Journal of Constructional Steel Research 62 (2006) 231–239

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Structural parameter identification anddamage detection for a steelstructure using a two-stage finite element model updating metho

J.R. Wu, Q.S. Li∗

Department of Building and Construction, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong

Received 4 February 2005; accepted 1 July 2005

Abstract

A two-stage eigensensitivity-based finite element (FE) model updating procedure is developed for structural parameter identificdamage detection for the IASC-ASCE structural health monitoring benchmark steel structure on the basis of ambient vibration measIn the first stage, both the weighted least squares and Bayesian estimation methods are adopted for the identification of the connectioof beam–column joints and Young’s modulus of the structure; then the damage detection is conducted via the FE model updating procfor detecting damaged braces with different damage patterns of the structure. Comparisons between the FE model updated resexperimental data show that the eigensensitivity-based FE model updating procedure is an effective tool for structural parameter ideand damage detection for steel frame structures.© 2005 Elsevier Ltd. All rights reserved.

Keywords: Finite element model updating; Structural parameter identification; Damage detection; Structural health monitoring; Finite element method; Steelstructure

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1. Introduction

System identification and damage detection techniqueconstitute a promising field with widespread applicatioin civil engineering. Detecting actual structural damage dto earthquake, impacts or strong wind actions can provimportant information on the operating state and structusafety of the structures concerned. Various methodsstructural system identification and damage detection habeen developed in the past two decades. In the aresystem identification, the traditional fast Fourier transfo(FFT) method [2], autoregressive moving average modbased on discrete time data [1], natural excitation techniquewith an eigensystem realization algorithm [4], and stochasticsubspace identification [24] have been widely adopted fordetermining dynamics characteristics of civil structures.

Meanwhile, in recent years there has been great intein the development of damage detection techniquesstatistical model updating methodology was adopted by L

∗ Corresponding author. Tel.: +852 27844677; fax: +852 27887612.E-mail address:bcqsli@cityu.edu.hk (Q.S. Li).

0143-974X/$ - see front matter © 2005 Elsevier Ltd. All rights reserveddoi:10.1016/j.jcsr.2005.07.003

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et al. [18] to establish stiffness reductions of a structudue to damage. Their study showed that the selection oappropriate class of models is critical for successful damdetection. A general strategy was proposed by BernalGunes [3] for discussing where information on the locatiof the damage was first extracted by interrogating chanin synthesized flexibility matrices of a structure and a moupdate was used to quantify the damage. A least squapproach was adopted by Caicedo et al. [5] to determine thestiffness of a structure and possible damage to the structwas detected by an iterative approach for determiningstiffness of the damaged structure.

The idea of using dynamics characteristic datastructural parameter identification and damage detecis especially attractive because it allows for a gloevaluation of the condition of a structure. A newly developmethodology called the finite element (FE) model updamethod [14] emerged rapidly in this field. Through the Fmodel updating methodology, the differences in structparameters (including physical, material propertiesdynamics characteristics of a structure) between abaseline model and an updated FE model can be determ

232 J.R.Wu, Q.S. Li / Journal of Constructional Steel Research 62 (2006) 231–239

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based on measured data. Thus, exact values of strucparameters can be obtained and the corresponding locaand quantities of damage states may be detectedexamining the changes of structural or modal parametRepresentative papers on this topic include [8] and [23].

Various system identification and damage detectimethods have been proposed and applied to diffestructures, making side-by-side comparison of the methdifficult. Therefore, there is a need to apply newlydeveloped methods to a well-known benchmark structo examine the performance of the methods. Following tidea, a two-stage eigensensitivity-based FE model updaprocedure is developed in this paper for structural paramidentification and damage detection for Phase II ofIASC-ASCE benchmark steel frame structure which wspecially built for the purpose of investigating structuhealth monitoring (SHM). The procedure developincludes two parts. In the first stage, the connection stiffnof beam–column joints of the steel structure is identifiedthe FE model updating method. In the second stage, throcomparing the differences of the dynamics characterisbetween the damaged and undamaged structure, the csection areas of the braces are obtained by the FE mupdating methodology. In this way, the locations of tdamaged braces can be identifiedand quantifications carrieout. It will be shown that the procedure developed iseffective method for structural parameter identification anddamage detection for steel frame structures.

