1900100 100

380

380

380

380

380

150






Hpq(ω) 1113944n

r1

Apqr


Alowastpqr





Hpq(ω) 1113944m

r1

ψprψqr




1113890 1113891 (2)



MFD Dd minus Dr

11138681113868111386811138681113868111386811138681113868

Dr

times 100 (3)




(a) (b)






Load

(kN

)

0

50

100

150

200

250

300


(a)

ST04ST08ST12

ST16ST20ST24

Load

(kN

)

0

50

100

150

200

250

300

5000 10000 15000 20000 250000Strain (με)

(b)

C11C12C13

C14C15

Load

(kN

)

0

50

100

150

200

250

300

500 1000 1500 2000 25000Strain (με)

(c)







Girder 1

Girder 2

Girder 3

(a)

Girder 1

Girder 2

Girder 3

(b)

Girder 1

Girder 2

Girder 3

(c)


100

10ndash2

10ndash4

10ndash6

10ndash80 50 100


Log

mag

nitu

de (m

mN

)



500

0

ndash5001500

1000

Y X5000 0

10002000

3000

Z

(a)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(b)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(c)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(d)

15001000

Y X5000 0

10002000

3000

500

0

ndash500Z

(e)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(f)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(g)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(h)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z(i)













MFD

val

ue (

)

Case 1Case 2

0

20

40

60

80

100

120

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 181Points 1ndash18


Diff

eren

ce v

alue

()

Case 3Case 4Case 5

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 271Points 1ndash27

0

20

40

60

80

100






Support


Spring element



(a)

Support


Spring element


(b)

Support


Spring element




(c)


X

Y

Z




Beam-support 20 109


Connection 10 0









σ

εtrεcr

ftr

εcu ε0

fcr

05fcr

(a)

σ

ε0

σu

εuεy

σy

(b)


ndashS2

ndashS3

ndashS1

(T M)

(C M)Kc = 23

Kc = 1

(a)

Uniaxialcompression

σ2

σ1

Biaxialcompression

Uniaxialtension

Biaxialtension

(b)





000102030405

Disp

lace

men

t (m

m)

2 3 4 5 6 7 8 9 1011121314151617181920212223242526271Points 1ndash27




objModal(x) 1113936

ni1 fi(x) + 1 minus MACi( 1113857( 1113857

n (4)


11138681113868111386811138681113868

111386811138681113868111386811138682

ϕTAi(x)ϕAi(x)1113872 1113873 ϕT

EiϕEi1113872 1113873 (5)

fi(x) f

j

E minus fj

A(x)

fj

E

(6)









Lowerlimit

Upperlimit














Ne1 x1( 1113857

1 minus x1

2

Ne2 x1( 1113857

1 + x1

2

(7)



018

020

022

024

026

028

030O

bjec

tive f

unct

ion


(a)

018

020

022

024

026

028

030

Obj

ectiv

e fun

ctio

n

09 10 11 12 13 14 1508Density ratio (ρρ0)

(b)


015

020

025

030

035

040

045

050

055

Obj

ectiv

e fun

ctio

n


(c)




ae

1113944ni

i1piNi xe

( 1113857 (8)



Ee

1 minus ae

( 1113857E0 (9)




08

10

12

14

16

Itera

tive r

esul

ts

10 20 30 40 50 60 700Iterations (n)

(a)

016

018

020

022

024

026

028

030

Obj

ectiv

e fun

ctio

n

10 20 30 40 50 60 700Iterations (n)

(b)







000102030405

Disp

lace

men

t (m

m)









5

3

11

10

19 20 21 22 23 24 25 26 27

11 12 13 14 15 16 17 18

2 3 4 5 6 7 8 9

6

4

2

Girder 3

Girder 2

Girder 1


Case 1

00

02

04

06

08

10

12

14

16

Stiff

ness

(nor

mal

ized

)


(a)

Case 2

00

02

04

06

08

10

12

14

16

Stiff

ness

(nor

mal

ized

)


(b)







a1

a2

⋮a9

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

N1 x1

1113872 1113873 N2 x1

1113872 1113873 N3 x1

1113872 1113873 N4 x1

1113872 1113873 N5 x1

1113872 1113873

N1 x2

1113872 1113873 N2 x2

1113872 1113873 N3 x2

1113872 1113873 N4 x2

1113872 1113873 N5 x2

1113872 1113873

⋮ ⋮ ⋮ ⋮ ⋮N1 x

91113872 1113873 N2 x

91113872 1113873 N3 x

91113872 1113873 N4 x

91113872 1113873 N5 x

91113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

P1

P2

P3

P4

P5

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎭

(11)



6 Conclusions



4000

19 20 21 22 23 24 25 26 27

10 11 12 13 14 15 16 17 18

1 2 3 4 5 6 7 8 9

530

760

760

(a)

1 2 3 4 5 6 7 8 9

1

05

N1 N3 N5 N7 N9

(b)



Mode Exp Freq(Hz)

UpdatedFreq (Hz)

Diff() MAC Exp Freq


Diff() MAC Exp Freq


Diff() MAC


1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627

00

01

02

03

04

05

Stiff

ness

deg

ener

atio

n co

effic

ient

Points 1ndash27

Case 3Case 4Case 5






Data Availability




Acknowledgments


References
































































150 760 760 380


60

30

(a)

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18

19 20 21 22 23 24 25 26 27

1900 1001900100

Girder 3

Girder 2




760

150

760

380


(b)


Headed stud

Coating oil

Bolt

Bolt

(a)

Concrete slab

Bolt


Steel grider

(b)








(a)

(b)


ST05-0817-20

ST09-1221-24

ST01-0413-16

Girder 3

Girder 2

Girder 1C01

C02

C03

C04

C05

D02

D13

C06

C07

C08

C09

C10

D08

C11

C12

C13

C14

C15

D03

D14

D09

D04

D15

D10

D05

D16

D11

D01

D12

D07

1900 1900 1900 1900 1900100 100

380

380

380

380

380

150






Hpq(ω) 1113944n

r1

Apqr


Alowastpqr





Hpq(ω) 1113944m

r1

ψprψqr




1113890 1113891 (2)



MFD Dd minus Dr

11138681113868111386811138681113868111386811138681113868

Dr

times 100 (3)




(a) (b)






Load

(kN

)

0

50

100

150

200

250

300


(a)

ST04ST08ST12

ST16ST20ST24

Load

(kN

)

0

50

100

150

200

250

300

5000 10000 15000 20000 250000Strain (με)

(b)

C11C12C13

C14C15

Load

(kN

)

0

50

100

150

200

250

300

500 1000 1500 2000 25000Strain (με)

(c)







Girder 1

Girder 2

Girder 3

(a)

Girder 1

Girder 2

Girder 3

(b)

Girder 1

Girder 2

Girder 3

(c)


100

10ndash2

10ndash4

10ndash6

10ndash80 50 100


Log

mag

nitu

de (m

mN

)



500

0

ndash5001500

1000

Y X5000 0

10002000

3000

Z

(a)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(b)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(c)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(d)

15001000

Y X5000 0

10002000

3000

500

0

ndash500Z

(e)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(f)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(g)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(h)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z(i)













MFD

val

ue (

)

Case 1Case 2

0

20

40

60

80

100

120

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 181Points 1ndash18


Diff

eren

ce v

alue

()

Case 3Case 4Case 5

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 271Points 1ndash27

0

20

40

60

80

100






Support


Spring element



(a)

Support


Spring element


(b)

Support


Spring element




(c)


X

Y

Z




Beam-support 20 109


Connection 10 0









σ

εtrεcr

ftr

εcu ε0

fcr

05fcr

(a)

σ

ε0

σu

εuεy

σy

(b)


ndashS2

ndashS3

ndashS1

(T M)

(C M)Kc = 23

Kc = 1

(a)

Uniaxialcompression

σ2

σ1

Biaxialcompression

Uniaxialtension

Biaxialtension

(b)





000102030405

Disp

lace

men

t (m

m)

2 3 4 5 6 7 8 9 1011121314151617181920212223242526271Points 1ndash27




objModal(x) 1113936

ni1 fi(x) + 1 minus MACi( 1113857( 1113857

n (4)


11138681113868111386811138681113868

111386811138681113868111386811138682

ϕTAi(x)ϕAi(x)1113872 1113873 ϕT

EiϕEi1113872 1113873 (5)

fi(x) f

j

E minus fj

A(x)

fj

E

(6)









Lowerlimit

Upperlimit














Ne1 x1( 1113857

1 minus x1

2

Ne2 x1( 1113857

1 + x1

2

(7)



018

020

022

024

026

028

030O

bjec

tive f

unct

ion


(a)

