Identification of dynamic models of Metsovo (Greece) Bridge using ambient vibration measurements

Transcription

1 Loughboough Univesity Institutional Repositoy Identification of dynamic models of Metsovo (Geece) Bidge using ambient vibation measuements This item was submitted to Loughboough Univesity's Institutional Repositoy by the/an autho. Citation: PANETSOS, P.... et al, 009. Identification of dynamic models of Metsovo (Geece) Bidge using ambient vibation measuements. IN: Papadakakis, M., Lagaos, N.D., Fagiadakis, M. (eds). nd ECCOMAS Thematic Confeence on Computational Methods in Stuctual Dynamics and Eathquake Engineeing (COMPDYN 009), Rhodes, Geece, -4 June, pp Additional Infomation: This is a confeence pape. Metadata Recod: Vesion: Accepted fo publication Publishe: National Technical Univesity of Athens, Geece ( c The authos) Please cite the published vesion.

2 This item was submitted to Loughboough s Institutional Repositoy ( by the autho and is made available unde the following Ceative Commons Licence conditions. Fo the full text of this licence, please go to:

3 COMPDYN 009 ECCOMAS Thematic Confeence on Computational Methods in Stuctual Dynamics and Eathquake Engineeing M. Papadakakis, N.D. Lagaos, M. Fagiadakis (eds.) Rhodes, Geece, 4 June 009 IDENTIFICATION OF DYNAMIC MODELS OF METSOVO (GREECE) BRIDGE USING AMBIENT VIBRATION MEASUREMENTS Panagiotis Panetsos 1, Evaggelos Ntotsios, Nikolaos-Aggelos Liokos, Costas Papadimitiou 1 Egnatia Odos S.A. Capital Maintenance Depatment, Themi 57001, Geece ppane@egnatia.g Univesity of Thessaly Depatment of Mechanical Engineeing, Volos 38334, Geece entotsio@uth.g, nickliokos@yahoo.com, costasp@uth.g Keywods: Bidges, Model Updating, Stuctual Identification, Multi-Objective Optimization, Stuctual Dynamics. Abstact. Available methods fo stuctual model updating ae employed to develop high fidelity models of the Metsovo bidge using ambient vibation measuements. The Metsovo bidge, the highest bidge of the Egnatia Odos Motoway, is a two-banch balanced cantileve avine bidge. It has a total length of 357m, a vey long cental span of 35m, and a height of 110m fo the talle pie. Ambient vibation measuements ae available duing diffeent constuction phases of the bidge. Opeational modal analysis softwae is used to obtain the modal chaacteistics of the bidge. The modal chaacteistics ae then used to update two model classes of finite element models of the bidge. These model classes ae based on beam and solid elements. A multi-objective stuctual identification method is used fo estimating the paametes of the finite element stuctual models based on minimising the modal esiduals. The method esults in multiple Paeto optimal stuctual models that ae consistent with the measued modal data and the modal esiduals used to measue the discepancies between the measued modal values and the modal values pedicted by the model. Single objective stuctual identification methods ae also evaluated as special cases of the poposed multi-objective identification method. The effectiveness of the updated models fom the two model classes and thei pedictive capabilities ae assessed. It is demonstated that the Paeto optimal stuctual models may vay consideably, depending on the fidelity of the model classes employed and the size of measuement eos.

