Hierarchical Bayesian Model Updating for Structural Identification

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1 Hirarchical Baysian Modl Updaing for Srucural Idnificaion Iman Bhmansh a, Babak Moavni a, Gr Lombar b, and Cosas Papadimiriou c a Dp. of Civil and Environmnal Enginring, Tufs Univrsiy, MA, USA b Dp. of Civil Enginring, KU Luvn, Hvrl, Blgium c Dp. of Mchanical Enginring, Univrsiy of Thssaly, Grc ABSTRACT A nw probabilisic fini lmn (FE modl updaing chniqu basd on Hirarchical Baysian modling is proposd for idnificaion of civil srucural sysms undr changing ambin/nvironmnal condiions. Th prformanc of h proposd chniqu is invsigad for (1 uncrainy quanificaion of modl updaing paramrs, and ( probabilisic damag idnificaion of h srucural sysms. Accura simaion of h uncrainy in modling paramrs such as mass or siffnss is a challnging ask. Svral Baysian modl updaing framworks hav bn proposd in h liraur ha can succssfully provid h paramr simaion uncrainy of modl paramrs wih h assumpion ha hr is no undrlying inhrn variabiliy in h updaing paramrs. Howvr, his assumpion may no b valid for civil srucurs whr srucural mass and siffnss hav inhrn variabiliy du o diffrn sourcs of uncrainy such as changing ambin mpraur, mpraur gradin, wind spd, and raffic loads. Hirarchical Baysian modl updaing is capabl of prdicing h ovrall uncrainy/variabiliy of updaing paramrs by assuming im-variabiliy of h undrlying linar sysm. A gnral soluion basd on Gibbs Samplr is proposd o sima h join probabiliy disribuions of h updaing paramrs. Th prformanc of h proposd Hirarchical approach is valuad numrically for uncrainy quanificaion and damag idnificaion of a 3-sory shar building modl. Effcs of modling rrors and incompl modal daa ar considrd in h numrical sudy. Kywords: Hirarchical Baysian Modl Updaing, Damag Idnificaion, Uncrainy Quanificaion, Coninuous Srucural Halh Monioring, Prdicion Error Corrlaion, Environmnal Condiion Effcs 1. INTRODUCTION Modl updaing chniqus basd on vibraion daa (.g., modal paramrs hav providd promising rsuls for damag idnificaion of civil srucurs. Modal paramrs such as naural frquncis and mod shaps can b accuraly idnifid from ambin vibraion or forcd vibraion ss. Howvr, h formr is mor araciv for opraional full-scal civil srucurs han h lar bcaus h forcd vibraion ss ofn rquir suspnding h srucur s opraion. In addiion, forcd vibraion ss ar no pracical for coninuous srucural halh monioring applicaions. Th fini lmn (FE modl updaing chniqus using h idnifid modal paramrs can ponially prdic h xisnc, locaion, and svriy of damag which is commonly dfind as a chang in h srucurs physical propris [1]. Rviws on vibraionbasd modl updaing and damag idnificaion of srucural sysms hav bn providd in [- 1

2 5]. Th FE modl updaing mhods can b dividd ino wo broad cagoris of drminisic and probabilisic approachs. Th drminisic FE modl updaing mhods ar wll sablishd in h liraur [6-9], wih svral succssful applicaions o civil srucurs [10-16]. Th qualiy of srucural idnificaion rsuls obaind from h drminisic FE modl updaing mhods dpnds on (1 h accuracy and informaivnss of masurd vibraion daa (.g., idnifid modal paramrs, and ( h accuracy of h iniial FE modl. In pracic, h idnifid modal paramrs of opraional srucurs show significan variaions from s o s, spcially if h srucur is bing moniord ovr a long priod of im. Ths variaions can b du o masurmn nois, simaion rrors, and mos imporanly changing nvironmnal/ ambin condiions [17-4]. Modling rrors also add o h simaion uncrainis of idnificaion rsuls spcially for complx civil srucurs ha ar usually modld wih many idalizaions and simplificaions [5, 5-7]. Ths sourcs of variabiliy moivad rsarchrs o incorpora h undrlying srucural uncrainis hrough h probabilisic FE modl updaing approachs, which can idnify h updaing modl paramrs and hir simaion uncrainis. For rliabl and robus srucural halh monioring, i is imporan o provid a masur of confidnc (uncrainy on h damag idnificaion rsuls. Diffrn probabilisic damag idnificaion mhods basd on FE modl updaing hav bn usd in h liraur including Baysian mhods [8-31] and prurbaion basd mhods [3-35]. Filring mhods (.g., Kalman filrs hav also bn applid for onlin paramr idnificaion of srucural sysms basd on masurd inpu-oupu im hisoris [36, 37]. Th availabl Baysian modl updaing framworks can succssfully prdic h simaion uncrainis of h updaing paramrs (.g., srucural siffnss or mass, bu do no considr h inhrn variabiliy of hs paramrs du o diffrn sourcs of uncrainis such as changing ambin mpraur, mpraur gradin, wind spd, and raffic load. This papr implmns h concp of Hirarchical Baysian modling [38-40] o dvlop a nw probabilisic FE modl updaing procdur ha can prdic h oal uncrainy of h updaing modl paramrs, including h paramr simaion uncrainy and mor imporanly h inhrn variabiliy of updaing srucural paramrs. This proposd framwork is xndd for probabilisic damag idnificaion of civil srucurs. Scion rviws h mos commonly usd Baysian modl updaing framwork in h liraur, rfrrd o as classical Baysian modl updaing in his papr. Scion 3 inroducs h proposd Hirarchical Baysian modl updaing procdur. Th prformanc of h proposd mhod for srucural uncrainy quanificaion and damag idnificaion is valuad hrough a numrical applicaion in Scion 4. Th ffcs of modling rrors, incomplnss of modal daa, rror funcion corrlaions, and h numbr of daa ss usd in h updaing procss on h idnificaion rsuls ar invsigad. Finally, Scion 5 provids h conclusions of his work.. CLASSICAL BAYESIAN FE MODEL UPDATING FRAMEWORK.1. Rviw of h Framwork Daild and in-dph rviws on h framwork can b found in [41-43]. Th paprs by Bck [8], Bck and Kaafygiois [9], and Sohn and Law [30] ar h pionring ffors in h probabilisic FE modl updaing using h Baysian infrnc schm. In h pas dcad, his mhod has bn implmnd for idnificaion of svral srucural sysms [7, 31, 44-50].