018

020

022

024

026

028

030

Obj

ectiv

e fun

ctio

n

09 10 11 12 13 14 1508Density ratio (ρρ0)

(b)


015

020

025

030

035

040

045

050

055

Obj

ectiv

e fun

ctio

n


(c)




ae

1113944ni

i1piNi xe

( 1113857 (8)



Ee

1 minus ae

( 1113857E0 (9)




08

10

12

14

16

Itera

tive r

esul

ts

10 20 30 40 50 60 700Iterations (n)

(a)

016

018

020

022

024

026

028

030

Obj

ectiv

e fun

ctio

n

10 20 30 40 50 60 700Iterations (n)

(b)







000102030405

Disp

lace

men

t (m

m)









5

3

11

10

19 20 21 22 23 24 25 26 27

11 12 13 14 15 16 17 18

2 3 4 5 6 7 8 9

6

4

2

Girder 3

Girder 2

Girder 1


Case 1

00

02

04

06

08

10

12

14

16

Stiff

ness

(nor

mal

ized

)


(a)

Case 2

00

02

04

06

08

10

12

14

16

Stiff

ness

(nor

mal

ized

)


(b)







a1

a2

⋮a9

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

N1 x1

1113872 1113873 N2 x1

1113872 1113873 N3 x1

1113872 1113873 N4 x1

1113872 1113873 N5 x1

1113872 1113873

N1 x2

1113872 1113873 N2 x2

1113872 1113873 N3 x2

1113872 1113873 N4 x2

1113872 1113873 N5 x2

1113872 1113873

⋮ ⋮ ⋮ ⋮ ⋮N1 x

91113872 1113873 N2 x

91113872 1113873 N3 x

91113872 1113873 N4 x

91113872 1113873 N5 x

91113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

P1

P2

P3

P4

P5

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎭

(11)



6 Conclusions



4000

19 20 21 22 23 24 25 26 27

10 11 12 13 14 15 16 17 18

1 2 3 4 5 6 7 8 9

530

760

760

(a)

1 2 3 4 5 6 7 8 9

1

05

N1 N3 N5 N7 N9

(b)



Mode Exp Freq(Hz)

UpdatedFreq (Hz)

Diff() MAC Exp Freq


Diff() MAC Exp Freq


Diff() MAC


1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627

00

01

02

03

04

05

Stiff

ness

deg

ener

atio

n co

effic

ient

Points 1ndash27

Case 3Case 4Case 5






Data Availability




Acknowledgments


References



























































150 760 760 380


60

30

(a)

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18

19 20 21 22 23 24 25 26 27

1900 1001900100

Girder 3

Girder 2




760

150

760

380


(b)


Headed stud

Coating oil

Bolt

Bolt

(a)

Concrete slab

Bolt


Steel grider

(b)








(a)

(b)


ST05-0817-20

ST09-1221-24

ST01-0413-16

Girder 3

Girder 2

Girder 1C01

C02

C03

C04

C05

D02

D13

C06

C07

C08

C09

C10

D08

C11

C12

C13

C14

C15

D03

D14

D09

D04

D15

D10

D05

D16

D11

D01

D12

D07

1900 1900 1900 1900 1900100 100

380

380

380

380

380

150






Hpq(ω) 1113944n

r1

Apqr


Alowastpqr





Hpq(ω) 1113944m

r1

ψprψqr




1113890 1113891 (2)



MFD Dd minus Dr

11138681113868111386811138681113868111386811138681113868

Dr

times 100 (3)




(a) (b)






Load

(kN

)

0

50

100

150

200

250

300


(a)

ST04ST08ST12

ST16ST20ST24

Load

(kN

)

0

50

100

150

200

250

300

5000 10000 15000 20000 250000Strain (με)

(b)

C11C12C13

C14C15

Load

(kN

)

0

50

100

150

200

250

300

500 1000 1500 2000 25000Strain (με)

(c)







Girder 1

Girder 2

Girder 3

(a)

Girder 1

Girder 2

Girder 3

(b)

Girder 1

Girder 2

Girder 3

(c)


100

10ndash2

10ndash4

10ndash6

10ndash80 50 100


Log

mag

nitu

de (m

mN

)



500

0

ndash5001500

1000

Y X5000 0

10002000

3000

Z

(a)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(b)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(c)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(d)

15001000

Y X5000 0

10002000

3000

500

0

ndash500Z

(e)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(f)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(g)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(h)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z(i)













MFD

val

ue (

)

Case 1Case 2

0

20

40

60

80

100

120

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 181Points 1ndash18


Diff

eren

ce v

alue

()

Case 3Case 4Case 5

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 271Points 1ndash27

0

20

40

60

80

100






Support


Spring element



(a)

Support


Spring element


(b)

Support


Spring element




(c)


X

Y

Z




Beam-support 20 109


Connection 10 0









σ

εtrεcr

ftr

εcu ε0

fcr

05fcr

(a)

σ

ε0

σu

εuεy

σy

(b)


ndashS2

ndashS3

ndashS1

(T M)

(C M)Kc = 23

Kc = 1

(a)

Uniaxialcompression

σ2

σ1

Biaxialcompression

Uniaxialtension

Biaxialtension

(b)





000102030405

Disp

lace

men

t (m

m)

2 3 4 5 6 7 8 9 1011121314151617181920212223242526271Points 1ndash27




objModal(x) 1113936

ni1 fi(x) + 1 minus MACi( 1113857( 1113857

n (4)


11138681113868111386811138681113868

111386811138681113868111386811138682

ϕTAi(x)ϕAi(x)1113872 1113873 ϕT

EiϕEi1113872 1113873 (5)

fi(x) f

j

E minus fj

A(x)

fj

E

(6)









Lowerlimit

Upperlimit














Ne1 x1( 1113857

1 minus x1

2

Ne2 x1( 1113857

1 + x1

2

(7)



018

020

022

024

026

028

030O

bjec

tive f

unct

ion


(a)

018

020

022

024

026

028

030

Obj

ectiv

e fun

ctio

n

09 10 11 12 13 14 1508Density ratio (ρρ0)

(b)


015

020

025

030

035

040

045

050

055

Obj

ectiv

e fun

ctio

n


(c)




ae

1113944ni

i1piNi xe

( 1113857 (8)



Ee

1 minus ae

( 1113857E0 (9)




08

10

12

14

16

Itera

tive r

esul

ts

10 20 30 40 50 60 700Iterations (n)

(a)

016

018

020

022

024

026

028

030

Obj

ectiv

e fun

ctio

n

10 20 30 40 50 60 700Iterations (n)

(b)







000102030405

Disp

lace

men

t (m

m)









5

3

11

10

19 20 21 22 23 24 25 26 27

11 12 13 14 15 16 17 18

2 3 4 5 6 7 8 9

6

4

2

Girder 3

Girder 2

Girder 1


Case 1

00

02

04

06

08

10

12

14

16

Stiff

ness

(nor

mal

ized

)


(a)

Case 2

00

02

04

06

08

10

12

14

16

Stiff

ness

(nor

mal

ized

)


(b)







a1

a2

⋮a9

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

N1 x1

1113872 1113873 N2 x1

1113872 1113873 N3 x1

1113872 1113873 N4 x1

1113872 1113873 N5 x1

1113872 1113873

N1 x2

1113872 1113873 N2 x2

1113872 1113873 N3 x2

1113872 1113873 N4 x2

1113872 1113873 N5 x2

1113872 1113873

⋮ ⋮ ⋮ ⋮ ⋮N1 x

91113872 1113873 N2 x

91113872 1113873 N3 x

91113872 1113873 N4 x

91113872 1113873 N5 x

91113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

P1

P2

P3

P4

P5

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎭

(11)



6 Conclusions



4000

19 20 21 22 23 24 25 26 27

10 11 12 13 14 15 16 17 18

1 2 3 4 5 6 7 8 9

530

760

760

(a)

1 2 3 4 5 6 7 8 9

1

05

N1 N3 N5 N7 N9

(b)



Mode Exp Freq(Hz)

UpdatedFreq (Hz)

Diff() MAC Exp Freq


Diff() MAC Exp Freq


Diff() MAC


1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627

00

01

02

03

04

05

Stiff

ness

deg

ener

atio

n co

effic

ient

Points 1ndash27

Case 3Case 4Case 5






Data Availability




Acknowledgments


References


















































150 760 760 380


60

30

(a)

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18

19 20 21 22 23 24 25 26 27

1900 1001900100

Girder 3

Girder 2




760

150

760

380


(b)


Headed stud

Coating oil

Bolt

Bolt

(a)

Concrete slab

Bolt


Steel grider

(b)








(a)

(b)