4 1 INTRODUCTION Stuctual model updating methods have been poposed in the past to econcile mathematical models, usually discetized finite element models, with expeimental data. The estimate of the optimal model fom a paameteized class of models is sensitive to uncetainties that ae due to limitations of the mathematical models used to epesent the behavio of the eal stuctue, the pesence of measuement and pocessing eo in the data, the numbe and type of measued modal o esponse time histoy data used in the econciling pocess, as well as the noms used to measue the fit between measued and model pedicted chaacteistics. The optimal stuctual models esulting fom such methods can be used fo impoving the model esponse and eliability pedictions [1], stuctual health monitoing applications [-5] and stuctual contol [6]. Stuctual model paamete estimation poblems based on measued data, such as modal chaacteistics (e.g. [-5]) o esponse time histoy chaacteistics [7], ae often fomulated as weighted least-squaes poblems in which metics, measuing the esiduals between measued and model pedicted chaacteistics, ae build up into a single weighted esiduals metic fomed as a weighted aveage of the multiple individual metics using weighting factos. Standad optimization techniques ae then used to find the optimal values of the stuctual paametes that minimize the single weighted esiduals metic epesenting an oveall measue of fit between measued and model pedicted chaacteistics. Due to model eo and measuement noise, the esults of the optimization ae affected by the values assumed fo the weighting factos. The model updating poblem has also been fomulated in a multi-objective context [8] that allows the simultaneous minimization of the multiple metics, eliminating the need fo using abitay weighting factos fo weighting the elative impotance of each metic in the oveall measue of fit. The multi-objective paamete estimation methodology povides multiple Paeto optimal stuctual models consistent with the data and the esiduals used in the sense that the fit each Paeto optimal model povides in a goup of measued modal popeties cannot be impoved without deteioating the fit in at least one othe modal goup. Theoetical and computational issues aising in multi-objective identification have been addessed and the coespondence between the multi-objective identification and the weighted esiduals identification has been established [9-10]. Emphasis was given in addessing issues associated with solving the esulting multi-objective and single-objective optimization poblems. Fo this, efficient methods wee also poposed fo estimating the gadients and the Hessians [11] of the objective functions using the Nelson s method [1] fo finding the sensitivities of the eigenpopeties to model paametes. In this wok, the stuctual model updating poblem using modal esiduals is fomulated as single- and multi-objective optimization poblems with the objective fomed as a weighted aveage of the multiple objectives using weighting factos. Theoetical and computational issues ae then eviewed and the model updating methodologies ae applied to update two diffeent model classes of finite element models of the Metsovo bidge using ambient vibation measuements. Emphasis is given in investigating the vaiability of the Paeto optimal models fom each model class. MODEL UPDATING BASED ON MODAL RESIDUALS ( k) ( ) 0 Let { ˆ, ˆ k N D R, 1,, m, k 1,, N } be the measued modal data fom a D stuctue, consisting of modal fequencies ( ) ˆ k and modeshape components measued degees of feedom (DOF), whee m is the numbe of obseved modes and ˆ( k ) at N 0 N D is

5 the numbe of modal data sets available. Conside a paameteized class of linea stuctual N models used to model the dynamic behavio of the stuctue and let R be the set of fee stuctual model paametes to be identified using the measued modal data. The objective in a modal-based stuctual identification methodology is to estimate the values of the paamete N set so that the modal data { ( ), ( ) d R, 1,, m }, whee N d is the numbe of model DOF, pedicted by the linea class of models best matches, in some sense, the expeimentally obtained modal data in D. Fo this, let ( ) ˆ ( ) and ( ) ˆ ( ) L ( ) ˆ ˆ 1,, m, be the measues of fit o esiduals between the measued modal data and the model pedicted modal data fo the -th modal fequency and modeshape components, espectively, whee z z z is the usual Euclidean nom, and ( ) ˆT L ( ) / L ( T ) is a nomalization constant that guaanties that the measued modeshape (1) ˆ at the measued DOFs is closest to the model modeshape ( ) L ( ) pedicted by the paticula value of. N0 Nd The matix L R is an obsevation matix compised of zeos and ones that maps the N d model DOFs to the N 0 obseved DOFs. In ode to poceed with the model updating fomulation, the measued modal popeties ae gouped into two goups [10]. The fist goup contains the modal fequencies while the second goup contains the modeshape components fo all modes. Fo each goup, a nom is intoduced to measue the esiduals of the diffeence between the measued values of the modal popeties involved in the goup and the coesponding modal values pedicted fom the model class fo a paticula value of the paamete set. Fo the fist goup the measue of fit J ( ) 1 is selected to epesent the diffeence between the measued and the model pedicted fequencies fo all modes. Fo the second goup the measue of fit J ( ) is selected to epesents the diffeence between the measued and the model pedicted modeshape components fo all modes. Specifically, the two measues of fit ae given by m m J ( ) ( ) and J ( ) ( ) () The afoementioned gouping scheme is used in the next subsections fo demonstating the featues of the poposed model updating methodologies..1 Multi-objective identification The poblem of identifying the model paamete values that minimize the modal o esponse time histoy esiduals can be fomulated as a multi-objective optimization poblem stated as follows [8]. Find the values of the stuctual paamete set that simultaneously minimizes the objectives y J( ) ( J1( ), J ( )) (3) subject to paamete constains, whee low uppe ( 1,, N ) is the paamete vecto, is the paamete space, y ( y1,, yn) Y is the objective vecto, Y is the objec- 3