3 Thr ar also a fw sudis ha applid his Baysian modl updaing procdur on full-scal civil srucurs [51-53]. Basd on h Bays horm h posrior (updad probabiliy disribuion funcion (PDF of h updaing srucural paramrs θ, and h modl rror paramrs, givn a singl daa s D can b xprssd as: p, p, p, whr pd θσ, is h so-calld liklihood funcion and, θσ D D θσ θσ (1 p θσ is h prior probabiliy. A common yp of masurd daa D in srucural idnificaion applicaions includs h idnifid sysm ignvalus (squars of circular naural frquncis and mod shaps. To formula h liklihood funcion, h rror funcions for a mod m ar dfind in Equaions ( and (3, and hy ar assumd o hav zro-man Gaussian disribuions: m m m( θ N 0, ( m Φ m amφm( θ Φ N 0, Σ Φ (3 m m Th idnifid ignvalus and mod shaps ar shown as λ and Φ, rspcivly. Modl calculad ignvalus and mod shaps ar shown by λθ ( and Φθ (. Plas no ha h idnifid mod shaps and h modl-calculad mod shaps a h masurd DOFs ar normalizd o hir uni lngh (i.., uni L norm. In all quaions of his papr, h modl calculad mod shaps Φθ ( conain only h componns a h masurd DOFs. am is h scaling facor of mod m and is s o b h do produc of h wo uni-normalizd mod shaps T Φ m. Φm( θ. Th liklihood funcion can b wrin as Equaion (4 by assuming ha h idnifid modal paramrs ar saisically indpndn, i.., knowing h valu of any obsrvd modal paramr dos no provid any informaion rgarding h probabiliy of obsrving ohr modal paramrs. Nm N m λφ, θσ,,, (, m θ Φ (, m m θσ Φ m m m θ Φ m m Φm θ ΣΦm p p p N N (4 m1 m1 In his quaion, Nm is h oal numbr of idnifid mods, N ( θ, is h valu of a Gaussian PDF wih h man ( θ and h sandard dviaion a, and similarly N Φ Φ( θ, ΣΦ is h vcor of a mulidimnsional Gaussian PDF wih h man Φθ ( and h covarianc marix Σ Φ a Φ. In mos of h Baysian FE modl updaing applicaions, no corrlaion is considrd for h rror funcions of Equaions ( and (3. In [41, 54], his saisical indpndnc assumpion is rgardd as h indpndncy of informaion and no as an inhrn propry of h sysm. Howvr, as mniond in [55], accouning for h corrlaions bwn h rror funcions can affc h idnificaion rsuls. In h cas of having N indpndn numbr of masurd daa ss, h liklihood funcion of Equaion (4 can b xndd as: 3

4 N Nm λ 1... λ N, Φ1... Φ, (, (, N θ σ m m θ Φ m m Φm θ ΣΦ m p N N (5 1 m1 whr sub-indx m indicas h idnifid modal paramrs of mod m from s. Th posrior probabiliy disribuions of h updaing paramrs can b obaind numrically by gnraing Markov Chain Mon Carlo (MCMC sampls from h posrior PDF [44, 45, 48, 50, 53, 56] or analyically hrough asympoic approximaions [7, 31, 46, 47, 49, 51]. If MCMC sampling chniqus ar usd, h disribuion of h sampls from h posrior PDF can provid a masur of paramr simaion uncrainy. If an asympoic approximaion is usd, h covarianc marix of updaing modl paramrs can b simad as h invrs Hssian of Log p θσ, D a h opimum of h updaing paramrs... Inrpraion of h Esimad Covarianc Marix Alhough h classical Baysian modl updaing procdur has bn succssfully implmnd for idnificaion of modl paramrs and opimal modl class slcion, h simad covarianc marix dos no provid h oal variabiliy of updaing modl paramrs. This covarianc marix only rprsns h paramr simaion uncrainy which will dcras wih incrasing numbr of daa ss [30, 53, 57], rfrrd o as nois miigaion in [31]. Th simad uncrainis of h modl paramrs from a frqunis approach, howvr, can provid h ovrall variabiliy of updaing paramrs and will convrg wih incrasing numbr of daa ss. A frqunis framwork can considr h inhrn variabiliy of h updaing modl paramrs and hus h propris of h corrsponding disribuion can b simad. Saisics of h updaing paramrs in a frqunis approach ar simad from h drminisically idnifid paramrs corrsponding o diffrn s daa [, 58, 59]. In [58], i is shown ha in h absnc of modling rrors h simad man and covarianc marix from h Baysian and h frqunis framworks can b rlad. In applicaions o civil srucurs, h physical modling paramrs such as mass, damping, siffnss, or boundary condiions show variaions du o h changing nvironmnal and ambin condiions [17-3]. For xampl, mpraur variaions affc h srucural boundary condiions and h marial propris, highr wind spd ofn rsuls in rducd ffciv siffnss, or variaions of raffic load (gnrally liv loads will affc h mass of h srucur. Ths variaions will rsul in diffrn modal paramrs from s o s. In ohr words, h N daa ss ha ar usd in Equaion (5 ar collcd from a srucur wih im-varying propris. Th following singl-dgr-of-frdom (SDOF xampl is dsignd o illusra ha h simad sandard dviaion in h classical Baysian approach undrsimas h oal uncrainy of updaing paramrs. Th considrd SDOF sysm consiss of a known drminisic mass and an unknown variabl siffnss ~ N,. Th masurd daa includ N idnifid naural frquncis of h sysm a diffrn ss which is similar o srucural idnificaion applicaions. Basd on h rror funcion of Equaion (, h posrior join probabiliy disribuion of h siffnss and h varianc of h rror funcion, can b xprsss as: 4