ST05-0817-20

ST09-1221-24

ST01-0413-16

Girder 3

Girder 2

Girder 1C01

C02

C03

C04

C05

D02

D13

C06

C07

C08

C09

C10

D08

C11

C12

C13

C14

C15

D03

D14

D09

D04

D15

D10

D05

D16

D11

D01

D12

D07

1900 1900 1900 1900 1900100 100

380

380

380

380

380

150






Hpq(ω) 1113944n

r1

Apqr


Alowastpqr





Hpq(ω) 1113944m

r1

ψprψqr




1113890 1113891 (2)



MFD Dd minus Dr

11138681113868111386811138681113868111386811138681113868

Dr

times 100 (3)




(a) (b)






Load

(kN

)

0

50

100

150

200

250

300


(a)

ST04ST08ST12

ST16ST20ST24

Load

(kN

)

0

50

100

150

200

250

300

5000 10000 15000 20000 250000Strain (με)

(b)

C11C12C13

C14C15

Load

(kN

)

0

50

100

150

200

250

300

500 1000 1500 2000 25000Strain (με)

(c)







Girder 1

Girder 2

Girder 3

(a)

Girder 1

Girder 2

Girder 3

(b)

Girder 1

Girder 2

Girder 3

(c)


100

10ndash2

10ndash4

10ndash6

10ndash80 50 100


Log

mag

nitu

de (m

mN

)



500

0

ndash5001500

1000

Y X5000 0

10002000

3000

Z

(a)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(b)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(c)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(d)

15001000

Y X5000 0

10002000

3000

500

0

ndash500Z

(e)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(f)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(g)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(h)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z(i)













MFD

val

ue (

)

Case 1Case 2

0

20

40

60

80

100

120

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 181Points 1ndash18


Diff

eren

ce v

alue

()

Case 3Case 4Case 5

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 271Points 1ndash27

0

20

40

60

80

100






Support


Spring element



(a)

Support


Spring element


(b)

Support


Spring element




(c)


X

Y

Z




Beam-support 20 109


Connection 10 0









σ

εtrεcr

ftr

εcu ε0

fcr

05fcr

(a)

σ

ε0

σu

εuεy

σy

(b)


ndashS2

ndashS3

ndashS1

(T M)

(C M)Kc = 23

Kc = 1

(a)

Uniaxialcompression

σ2

σ1

Biaxialcompression

Uniaxialtension

Biaxialtension

(b)





000102030405

Disp

lace

men

t (m

m)

2 3 4 5 6 7 8 9 1011121314151617181920212223242526271Points 1ndash27




objModal(x) 1113936

ni1 fi(x) + 1 minus MACi( 1113857( 1113857

n (4)


11138681113868111386811138681113868

111386811138681113868111386811138682

ϕTAi(x)ϕAi(x)1113872 1113873 ϕT

EiϕEi1113872 1113873 (5)

fi(x) f

j

E minus fj

A(x)

fj

E

(6)









Lowerlimit

Upperlimit














Ne1 x1( 1113857

1 minus x1

2

Ne2 x1( 1113857

1 + x1

2

(7)



018

020

022

024

026

028

030O

bjec

tive f

unct

ion


(a)

018

020

022

024

026

028

030

Obj

ectiv

e fun

ctio

n

09 10 11 12 13 14 1508Density ratio (ρρ0)

(b)


015

020

025

030

035

040

045

050

055

Obj

ectiv

e fun

ctio

n


(c)




ae

1113944ni

i1piNi xe

( 1113857 (8)



Ee

1 minus ae

( 1113857E0 (9)




08

10

12

14

16

Itera

tive r

esul

ts

10 20 30 40 50 60 700Iterations (n)

(a)

016

018

020

022

024

026

028

030

Obj

ectiv

e fun

ctio

n

10 20 30 40 50 60 700Iterations (n)

(b)







000102030405

Disp

lace

men

t (m

m)









5

3

11

10

19 20 21 22 23 24 25 26 27

11 12 13 14 15 16 17 18

2 3 4 5 6 7 8 9

6

4

2

Girder 3

Girder 2

Girder 1


Case 1

00

02

04

06

08

10

12

14

16

Stiff

ness

(nor

mal

ized

)


(a)

Case 2

00

02

04

06

08

10

12

14

16

Stiff

ness

(nor

mal

ized

)


(b)







a1

a2

⋮a9

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

N1 x1

1113872 1113873 N2 x1

1113872 1113873 N3 x1

1113872 1113873 N4 x1

1113872 1113873 N5 x1

1113872 1113873

N1 x2

1113872 1113873 N2 x2

1113872 1113873 N3 x2

1113872 1113873 N4 x2

1113872 1113873 N5 x2

1113872 1113873

⋮ ⋮ ⋮ ⋮ ⋮N1 x

91113872 1113873 N2 x

91113872 1113873 N3 x

91113872 1113873 N4 x

91113872 1113873 N5 x

91113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

P1

P2

P3

P4

P5

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎭

(11)



6 Conclusions



4000

19 20 21 22 23 24 25 26 27

10 11 12 13 14 15 16 17 18

1 2 3 4 5 6 7 8 9

530

760

760

(a)

1 2 3 4 5 6 7 8 9

1

05

N1 N3 N5 N7 N9

(b)



Mode Exp Freq(Hz)

UpdatedFreq (Hz)

Diff() MAC Exp Freq


Diff() MAC Exp Freq


Diff() MAC


1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627

00

01

02

03

04

05

Stiff

ness

deg

ener

atio

n co

effic

ient

Points 1ndash27

Case 3Case 4Case 5






Data Availability




Acknowledgments


References

















































(a)

(b)


ST05-0817-20

ST09-1221-24

ST01-0413-16

Girder 3

Girder 2

Girder 1C01

C02

C03

C04

C05

D02

D13

C06

C07

C08

C09

C10

D08

C11

C12

C13

C14

C15

D03

D14

D09

D04

D15

D10

D05

D16

D11

D01

D12

D07

1900 1900 1900 1900 1900100 100

380

380

380

380

380

150






Hpq(ω) 1113944n

r1

Apqr


Alowastpqr





Hpq(ω) 1113944m

r1

ψprψqr




1113890 1113891 (2)



MFD Dd minus Dr

11138681113868111386811138681113868111386811138681113868

Dr

times 100 (3)




(a) (b)






Load

(kN

)

0

50

100

150

200

250

300


(a)

ST04ST08ST12

ST16ST20ST24

Load

(kN

)

0

50

100

150

200

250

300

5000 10000 15000 20000 250000Strain (με)

(b)

C11C12C13

C14C15

Load

(kN

)

0

50

100

150

200

250

300

500 1000 1500 2000 25000Strain (με)

(c)







Girder 1

Girder 2

Girder 3

(a)

Girder 1

Girder 2

Girder 3

(b)

Girder 1

Girder 2

Girder 3

(c)


100

10ndash2

10ndash4

10ndash6

10ndash80 50 100


Log

mag

nitu

de (m

mN

)



500

0

ndash5001500

1000

Y X5000 0

10002000

3000

Z

(a)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(b)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(c)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(d)

15001000

Y X5000 0

10002000

3000

500

0

ndash500Z

(e)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(f)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(g)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(h)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z(i)













MFD

val

ue (

)

Case 1Case 2

0

20

40

60

80

100

120

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 181Points 1ndash18


Diff

eren

ce v

alue

()

Case 3Case 4Case 5

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 271Points 1ndash27

0

20

40

60

80

100






Support


Spring element



(a)

Support


Spring element


(b)

Support


Spring element




(c)


X

Y

Z




Beam-support 20 109


Connection 10 0









σ

εtrεcr

ftr

εcu ε0

fcr

05fcr

(a)

σ

ε0

σu

εuεy

σy

(b)


ndashS2

ndashS3

ndashS1

(T M)

(C M)Kc = 23

Kc = 1

(a)

Uniaxialcompression

σ2

σ1

Biaxialcompression

Uniaxialtension

Biaxialtension

(b)





000102030405

Disp

lace

men

t (m

m)

2 3 4 5 6 7 8 9 1011121314151617181920212223242526271Points 1ndash27




objModal(x) 1113936

ni1 fi(x) + 1 minus MACi( 1113857( 1113857

n (4)


11138681113868111386811138681113868

111386811138681113868111386811138682

ϕTAi(x)ϕAi(x)1113872 1113873 ϕT

EiϕEi1113872 1113873 (5)

fi(x) f

j

E minus fj

A(x)

fj

E

(6)









Lowerlimit

Upperlimit














Ne1 x1( 1113857

1 minus x1

2

Ne2 x1( 1113857

1 + x1

2

(7)



018

020

022

024

026

028

030O

bjec

tive f

unct

ion


(a)

018

020

022

024

026

028

030

Obj

ectiv

e fun

ctio

n

09 10 11 12 13 14 1508Density ratio (ρρ0)