6 tive space and and low uppe ae espectively the lowe and uppe bounds of the paamete vecto. Fo conflicting objectives J ( ) and 1 J ( ) thee is no single optimal solution, but athe a set of altenative solutions, known as Paeto optimal solutions, that ae optimal in the sense that no othe solutions in the paamete space ae supeio to them when both objectives ae consideed. The set of objective vectos y J ( ) coesponding to the set of Paeto optimal solutions is called Paeto optimal font. The chaacteistics of the Paeto solutions ae that the esiduals cannot be impoved in one goup without deteioating the esiduals in the othe goup. The multiple Paeto optimal solutions ae due to modelling and measuement eos. The level of modelling and measuement eos affect the size and the distance fom the oigin of the Paeto font in the objective space, as well as the vaiability of the Paeto optimal solutions in the paamete space. The vaiability of the Paeto optimal solutions also depends on the oveall sensitivity of the objective functions o, equivalently, the sensitivity of the modal popeties, to model paamete values. Such vaiabilities wee demonstated fo the case of two-dimensional objective space and one-dimensional paamete space in the wok by Chistodoulou and Papadimitiou [9].. Weighted modal esiduals identification The paamete estimation poblem is taditionally solved by minimizing the single objective J( ; w) w1 J1( ) w J ( ) (4) fomed fom the multiple objectives J i( ) using the weighting factos w i 0, i 1,, with w1 w 1. The objective function J( ; w ) epesents an oveall measue of fit between the measued and the model pedicted chaacteistics. The elative impotance of the esidual eos in the selection of the optimal model is eflected in the choice of the weights. The esults of the identification depend on the weight values used. Conventional weighted least squaes methods assume equal weight values, w1 w 1. This conventional method is efeed heein as the equally weighted modal esiduals method. The single objective is computationally attactive since conventional minimization algoithms can be applied to solve the poblem. Howeve, a sevee dawback of geneating Paeto optimal solutions by solving the seies of weighted single-objective optimization poblems by unifomly vaying the values of the weights is that this pocedue often esults in cluste of points in pats of the Paeto font that fail to povide an adequate epesentation of the entie Paeto shape. Thus, altenative algoithms dealing diectly with the multi-objective optimization poblem and geneating unifomly spead points along the entie Paeto font should be pefeed. Fomulating the paamete identification poblem as a multi-objective minimization poblem, the need fo using abitay weighting factos fo weighting the elative impotance of the esiduals J i( ) of a modal goup to an oveall weighted esiduals metic is eliminated. An advantage of the multi-objective identification methodology is that all admissible solutions in the paamete space ae obtained. Special algoithms ae available fo solving the multi-objective optimization poblem. Computational algoithms and elated issues fo solving the single-objective and the multi-objective optimization poblems ae biefly discussed in the next Section. 4

7 3 COMPUTATIONAL ISSUES IN MODEL UPDATING The poposed single and multi-objective identification poblems ae solved using available single- and multi-objective optimization algoithms. These algoithms ae biefly eviewed and vaious implementation issues ae addessed, including estimation of global optima fom multiple local/global ones, as well as convegence poblems. 3.1 Single-objective identification The optimization of J( ; w ) in (4) with espect to fo given w can eadily be caied out numeically using any available algoithm fo optimizing a nonlinea function of seveal vaiables. These single objective optimization poblems may involve multiple local/global optima. Conventional gadient-based local optimization algoithms lack eliability in dealing with the estimation of multiple local/global optima obseved in stuctual identification poblems [9,13], since convegence to the global optimum is not guaanteed. Evolution stategies (ES) [14] ae moe appopiate and effective to use in such cases. ES ae andom seach algoithms that exploe bette the paamete space fo detecting the neighbohood of the global optimum, avoiding pematue convegence to a local optimum. A disadvantage of ES is thei slow convegence at the neighbohood of an optimum since they do not exploit the gadient infomation. A hybid optimization algoithm should be used that exploits the advantages of ES and gadient-based methods. Specifically, an evolution stategy is used to exploe the paamete space and detect the neighbohood of the global optimum. Then the method switches to a gadient-based algoithm stating with the best estimate obtained fom the evolution stategy and using gadient infomation to acceleate convegence to the global optimum. 3. Multi-Objective Identification The set of Paeto optimal solutions can be obtained using available multi-objective optimization algoithms. Among them, the evolutionay algoithms, such as the stength Paeto evolutionay algoithm [15], ae well-suited to solve the multi-objective optimization poblem. The stength Paeto evolutionay algoithm, although it does not equie gadient infomation, it has the disadvantage of slow convegence fo objective vectos close to the Paeto font [8] and also it does not geneate an evenly spead Paeto font, especially fo lage diffeences in objective functions. Anothe vey efficient algoithm fo solving the multi-objective optimization poblem is the Nomal-Bounday Intesection (NBI) method [16]. It poduces an evenly spead of points along the Paeto font, even fo poblems fo which the elative scaling of the objectives ae vastly diffeent. The NBI optimization method involves the solution of constained nonlinea optimization poblems using available gadient-based constained optimization methods. The NBI uses the gadient infomation to acceleate convegence to the Paeto font. 3.3 Computations of gadients In ode to guaantee the convegence of the gadient-based optimization methods fo stuctual models involving a lage numbe of DOFs with seveal contibuting modes, the gadients of the objective functions with espect to the paamete set has to be estimated accuately. It has been obseved that numeical algoithms such as finite diffeence methods fo gadient evaluation does not guaantee convegence due to the fact that the eos in the numeical estimation may povide the wong diections in the seach space and convegence to the local/global minimum is not achieved, especially fo intemediate paamete values in the vicinity of a local/global optimum. Thus, the gadients of the objective functions should 5

