STRUCTURAL FAULT DETECTION OF BRIDGES BASED ON LINEAR SYSTEM PARAMETER AND MTS METHOD

Transcription

1 Joural of JSCE, Vol., 3-43, 03 STRUCTURAL FAULT DETECTION OF BRIDGES BASED ON LINEAR SYSTEM PARAMETER AND MTS METHOD Chul-Woo KIM, Ro ISEMOTO, Kuomo SUGIURA 3 ad Msuo KAWATANI 4 Mmbr of JSCE, Profssor, Dp. of Cvl ad Earh Rsourcs Eg., Koo Uvrs (Koo-Dagau-Kasura, Nsho-u, Koo , Japa E-mal: m.chulwoo.5u@oo-u.ac.jp Sud Mmbr of JSCE, Dp. of Cvl ad Earh Rsourcs Eg., Koo Uvrs (Koo-Dagau-Kasura, Nsho-u, Koo , Japa E-mal: ro.smoo@f8.cs.oo-u.ac.jp 3 Mmbr of JSCE, Profssor, Dp. of Cvl ad Earh Rsourcs Eg., Koo Uvrs (Koo-Dagau-Kasura, Nsho-u, Koo , Japa E-mal: sugura.uomo.4@oo-u.ac.jp 4 Fllow of JSCE, Profssor, Dp. of Cvl Eg., Kob Uvrs (-, Rooda, Nada-u, Kob , Japa E-mal: m-awa@ob-u.ac.jp Ths sud vsgad h fasbl of brdg halh moorg (BHM usg a lar ssm paramr of a m srs modl dfd from raffc-ducd vbrao daa of brdgs, whch daa wr obad hrough a movg-vhcl prm o scald modl brdgs. I ordr o dc possbl aomals brdgs, hs sud adopd a paramr from auorgrssv (AR coffcs. Cosdrao was gv o dagoss of h brdg codo from par chags of dfd ssm paramrs du o damag. Th Mahalaobs-Taguch ssm (MTS was succssfull appld o ma dcsos o h brdg halh codo. Obsrvaos dmosra h fasbl of srucural dagoss of brdgs from h dfd ssm paramr. K Words : brdg halh moorg (BHM, lar ssm modl, Mahalaobs-Taguch ssm (MTS, raffc-ducd vbraos. INTRODUCTION Maag ad mprovg cvl frasrucur, cludg brdg srucurs, ar mpora chcal ssus. Morovr, a ffcv maac srag srogl dpds o ml dcsos o h halh codo of h srucur. I has b hough ha srucural halh moorg (SHM usg vbrao daa s o of h ffcv chologs ha ad dcso mag o brdg maac. Mos prcd suds o SHM spcfcall am h chag modal proprs of srucurs. Th fudamal cocp of hs cholog s ha modal paramrs ar fucos of a srucur s phscal proprs. Thrfor, a chag phscal proprs, such as rducd sffss rsulg from damag, wll chag hs modal proprs -5. A challg for brdg halh moorg (BHM usg vbrao masurms s how o c h brdg coomcall, rlabl ad rapdl. Amb vbraos ducd b raffc ad wd, hus, ar adopd as damc daa for BHM. Howvr, for shor-spa brdgs, whch covr major of brdg srucurs, h wd-ducd vbraos ar usuall oo wa o us h BHM of such brdgs. O h ohr had, raffc-ducd vbraos ar doma for shor-spa brdgs. I s oworh ha h raffc-ducd vbrao of brdgs s a d of osaoar vbrao 6, 7. Dsp h osaoar propr of raffc-ducd vbraos of brdgs, h da bhd usg raffc-ducd vbrao daa of shor-spa brdgs BHM s ha paramrs dfd rpadl udr movg vhcls could provd a par ad gv usful formao o ma a dcso o h brdg codo. 3