2. Introduction on the benchmark steel structure

The structure considered in the present study is phII of the IASC-ASCE benchmark steel frame structuIt is a four-story, two-bay by two-bay steel frame scastructure as shown inFig. 1 [13]. This structure wasbuiltby the Earthquake Engineering Research Laboratory aUniversity of British Columbia (UBC). It is 2.5 m× 2.5 min plan and 3.6 m in height. The members of the structuare made of hot rolled grade 300W steel (normal yield st300 MPa). The columns are B100× 9 sections and thefloor beams are S75× 11 sections. The frame is bracewith two 12.7 mm diameter threaded steel rods placedparallel along the diagonal. The beam–column joints arebolt connected and the support conditions of the benchmstructure are regarded as fully rigid supported. Four splates are attached to each floor to represent the mathe structure. Each plate on floors 1 to 3 has a nominalmass of 1000 kg, and the mass on floor 4 is 750 kg.placement of these plates is identical on each floor, wmasses being distributed asymmetrically so that the motare coupled. A total of 16 unaxial accelerometers wplaced on the structure to measure its responses for allcases and damage scenarios. Each floor was equippedthree sensors, two of which measured accelerations innorth–south direction at opposite sides of the structure,third measuring the east–west accelerations. An examp

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Fig. 1. Steel frame scaled benchmark structure.

a configuration of mass and accelerometer distributionthe first floor is shown inFig. 2 which was downloadedfrom webpagehttp://wusceel.cive.wustl.edu./asce.shm/. Adetailed description of the benchmark problem,experiment data records and more related information cafound at the websitehttp://wusceel.cive.wustl.edu/asce.sh.A three-dimensional FE baseline model for the benchmsteel structure was established in this study basedthe previous 120-degrees-of-freedom (DOFs) FE moobtained by Johnson et al. [21].

Extensive research work has been conducted to intigate the structural healthmonitoring of the IASC-ASCEbenchmark steel structure. A two-stage approach wasployed by Yuen et al. [28] to investigate the SHM problemand a Bayesian model updating method was used to dmine the most probable values of the stiffness parameand the probability that damage in substructure exceespecified level. A Hilbert–Huang-based approach was deoped by Yang et al. [27] for damage detection for the bencmark structure. The time variation of the frequency was uto detect the instant of the damage and the damage loction. A wavelet-based approach was used by Hera and[15] for damage diction and locating damage regions forbenchmark structure.

J.R.Wu, Q.S. Li / Journal of Constructional Steel Research 62 (2006) 231–239 233

.

Fig. 2. Configuration of the mass and accelerometer distributions in the first floor

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A literature review revealed that there are certdifferences in natural frequencies of the benchmstructure, especially for the first four modes, betweenresults from thefinite element method analysis and frothe ambient vibration tests [13]. Therefore, this presenan important issue: how to update the FE model usexperimental results sothat the predicted modal parametematch the measurements. In the benchmark steel structhe floor beams are connected with the flanges ofcolumns by strong-axis connections in the south–ndirection, and weak-axis connected with the columns inwest–east direction. Therefore, the connection stiffness fothe beam–column joints in the benchmark structure shohave different values in two separate directions. Especin the west–east direction, the connection stiffnessthe beam–column joints should be smaller than thatfully rigid connected joints. Actually all the beam–columjoints in the benchmark structure should be regardedsemi-rigid connection joints with a certain connectstiffness. However, in the previous studies for the benchmastructure, they were modeled as fully rigid connectedtaking an arbitrary value for the connection stiffness. Inthis paper, taking the benchmark structure as an examan integrated two-stage FE model updating procedis developed for identification of structural paramet(including the connection stiffness of beam–column joinand damage detection for steel frame structures.