5 p (6 1 N 1 1, 1,,..., N xp ( N Th sub-indx rfrs o a singl s and h condiion of h sysm during ha s. No ha in his formulaion, no inhrn variabiliy is considrd for paramr θ. Th mos probabl θ corrsponds o h avrag of h idnifid sysm ignvalus, i.., ˆ m. Th mos probabl varianc of h rror funcion can b calculad as: N ˆ 1 1 (7 N ˆ ( N 1 N 1 In his simpl xampl, h sandard dviaion ˆ can b rlad o h ru sandard dviaion of h sysm siffnss, as shown in Equaion (8. N ˆ N ˆ ˆ (8 N m m m N m Th varianc of h siffnss paramr ˆ can b wrin as: ˆ m ˆ (9 N N which is diffrn from h ru sandard dviaion of θ or h simad sandard dviaion hrough h frqunis approach. In h prsnc of variabiliy in h srucural paramrs, a diffrn approach should b implmnd ha can prdic boh h inhrn variabiliy and h paramr simaion uncrainis of h updaing paramrs. Th Hirarchical Baysian FE modl updaing is proposd in his papr o achiv his goal. 3. HIERARCHICAL BAYESIAN FE MODEL UPDATING Rsarchrs in h social and bhavioral scincs hav invsigad mhods o ra hirarchical daa wih diffrn lvls of variabls in h sam saisical modl. For xampl, h hirarchical daa for sociological survy analysis includ masurmns from individuals wih diffrn hisorical, gographic, or conomic variabls. To his nd, h hirarchical modling was proposd o accoun for h diffrn grouping or ims a which daa ar masurd [40]. Similar analogy can b mad for h collcd masurmns from a srucur undr diffrn ambin and nvironmnal condiions. This framwork has bn rcnly implmnd for an uncrainy quanificaion applicaion in srucural dynamics [60]. In h proposd Hirarchical Baysian modl updaing, h srucural paramrs ar assumd o b disribud according o a probabiliy modl; a runcad Gaussian disribuion (no ngaiv siffnss is assumd in his sudy, i.., θ N μθ, Σ θ. Th rror funcion vcor, dfind as h diffrnc bwn h idnifid and modl-calculad modal paramrs, can b rprsnd as a mulivaria Gaussian disribuion: 5

6 λ N μ, Σ Φ whr is h ignvalu rror vcor of siz Nm, and λ (10 Φ is h mod shap rror vcor of siz Nm Ns. Ns is h numbr of modl shap componns. Ths rrors ar dfind as h following for mod m: m m( θ 1 m Φ a Φ ( θ (1 Φm m m m Givn h rror funcion of Equaion (10, h posrior probabiliy disribuion of h updaing paramrs can b xprssd by Equaions (13 for a singl daa s from s. μθ, Σθ, θ, μ, Σ λ, Φ λ, Φi θ, μ, Σ θ μθ, Σθ μθ, Σθ, μ, Σ p p p p (13 No h diffrncs bwn h rror funcions givn by Equaion (10 and Equaions (-3. Th firs diffrnc is h sub-indx for h srucural paramrs and h idnifid modal paramrs, which spcifis h valus of updaing srucural paramr during h collcion of daa s. Also, h rror funcions ar no assumd uncorrlad and wih zro mans. In h prsnc of modling rrors, prior assumpions for h prdicion rror corrlaions [55] and hir variancs [47, 54] affc h idnificaion rsuls. Thrfor, updaing h prdicion rror paramrs provids a mor robus idnificaion. Th prior probabiliis ar spcifid in wo hirarchical sags of pθ μθ, Σ θ and hypr-prior probabiliy disribuion pμθ, Σθ, μ, Σ. In h cas of having N indpndn daa ss, h join posrior PDF can b sad as N Θ, μθ, Σθ, μ, Σ λ, Φ λ, Φ θ, μ, Σ θ μθ, Σθ μθ, Σθ, μ, Σ p p p p (14 whr N 1 Θ θ,...,,..., 1 θ θ. Th graphical rprsnaion of h proposd Hirarchical Baysian modling is shown in Figur 1. Non-informaiv priors ar assumd for μ θ and p μθ, μ 1. Dpnding on h slcion of updaing srucural paramrs, i is ofn rasonabl o assum no corrlaion bwn hs paramrs and hrfor h covarianc marix Σ θ can b prsnd as a diagonal marix: 1 p Np μ, i.., Σ Diag θ,,...,,..., (15 wih Np = numbr of updaing srucural paramrs in θ. No ha h formulaions can b xndd for corrlad srucural paramrs by updaing all componns of h full covarianc marix Σ. An Invrs Gamma probabiliy disribuion is assumd for h prior probabiliy of h p : θ (11 6

7 p ~ InvrsGamma,1 p (16 whr α and β can b akn idnically for all h updaing srucural paramrs. Th prior probabiliy disribuion of Σ is assumd as Invrs Wishar disribuion [61]: p Σ ~ InvrsWishar I N, N (17 whr N=Nm (Ns+1 is h siz of h rror funcion vcor and is a consan. Basd on h considrd priors, h join posrior probabiliy disribuion of all h updaing paramrs can b sad as in Equaion (18. p Θμ,, Σ, μ, Σ λφ, Σ θ θ 1 1 N N J θ,,,, λ Φ μ Σ N p p p 1 1 xp NN1 N p N 1 p1 p p p1 whr J T 1 θ, λ, Φ, μ, Σ μ Σ μ (19 Th mos common chniqu o solv Equaion (18 is h Gibbs Samplr [6, 63] Esimaion of Posrior Probabiliis Using Gibbs Samplr In h applicaion of Gibbs sampling chniqus, sampls ar gnrad from h full condiional probabiliy disribuion of ach paramr unil convrgnc is rachd. Convrgnc is achivd whn h changs in saisics of gnrad sampls bcoms smallr han a prscribd hrshold. Th full condiional posrior probabiliy disribuions of all h updaing paramrs ar prsnd in Equaions (0 o (4. p p N p p θ μθ, Σθ, μ, Σ, λ, Φ xp J θ, λ, Φ, μ, Σ, 1,..., N p1 p (0 N p N p p p μθ Θ, Σθ, μ, Σ, λ, Φ xp p1 1 p (1 N p N 1 1 Np p p 1 p Σθ Θ, μθ, μ, Σ, λ, Φ xp N p N 1 p1 1 p1 p p ( p1 p (18 7