2 Joural of JSCE, Vol., 3-43, 03 Ma suds focus o chags ssm frqucs ad srucural dampg cosas for h srucural dagoss of brdgs b ulzg a lar m srs modl 8-5. Howvr, hr s drawbacs modal paramr basd brdg dagoss usg m srs modls;.g., h opmal m srs modl for vbrao rsposs of brdg srucurs usuall comprss a hghr-ordr rm, ad as a rsul h opmal modl dfs v spurous modal paramrs, whch causs fals ssm frqucs ad dampg cosas 6. Thos fals modal paramrs ma dffcul o choos h propr modal paramrs affcd b srucural damag. A rsg approach ulzg a mcro shar for damag dfcao of a sl russ brdg has also b rpord 7. Howvr, h accurac of h damag dfcao ma var accordg o raffc o h brdg. Th drawbac of h classcal mhod s h drvg forc bhd hs sud. Ths sud cosdrd a alrav paramr from auorgrssv (AR coffcs as a damag-ssv faur for h vbrao-basd BHM bcaus boh ssm frquc ad dampg cosa ar rlad o AR coffcs 6, 8. Th Mahalaobs-Taguch ssm (MTS 9, whch s o of h suprvsd larg schms, was also adopd for mag srucural dagoss of brdgs.. AR MODEL AND DAMAGE INDICATOR Th lar damc ssm ca b dfd usg h AR modl 3, 4 as p ( a ( ( ( whr ( dos h oupu of h ssm, a s h -h ordr AR coffc ad ( dcas h os rm. To sma AR paramrs, h auocorrlao fuco of (, whch s obaabl b mulplg ach rm of Eq. ( wh (- ad ag h mahmacal pcao, s usd. Ths procss lds h followg Yul-Walr quao 3, 4, 0 : Ra r ( whr R s a Toplz mar abou R -s = E[(-s(-], whch s h auocorrlao fuco of h sgal; a= [a ; ;a p ] ad r= [R ; ;R p ]; ad p dcas h AR ordr. Th Lvso-Durb algorhm 0 s adopd o solv Eq. (. I s oworh ha h coffc a p s a pol of h ssm bcaus h z-rasformao of Eq. ( ca b wr as Y ( z H ( z E( z p a z - E( z (3 whr Y(z ad E(z ar z-rasformaos of ( ad (, H(z s h rasfr fuco of h ssm h dscr-m compl doma, ad z - dos h forward shf opraor. Valus of z whch h lms of h rasfr fuco mar show f valus ar h pols. Ths mas ha h domaor of h rasfr fuco s h characrsc quao of h damc ssm, gv as p p p z a z a z a z a 0 (4 p p Th pols o h compl pla ar assocad wh h frquc ad dampg cosa of h damc ssm of srucurs, as follows: z p h j h (5 whr h ad ar h dampg cosa ad crcular frquc, rspcvl, of h -h mod of h ssm, ad j rprss h magar u. Th AR procss wh h modl ordr p Eq. ( ca b prssd h z-pla as alrad gv Eq. (3. Th H(z Eq. (3 s dfd as h AR polomal of h modl rasfr fuco rlag h pu o h oupu. Th pols z of Eq. (3 ar obad b fdg h roos of h AR coffc polomal h domaor of H(z. Sc h valus of h coffc of H(z ar ral, h roos mus b ral or compl cojuga pars. Th umbr of pols h z-pla quals h AR modl ordr. Thrfor z show Eq. (5 ad AR coffcs hav h followg rlaoshps accordg o Va s formula : z a ; z z a ; (6, j Eq. (5 ad Eq. (6 show ha AR coffcs ar drcl assocad wh ssm frqucs ad h dampg cosa. Thrfor h paramr from AR coffcs s adopd as a damag-ssv faur ad dfd as 6 a DI j j ; DI a j (7 a whr a dos h -h AR coffc ad DI j s h damag dcaor ha cosdrs up o h j-h AR coffc. Grall, damag srucurs chags h modal paramrs such as ω ad h Eq. (5, ad as a rsul chags z. Morovr, accordg o Eqs. (5 ad (6, AR coffcs ar affcd b damag, ad srucural damag also chags h DI j valus dfd Eq. (7. j 33