Although FE model updating is being actively studin various areas, studies on FE model updating of cstructures using ambient vibration measurements are sel

,

reported. So it is necessary to conduct comprehenstudies on this topic.

3. Ambient vibration test of the benchmark steel framestructure

The ambient vibration test of the benchmark steel stture was conducted at the University of British Columon August 2002 [13]; the test results can be extracted frothe website athttp://wusceel.cive.wustl.edu/asce.shm[17].A series of tests were conducted for the structure withious damage scenarios. In the tests, damage was simuby removing braces from the structure or by looseningbolts at beam–column connection joints. Only the dampatterns with braces removed on the eastern side of the sture are considered in this paper. The various test casedescribed inTable 1. In the first test specified as Case 1, thenominal structure was configured with all braces in plaOther damaged cases investigated in this study are alsoin Table 1. Thenatural frequencies of the first five vibratiomodes identified for Cases 1–5 from the ambient vibratest are listed inTable 2. Their corresponding mode shapwereobtained in this study.

4. Finite element baseline model of the benchmark steelstructure

A three-dimensional FE baseline model for the benmark steel structure was established in this study. In the

234 J.R.Wu, Q.S. Li / Journal of Constructional Steel Research 62 (2006) 231–239

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Table 1Damage patterns investigated in this study

Case no. Description

Case 1 Undamaged structure. All braces presentCase 2 All braces on the east side are removedCase 3 Braces on the south half of the east side are removedCase 4 Braces on the first and fourth floor of the south half of the

east side areremovedCase 5 Braces on the first floor of the south half of the east side

removed

Table 2Identified natural frequencies (Hz) of the first five modes, based on thambient test

Case Mode 1 Mode 2 Mode 3 Mode 4 Mode 5

Case 1 7.495 7.764 14.475 19.898 21.025Case 2 5.653 7.732 12.742 15.018 20.132Case 3 6.647 7.653 13.475 18.905 20.026Case 4 7.348 7.603 13.974 19.713 20.126Case 5 7.452 7.560 14.028 19.895 20.632

baseline model, the initial Young’s modulus of the strutural members was selected as 210 N/mm2. The steel beamsand columns are assumed to be connected at the juncof their centroidal axis. Due to the panel zone effectbeam–column connection joints, the equivalent lengthbeams and columns for determining the element stiffnesmatrix may be smaller than the length between the centroidaaxes. The initial equivalent length coefficients (the ratiothe equivalent length to the length between the centroidaxes) for all structural members were all set to 1. The iniconnection stiffness of the beam–column semi-rigid joiin the strong-axis direction of the column members is takto be1020 N m/rad, and 6/33 × 1020 N m/rad was takenfor the weak-axis direction,according to Ventura et al. [25].The element stiffness matrix considering semi-rigid connection joints was formed based on the method proposedHo and Chan [16], and Chan and Chui [6]. Table 3 liststhe difference in natural frequencies of the first five modbetween the results from the ambient test and those fromFE baseline model. The natural frequencies obtained frothe FE baseline model are those of the undamped systwhile the natural frequencies identified from the ambient teare actually the natural frequencies of the damped sysAs reported by Yuen et al. [28], the identified fundamental damping ratios along the twomajor axes of the structurare about 1.0%. According to Clough and Penzien [10], thenatural frequencies of the damped system can be calcuusing the equationωD = ω

√1 − ξ2 (whereωD, ω, ξ are the

damped natural frequency, undamped natural frequency andamping ratio, respectively). If the damping ratio is 1.0the relative difference betweenωD andω is lessthan 0.0001,which is consistent with the observation made by Li et[20] and Li [19]. Obviously the damping effect is negligble. Therefore, the natural frequencies of the damped sy