(b)


015

020

025

030

035

040

045

050

055

Obj

ectiv

e fun

ctio

n


(c)




ae

1113944ni

i1piNi xe

( 1113857 (8)



Ee

1 minus ae

( 1113857E0 (9)




08

10

12

14

16

Itera

tive r

esul

ts

10 20 30 40 50 60 700Iterations (n)

(a)

016

018

020

022

024

026

028

030

Obj

ectiv

e fun

ctio

n

10 20 30 40 50 60 700Iterations (n)

(b)







000102030405

Disp

lace

men

t (m

m)









5

3

11

10

19 20 21 22 23 24 25 26 27

11 12 13 14 15 16 17 18

2 3 4 5 6 7 8 9

6

4

2

Girder 3

Girder 2

Girder 1


Case 1

00

02

04

06

08

10

12

14

16

Stiff

ness

(nor

mal

ized

)


(a)

Case 2

00

02

04

06

08

10

12

14

16

Stiff

ness

(nor

mal

ized

)


(b)







a1

a2

⋮a9

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

N1 x1

1113872 1113873 N2 x1

1113872 1113873 N3 x1

1113872 1113873 N4 x1

1113872 1113873 N5 x1

1113872 1113873

N1 x2

1113872 1113873 N2 x2

1113872 1113873 N3 x2

1113872 1113873 N4 x2

1113872 1113873 N5 x2

1113872 1113873

⋮ ⋮ ⋮ ⋮ ⋮N1 x

91113872 1113873 N2 x

91113872 1113873 N3 x

91113872 1113873 N4 x

91113872 1113873 N5 x

91113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

P1

P2

P3

P4

P5

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎭

(11)



6 Conclusions



4000

19 20 21 22 23 24 25 26 27

10 11 12 13 14 15 16 17 18

1 2 3 4 5 6 7 8 9

530

760

760

(a)

1 2 3 4 5 6 7 8 9

1

05

N1 N3 N5 N7 N9

(b)



Mode Exp Freq(Hz)

UpdatedFreq (Hz)

Diff() MAC Exp Freq


Diff() MAC Exp Freq


Diff() MAC


1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627

00

01

02

03

04

05

Stiff

ness

deg

ener

atio

n co

effic

ient

Points 1ndash27

Case 3Case 4Case 5






Data Availability




Acknowledgments


References













































Hpq(ω) 1113944n

r1

Apqr


Alowastpqr





Hpq(ω) 1113944m

r1

ψprψqr




1113890 1113891 (2)



MFD Dd minus Dr

11138681113868111386811138681113868111386811138681113868

Dr

times 100 (3)




(a) (b)






Load

(kN

)

0

50

100

150

200

250

300


(a)

ST04ST08ST12

ST16ST20ST24

Load

(kN

)

0

50

100

150

200

250

300

5000 10000 15000 20000 250000Strain (με)

(b)

C11C12C13

C14C15

Load

(kN

)

0

50

100

150

200

250

300

500 1000 1500 2000 25000Strain (με)

(c)







Girder 1

Girder 2

Girder 3

(a)

Girder 1

Girder 2

Girder 3

(b)

Girder 1

Girder 2

Girder 3

(c)


100

10ndash2

10ndash4

10ndash6

10ndash80 50 100


Log

mag

nitu

de (m

mN

)



500

0

ndash5001500

1000

Y X5000 0

10002000

3000

Z

(a)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(b)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(c)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(d)

15001000

Y X5000 0

10002000

3000

500

0

ndash500Z

(e)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(f)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(g)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(h)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z(i)













MFD

val

ue (

)

Case 1Case 2

0

20

40

60

80

100

120

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 181Points 1ndash18


Diff

eren

ce v

alue

()

Case 3Case 4Case 5

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 271Points 1ndash27

0

20

40

60

80

100






Support


Spring element



(a)

Support


Spring element


(b)

Support


Spring element




(c)


X

Y

Z




Beam-support 20 109


Connection 10 0









σ

εtrεcr

ftr

εcu ε0

fcr

05fcr

(a)

σ

ε0

σu

εuεy

σy

(b)


ndashS2

ndashS3

ndashS1

(T M)

(C M)Kc = 23

Kc = 1

(a)

Uniaxialcompression

σ2

σ1

Biaxialcompression

Uniaxialtension

Biaxialtension

(b)





000102030405

Disp

lace

men

t (m

m)

2 3 4 5 6 7 8 9 1011121314151617181920212223242526271Points 1ndash27




objModal(x) 1113936

ni1 fi(x) + 1 minus MACi( 1113857( 1113857

n (4)


11138681113868111386811138681113868

111386811138681113868111386811138682

ϕTAi(x)ϕAi(x)1113872 1113873 ϕT

EiϕEi1113872 1113873 (5)

fi(x) f

j

E minus fj

A(x)

fj

E

(6)









Lowerlimit

Upperlimit














Ne1 x1( 1113857

1 minus x1

2

Ne2 x1( 1113857

1 + x1

2

(7)



018

020

022

024

026

028

030O

bjec

tive f

unct

ion


(a)

018

020

022

024

026

028

030

Obj

ectiv

e fun

ctio

n

09 10 11 12 13 14 1508Density ratio (ρρ0)

(b)


015

020

025

030

035

040

045

050

055

Obj

ectiv

e fun

ctio

n


(c)




ae

1113944ni

i1piNi xe

( 1113857 (8)



Ee

1 minus ae

( 1113857E0 (9)




08

10

12

14

16

Itera

tive r

esul

ts

10 20 30 40 50 60 700Iterations (n)

(a)

016

018

020

022

024

026

028

030

Obj

ectiv

e fun

ctio

n

10 20 30 40 50 60 700Iterations (n)

(b)







000102030405

Disp

lace

men

t (m

m)









5

3

11

10

19 20 21 22 23 24 25 26 27

11 12 13 14 15 16 17 18

2 3 4 5 6 7 8 9

6

4

2

Girder 3

Girder 2

Girder 1


Case 1

00

02

04

06

08

10

12

14

16

Stiff

ness

(nor

mal

ized

)


(a)

Case 2

00

02

04

06

08

10

12

14

16

Stiff

ness

(nor

mal

ized

)


(b)







a1

a2

⋮a9

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

N1 x1

1113872 1113873 N2 x1

1113872 1113873 N3 x1

1113872 1113873 N4 x1

1113872 1113873 N5 x1

1113872 1113873

N1 x2

1113872 1113873 N2 x2

1113872 1113873 N3 x2

1113872 1113873 N4 x2

1113872 1113873 N5 x2

1113872 1113873

⋮ ⋮ ⋮ ⋮ ⋮N1 x

91113872 1113873 N2 x

91113872 1113873 N3 x

91113872 1113873 N4 x

91113872 1113873 N5 x

91113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

P1

P2

P3

P4

P5

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎭

(11)



6 Conclusions



4000

19 20 21 22 23 24 25 26 27

10 11 12 13 14 15 16 17 18

1 2 3 4 5 6 7 8 9

530

760

760

(a)

1 2 3 4 5 6 7 8 9

1

05

N1 N3 N5 N7 N9

(b)



Mode Exp Freq(Hz)

UpdatedFreq (Hz)

Diff() MAC Exp Freq


Diff() MAC Exp Freq


Diff() MAC


1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627

00

01

02

03

04

05

Stiff

ness

deg

ener

atio

n co

effic

ient

Points 1ndash27

Case 3Case 4Case 5






Data Availability




Acknowledgments


References















































Load

(kN

)

0

50

100

150

200

250

300


(a)

ST04ST08ST12

ST16ST20ST24

Load

(kN

)

0

50

100

150

200

250

300

5000 10000 15000 20000 250000Strain (με)

(b)

C11C12C13

C14C15

Load

(kN

)

0

50

100

150

200

250

300

500 1000 1500 2000 25000Strain (με)

(c)







Girder 1

Girder 2

Girder 3

(a)

Girder 1

Girder 2

Girder 3

(b)

Girder 1

Girder 2

Girder 3

(c)


100

10ndash2

10ndash4

10ndash6

10ndash80 50 100


Log

mag

nitu

de (m

mN

)



500

0

ndash5001500

1000

Y X5000 0

10002000

3000

Z

(a)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(b)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(c)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(d)

15001000

Y X5000 0

10002000

3000

500

0

ndash500Z

(e)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(f)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(g)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(h)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z(i)













MFD

val

ue (

)

Case 1Case 2

0

20

40

60

80

100

120

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 181Points 1ndash18


Diff

eren

ce v

alue

()

Case 3Case 4Case 5

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 271Points 1ndash27

0

20

40

60

80

100






Support


Spring element



(a)

Support


Spring element


(b)

Support


Spring element




(c)