8 be povided analytically. Moeove, gadient computations with espect to the paamete set using the finite diffeence method equies the solution of as many eigenvalue poblems as the numbe of paametes. The gadients of the modal fequencies and modeshapes, equied in the estimation of the gadient of J( ; w ) in (4) o the gadients of the objectives J i( ) in (3) ae computed by expessing them exactly in tems of the modal fequencies, modeshapes and the gadients of the stuctual mass and stiffness matices with espect to using Nelson s method [1]. Special attention is given to the computation of the gadients and the Hessians of the objective functions fo the point of view of the eduction of the computational time equied. Analytical expessions fo the gadient of the modal fequencies and modeshapes ae used to ovecome the convegence poblems. In paticula, Nelson s method [1] is used fo computing analytically the fist deivatives of the eigenvalues and the eigenvectos. The advantage of the Nelson s method compaed to othe methods is that the gadient of the eigenvalue and the eigenvecto of one mode ae computed fom the eigenvalue and the eigenvecto of the same mode and thee is no need to know the eigenvalues and the eigenvectos fom othe modes. Fo each paamete in the set this computation is pefomed by solving a linea system of the same size as the oiginal system mass and stiffness matices. Nelson s method has also been extended to compute the second deivatives of the eigenvalues and the eigenvectos. The fomulation fo the gadient and the Hessian of the objective functions ae pesented in efeences [11, 17]. The computation of the gadients and the Hessian of the objective functions is shown to involve the solution of a single linea system, instead of N linea systems equied in usual computations of the gadient and N N 1 linea systems equied in the computation of the Hessian. This educes consideably the computational time, especially as the numbe of paametes in the set incease. 4 APPLICATION TO METSOVO BRIDGE 4.1 Desciption of Metsovo bidge and instumentation The new unde constuction avine bidge of Metsovo (Figue 1, Novembe 008), in section 3. (Anthohoi tunnel-anilio tunnel) of Egnatia Motoway, is cossing the deep avine of Metsovitikos ive, 150 m ove the ivebed. This is the highe bidge of Egnatia Motoway, with the height of the talle pie M equal to 110 m. The total length of the bidge is 357 m. As a consequence of the stong inequality of the heights of the two basic pies of the bidge, Μ and M3 (110 m to 35 m), the vey long cental span of 35 m, is even longe duing constuction, as the pie M balanced cantileve is 50 m long, due to the eccentical position of the key segment. The key of the cental span is not in midspan due to the diffeent heights of the supestuctue at its suppots to the adjacent pies (13.0 m in pie Μ and m in pie M3) fo edistibuting mass and load in favo of the shot pie M3 and thus elaxing stong stuctual abnomality. The last was the main eason of this bidge to be designed to esist eathquakes fully elastic (q facto equal to 1). The longitudinal section of the left banch of Metsovo avine bidge is shown in Figue. The bidge has 4 spans of length m / m /35.00 m/ m and thee pies of which Μ1, 45 m high, suppots the boxbeam supestuctue though pot beaings (movable in both hoizontal diections), while Μ, Μ3 pies connect monolithically to the supestuctue. The bidge was being constucted by the balanced cantileve method of constuction and accoding to the constuctional phases shown in Figues 3(a) and 3(b). The total width of the deck is m, fo each caiageway. The supestuctue is limited pestessed 6