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identified from the ambient test can be directly used forFE updating procedure without further modifications. Tnatural frequencies calculated from the FE baseline mowhich are listed in the third row inTable 3, show that if allthe beam–column joints are regarded as rigidly connectlarge differences exist between the computational resand the ambient measurementsfor the first three vibrationmodes. This implies that the initial values of the physiproperties and/or parameters, and the initial connection snesses of the semi-rigid joints may deviate from their acvaluesto some extent. Therefore, the first stage of this studwill focus on the structural parameter identification for tbenchmark structure via the FE model updating approac

5. Selection of updating parameters in the FE modelupdating procedure

In FE updating procedures, it is often assumedmeasured natural frequencies are more reliablemeasured mode shapes [26]. Thus, the measured naturfrequencies of the first five vibration modes of tbenchmark structure were selected as the tuned mparameters, while the tuning of the measured mode shwas incorporated into the constraint condition at eaiteration step in the present FE model updating procedAs thephysical dimension of the beam and column membin the benchmark steel structure are specially designedthis scaled model, their physical properties (cross-secarea, inertia moment, polar moment of inertia of the crosectional area) can be obtained with much more accuthan their material properties. As illustrated from the secto sixth rows in Table 3, the natural frequencies of thfirst three vibration modesobtained from the FE modelare much smaller than those from the ambient test eif all the beam–column joints were assumed to be fulrigidly connected. This implies that the value of Youngmodulus (E) adopted in the FE baseline model mayunderestimated. Actually the beam–column joints shobe treated as neither fully rigid nor hinge connectioTherefore, semi-rigid joints should be adopted forbeam–column connections in the present FE model updaprocedure. On the other hand, the equivalent length forbeam and/or column members will be somewhat differfrom the length between the junctions of centroidal axesthe beams and columns. Hence, in this study, six strucparameters were selected as the updating parametersfirst stage of the FE model updating procedure forfully braced and undamaged configuration of the benchmstructure. They are the Young’s modulus(E) of the beamand column members, the connection stiffness ofbeam–column joints in the strong-axis(S1) and weak-axis(S2) directions of the column members, the equivallength coefficient for the beams in the south–north direc(C1) and the east–west direction(C2), and the equivalentlength coefficient for the column members(C3). Once theabove six updating parameters had been determined in

J.R.Wu, Q.S. Li / Journal of Constructional Steel Research 62 (2006) 231–239 235

1%

45%

Table 3Difference of natural frequencies (Hz) of the first five modes between the ambient test and the FE baseline model

Case Mode 1 Mode 2 Mode 3 Mode 4 Mode 5Case 1 7.495 7.764 14.475 19.898 21.025

Initial baseline modelCalculated results 6.301 6.703 11.310 18.103 19.70Difference from Case 1 15.93% 13.67% 21.86% 9.01% 6.29

FE model assuming all beam–column joints fully rigidly connectedCalculated results 6.702 7.201 11.803 19.201 21.20Difference from Case 1 10.56% 7.7% 18.22% 3.6% 0.37

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first stage of the FE model updating procedure, they wregarded as fixed values in the second stage. Moreoanother eight parameters representing the cross-sectionof eight braces installed in the eastern side of the benchmsteel structure were chosen as the updating parametethe second stage of the FE model updating procedureCases 2–5.

6. Methodology of the eigensensitivity-based FE modelupdating

Among various FE model updating approaches,eigensensitivity-based finite element model updating pcedure has been recognized as an effective approacimproving FE models as it selects thegeometric and/ormaterial properties of the FE structural models as uping parameters and it can be easily implemented in mFEM-based codes. For the sensitivity-based approachthe term “sensitivity” refers to the rate of change of themodal parameters with structural parameters. Examof sensitivity-based approaches include those develoby Collins et al. [11] and Chen and Garba [7]. Forthe eigensensitivity-based FE model updating meththe relationship between the perturbation in the updatparameters and the difference between the measuredand calculation results from theFE model can be representeby a sensitivity matrix [14]:

δR = SδP (1)

where δP = P − P(i ) is the vector representing thperturbation in the updating parameters, in whichP andP( j )

are updated and current vectors of the updating paramerespectively;δR = R(m)−R( j ) is the vector of the differencebetween the measured data and the computational refrom the FE model, in which R(m) andR( j ) are vectors ofthe measured and calculation data in the current iteratrespectively;S is the sensitivity matrix whose entries candetermined by the following formula:

Si j = ∂Ri

∂P j(2)

in which Ri is the i -th component of the modal vectoand P j is the j -th component of the updating paramevector.

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Table 4The lower and upper limits of the updated parameters for Cases 1–5 (ware represented by the ratios between the updated and initial values)

FE model Updating Lower Upperupdating cases parameters limits limits

Case 1

E −0.02 0.15S1 −0.15 1.0S2 −0.97 1.0C1 −0.10 0C2 −0.10 0C3 −0.15 0

Cases 2–5 C1–C8 −1.0 0

6.1. Formulation of the sensitivity matrix

A formula for eigenvalue derivation with respectupdating parameters was proposed by Jung [22] throughdifferentiating the eigenequation(Kϕ = λMϕ) of astructural system with respect to updating parameters.derived formula for natural frequencies is as follows:

∂λr

∂ Pi= {ϕr}T

a∂[K]∂ Pi

{ϕr}a − λra{ϕr}Ta∂[M]∂ Pi

{ϕr}a (3)

where ∂λr∂ Pi

is the notation for the sensitivity of ther -theigenvalue(λr ) with respect to updating parameter(Pj );λra is the currentr -th analytical eigenvalue;[K] and [M]are current analytical global stiffness and mass matrof the structural system, respectively;{ϕr}a is the currentr -th analytical mode shape which is normalized to the masmatrix [M].

6.2. Weighting matrix for measured modal data aupdating parameters

In ambient tests, higher natural frequencies are oobtained with less accuracy than the lower order oTherefore, a weighting matrixWR, whose entries aroften obtained from the reciprocals of the variance ofcorresponding modal data [14], is introduced in the FEmodel updating algorithm. As the ambient tests forbenchmark steel structure were conducted in laboraconditions, the weighting entries for the first five natufrequencies for all the FE updating cases considered instudy are all selected with the same value of 100.

236 J.R.Wu, Q.S. Li / Journal of Constructional Steel Research 62 (2006) 231–239

od

5

Table 5The structural parameters of the benchmark structure identified via the FE model updating procedure for Case 1

FE model updating case Structural parameters identified Identified values ofstructural parametersWeighted least squares method Bayesian estimation meth

Case 1

E 230 N/mm2 230 N/mm2

S1 1.5455× 1020 N m/rad 1.763× 1020 N m/radS2 1.8259× 1019 N m/rad 1.942× 1019 N m/radC1 0.9 0.9C2 0.9 1.0C3 1.0 1.0

Table 6The natural frequencies (Hz) of the first five modes via the FE model updating for Case 1 and the corresponding MAC values

Case Mode 1 Mode 2 Mode 3 Mode 4 Mode 5Ambient test results for Case 1 7.495 7.764 14.475 19.898 21.02

Updated FE model for Case 1(Bayesian estimation method)

Updated results 7.384 7.968 13.100 21.257 23.466Difference from the ambient test results −1.48% 2.83% −9.51% 6.84% 11.63%MAC value 0.821 0.912 0.955 0.851 0.935

Updated FE model for Case 1(Weighted least squares method)

Updated results 7.289 7.976 13.050 21.002 23.469Difference from the ambient test results −2.75% 2.73% −9.86% 5.55% 11.65%MAC value 0.818 0.917 0.955 0.853 0.923

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Meanwhile, some updating parameters can be estimwith more accuracy than others. The weighting matrixupdating parametersWP can be incorporated into the Fmodel updating procedure in the same manner as that omeasured modal data. However, sometimes it is difficulobtain the statistical variance of both the measured mdata and updating parameters; an alternative solutionset an arbitrary value according to their confidence levelslarger weighting value is used for updating parameters wa lowerconfidence level, and vice versa.