X

Y

Z




Beam-support 20 109


Connection 10 0









σ

εtrεcr

ftr

εcu ε0

fcr

05fcr

(a)

σ

ε0

σu

εuεy

σy

(b)


ndashS2

ndashS3

ndashS1

(T M)

(C M)Kc = 23

Kc = 1

(a)

Uniaxialcompression

σ2

σ1

Biaxialcompression

Uniaxialtension

Biaxialtension

(b)





000102030405

Disp

lace

men

t (m

m)

2 3 4 5 6 7 8 9 1011121314151617181920212223242526271Points 1ndash27




objModal(x) 1113936

ni1 fi(x) + 1 minus MACi( 1113857( 1113857

n (4)


11138681113868111386811138681113868

111386811138681113868111386811138682

ϕTAi(x)ϕAi(x)1113872 1113873 ϕT

EiϕEi1113872 1113873 (5)

fi(x) f

j

E minus fj

A(x)

fj

E

(6)









Lowerlimit

Upperlimit














Ne1 x1( 1113857

1 minus x1

2

Ne2 x1( 1113857

1 + x1

2

(7)



018

020

022

024

026

028

030O

bjec

tive f

unct

ion


(a)

018

020

022

024

026

028

030

Obj

ectiv

e fun

ctio

n

09 10 11 12 13 14 1508Density ratio (ρρ0)

(b)


015

020

025

030

035

040

045

050

055

Obj

ectiv

e fun

ctio

n


(c)




ae

1113944ni

i1piNi xe

( 1113857 (8)



Ee

1 minus ae

( 1113857E0 (9)




08

10

12

14

16

Itera

tive r

esul

ts

10 20 30 40 50 60 700Iterations (n)

(a)

016

018

020

022

024

026

028

030

Obj

ectiv

e fun

ctio

n

10 20 30 40 50 60 700Iterations (n)

(b)







000102030405

Disp

lace

men

t (m

m)









5

3

11

10

19 20 21 22 23 24 25 26 27

11 12 13 14 15 16 17 18

2 3 4 5 6 7 8 9

6

4

2

Girder 3

Girder 2

Girder 1


Case 1

00

02

04

06

08

10

12

14

16

Stiff

ness

(nor

mal

ized

)


(a)

Case 2

00

02

04

06

08

10

12

14

16

Stiff

ness

(nor

mal

ized

)


(b)







a1

a2

⋮a9

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

N1 x1

1113872 1113873 N2 x1

1113872 1113873 N3 x1

1113872 1113873 N4 x1

1113872 1113873 N5 x1

1113872 1113873

N1 x2

1113872 1113873 N2 x2

1113872 1113873 N3 x2

1113872 1113873 N4 x2

1113872 1113873 N5 x2

1113872 1113873

⋮ ⋮ ⋮ ⋮ ⋮N1 x

91113872 1113873 N2 x

91113872 1113873 N3 x

91113872 1113873 N4 x

91113872 1113873 N5 x

91113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

P1

P2

P3

P4

P5

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎭

(11)



6 Conclusions



4000

19 20 21 22 23 24 25 26 27

10 11 12 13 14 15 16 17 18

1 2 3 4 5 6 7 8 9

530

760

760

(a)

1 2 3 4 5 6 7 8 9

1

05

N1 N3 N5 N7 N9

(b)



Mode Exp Freq(Hz)

UpdatedFreq (Hz)

Diff() MAC Exp Freq


Diff() MAC Exp Freq


Diff() MAC


1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627

00

01

02

03

04

05

Stiff

ness

deg

ener

atio

n co

effic

ient

Points 1ndash27

Case 3Case 4Case 5






Data Availability




Acknowledgments


References
















































Girder 1

Girder 2

Girder 3

(a)

Girder 1

Girder 2

Girder 3

(b)

Girder 1

Girder 2

Girder 3

(c)


100

10ndash2

10ndash4

10ndash6

10ndash80 50 100


Log

mag

nitu

de (m

mN

)



500

0

ndash5001500

1000

Y X5000 0

10002000

3000

Z

(a)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(b)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(c)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(d)

15001000

Y X5000 0

10002000

3000

500

0

ndash500Z

(e)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(f)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(g)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(h)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z(i)













MFD

val

ue (

)

Case 1Case 2

0

20

40

60

80

100

120

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 181Points 1ndash18


Diff

eren

ce v

alue

()

Case 3Case 4Case 5

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 271Points 1ndash27

0

20

40

60

80

100






Support


Spring element



(a)

Support


Spring element


(b)

Support


Spring element




(c)


X

Y

Z




Beam-support 20 109


Connection 10 0









σ

εtrεcr

ftr

εcu ε0

fcr

05fcr

(a)

σ

ε0

σu

εuεy

σy

(b)


ndashS2

ndashS3

ndashS1

(T M)

(C M)Kc = 23

Kc = 1

(a)

Uniaxialcompression

σ2

σ1

Biaxialcompression

Uniaxialtension

Biaxialtension

(b)





000102030405

Disp

lace

men

t (m

m)

2 3 4 5 6 7 8 9 1011121314151617181920212223242526271Points 1ndash27




objModal(x) 1113936

ni1 fi(x) + 1 minus MACi( 1113857( 1113857

n (4)


11138681113868111386811138681113868

111386811138681113868111386811138682

ϕTAi(x)ϕAi(x)1113872 1113873 ϕT

EiϕEi1113872 1113873 (5)

fi(x) f

j

E minus fj

A(x)

fj

E

(6)









Lowerlimit

Upperlimit














Ne1 x1( 1113857

1 minus x1

2

Ne2 x1( 1113857

1 + x1

2

(7)



018

020

022

024

026

028

030O

bjec

tive f

unct

ion


(a)

018

020

022

024

026

028

030

Obj

ectiv

e fun

ctio

n

09 10 11 12 13 14 1508Density ratio (ρρ0)

(b)


015

020

025

030

035

040

045

050

055

Obj

ectiv

e fun

ctio

n


(c)




ae

1113944ni

i1piNi xe

( 1113857 (8)



Ee

1 minus ae

( 1113857E0 (9)




08

10

12

14

16

Itera

tive r

esul

ts

10 20 30 40 50 60 700Iterations (n)

(a)

016

018

020

022

024

026

028

030

Obj

ectiv

e fun

ctio

n

10 20 30 40 50 60 700Iterations (n)

(b)







000102030405

Disp

lace

men

t (m

m)









5

3

11

10

19 20 21 22 23 24 25 26 27

11 12 13 14 15 16 17 18

2 3 4 5 6 7 8 9

6

4

2

Girder 3

Girder 2

Girder 1


Case 1

00

02

04

06

08

10

12

14

16

Stiff

ness

(nor

mal

ized

)


(a)

Case 2

00

02

04

06

08

10

12

14

16

Stiff

ness

(nor

mal

ized

)


(b)







a1

a2

⋮a9

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

N1 x1

1113872 1113873 N2 x1

1113872 1113873 N3 x1

1113872 1113873 N4 x1

1113872 1113873 N5 x1

1113872 1113873

N1 x2

1113872 1113873 N2 x2

1113872 1113873 N3 x2

1113872 1113873 N4 x2

1113872 1113873 N5 x2

1113872 1113873

⋮ ⋮ ⋮ ⋮ ⋮N1 x

91113872 1113873 N2 x

91113872 1113873 N3 x

91113872 1113873 N4 x

91113872 1113873 N5 x

91113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

P1

P2

P3

P4

P5

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎭

(11)



6 Conclusions



4000

19 20 21 22 23 24 25 26 27

10 11 12 13 14 15 16 17 18

1 2 3 4 5 6 7 8 9

530

760

760

(a)

1 2 3 4 5 6 7 8 9

1

05

N1 N3 N5 N7 N9

(b)



Mode Exp Freq(Hz)

UpdatedFreq (Hz)

Diff() MAC Exp Freq


Diff() MAC Exp Freq


Diff() MAC


1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627

00

01

02

03

04

05

Stiff

ness

deg

ener

atio

n co

effic

ient

Points 1ndash27

Case 3Case 4Case 5






Data Availability




Acknowledgments


References












































500

0

ndash5001500

1000

Y X5000 0

10002000

3000

Z

(a)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(b)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(c)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(d)

15001000

Y X5000 0

10002000

3000

500

0

ndash500Z

(e)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(f)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(g)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z

(h)

15001000

Y X5000 0

10002000

3000

500

0

ndash500

Z(i)













MFD

val

ue (

)

Case 1Case 2

0

20

40

60

80

100

120

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 181Points 1ndash18


Diff

eren

ce v

alue

()

Case 3Case 4Case 5

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 271Points 1ndash27

0

20

40

60

80

100






Support


Spring element



(a)

Support


Spring element


(b)