8 p (3 1 N 1 T 1 μ Θ, μθ, Σθ, Σ, λ, Φxp μ Σ μ N 1 1 T pσ Θ, μθ, Σθ, μ, λ, Φ xp N 1 N r μ Σ μ Σ (4 1 Σ whr r dnos h rac of a marix. Equaions (1 o (4 can b alrnaivly wrin as N 1 1 pμθ. N, N θ Σ θ (5 1 N p N N 1 1. InvrsGamma, p p p (6 1 N 1 1 pμ. N, N Σ (7 1 N N T pσ. InvrsWishar IN μ μ, N N (8 1 Th Invrs Wishard disribuion in Equaion (8 can b jusifid by r-wriing h firs xponnial rm of Equaion (4 as: r N N N T 1 T 1 T r μ Σ μ μ Σ μ μ μ Σ (9 I can b obsrvd ha h full condiional probabiliy disribuions of all h updaing paramrs xcp θ ar sandard disribuions. Th posrior join probabiliy disribuion of updaing paramr can b accuraly simad if an adqua numbr of sampls has bn gnrad. Gnraing sampls from Equaions (5 o (8 is rivial du o hir known disribuion funcions; howvr, gnraing sampls from h condiional probabiliy disribuions of θ, Equaion (0, rquirs using advancd sampling chniqus such as Mropolis-Hasing [64, 65], adapiv Mropolis-Hasings [44, 66], or Translaional Markov Chain Mon Carlo algorihm [48, 56]. I is worh noing ha h θ sampls can b gnrad indpndnly for ach s daa which is idal for paralll compuing o rduc h compuaional im [67]. Alrnaivly, h condiional probabiliy disribuion of θ in Equaion (0 can b approximad as a Gaussian disribuion using Laplac asympoic approximaion o simplify h sampling procss. Th sandard dviaions of h gnrad sampls rflc h paramr simaion uncrainis which will b rducd by incrasing h numbr of daa ss usd in h updaing procss. Th rquird numbr of sampls for convrging o h join probabiliy disribuion of Equaion (18 can bcom vry larg in complx civil srucurs wih a larg numbr of updaing 8

9 srucural paramrs. Thrfor, a simplifid and mor compuaionally fficin procdur will b inroducd in h nx scion for h MAP simaions of h updaing paramrs. 3.. Proposd Simplifid Approach for MAP Esimaions In his subscion, a simplifid procdur is proposd which is basd on h MAP valus of h full condiional PDFs of ach updaing paramr in Equaions (0, (5 o (8. This approach is similar o h Empirical Bays mhod, which can approximaly solv h Hirarchical Bays modls. Th Empirical Bays mhods wr mosly usd bfor h advn of Markov Chain Mon Carlo simulaion chniqus [61]. In his procdur, hr is no nd o draw random sampls from h full condiional probabiliy funcions. This simplifid procdur nglcs h paramr simaion uncrainis and provids h mos probabl posrior probabiliy disribuion of h updaing srucural paramrs (basd on h MAP of μ θ and Σ θ. This procdur is basd on h assumpions ha (1 θ is globally idnifiabl for ach daa s, and ( a larg numbr of daa ss is availabl. Th proposd algorihm has h following sps: (a Sar wih iniial simas for 0 μ ˆ θ, 0 μ ˆ, 0 Σ ˆ and 0 Σ ˆ θ (b A iraion j i. Find h MAP of θ in Equaion (0 givn h j1 μ ˆ θ, ˆ j 1, and Σ : j1 ˆ θ j1 ˆ p p θ μ Σ μ (30 N p jˆ j1 T j1 1 j1 arg min, 1,..., N j1 θ p1 p ii. Find h MAP of μ θ, μ and man of j Σ in Equaions (5, (7, and (8 givn θ ˆ : j μˆ θ N 1 j θ ˆ N 1 (31 j j N 1 j μˆ ˆ (3 N 1 N j j j j T ˆ ˆ ˆ ˆ I N μ μ 1 Σ ˆ (33 N 1 whr j j ˆ ar h prdicion rrors a h opimum θ ˆ valus. iii. Find h MAP of j Σ in Equaion (6 givn θ ˆ j and μ ˆ : θ θ 9