6.3. Estimation methods for updating parameters

The measured natural frequencies of the first fivibration modes of the benchmark steel structure wselected as modal data to be tuned in the FE model updaprocedure, while the updating parameters are physparametersin which E, S1, S2, C1, C2, C3were selectedfor FE updating for Case 1, and eight physical paramerepresenting the cross-section area of the eight braces ieastern side of the structure were adopted for the updaCases 2–5. Therefore, the number of equations is less ththat of the updating parameters for all the updating caseonly the weighting matrix of the updating parametersWP isconsidered, the best estimation for the updating paramecan be obtained through the weighted least squares meIn this way, the solution for the simultaneous Eq. (1) can beobtained by considering a constrained optimization probas follows:

Minimize δPTWPδP subject toδR = SδP. (4)

Its corresponding solution is

δP = W−1P ST(SW−1

P ST)−1δR. (5)

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If both the weighting matricesWP andWR are included,the best estimation of the updating parameters canobtained by the Bayesian estimation technique [14,12].The associated FE model updating procedure canregarded as seeking the solution of the following constraioptimization problem:

Minimize (δR − SδP)TWR(δR − SδP) + δPTWPδP

subject toδR = SδP. (6)

The corresponding solution can be obtained bymethod developed by Dascotte et al. [12]:

δP = W−1P ST(W−1

R + SW−1P ST)−1δR. (7)

6.4. Bounds of the updating parameters and convergecriteria

In order to avoid the updated results being physicameaningless, the lower and upper limits for the updatparameters are necessarily set in the FE model updaprocedure; these are listed in Table 4. The convergencecriteria were also set in each iteration loop as follows:

| f (k)a − fe| ≤specified limit of naturalfrequency difference (8

MAC (ai , ei )i=1,5 ≥ 0.8 (9)

BL ≤ P(k) ≤ BU (10)

where f (k)a , fe are the current analytical and correspond

experimental values of the natural frequency, respectivBL, BU are the lower and upper limits of the updatinparameters, respectively; MAC(ai , ei )i=1,5 are the modalassurance criterion (MAC) indices for between the

J.R.Wu, Q.S. Li / Journal of Constructional Steel Research 62 (2006) 231–239 237

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computational and experimental mode shapes, whindicate how well the FE mode shapes fit the correspondmeasured ones. The formula for obtaining a MAC indexas follows [14]:

MAC (ai , ei ) =

(n∑

j =1φa

j i φej i

)2

n∑j =1

(φaj i )

2n∑

j =1(φe

j i )2

(11)

in which φaj i , φ

ej i are the j -th coordinates of thei -th

analytical and measured mode shapes, respectively;n is thetotal degree of freedom of the structural system.

Once all the conditions listed in Eqs. (8)–(11) aresatisfied, the iteration process ends, and the final FE mupdated results are obtained.

7. Two-stage FE model updating procedure for thebenchmark steel structure: Structural parameter iden-tification and damage detection

Based on the eigensensitivity-based FE model updatprocedure described in the previous sections, a two-sFE model updating methodology is developed and appto the benchmark steel structure for structural paramidentification and damage detection. Case 1 and Cases 2described inTable 1were considered in this study, and somselected results will be presented in the following.

7.1. Stage I: Identification of the connection stiffness ofbeam–column semi-rigid joints

As is well known, the concept of the semi-rigidbeam–column joint has been accepted in the designanalysis of steel structures. However, how to determthe exact value of the connection stiffness of a semi-ribeam–column joint is still a problem which has not befully solved. Although there are many existing experimenresults and empirical formulasavailable for predicting theconnection stiffness of semi-rigid beam–column joints [9],they were mainly concerned with standard connection types.The most reliable method for obtaining the connectstif fness of beam–column joints is conducting experimentests of prototype joints. However, such tests are usutime-consuming and expensive. It is believed that tidentification of the connection stiffness of semi-rigid jointhrough the FE model updating procedure is an alternaand effective approach.