9 of single boxbeam section, of height vaying fom the maximum 13.5 m in its suppot to pie M to the minimum 4.00 m in key section. Figue 1: Geneal view of Metsovo avine bidge. Figue : Longitudinal section of Metsovo avine bidge of Egnatia Motoway. The pie M3 balanced cantileve was instumented afte the constuction of all its segments and befoe the constuction of the key segment that will join with the balanced cantileve of pie M (Figue 3). The total length of M3 cantileve was at the time of its instumentation 15 m while its total height is 35 m. Pies Μ, Μ3 ae founded on huge cicula Ø1.0 m ock sockets in the steep slopes of the avine of the Metsovitikos ive, in a depth of 5 m and 15 m, espectively. 7

10 Figue 3: Views of unde constuction Metsovo avine bidge, (a) Geneal view, (b) key of cental span. Six uniaxial acceleometes wee installed inside the box beam cantileve M3 of the left caiageway of Metsovo avine bidge. The acceleomete aays, ae shown in Figue 4. Due to the symmety of the constuction method (balanced cantileveing) and as the same numbe of segments wee completed on both sides of pie M3, the instumentation was limited to the ight cantileve of pie M3, following two basic aangements. Accoding to the 1 st aangement two () sensos wee suppoted on the head of pie M3, one measuing longitudinal and the othe tansvese acceleations (Μ3L, Μ3T), while the emaining fou (4) acceleometes wee suppoted on the ight and the left intenal sides of the box beam s webs, two () distant 46m and two () distant 68m fom M3 axis, espectively (LV3, RV4, LV5, RV6). All fou wee measuing vetical acceleation. Accoding to the nd aangement the last two sensos of the 1 st aangement wee fixed in a section nea the cantileve edge, distant 93m fom M3 axis, while the othe fou emain in the same positions. In both aangements the sixth senso was adjusted to altenatively measue both in vetical and in tansvese hoizontal diection (RV6 o RT6). RV6_1 RΤ6-1 LV5_1 RΤ6 RV6 LV5 RV4 LV3 Μ3Τ Μ3L 5ος 19ος 14ος σπόνδυλ σπόνδυλ σπόνδυλ ος 6 ος 6 6 ος 4 ' RV6_1 RΤ6 RV6 RV LV5_1 ' LV5 LV3 Μ3Τ RΤ6_ m m 93 m Figue 4: Acceleomete installation aangements. 1 Μ3L. 3 m 0.93 m 8

11 4. Modal identification The esponse of the cantileve stuctue subjected to ambient loads such as the wind, and loads induced by constuction activities such as the cossing of light vehicles placing the pestessing cables inside the tendon tubes, was as expected of vey low intensity (0,6% of the acceleation of gavity). Fom acceleation esponse time histoies, measued fom the 6 channel aays, the coss powe spectal density (CPSD) functions wee estimated and used to estimate the modal popeties using the Modal Identification Toolbox (MITool) [18] developed by the System Dynamics Laboatoy in the Univesity of Thessaly. All the basic modal fequencies, modal damping atios and modeshape components of the bidge wee identified. The identified values of the modal fequencies, thei type and the coesponding values of the damping atios ae shown in Table 1. The modal damping atios epoted in Table 1 show that the values of damping ae of the ode of 0.% to.9%. Due to the small numbe of available sensos (six) the type of some of the modeshapes wee not identified with confidence. The fist six identified modeshapes ae pesented in Figue 5. The aows ae placed in measuing points and thei length is popotional to the espective value of the nomalized modal component. The accuacy in the estimation of the modal chaacteistics is shown in Figue 6 compaing the measued with the modal model pedicted CPSD. As it is seen, the fit of the measued powe spectal density is vey good which validates the effectiveness of the poposed modal identification softwae based on ambient vibations. No Identified Modes Hz Damping % FEM (beam) FEM (solid) 1 1 st otational, z axis st longitudinal st tansvese nd longitudinal st bending (deck) nd tansvese tansvese nd bending (deck) nd otational, z axis d bending (deck) d tansvese st otational, x axis th bending (deck) Table 1: Identified modes of the Metsovo avine bidge 9

12 10

13 Figue 5: The fist six identified modeshapes of Metsovo bidge. 10 Channel: 1 Channel: Channel: Modal Fit Figue 6: Compaison between measued and modal model pedicted CPSD. 11