10 j ˆ p 1 N 1 j ˆ j ˆ p p N 1 (c Chck for h convrgnc of h updaing paramrs. Convrgnc is achivd whn h rlaiv changs in μ θ and p bcoms lss han a prscribd hrshold. Th MAP simas from Equaions (30 o (3, and (34 will maximiz h posrior condiional PDFs of Equaions (0, and (5 o (7. MAP sima of h condiional disribuion of Equaion (8 canno b asily compud; hrfor, h man sima is usd in Equaion (33 which should b clos o h MAP if sufficin numbr of daa ss is availabl. Afr convrgnc, h MAP simas of all h updaing paramrs provid h global maxima of h posrior join PDF of Equaion ( APPLICATION TO A NUMERICAL CASE STUDY A hr-sory shar building modl is usd as a numrical cas sudy o valua h prformanc of h proposd Hirarchical Baysian FE modl updaing procdur. This 3-DOF srucur is shown in Figur, whr h mass of ach sory is s o 1. mric on, and h siffnss of soris on o hr ar assumd o b random variabls wih runcad Gaussian disribuions of N(000kN/m, 100 kn /m, N(1000kN/m, 50 kn /m, and N(1000kN/m, 0 kn /m, rspcivly. Th naural frquncis of h building a h man valus of sory siffnsss ar.378 Hz, Hz, and Hz. In Scion 4.1, h prformanc of h proposd Hirarchical Baysian FE modl updaing framwork is valuad and h rsuls ar compard wih hos of h classical Baysian modl updaing framwork. In Scion 4., h proposd framwork is xndd for probabilisic damag idnificaion whr h simad variabiliy/uncrainy of updaing srucural paramrs is propagad in h damag idnificaion rsuls Prformanc Evaluaion of h Hirarchical Baysian FE Modl Updaing Th simulad masurd daa includ 400 ss of naural frquncis and mod shaps which ar gnrad by sampling h sory siffnss valus from hir considrd probabiliy disribuions. Th hisograms of h gnrad naural frquncis ar shown in Figur 3. Ths gnrad modal paramrs will b usd as h masurd daa in h modl updaing procsss. In Scion 4.1.1, h prformanc of h proposd framwork is valuad wih no modling rrors considrd whil in Scion 4.1. h Hirarchical approach is implmnd in h prsnc of modling rrors. Th ffcs of modal daa incomplnss ar sudid in Scion Th proposd simplifid approach of Scion 3. is implmnd for MAP simaions. Th iniial poins for h man and sandard dviaion of h updaing srucural paramrs for all hr soris ar akn as 800 and 1000, rspcivly. Th paramrs and in Equaion (16 ar akn as 1 and, rspcivly. (34 10

11 Compl Modal Daa and No Modling Errors In h absnc of modling rrors, h prdicion rror paramrs will all b zros, sinc hr xiss a θ ha maks h rror funcions of Equaion (11 and (1 zro. Thrfor a opimal θ valus, w can assum ha μ is known and qual o zro, and Σ is a diagonal marix (no corrlaion wih variancs for ignvalu rrors and w variancs for mod shap rrors. w is h considrd raio bwn h mod shap rror and ignvalu rror variancs. Givn hs assumpions, h liklihood funcion bcoms similar o h liklihood funcions ha hav bn commonly usd in h liraur [30, 31, 47, 5-54, 68, 69]. In his cas, h objciv funcion of Equaion (19 will b simplifid o Equaion (35 and h condiional probabiliy disribuion of can b wrin as Equaion (36 if is prior is assumd o b uniform [70]: ( θ 1 J a a m1 m w N m T T m θ, λ, Φ 1 Φ m mφm( θ Φ m mφm( θ NN ( N 1 1 p InvrsGamma J 1 N m s. 1, θ, λ, Φ 11 (35 (36 Th iniial rror sandard dviaion is considrd o b Log 40, and w is considrd as 1/Ns. Svn diffrn subss of daa wih n = {, 5, 0, 50, 100, 00, and 400} daa s numbrs ar usd for modl updaing. Tabl 1 rpors h modl updaing rsuls for h considrd daa subss. Th rsuls includ h MAP simas of h man and sandard dviaion of h hr updaing srucural paramrs and h rror sandard dviaion. Th mans and h sandard dviaions of h hr srucural paramrs ar simad accuraly xcp for h firs wo cass wih n = or n = 5, whr insufficin numbr of daa ss is usd in h updaing procss. Th mans and sandard dviaions of updaing srucural paramrs ar also simad hrough a frqunis approach (i.., minimizing h objciv funcion of Equaion (36 for ach s of daa and mach h valus rpord in Tabl 1. In h absnc of modling rrors and dominan prior assumpions, h Hirarchical and frqunis approachs provid idnical simas for h mans and sandard dviaions of srucural paramrs. Th las column of Tabl 1 shows h sandard dviaion of h rror funcions, which is vry clos o zro implying good machs bwn h updad modl and h daa. This is du o h fac ha no modling rror is considrd in his scion ( is no xacly zro bcaus of numrical round-off rror [71, 7]. As dsird, h saisical propris of h updaing srucural paramrs convrg o hir ru valus by adding mor daa ss. Th mos probabl posrior PDFs of h hr updaing srucural paramrs ar shown in Figur 4 for h cass of using 5, 0, 50, and 400 daa ss. I should b nod ha hs disribuions corrspond o h MAP simas of mans and sandard dviaions, i.., N ˆ, ˆ p. Howvr, h simad mans and sandard dviaions ar also p associad wih uncrainis ha ar no includd in h rsuls rpord in his scion. Th numbr of daa ss rquird for accura simaion of updaing paramrs can b chckd from h convrgnc of h saisics of h masurd daa (i.., idnifid modal