Support


Spring element




(c)


X

Y

Z




Beam-support 20 109


Connection 10 0









σ

εtrεcr

ftr

εcu ε0

fcr

05fcr

(a)

σ

ε0

σu

εuεy

σy

(b)


ndashS2

ndashS3

ndashS1

(T M)

(C M)Kc = 23

Kc = 1

(a)

Uniaxialcompression

σ2

σ1

Biaxialcompression

Uniaxialtension

Biaxialtension

(b)





000102030405

Disp

lace

men

t (m

m)

2 3 4 5 6 7 8 9 1011121314151617181920212223242526271Points 1ndash27




objModal(x) 1113936

ni1 fi(x) + 1 minus MACi( 1113857( 1113857

n (4)


11138681113868111386811138681113868

111386811138681113868111386811138682

ϕTAi(x)ϕAi(x)1113872 1113873 ϕT

EiϕEi1113872 1113873 (5)

fi(x) f

j

E minus fj

A(x)

fj

E

(6)









Lowerlimit

Upperlimit














Ne1 x1( 1113857

1 minus x1

2

Ne2 x1( 1113857

1 + x1

2

(7)



018

020

022

024

026

028

030O

bjec

tive f

unct

ion


(a)

018

020

022

024

026

028

030

Obj

ectiv

e fun

ctio

n

09 10 11 12 13 14 1508Density ratio (ρρ0)

(b)


015

020

025

030

035

040

045

050

055

Obj

ectiv

e fun

ctio

n


(c)




ae

1113944ni

i1piNi xe

( 1113857 (8)



Ee

1 minus ae

( 1113857E0 (9)




08

10

12

14

16

Itera

tive r

esul

ts

10 20 30 40 50 60 700Iterations (n)

(a)

016

018

020

022

024

026

028

030

Obj

ectiv

e fun

ctio

n

10 20 30 40 50 60 700Iterations (n)

(b)







000102030405

Disp

lace

men

t (m

m)









5

3

11

10

19 20 21 22 23 24 25 26 27

11 12 13 14 15 16 17 18

2 3 4 5 6 7 8 9

6

4

2

Girder 3

Girder 2

Girder 1


Case 1

00

02

04

06

08

10

12

14

16

Stiff

ness

(nor

mal

ized

)


(a)

Case 2

00

02

04

06

08

10

12

14

16

Stiff

ness

(nor

mal

ized

)


(b)







a1

a2

⋮a9

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

N1 x1

1113872 1113873 N2 x1

1113872 1113873 N3 x1

1113872 1113873 N4 x1

1113872 1113873 N5 x1

1113872 1113873

N1 x2

1113872 1113873 N2 x2

1113872 1113873 N3 x2

1113872 1113873 N4 x2

1113872 1113873 N5 x2

1113872 1113873

⋮ ⋮ ⋮ ⋮ ⋮N1 x

91113872 1113873 N2 x

91113872 1113873 N3 x

91113872 1113873 N4 x

91113872 1113873 N5 x

91113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

P1

P2

P3

P4

P5

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎭

(11)



6 Conclusions



4000

19 20 21 22 23 24 25 26 27

10 11 12 13 14 15 16 17 18

1 2 3 4 5 6 7 8 9

530

760

760

(a)

1 2 3 4 5 6 7 8 9

1

05

N1 N3 N5 N7 N9

(b)



Mode Exp Freq(Hz)

UpdatedFreq (Hz)

Diff() MAC Exp Freq


Diff() MAC Exp Freq


Diff() MAC


1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627

00

01

02

03

04

05

Stiff

ness

deg

ener

atio

n co

effic

ient

Points 1ndash27

Case 3Case 4Case 5






Data Availability




Acknowledgments


References
















































MFD

val

ue (

)

Case 1Case 2

0

20

40

60

80

100

120

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 181Points 1ndash18


Diff

eren

ce v

alue

()

Case 3Case 4Case 5

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 271Points 1ndash27

0

20

40

60

80

100






Support


Spring element



(a)

Support


Spring element


(b)

Support


Spring element




(c)


X

Y

Z




Beam-support 20 109


Connection 10 0









σ

εtrεcr

ftr

εcu ε0

fcr

05fcr

(a)

σ

ε0

σu

εuεy

σy

(b)


ndashS2

ndashS3

ndashS1

(T M)

(C M)Kc = 23

Kc = 1

(a)

Uniaxialcompression

σ2

σ1

Biaxialcompression

Uniaxialtension

Biaxialtension

(b)





000102030405

Disp

lace

men

t (m

m)

2 3 4 5 6 7 8 9 1011121314151617181920212223242526271Points 1ndash27




objModal(x) 1113936

ni1 fi(x) + 1 minus MACi( 1113857( 1113857

n (4)


11138681113868111386811138681113868

111386811138681113868111386811138682

ϕTAi(x)ϕAi(x)1113872 1113873 ϕT

EiϕEi1113872 1113873 (5)

fi(x) f

j

E minus fj

A(x)

fj

E

(6)









Lowerlimit

Upperlimit














Ne1 x1( 1113857

1 minus x1

2

Ne2 x1( 1113857

1 + x1

2

(7)



018

020

022

024

026

028

030O

bjec

tive f

unct

ion


(a)

018

020

022

024

026

028

030

Obj

ectiv

e fun

ctio

n

09 10 11 12 13 14 1508Density ratio (ρρ0)

(b)


015

020

025

030

035

040

045

050

055

Obj

ectiv

e fun

ctio

n


(c)




ae

1113944ni

i1piNi xe

( 1113857 (8)



Ee

1 minus ae

( 1113857E0 (9)




08

10

12

14

16

Itera

tive r

esul

ts

10 20 30 40 50 60 700Iterations (n)

(a)

016

018

020

022

024

026

028

030

Obj

ectiv

e fun

ctio

n

10 20 30 40 50 60 700Iterations (n)

(b)







000102030405

Disp

lace

men

t (m

m)









5

3

11

10

19 20 21 22 23 24 25 26 27

11 12 13 14 15 16 17 18

2 3 4 5 6 7 8 9

6

4

2

Girder 3

Girder 2

Girder 1


Case 1

00

02

04

06

08

10

12

14

16

Stiff

ness

(nor

mal

ized

)


(a)

Case 2

00

02

04

06

08

10

12

14

16

Stiff

ness

(nor

mal

ized

)


(b)







a1

a2

⋮a9

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

N1 x1

1113872 1113873 N2 x1

1113872 1113873 N3 x1

1113872 1113873 N4 x1

1113872 1113873 N5 x1

1113872 1113873

N1 x2

1113872 1113873 N2 x2

1113872 1113873 N3 x2

1113872 1113873 N4 x2

1113872 1113873 N5 x2

1113872 1113873

⋮ ⋮ ⋮ ⋮ ⋮N1 x

91113872 1113873 N2 x

91113872 1113873 N3 x

91113872 1113873 N4 x

91113872 1113873 N5 x

91113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

P1

P2

P3

P4

P5

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎭

(11)



6 Conclusions



4000

19 20 21 22 23 24 25 26 27

10 11 12 13 14 15 16 17 18

1 2 3 4 5 6 7 8 9

530

760

760

(a)

1 2 3 4 5 6 7 8 9

1

05

N1 N3 N5 N7 N9

(b)



Mode Exp Freq(Hz)

UpdatedFreq (Hz)

Diff() MAC Exp Freq


Diff() MAC Exp Freq


Diff() MAC


1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627

00

01

02

03

04

05

Stiff

ness

deg

ener

atio

n co

effic

ient

Points 1ndash27

Case 3Case 4Case 5






Data Availability




Acknowledgments


References















































Support


Spring element



(a)

Support


Spring element


(b)

Support


Spring element




(c)


X

Y

Z




Beam-support 20 109


Connection 10 0









σ

εtrεcr

ftr

εcu ε0

fcr

05fcr

(a)

σ

ε0

σu

εuεy

σy

(b)


ndashS2

ndashS3

ndashS1

(T M)

(C M)Kc = 23

Kc = 1

(a)

Uniaxialcompression

σ2

σ1

Biaxialcompression

Uniaxialtension

Biaxialtension

(b)





000102030405

Disp

lace

men

t (m

m)

2 3 4 5 6 7 8 9 1011121314151617181920212223242526271Points 1ndash27




objModal(x) 1113936

ni1 fi(x) + 1 minus MACi( 1113857( 1113857

n (4)


11138681113868111386811138681113868

111386811138681113868111386811138682

ϕTAi(x)ϕAi(x)1113872 1113873 ϕT

EiϕEi1113872 1113873 (5)

fi(x) f

j

E minus fj

A(x)

fj

E

(6)