14 4.3 Updating of finite element model classes Initially, thee diffeent analytical dynamic models of the bidge cantileve M3 wee constucted, using Eule beam elements epesenting mateial and geomety of the stuctue, as consideed by the design. Fo bidge modeling the softwae package COMSOL Multiphysics [19] was used. Thee dimension Eule beam finite elements wee used fo the constuction of this model, coincided with the axis connecting the centoids of the deck and pie sections. Fo bette gaphical epesentation of the highe modeshapes, on the measued points of the bidge cantileve (positions of sensos), additional igid tansvese extensions of no mass wee added to both sides of its centoid axis. Fo epesenting igid connection of the supestuctue to the pie, igid elements of no mass wee used. The beam model shown in Figue 7 has 306 degees of feedom. Pie tansvese webs wee simulated by beam elements accoding to design dawings. Fo compaison puposes, Table 1 lists the values of the modal fequencies pedicted by the nominal finite element beam model constucted in the COMSOL Multiphysics softwae. Compaing with the identified modal fequency values it can be seen that the nominal FEM-based modal fequencies fo the beam model ae faily close to the expeimental ones. Figue 7: Dynamic model of cantileve Μ3 of Metsovo bidge with Eule beam finite elements. Next, detailed finite element models wee ceated using thee dimensional solid elements. Fo this the stuctue was fist designed in CAD envionment and then impoted in COMSOL Multiphysics [19] modelling envionment. The models wee constucted based on the mateial popeties and the geometic details of the stuctue. The finite element models fo the bidge wee ceated using thee dimensional tetahedon solid finite elements to model the whole stuctue. The entie simulation and the model updating is pefomed within the COMSOL Multiphysics modelling envionment using Matlab-based model updating softwae designed by the System Dynamics Laboatoy of the Univesity of Thessaly. In ode to investigate the sensitivity of the model eo due to the finite element discetization, seveal models wee ceated deceasing the size of the elements in the finite element mesh. The esulted twelve finite element models consist of 6683 to tiangula shell elements coesponding to 3991 to DOF. The convegence in the fist six modefequencies pedicted by the finite element models with espect to the numbe of models DOF is given in Figue 8. Accoding to the esults in Figue 8, a model of 6683 finite elements having 3991 DOF was chosen fo the adequate modelling of the expeimental vehicle. This model is shown in Figue 9 and fo compaison puposes, Table 1 lists the values of the modal fequencies pedicted by the nominal finite element model. Compaing with the identified modal 1

15 elative eo (%) Panagiotis Panetsos, Evangelos Ntotsios, Nikolaos-Aggelos Liokos and Costas Papadimitiou fequency values it can be seen that the nominal FEM-based modal fequencies fo the solid element model, ae close to the expeimental ones than the values fo the modal fequencies pedicted by the beam model. Repesentative modeshapes pedicted by the finite solid element model ae also shown in Figue 5 fo the fist six modes modal fequency 1 modal fequency modal fequency 3 modal fequency 4 modal fequency 5 modal fequency numbe of degees of feedom x 10 5 Figue 8. Relative eo of the modal fequencies pedicted by the finite element models with espect to the models numbe of degees of feedom. Figue 9. Finite element model of Metsovo bidge consisted of 6683 thee dimensional tetahedon elements. 13

16 The two finite element models of Metsovo bidge wee employed in ode to demonstate the applicability of the poposed finite element model updating methodologies, and point out issues associated with the multi-objective identification. Both models wee paameteized using thee paametes. The paameteization is shown in Figue 10. The fist paamete accounts fo the modulus of elasticity of the deck of the bidge, the second paamete 1 accounts fo the modulus of elasticity of the head of the pie M3 of the bidge, while the thid paamete 3 accounts fo the modulus of elasticity of the pie M3 of the bidge. The nominal finite element model coesponds to values of The paameteized finite element model classes ae updated using lowest five modal fequencies and modeshapes (modes 1 to 5 in Table 1) obtained fom the modal analysis, and the two modal goups with modal esiduals given by (). Figue 10: Paameteization of finite element model classes of Metsovo bidge. The esults fom the multi-objective identification methodology fo both beam and solid element models ae shown in Figue 11. The nomal bounday intesection algoithm was used to estimate the Paeto solutions. Fo each model class and associated stuctual configuation, the Paeto font, giving the Paeto solutions in the two-dimensional objective space, is shown in Figue 11(a). The non-zeo size of the Paeto font and the non-zeo distance of the Paeto font fom the oigin ae due to modeling and measuement eos. Specifically, the distance of the Paeto points along the Paeto font fom the oigin is an indication of the size of the oveall measuement and modeling eo. The size of the Paeto font depends on the size of the model eo and the sensitivity of the modal popeties to the paamete values [18]. Figues 11(b-d) show the coesponding Paeto optimal solutions in the theedimensional paamete space. Specifically, these figues show the pojection of the Paeto solutions in the two-dimensional paamete spaces ( 1, ), ( 1, 3) and (, 3). It should be noted that the equally weighted solution is also computed and is shown in Figue 11. The optimal stuctual models coesponding to the equally weighted (EWM) esiduals methods fo the beam and the solid finite element model classes ae also shown in Figue 9. It can be seen that these optimal models ae points along the Paeto font, as it should be expected. 14