12 paramr. Figur 5 plos h mans and sandard dviaions of h idnifid naural frquncis for incrasing numbr of daa ss. From his plo, i can b concludd ha in h prsnc of h considrd siffnss variaions, approximaly 100 daa ss or mor ar ndd for unbiasd simaions. Such larg numbr of daa ss can b collcd in moniord srucurs in a mar of fw days [10, 16, 3, 53] alhough a longr priod of daa collcion is rcommndd o obsrv h full rang of nvironmnal and ambin variaions. No ha his sp for prdicing h rquird amoun of masurd daa can b prformd indpndnly from h modl updaing. For comparison purposs, h corrsponding modl updaing rsuls from h classical Baysian modl updaing framwork ar also providd in Tabl. Th Adapiv Mropolis- Hasings algorihm of [66] is usd o sampl h posrior probabiliy disribuions of updaing paramrs. In his algorihm, h adapion is prformd on h proposal probabiliy disribuion funcions of h sandard Mropolis-Hasings algorihm. Tabl prsns h simad MAP and paramr simaion uncrainy of updaing srucural paramrs for diffrn numbr of daa ss. Figur 6 also shows h posrior PDF of srucural paramrs in hr cass of n = {5, 50, 400}. From Tabl and Figur 6, i can b sn ha (1 h simad MAP valus ar in good agrmn wih h man simas of h Hirarchical framwork and h xac valus, and ( h paramr simaion uncrainis do no rprsn h oal uncrainy of srucural paramrs and always dcras wih addiion of daa. In addiion, h rror sandard dviaion conains boh h modling rrors and h variabiliy of srucural paramrs. Thrfor, i should no b usd as a masur of goodnss of fi. In h Hirarchical Baysian FE modl updaing, paramr shows h lvl of mismach bwn h modl and h daa du o only h ffcs of modling rrors, i.., h rror sandard dviaion gs clos o zro in h absnc of modling rror vn if h idnifid modal paramrs hav larg variaions. This comparison highlighs h bnfis of h proposd Hirarchical Baysian modl updaing framwork Compl Modal Daa and Considring Modling Errors Th accuracy of FE modl updaing rsuls can b significanly affcd by modling rrors [5-7, 53, 57]. To rprsn h ffcs of modling rrors, h 3-sory shar building is assumd o b on a flxibl bas wih roaional and horizonal springs as shown in Figur 7. Th idnifid modal paramrs ar gnrad from h srucur wih h flxibl bas, bu h FE modl usd in h updaing procss is assumd o b on a rigid bas as in Figur. Siffnsss of h horizonal and roaional springs ar consan and ar assumd o b 0,000 [kn/m] and 00,000 [kn-m/rad]. Th naural frquncis of h sysm a h man sory siffnss valus ar.85hz, 6.387Hz, 8.74Hz, and Hz. In his scion, wo cass of idnificaions ar prformd, (1 basd on h simplifid assumpion ha h rror rms ar uncorrlad wih zro-mans and hrfor, Σ is a diagonal marix (s Subscion 4.1.1, and ( basd on h gnral updaing procss of Scion 3 ha dos no considr any simplifying assumpion for Σ and μ. No ha h paramr in Equaion (17 is akn as Th modl updaing rsuls for h firs cas of idnificaion (uncorrlad rror rms ar providd in Tabl 3 for diffrn numbr of daa ss. From his abl, i is obsrvd ha h man and sandard dviaion of h firs sory s siffnss ar undrsimad whil h saisics of highr sory siffnss valus ar accuraly idnifid. Th bias in h prdicd man of h firs sory siffnss can b du o h compnsaion ffcs of modling rrors. Th simplifid srucural modl of Figur, which is usd in h FE modl updaing procss, canno simula 1

13 h flxibiliy of h srucural bas and hrfor, h siffnss of h firs sory will b undrsimad o compnsa for h bas flxibiliy. Th simad valus ar largr han hos obaind in Tabl 1, implying highr modling rrors. Th undrsimad sandard dviaion of h firs sory siffnss can b du o h fac ha h uncrainis of unmasurd daa (modal paramrs of mod 4 and mod shap componns a h bas for mods 1-3 ar no accound for in h updaing procss. This rror is inviabl in ral-world applicaions whn h masurd daa is incompl and h srucural modls ar discrizd and simplifid. Thrfor in ral-world applicaions, h prdicd uncrainis ar xpcd o b undrsimad. Tabl 4 rpors h idnificaion rsuls whn h full covarianc marix of h rror funcions is considrd as an updaing paramr (idnificaion cas. Th las wo columns show h L norm of h μ ˆ dividd by N (1 in his scion ha rprsn h avrag 1 Log r Σ ˆ / N which is an quivaln masur rror biass a h opimal paramrs, and Log in Tabl 3. I can b sn ha h simad MAP valus ar vry clos o h o h ˆ rsuls of Tabls 3-4 and h avrag rror variancs ar much smallr han hir quivaln valus in Tabl 3. In ohr words, h modl uncrainis can b rducd by considring boh μ and Σ as updaing paramrs. Th updaing srucural paramrs ar also simad hrough a frqunis approach. Th MAP valus ar rpord in Tabl 5, which ar in good agrmn wih h MAP valus from h proposd Hirarchical framwork (Tabl 3 alhough h bias in h man simas is smallr in h Hirarchical updaing procss. Basd on Tabls 3-5, i can b obsrvd ha MAP simas from h Hirarchical Baysian and frqunis approachs ar no idnical bu clos in h prsnc of modling rrors Incompl Modal Daa and Considring Modling Errors In his scion, in addiion o h considrd modling rrors, modl updaing is prformd whn using only h modal paramrs of h firs wo vibraion mods. Similar o Scion 4.1., wo cass of idnificaions ar prformd using (1 uncorrlad zro-man rror rms, and ( corrlad rror rms wih unknown μ and Σ. Tabl 6 shows h rsuls for h firs idnificaion cas, which ar vry similar o h rsuls obaind in prvious subscion, xcp for h undrsimad man and sandard dviaion of. Plas no ha h bias and undrsimaion in h mans and sandard dviaions ar no obsrvd whn using incompl modal daa in h absnc of modling rrors, providd ha h problm is globally idnifiabl. Tabl 7 rpors h scond cas of idnificaion rsuls. Unlik h prvious scion, hs updaing rsuls hav smallr bias compard o hos lisd in Tabl 6, spcially for. 4.. Hirarchical Baysian FE Modl Updaing for Damag Idnificaion Th proposd Hirarchical Baysian FE modl updaing framwork is xndd o b usd for probabilisic damag idnificaion of civil srucurs. In his scion, h posrior probabiliy disribuions of updaing paramrs hav bn simad using h Gibbs Samplr chniqu ha was rviwd in Scion 3.1. Thrfor, h paramr simaion uncrainis ar also availabl 13