Lowerlimit

Upperlimit














Ne1 x1( 1113857

1 minus x1

2

Ne2 x1( 1113857

1 + x1

2

(7)



018

020

022

024

026

028

030O

bjec

tive f

unct

ion


(a)

018

020

022

024

026

028

030

Obj

ectiv

e fun

ctio

n

09 10 11 12 13 14 1508Density ratio (ρρ0)

(b)


015

020

025

030

035

040

045

050

055

Obj

ectiv

e fun

ctio

n


(c)




ae

1113944ni

i1piNi xe

( 1113857 (8)



Ee

1 minus ae

( 1113857E0 (9)




08

10

12

14

16

Itera

tive r

esul

ts

10 20 30 40 50 60 700Iterations (n)

(a)

016

018

020

022

024

026

028

030

Obj

ectiv

e fun

ctio

n

10 20 30 40 50 60 700Iterations (n)

(b)







000102030405

Disp

lace

men

t (m

m)









5

3

11

10

19 20 21 22 23 24 25 26 27

11 12 13 14 15 16 17 18

2 3 4 5 6 7 8 9

6

4

2

Girder 3

Girder 2

Girder 1


Case 1

00

02

04

06

08

10

12

14

16

Stiff

ness

(nor

mal

ized

)


(a)

Case 2

00

02

04

06

08

10

12

14

16

Stiff

ness

(nor

mal

ized

)


(b)







a1

a2

⋮a9

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

N1 x1

1113872 1113873 N2 x1

1113872 1113873 N3 x1

1113872 1113873 N4 x1

1113872 1113873 N5 x1

1113872 1113873

N1 x2

1113872 1113873 N2 x2

1113872 1113873 N3 x2

1113872 1113873 N4 x2

1113872 1113873 N5 x2

1113872 1113873

⋮ ⋮ ⋮ ⋮ ⋮N1 x

91113872 1113873 N2 x

91113872 1113873 N3 x

91113872 1113873 N4 x

91113872 1113873 N5 x

91113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

P1

P2

P3

P4

P5

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎭

(11)



6 Conclusions



4000

19 20 21 22 23 24 25 26 27

10 11 12 13 14 15 16 17 18

1 2 3 4 5 6 7 8 9

530

760

760

(a)

1 2 3 4 5 6 7 8 9

1

05

N1 N3 N5 N7 N9

(b)



Mode Exp Freq(Hz)

UpdatedFreq (Hz)

Diff() MAC Exp Freq


Diff() MAC Exp Freq


Diff() MAC


1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627

00

01

02

03

04

05

Stiff

ness

deg

ener

atio

n co

effic

ient

Points 1ndash27

Case 3Case 4Case 5






Data Availability




Acknowledgments


References















































σ

εtrεcr

ftr

εcu ε0

fcr

05fcr

(a)

σ

ε0

σu

εuεy

σy

(b)


ndashS2

ndashS3

ndashS1

(T M)

(C M)Kc = 23

Kc = 1

(a)

Uniaxialcompression

σ2

σ1

Biaxialcompression

Uniaxialtension

Biaxialtension

(b)





000102030405

Disp

lace

men

t (m

m)

2 3 4 5 6 7 8 9 1011121314151617181920212223242526271Points 1ndash27




objModal(x) 1113936

ni1 fi(x) + 1 minus MACi( 1113857( 1113857

n (4)


11138681113868111386811138681113868

111386811138681113868111386811138682

ϕTAi(x)ϕAi(x)1113872 1113873 ϕT

EiϕEi1113872 1113873 (5)

fi(x) f

j

E minus fj

A(x)

fj

E

(6)









Lowerlimit

Upperlimit














Ne1 x1( 1113857

1 minus x1

2

Ne2 x1( 1113857

1 + x1

2

(7)



018

020

022

024

026

028

030O

bjec

tive f

unct

ion


(a)

018

020

022

024

026

028

030

Obj

ectiv

e fun

ctio

n

09 10 11 12 13 14 1508Density ratio (ρρ0)

(b)


015

020

025

030

035

040

045

050

055

Obj

ectiv

e fun

ctio

n


(c)




ae

1113944ni

i1piNi xe

( 1113857 (8)



Ee

1 minus ae

( 1113857E0 (9)




08

10

12

14

16

Itera

tive r

esul

ts

10 20 30 40 50 60 700Iterations (n)

(a)

016

018

020

022

024

026

028

030

Obj

ectiv

e fun

ctio

n

10 20 30 40 50 60 700Iterations (n)

(b)







000102030405

Disp

lace

men

t (m

m)









5

3

11

10

19 20 21 22 23 24 25 26 27

11 12 13 14 15 16 17 18

2 3 4 5 6 7 8 9

6

4

2

Girder 3

Girder 2

Girder 1


Case 1

00

02

04

06

08

10

12

14

16

Stiff

ness

(nor

mal

ized

)


(a)

Case 2

00

02

04

06

08

10

12

14

16

Stiff

ness

(nor

mal

ized

)


(b)







a1

a2

⋮a9

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

N1 x1

1113872 1113873 N2 x1

1113872 1113873 N3 x1

1113872 1113873 N4 x1

1113872 1113873 N5 x1

1113872 1113873

N1 x2

1113872 1113873 N2 x2

1113872 1113873 N3 x2

1113872 1113873 N4 x2

1113872 1113873 N5 x2

1113872 1113873

⋮ ⋮ ⋮ ⋮ ⋮N1 x

91113872 1113873 N2 x

91113872 1113873 N3 x

91113872 1113873 N4 x

91113872 1113873 N5 x

91113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

P1

P2

P3

P4

P5

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎭

(11)



6 Conclusions



4000

19 20 21 22 23 24 25 26 27

10 11 12 13 14 15 16 17 18

1 2 3 4 5 6 7 8 9

530

760

760

(a)

1 2 3 4 5 6 7 8 9

1

05

N1 N3 N5 N7 N9

(b)



Mode Exp Freq(Hz)

UpdatedFreq (Hz)

Diff() MAC Exp Freq


Diff() MAC Exp Freq


Diff() MAC


1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627

00

01

02

03

04

05

Stiff

ness

deg

ener

atio

n co

effic

ient

Points 1ndash27

Case 3Case 4Case 5






Data Availability




Acknowledgments


References













































objModal(x) 1113936

ni1 fi(x) + 1 minus MACi( 1113857( 1113857

n (4)


11138681113868111386811138681113868

111386811138681113868111386811138682

ϕTAi(x)ϕAi(x)1113872 1113873 ϕT

EiϕEi1113872 1113873 (5)

fi(x) f

j

E minus fj

A(x)

fj

E

(6)









Lowerlimit

Upperlimit














Ne1 x1( 1113857

1 minus x1

2

Ne2 x1( 1113857

1 + x1

2

(7)



018

020

022

024

026

028

030O

bjec

tive f

unct

ion


(a)

018

020

022

024

026

028

030

Obj

ectiv

e fun

ctio

n

09 10 11 12 13 14 1508Density ratio (ρρ0)

(b)


015

020

025

030

035

040

045

050

055

Obj

ectiv

e fun

ctio

n


(c)




ae

1113944ni

i1piNi xe

( 1113857 (8)



Ee

1 minus ae

( 1113857E0 (9)




08

10

12

14

16

Itera

tive r

esul

ts

10 20 30 40 50 60 700Iterations (n)

(a)

016

018

020

022

024

026

028

030

Obj

ectiv

e fun

ctio

n

10 20 30 40 50 60 700Iterations (n)

(b)







000102030405

Disp

lace

men

t (m

m)









5

3

11

10

19 20 21 22 23 24 25 26 27

11 12 13 14 15 16 17 18

2 3 4 5 6 7 8 9

6

4

2

Girder 3

Girder 2

Girder 1


Case 1

00

02

04

06

08

10

12

14

16

Stiff

ness

(nor

mal

ized

)


(a)

Case 2

00

02

04

06

08

10

12

14

16

Stiff

ness

(nor

mal

ized

)


(b)







a1

a2

⋮a9

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

N1 x1

1113872 1113873 N2 x1

1113872 1113873 N3 x1

1113872 1113873 N4 x1

1113872 1113873 N5 x1

1113872 1113873

N1 x2

1113872 1113873 N2 x2

1113872 1113873 N3 x2

1113872 1113873 N4 x2

1113872 1113873 N5 x2

1113872 1113873

⋮ ⋮ ⋮ ⋮ ⋮N1 x

91113872 1113873 N2 x

91113872 1113873 N3 x

91113872 1113873 N4 x

91113872 1113873 N5 x

91113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

P1

P2

P3

P4

P5

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎭

(11)



6 Conclusions



4000

19 20 21 22 23 24 25 26 27

10 11 12 13 14 15 16 17 18

1 2 3 4 5 6 7 8 9

530

760

760

(a)