Therefore, in the first stage of the FE model updatprocedure, Case 1 was considered to identify the probstiffness values for semi-rigid beam–column joints. Thweighted least squares and Bayesian estimation metoutlined in Eqs. (5) and (7) were adopted for estimating thupdating parameters. The final updated structural paramE, S1, S2, C1, C2andC3 are presented inTable 5, while the

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Table 7The cross-section areas of braces identified with different damage patteof the benchmark structure via the FE model updating procedure

FEmodel Brace no Updated cross-section areas of braces(m2)

updating Updated Updated Detected Actualcase value value/ damaged damage

(10−3 m2) original state statevalue (yes/no) (yes/no)

Case 2

1 0 0 Yes Yes2 0 0 Yes Yes3 0 0 Yes Yes4 0 0 Yes Yes5 0.102 0.723 No Yes6 0 0 Yes Yes7 0 0 Yes Yes8 0.032 0.22 No Yes

Case 3

1 0 0 Yes Yes2 0 0 Yes No3 0 0 Yes Yes4 0.141 1 No No5 0.141 1 No Yes6 0.141 1 No No7 0 0 Yes Yes8 0.141 1 No No

Case 4

1 0.1410 1 No Yes2 0.1410 1 No No3 0.1410 1 No No4 0 0 Yes Yes5 0.1410 1 No No6 0.1410 1 No No7 0 0 Yes No8 0.1410 1 No No

Case 5

1 0 0 Yes Yes2 0.1410 1 No No3 0 0 Yes No4 0.1410 1 No No5 0.1410 1 No No6 0.1255 0.891 No No7 0.1158 0.821 No No8 0.1258 0.892 No No

dynamic characteristics of the updated FE model and Mvaluesare listed inTable 6.

It is shown fromTable 5 that the updated results fromthe weighted least squares estimation method and Bayeestimation methods are very similar. If the connectstiffness of beam–column joints is infinite, the beam–columjoint is fully rigidly connected. The updated results listin Table 5 show the actual values of the connectiostiffnesses of the beam–column semi-rigid joints. The ratof connection stiffness of the beam–column joints inweak-axis direction of the column(S2) to that in the strong-axis direction (S1) is about 0.18259/1.5455 = 0.118(obtained from the weighted least squares method)0.1942/1.763 = 0.110 (determined from the Bayesiaestimation method). Experimental results for similar serigid joints [25] showed that the corresponding ratio w6/33 = 0.1818, which wasa little greater than thoseobtained from the FE model updated results. The satisfac

238 J.R.Wu, Q.S. Li / Journal of Constructional Steel Research 62 (2006) 231–239

55

Table 8The natural frequencies (Hz) of the first five modes for FE model updating for Cases 2–5 and the corresponding MAC values

Case No Mode 1 Mode 2 Mode 3 Mode 4 Mode 5

Case 2

Ambient test 5.653 7.732 12.742 15.018 20.132Updated results 5.890 7.932 14.125 16.230 21.320Difference from the ambient test results 4.19% 2.589% 10.85% 8.07% 5.91%MAC value 0.873 0.943 0.834 0.901 0.945

Case 3

Ambient test 6.647 7.653 13.475 18.905 20.026Updated results 6.8786 7.4228 12.2789 20.9415 21.52Difference from the ambient test results 3.44% −2.97% −8.91% 10.74% 7.52%MAC value 0.9723 0.9725 0.9728 0.9249 0.9399

Case 4

Ambient test 7.348 7.603 13.974 19.713 20.126Updated results 7.209 7.458 12.605 20.973 22.064Difference from the ambient test results −1.91% −1.88% −9.77% 6.41% 9.66%MAC value 0.9051 0.8616 0.9710 0.9946 0.9922