14 from h Gibbs sampls. Damag facor (DF of h srucural componn i is dfind as h loss of siffnss from h rfrnc sa o h currn sa. r c i i DF i r i (37 whr suprscrip r rfrs o h rfrnc sa and suprscrip c rfrs o h currn sa of h srucur. Equaion (38 provids h probabiliy of damag xcding a givn damag facor df givn h masurd daa in boh h rfrnc and currn sas of h srucur. r r c ( r c r r c r r c r c 1 1 df i i i Pi i df i, 1 CDF df i, rf i i D D i i r c (38 ( ( i i whr CDF is h cumulaiv Gaussian disribuion funcion, and rf is Gauss Error Funcion. 400 ss of daa ar gnrad from h srucur in h halhy/rfrnc sa and anohr 400 ss of daa ar gnrad from h srucur a h damagd sa. Damag is considrd o b 5% loss of siffnss in h firs sory, i.., h man of h firs sory siffnss is rducd o 1900 [kn/m] whil is sandard dviaion is kp h sam as in h undamagd sa. Th naural frquncis of h damagd srucur a h man valus of sory siffnsss ar.357 Hz, Hz, and Hz. Th naural frquncy of h firs mod is affcd mos by h damag and i is rducd only by 0.9% from h rfrnc sa. This rducion is wihin h variaion rang of h naural frquncis of h srucur in h halhy sa as shown in Figur 3, which maks h damag idnificaion challnging. This scnario is ofn obsrvd in ral-world civil srucurs whr h variaions of modal paramrs du o changing nvironmnal and ambin condiion ar rlaivly larg and snsiiviy of modal paramrs o srucural damag is small [17-3]. As i can b obsrvd from Equaions (5, 6, h man and variancs of h updaing srucural paramrs ar associad wih paramr simaion uncrainis obaind from h Gibbs sampls. Figur 8 shows h posrior probabiliy disribuions of h man and h sandard dviaion of h firs sory siffnss basd on diffrn numbr of daa ss a h undamagd sa. I can b sn ha h simaion uncrainis will b significanly rducd by using mor daa ss in h updaing procss. By incrasing h numbr of daa ss, h posrior probabiliy disribuion of ach paramr bcoms closr o a Dirac Dla funcion a is ru valu in h absnc of masurmn nois and modling rrors. In h prsnc of paramr simaion uncrainis, h updaing srucural paramrs should b dfind as PDFs wih uncrainis in hir simad mans and sandard dviaions. Thrfor, in h damag idnificaion procss, h simaion uncrainis of hs paramrs should b propagad o h damag probabiliis in Equaion (38. Th damag idnificaion rsuls using h daa of h damagd sa (in h prsnc of 5% damag in h firs sory ar shown in Figur 9. This figur shows h damag xcdanc probabiliis for damag facor rang of [ ] in h firs sory using n = 5, 10, 50, 400 daa ss. I is obsrvd ha h mos probabl damag simas a 50% confidnc lvl ar clos o h xac damag valu (5% for diffrn numbr of daa ss whil h variabiliy of damag probabiliy simas dcras drasically wih incrasing numbr of daa ss. Th probabiliy disribuions of h mos probabl damag facors (corrsponding o 50% confidnc lvl considring simaion uncrainis ar shown is Figur 10. Th obaind rsuls undrlin 14

15 h fac ha damag can b accuraly prdicd vn whn h changs in h masurd daa du o damag ar smallr han hos du o nvironmnal condiions providd a sufficin numbr of masurd daa ss is availabl. Th proposd framwork is wll suid for damag assssmn of opraional civil srucurs whr changing nvironmnal condiions can significanly affc h idnifid modal paramrs. I should b nod ha damag can sill b simad basd on only fw masurd daa ss. Howvr, h corrsponding larg simaion uncrainis for h prdicd damags should hn b considrd in any dcision making or dcion analysis [73, 74]. Finally, i can b sn from Figur 9 ha vn wih ngligibl paramr simaion uncrainy (simaion uncrainy ha can b rducd by adding mor daa damag is sill simad probabilisically du o inhrn variabiliy of srucural paramrs [75, 76]. 5. SUMMARY AND CONCLUSION In his papr, a Hirarchical Baysian FE modl updaing procss is proposd for uncrainy quanificaion and damag idnificaion of srucural sysms. Th proposd framwork can prdic h ovrall uncrainis of h updaing paramrs accuraly in h absnc of modling rror and rasonably wll in h prsnc of modling rrors. Th inhrn variabiliy of h srucural mass or siffnss propris is ofn du o changing ambin mpraur, mpraur gradin, wind spd, and raffic load ha can affc h srucural mass or siffnss. This framwork can also sima h posrior disribuion of considrd rror funcions, which rprsn h misfi bwn h daa and h modl prdicd rsponss. Th analyical formulaion of h proposd framwork for modl updaing is prsnd firs and hn wo chniqus ar dscribd for simaing h join posrior probabiliy disribuion of updaing paramrs. Th firs chniqu is basd on a sandard Gibbs samplr whil h scond approach is mor simplifid and provids only h MAP simas of h updaing paramrs. Th prformanc of h proposd Hirarchical Baysian framwork for modl updaing and uncrainy quanificaion is valuad by mans of numrical applicaion o a hr-sory shar building modl. I is shown ha h idnifid saisical propris of updaing paramrs ar h sam as hos obaind using a frqunis approach in h absnc of modling rrors. Howvr, h wo approachs provid similar bu slighly diffrn simas in h prsnc of modling rrors. Th ffcs of modling rrors and modal daa incomplnss ar also invsigad and i is obsrvd ha hs facors will inroduc bias in h simad man and undrsimaion in h sandard dviaions of h updaing srucural paramrs. Thrfor, h prdicd uncrainis ar xpcd o b undrsimad in ral-world applicaions. In h prsnc of modling rrors, wo ss of idnificaions ar prformd using (1 zroman uncorrlad rror funcions, and ( corrlad rror funcions wih unknown mans and corrlaions (usd as updaing paramrs. Updaing h man vcor and covarianc marix of h rror funcions has improvd h idnificaion rsuls in h prsnc of modling rrors. Mor discussion on h opic rquirs furhr invsigaion on mor complicad srucurs using boh numrically simulad and xprimnal daa. In his cas, fficin sampling chniqus ar rquird o b abl o handl non-sandard condiional probabiliy disribuions. Finally, h proposd Hirarchical framwork is xndd for probabilisic damag idnificaion. Th idnifid damags ar associad wih inhrn uncrainis du o h variabiliy of srucural siffnss and paramr simaion uncrainy. I is obsrvd ha h lar can b rducd by using mor daa in h updaing procss. Th obaind rsuls undrlin 15