1 2 3 4 5 6 7 8 9

1

05

N1 N3 N5 N7 N9

(b)



Mode Exp Freq(Hz)

UpdatedFreq (Hz)

Diff() MAC Exp Freq


Diff() MAC Exp Freq


Diff() MAC


1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627

00

01

02

03

04

05

Stiff

ness

deg

ener

atio

n co

effic

ient

Points 1ndash27

Case 3Case 4Case 5






Data Availability




Acknowledgments


References















































Ne1 x1( 1113857

1 minus x1

2

Ne2 x1( 1113857

1 + x1

2

(7)



018

020

022

024

026

028

030O

bjec

tive f

unct

ion


(a)

018

020

022

024

026

028

030

Obj

ectiv

e fun

ctio

n

09 10 11 12 13 14 1508Density ratio (ρρ0)

(b)


015

020

025

030

035

040

045

050

055

Obj

ectiv

e fun

ctio

n


(c)




ae

1113944ni

i1piNi xe

( 1113857 (8)



Ee

1 minus ae

( 1113857E0 (9)




08

10

12

14

16

Itera

tive r

esul

ts

10 20 30 40 50 60 700Iterations (n)

(a)

016

018

020

022

024

026

028

030

Obj

ectiv

e fun

ctio

n

10 20 30 40 50 60 700Iterations (n)

(b)







000102030405

Disp

lace

men

t (m

m)









5

3

11

10

19 20 21 22 23 24 25 26 27

11 12 13 14 15 16 17 18

2 3 4 5 6 7 8 9

6

4

2

Girder 3

Girder 2

Girder 1


Case 1

00

02

04

06

08

10

12

14

16

Stiff

ness

(nor

mal

ized

)


(a)

Case 2

00

02

04

06

08

10

12

14

16

Stiff

ness

(nor

mal

ized

)


(b)







a1

a2

⋮a9

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

N1 x1

1113872 1113873 N2 x1

1113872 1113873 N3 x1

1113872 1113873 N4 x1

1113872 1113873 N5 x1

1113872 1113873

N1 x2

1113872 1113873 N2 x2

1113872 1113873 N3 x2

1113872 1113873 N4 x2

1113872 1113873 N5 x2

1113872 1113873

⋮ ⋮ ⋮ ⋮ ⋮N1 x

91113872 1113873 N2 x

91113872 1113873 N3 x

91113872 1113873 N4 x

91113872 1113873 N5 x

91113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

P1

P2

P3

P4

P5

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎭

(11)



6 Conclusions



4000

19 20 21 22 23 24 25 26 27

10 11 12 13 14 15 16 17 18

1 2 3 4 5 6 7 8 9

530

760

760

(a)

1 2 3 4 5 6 7 8 9

1

05

N1 N3 N5 N7 N9

(b)



Mode Exp Freq(Hz)

UpdatedFreq (Hz)

Diff() MAC Exp Freq


Diff() MAC Exp Freq


Diff() MAC


1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627

00

01

02

03

04

05

Stiff

ness

deg

ener

atio

n co

effic

ient

Points 1ndash27

Case 3Case 4Case 5






Data Availability




Acknowledgments


References













































ae

1113944ni

i1piNi xe

( 1113857 (8)



Ee

1 minus ae

( 1113857E0 (9)




08

10

12

14

16

Itera

tive r

esul

ts

10 20 30 40 50 60 700Iterations (n)

(a)

016

018

020

022

024

026

028

030

Obj

ectiv

e fun

ctio

n

10 20 30 40 50 60 700Iterations (n)

(b)







000102030405

Disp

lace

men

t (m

m)









5

3

11

10

19 20 21 22 23 24 25 26 27

11 12 13 14 15 16 17 18

2 3 4 5 6 7 8 9

6

4

2

Girder 3

Girder 2

Girder 1


Case 1

00

02

04

06

08

10

12

14

16

Stiff

ness

(nor

mal

ized

)


(a)

Case 2

00

02

04

06

08

10

12

14

16

Stiff

ness

(nor

mal

ized

)


(b)







a1

a2

⋮a9

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

N1 x1

1113872 1113873 N2 x1

1113872 1113873 N3 x1

1113872 1113873 N4 x1

1113872 1113873 N5 x1

1113872 1113873

N1 x2

1113872 1113873 N2 x2

1113872 1113873 N3 x2

1113872 1113873 N4 x2

1113872 1113873 N5 x2

1113872 1113873

⋮ ⋮ ⋮ ⋮ ⋮N1 x

91113872 1113873 N2 x

91113872 1113873 N3 x

91113872 1113873 N4 x

91113872 1113873 N5 x

91113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

P1

P2

P3

P4

P5

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎭

(11)



6 Conclusions



4000

19 20 21 22 23 24 25 26 27

10 11 12 13 14 15 16 17 18

1 2 3 4 5 6 7 8 9

530

760

760

(a)

1 2 3 4 5 6 7 8 9

1

05

N1 N3 N5 N7 N9

(b)



Mode Exp Freq(Hz)

UpdatedFreq (Hz)

Diff() MAC Exp Freq


Diff() MAC Exp Freq


Diff() MAC


1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627

00

01

02

03

04

05

Stiff

ness

deg

ener

atio

n co

effic

ient

Points 1ndash27

Case 3Case 4Case 5






Data Availability




Acknowledgments


References


















































5

3

11

10

19 20 21 22 23 24 25 26 27

11 12 13 14 15 16 17 18

2 3 4 5 6 7 8 9

6

4

2

Girder 3

Girder 2

Girder 1


Case 1

00

02

04

06

08

10

12

14

16

Stiff

ness

(nor

mal

ized

)


(a)

Case 2

00

02

04

06

08

10

12

14

16

Stiff

ness

(nor

mal

ized

)


(b)







a1

a2

⋮a9

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

N1 x1

1113872 1113873 N2 x1

1113872 1113873 N3 x1

1113872 1113873 N4 x1

1113872 1113873 N5 x1

1113872 1113873

N1 x2

1113872 1113873 N2 x2

1113872 1113873 N3 x2

1113872 1113873 N4 x2

1113872 1113873 N5 x2

1113872 1113873

⋮ ⋮ ⋮ ⋮ ⋮N1 x

91113872 1113873 N2 x

91113872 1113873 N3 x

91113872 1113873 N4 x

91113872 1113873 N5 x

91113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

P1

P2

P3

P4

P5

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎭

(11)



6 Conclusions



4000

19 20 21 22 23 24 25 26 27

10 11 12 13 14 15 16 17 18

1 2 3 4 5 6 7 8 9

530

760

760

(a)

1 2 3 4 5 6 7 8 9

1

05

N1 N3 N5 N7 N9

(b)



Mode Exp Freq(Hz)

UpdatedFreq (Hz)

Diff() MAC Exp Freq


Diff() MAC Exp Freq


Diff() MAC


1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627

00

01

02

03

04

05

Stiff

ness

deg

ener

atio

n co

effic

ient

Points 1ndash27

Case 3Case 4Case 5






Data Availability




Acknowledgments


References














































a1

a2

⋮a9

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

N1 x1

1113872 1113873 N2 x1

1113872 1113873 N3 x1

1113872 1113873 N4 x1

1113872 1113873 N5 x1

1113872 1113873

N1 x2

1113872 1113873 N2 x2

1113872 1113873 N3 x2

1113872 1113873 N4 x2

1113872 1113873 N5 x2

1113872 1113873

⋮ ⋮ ⋮ ⋮ ⋮N1 x

91113872 1113873 N2 x

91113872 1113873 N3 x

91113872 1113873 N4 x

91113872 1113873 N5 x

91113872 1113873

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

P1

P2

P3

P4

P5

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎭

(11)



6 Conclusions



4000

19 20 21 22 23 24 25 26 27

10 11 12 13 14 15 16 17 18

1 2 3 4 5 6 7 8 9

530

760

760

(a)

1 2 3 4 5 6 7 8 9

1

05

N1 N3 N5 N7 N9

(b)



Mode Exp Freq(Hz)

UpdatedFreq (Hz)

Diff() MAC Exp Freq


Diff() MAC Exp Freq


Diff() MAC


1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627

00

01

02

03

04

05

Stiff

ness

deg

ener

atio

n co

effic

ient

Points 1ndash27

Case 3Case 4Case 5






Data Availability




Acknowledgments


References















































Data Availability




Acknowledgments


References




































































Documents

Vibration-BasedDamageDetectionofaSteel ...downloads.hindawi.com/journals/ace/2020/8889277.pdfdeﬁnedamageprobabilityfunctions[2].Wheninvisible damageoccursonacomplicatedstructuresuchasacom-positebridgedeckwithshearconnectors,theVBDDmethod