17 It is obseved that a wide vaiety of Paeto optimal solutions ae obtained fo diffeent stuctual configuations that ae consistent with the measued data and the objective functions used. The Paeto optimal solutions ae concentated along a one-dimensional manifold in the thee-dimensional paamete space. Compaing the Paeto optimal solutions, it can be said that thee is no Paeto solution that impoves the fit in both modal goups simultaneously. Thus, all Paeto solutions coespond to acceptable compomise stuctual models tading-off the fit in the modal fequencies involved in the fist modal goup with the fit in the modeshape components involved in the second modal goups. Compaing the two diffeent element models, the solid element model seems to be able to fit bette to the expeimental esults and this is shown in Figue 11(a). Specifically, fo the highe fidelity model class of 6683 solid elements, the Paeto font moves close to the oigin of the objective space. In addition it is obseved that the sizes of the Paeto fonts fo the solid elements model classes educe to appoximately half the sizes of the Paeto fonts obseved fo the beam element model class. These esults cetify, as it should be expected based on the modeling assumptions, that the solid element model classes of 3991 DOF ae highe fidelity model classes than the beam model classes of 306 DOF. Also the esults in Figue 11(a) quantify the quality of fit, acceptance and degee of accuacy of a model class in elation to anothe model class based on the measue data. J Beam Paeto solutions Beam equally weighted Beam nominal model Solid Paeto solutions Solid equally weighted Solid nominal model J Figue 9: Paeto font and Paeto optimal solutions fo the thee paamete model classes in the (a) objective space and (b-d) paamete space. The vaiability in the values of the model paametes fo the beam element model class ae of the ode of 75%, 60% and 10% fo 1, and 3 espectively. It should be noted that the Paeto solutions 16 to 0 fom a one dimensional solution manifold in the paamete space that coespond to the non-identifiable solutions obtained by minimizing the second objective 15

18 function. The eason fo such solutions to appea in the Paeto optimal set has been discussed in efeence [10]. The vaiability in the values of the model paametes fo the solid element model class ae of the ode of 1%, 155% and 43% fo, and 1 3 espectively. Fom the esults in Figue 9(b-d) one should also obseve that the Paeto optimal values of the paametes pedicted by the solid finite element model class ae significant diffeent fom the Paeto optimal values pedicted by the simple beam finite element model class. Thus, the model updating esults and paamete estimates depend highly on the fidelity of the model class consideed. The pecentage eo between the expeimental (identified) values of the modal fequencies and the values of the modal fequencies pedicted by the thee paametes beam element model fo the nominal values of the paametes, the equally weighted solution and the Paeto optimal solutions 1, 5, 10, 15 and 0 ae epoted in Table. Table 3 epots the MAC values between the model pedicted and the expeimental modeshapes fo the nominal, the equally weighted and the Paeto optimal models 1, 5, 10, 15 and 0. It is obseved that fo the modal fequencies the diffeence between the expeimental values and the values pedicted by the Paeto optimal model vay between 0.% and 13.4%. Specifically, fo the Paeto solution 1 that coesponds to the one that minimizes the eos in the modal fequencies (fist objective function), the modal fequency eos vay fom 0.% to 3.9%. Highe modal fequency eos ae obseved as one moves towads Paeto solution 0 since such solutions ae based moe on minimizing the eos in the modeshapes than the eo in the modal fequencies. Howeve, these eos fom the Paeto solutions ae significantly smalle than the eos obseved fo the nominal model which ae as high as 0.1%. The MAC values between the expeimental modeshapes and the modeshapes pedicted by the Paeto optimal model ae vey close to one because of the small numbe of measued DOF available. The MAC values vay between 0.95 and Fo the Paeto solution 0, the lowest MAC value is appoximately Fo the thee paametes solid element model, the pecentage eo between the expeimental values of the modal fequencies and the values of the modal fequencies pedicted by the thee paamete solid element model fo the nominal values of the paametes, the equally weighted solution and the Paeto optimal solutions 1, 5, 10, 15 and 0 ae epoted in Table 4. Table 5 epots the MAC values between the model pedicted and the expeimental modeshapes fo the nominal, the equally weighted and the Paeto optimal models 1, 5, 10, 15 and 0. It is obseved that fo the modal fequencies the diffeence between the expeimental values and the values pedicted by the Paeto optimal model vay between 0.% and 9.%. Specifically, fo the Paeto solution 1 that coesponds to the one that minimizes the eos in the modal fequencies (fist objective function), the modal fequency eos vay fom 0.% to.9%. The ange of vaiability of these eos is smalle than the ange of vaiability of the eos obseved fom the beam model class. Highe modal fequency eos ae obseved as one moves towads Paeto solution 0 since such solutions ae based moe on minimizing the eos in the modeshapes than the eo in the modal fequencies. The MAC values between the expeimental modeshapes and the modeshapes pedicted by the Paeto optimal model ae vey close to one because of the small numbe of measued DOF available. The MAC values vay between 0.94 and 1. Compaing the esults in the Tables to 5 and the Figue 9 between the two model classes, it is evident that the Paeto optimal models fom the solid element model class give bette fit to the measued data than the fit povided by the beam element model class. This veifies that highe fidelity model classes tend to involve less model eo and povide bette fit to the measued quantities. 16