Case 5

Ambient test 7.452 7.560 14.028 19.895 20.632Updated results 7.108 7.571 12.626 21.002 22.578Difference from the ambient test results −4.59% 0.14% −10.0% 5.59% 9.44%MAC value 0.8573 0.8279 0.9752 0.9978 0.9857

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agreement between the FE model updated results anexperiment data from Ventura et al. [25] illustrates thatthe FE model updating method is effective and reliableidentification of the connection stiffness of the semi-rigjoints.

7.2. Stage II: Damage detection for the benchmark sstructure

In the ambient tests for Cases 2–5, different dampatterns were simulated by removing some or all brafrom the eastern side of the structure. For the undamapattern (Case 1), there are eight braces arranged oneastern side. For clarity of description in the followinsections, these braces are specified as Nos 1–8. Therethe damaged pattern of Case 2 is represented as that wbraces with Nos 1–8 are removed, while four braces (No3, 5 and 7) are removed to represent Case 3, two br(Nos 1 and 4) are removed for Case 4, and onlybrace (No 1) is removed to represent the damaged paof Case 5. In the second stage of the present FE mupdating procedure, the structural parameters identifiethe first stage by the Bayesian estimation method wadopted as the initial values. The updating parameters uconsideration in the second stage were the eight parammentioned previously, including the cross-section areathe above-mentioned braces in the eastern side. Ifupdated value for one brace is less than 10% of its acvalue forthe cross-section area, this brace is assumed tdamaged, which is represented by removing the brace fthe original undamaged structure. The Bayesian estimamethod outlined in Eq. (7) was adopted for estimatinthe eight updating parameters for Cases 2–5 describeTable 4. The final updated structural parameters whichthe cross-section areas of the eight braces in the easternare listed inTable 7. Meanwhile, the dynamic characteristi

e

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s

nl

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e

and MAC values for the updated FE models correspondto Cases 2–5 are listed inTable 8.

From the updated results inTable 7, the damaged bracewith Nos 1–4, 6–7 removed were detected in Case 2.damaged braces with Nos 1, 3 and 7 removed in Casthe damaged brace with No 4 removed in Case 4the damaged brace with No 1 removed were successfuldetected in the second stage of the FE model updaprocedure. Therefore, it can be concluded that the locatand quantifications of the damaged braces for Casescan be effectively detected for most cases by the medeveloped. However, there are some discrepancies betthe detected results and the actual damage patterns, as sin the fifth and sixth columns inTable 7. This demonstratedthat the present FE model updating procedure could balternative and effective method for damage detectionstructural health monitoring for steel frame structures.

8. Conclusion

A integrated two-stage eigensensitivity-based FE modupdating method was developed in this paperidentification of the structural parameters and damadetection for Phase II of the IASC-ASCE benchmasteel frame structure based on ambient vibration tesIdentifications of the connection stiffnesses of beam–columsemi-rigid joints and other structural parameters weconducted in the first stage of the FE model updaprocedure. Comparisons with the experimental resshowed that the identified connection stiffness valuesbeam–column semi-rigid joints were accurate. Furthermordamage detection for the benchmark steel structureconducted in the second stage of the FE model upprocedure; it was found that the damage patterns deteby the FE model updating procedure were consistent w

J.R.Wu, Q.S. Li / Journal of Constructional Steel Research 62 (2006) 231–239 239

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the actual damage situations for most cases considerethis study. The results from the two-stage FE model updatprocedure and the comparisons with the experimentaldemonstrated thatthe eigensensitivity-based FE modupdating methodology developed in this paper is an effecmethod for structural parameter identification and damagdetection for steel frame structures.

Acknowledgement

The work described in this paper was fully supporteda grant from the Research Grant Council of Hong KoSpecial Administrative Region, China (Project No CityU1093/02E).

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