16 h fac ha in h absnc of modling rrors, damag can b accuraly prdicd vn whn h changs in masurd daa du o damag ar smallr han hos du o nvironmnal/ambin condiions providd a sufficin numbr of masurd daa ss is availabl. Compard o h classical Baysian modl updaing mhods, h proposd framwork is br suid for damag assssmn of opraional civil srucurs whr changing nvironmnal condiions can significanly affc h idnifid modal paramrs. ACKNOWLEDGMENT Th auhors would lik o acknowldg parial suppor of his sudy by h Naional Scinc Foundaion Gran No which was awardd undr h Broadning Paricipaion Rsarch Iniiaion Grans in Enginring (BRIGE program. Th auhors would also lik o acknowldg Mr. Durwood Marshall a h Tufs Tchnology Srvics for his hlp and suppor in using Tufs High-prformanc compuing rsarch clusr. Th opinions, findings, and conclusions xprssd in h papr ar hos of h auhors and do no ncssarily rflc h viws of h individuals and organizaions involvd in his projc. REFERENCES [1] M.I. Friswll, J.E. Morshad, Fini lmn modl updaing in srucural dynamics, Kluwr Acadmic Publishrs, Boson; Dordrch, [] J.E. Morshad, M.I. Friswll, Modl Updaing In Srucural Dynamics - A Survy, J. Sound Vibr., 167 ( [3] H. Sohn, C.R. Farrar, F.M. Hmz, D.D. Shunk, D.W. Sinmas, B.R. Nadlr, J.J. Czarncki, A rviw of srucural halh monioring liraur: , Los Alamos Naional Laboraory Los Alamos, NM, 004. [4] E.P. Cardn, P. Fanning, Vibraion basd condiion monioring: A rviw, Sruc. Halh Moni., 3 ( [5] M.I. Friswll, Damag idnificaion using invrs mhods, Philosophical Transacions of h Royal Sociy A: Mahmaical, Physical and Enginring Scincs, 365 ( [6] K. Alvin, Fini lmn modl upda via Baysian simaion and minimizaion of dynamic rsiduals, Aiaa J., 35 ( [7] J.E. Morshad, M. Link, M.I. Friswll, Th snsiiviy mhod in fini lmn modl updaing: A uorial, Mch. Sys. Signal Proc., 5 ( [8] M. Baruch, Opimal corrcion of mass and siffnss marics using masurd mods, Aiaa J., 0 ( [9] C. Farha, F.M. Hmz, Updaing fini lmn dynamic modls using an lmn-by-lmn snsiiviy mhodology, Aiaa J., 31 ( [10] A. Tughls, G. D Rock, Srucural damag idnificaion of h highway bridg Z4 by FE modl updaing, J. Sound Vibr., 78 (

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22 Tabl 1. MAP simas from h Hirarchical framwork wih no modling rrors θ 1 θ θ 3 - Man ( ˆ STD ( ˆ Man ( ˆ STD ( ˆ Man ( ˆ STD ( ˆ Log ˆ Exac n = n = n = n = n = n = n = Tabl. Updaing rsuls from h classical Baysian framwork MAP θ 1 θ θ 3 - STD ( ˆ MAP STD ( ˆ MAP STD ( ˆ Log ˆ Exac n = n = n = n = n = n = n =

23 Tabl 3. MAP simas from h Hirarchical framwork wih modling rrors; Cas 1 θ 1 θ θ 3 - Man ( ˆ STD ( ˆ Man ( ˆ STD ( ˆ Man ( ˆ STD ( ˆ Log ˆ Exac n = n = n = n = n = n = n = Tabl 4. MAP simas from h Hirarchical framwork wih modling rrors; Cas Man ( ˆ θ 1 θ θ STD ( ˆ Man ( ˆ STD ( ˆ Man ( ˆ STD ( ˆ μ ˆ 1 r Σ ˆ 1 Log 1 Exac n = n = n = n = n =

24 Tabl 5. MAP simas from h frqunis framwork wih modling rrors θ 1 θ θ 3 Man ( ˆ STD ( ˆ Man ( ˆ STD ( ˆ Man ( ˆ STD ( ˆ Exac n = n = n = n = n = n = n = Tabl 6. MAP simas basd on using h firs wo mods in h updaing procss; Cas 1 θ 1 θ θ 3 - Man ( ˆ STD ( ˆ Man ( ˆ STD ( ˆ Man ( ˆ STD ( ˆ Log Exac n = n = n = n = n = n = n = ˆ 3

25 Man ( ˆ Tabl 7. MAP simas basd on using h firs wo mods; Cas θ 1 θ θ STD ( ˆ Man ( ˆ STD ( ˆ Man ( ˆ STD ( ˆ μ ˆ 1 r Σ ˆ 8 Log 8 Exac n = n = n = n = n =

26 Figur Figur 1. Graphical rprsnaion for h proposd Hirarchical Baysian modling 1

27 Figur. Thr-sory shar building modl

28 100 Mod Mod 100 Mod Frquncy [Hz] Figur 3. Hisogram of idnifid naural frquncis 3

29 θ θ Exac θ ,800,000, ,000 1, ,000 1,050 Figur 4. Mos probabl posrior PDFs using Hirarchical Baysian framwork 4

30 .40 Man naural frquncy of 1 s mod 0.0 Sandard dviaions of naural frquncis f 1 f f Man naural frquncy of nd mod Man naural frquncy of 3 rd mod Numbr of daa ss (n Numbr of daa ss (n Figur 5. Convrgnc of h idnifid naural frquncy saisics 5

31 θ θ Exac θ ,800,000, ,000 1, ,000 1,050 Figur 6. Posrior PDFs from h classical Baysian framwork 6

32 Figur 7. Flxibl-bas hr-sory shar modl building 7

33 n = p (μ θ1 D n = 50 p (σ θ1 D n = μ θ σ θ1 Figur 8. Paramr simaion uncrainis for h man and sandard dviaion of θ 1 in h undamagd sa 8

34 P = 50 % P = 50 % P = 50 % P = 50 % Figur 9. Probabiliy of damag givn h baslin daa and h daa in h damagd sa 9