19 Relative fequency eo (%) Mode Nominal Equally Paeto solution model weighted Table : Relative eo between expeimental and beam model pedicted modal fequencies. MAC value Mode Nominal Equally Paeto solution model weighted Table 3: MAC values between expeimental and beam model pedicted modeshapes. Relative fequency eo (%) Mode Nominal Equally Paeto solution model weighted Table 4: Relative eo between expeimental and solid model pedicted modal fequencies. MAC value Mode Nominal Equally Paeto solution model weighted Table 5: MAC values between expeimental and solid model pedicted modeshapes. 17

20 5 CONCLUSIONS Methods fo modal identification and stuctual model updating wee used to develop high fidelity finite element models of the Metsovo bidge using ambient acceleation measuements. A multi-objective stuctual identification method was used fo estimating the paametes of the finite element stuctual models based on minimizing two goups of modal esiduals, one associated with the modal fequencies and the othe with the modeshapes. The constuction of high fidelity models consistent with the data depends on the assumptions made to build the mathematical model, the finite elements selected to model the diffeent pats of the stuctue, the dicetization scheme contolling the size of the finite elements, as well as the paameteization scheme used to define the numbe and type of paametes to be updated by the methodology. The fidelity of two diffeent model classes was examined: a simple model class consisting of a elatively small numbe of beam elements and a highe fidelity model class consisting of a lage numbe of solid elements. The multi-objective identification method esulted in multiple Paeto optimal stuctual models that ae consistent with the measued (identified) modal data and the two goups of modal esiduals used to measue the discepancies between the measued modal values and the modal values pedicted by the model. A wide vaiety of Paeto optimal stuctual models was obtained that tade off the fit in vaious measued modal quantities. These Paeto optimal models ae due to uncetainties aising fom model and measuement eos. The size of obseved vaiations in the Paeto optimal solutions depends on the infomation contained in the measued data, as well as the size of model and measuement eos. It has been demonstated that highe fidelity model classes, tend to involve less model eo, move the Paeto font towads the oigin and educe the size of the Paeto font in the objective space, educe the vaiability of the Paeto optimal solutions, povide bette fit to the measued quantities, and give much bette pedictions coesponding to educed vaiability. ACKNOWLEDGEMENTS This eseach was co-funded 75% fom the Euopean Union (Euopean Social Fund), 5% fom the Geek Ministy of Development (Geneal Secetaiat of Reseach and Technology) and fom the pivate secto, in the context of measue 8.3 of the Opeational Pogam Competitiveness (3 d Community Suppot Famewok Pogam) unde gant 03-ΕΔ-54 (PENED 003). This suppot is gatefully acknowledged. REFERENCES [1] C. Papadimitiou, J.L. Beck, L.S. Katafygiotis, Updating obust eliability using stuctual test data. Pobabilistic Engineeing Mechanics, 16, , 001. [] C.P. Fitzen, D. Jennewein, T. Kiefe, Damage detection based on model updating methods. Mechanical Systems and Signal Pocessing, 1 (1), , [3] A. Teughels, G. De Roeck, Damage detection and paamete identification by finite element model updating. Achives of Computational Methods in Engineeing, 1 (), , 005. [4] M.W. Vanik, J.L. Beck, S.K. Au, Bayesian pobabilistic appoach to stuctual health monitoing. Jounal of Engineeing Mechanics (ASCE), 16, ,

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