3 Joural of JSCE, Vol., 3-43, 03 Fg. Eprmal sup. Fg. Cocp of damag scaro o lf flag. Fg. 3 Phoo of damag scaro D3. Aohr wa o pla h rlaoshp bw AR coffc, crcular frquc ad dampg cosa s dpcd Appd A. 3. LABORATORY MOVING-VEHICLE EXPERIMENT OF MODEL BRIDGE ( Modl brdg A laboraor movg-vhcl prm was prformd o vrf h vald of h proposd approach. Th prmal sup s show Fg.. Th prmal sup comprsd hr smpl bams rprsg h acclrag bam, obsrvao bam ad dclrag bam. Th aural frquc of h brdg was cosdrd as a facor h scalg of h brdg modl. Th brdg modld a sgl-spa brdg wh a spa of 40.4m ad a frs aural frquc of.35hz. Scal roadwa profls wr pavd o boh of h lf ad rgh whl pahs of h vhcl wh a lcrcal ap a h rval of 00mm, as show Fg.. Th hcss of h ap was 0.mm. Th roadwa profl was fac a from h sgl-spa brdg, wh bumps addd o ralz a roughr lvl of road profl. Ths sud cosdrd hr ds of damag: h lowr pars of boh flags bw 3L/8 ad L/ of h modl brdg, a dph of 5mm, wr rmovd for damag scaro D; for damag scaro D, a addoal 5mm cu was appld o h damagd pla of h D scaro ad rmovd; ad for damag scaro D3, aohr 5mm cu was appld o h pla damagd b damag scaro D ad was also rmovd. Th cocp ad phoo of h damag ar show Fg. ad Fg. 3. Th bdg rgd of h modl brdg dcrasd o aroud 34

4 Joural of JSCE, Vol., 3-43, 03 Fg. 4 Acclrao ad Fourr spcra obsrvd a Po I: a Iac; b D; c D; d D3 (SCN, vhcl V a 0.93m/s. Th frs aural frqucs smad from fr vbraos wr.66hz for h ac brdg,.6hz for damag scaro D,.57Hz for damag scaro D ad.5hz for damag scaro D3. Th hrd aural frqucs wr 3.8Hz for h ac brdg, 3.3Hz for D, 3.0Hz for D ad.6hz for D3. I should b od ha, hs fasbl sud, h focus s o vrfg h fasbl of h prs approach. Thrfor, h arfcal damag ps wr o dd o prfcl smula ral damag, bu o ma h brdgs srv as damagd sampls comparso o ac os, rms of bdg rgd rduco. Fg. 5 Road profl ad wavl rasform of acclrao rspos obsrvd a po I of ac brdg udr SCN. Tabl Scaros of laboraor movg-vhcl s. Scaro Vhcl p Spd SCN S=0.93m/s SCN V (M=.6g, f=.93hz S=.6m/s SCN 3 S3=.63m/s SCN 4 S=0.93m/s SCN 5 V (M=.6g, f=3.76hz S=.6m/s SCN 6 S3=.63m/s SCN 7 S=0.93m/s SCN 8 V3 (M=5.8g, f=3.03hz S=.6m/s SCN 9 S3=.63m/s 94% of h ac sa du o damag scaro D. For damag scaro D, h bdg rgd dcrasd o aroud 80% of h ac sa. Damag scaro D3 ld o a dcras h bdg rgd o aroud 65% of h ac sa. ( Modl vhcl Naural frquc ad spd paramr 3 ar cosdrd as facors h scalg of h vhcl modl. Th vhcl modl ca b adjusd o oba dffr damc proprs; h sprg sffss of h als ca b vard b chagg h sprgs whl h bod mass ca b vard usg sl plas. Thr vhcls V, V ad V3 wr cosdrd h prm. Naural frqucs for h bouc moo of h vhcl modls wr.93hz, 3.76Hz ad 3.03Hz rspcvl. Ths frqucs ar clos o 3Hz, whch s h frquc of h bouc moo of acual dump rucs. Thr dffr spds of 0.93m/s (hrafr, S,.6m/s (hrafr, S ad.63m/s (hrafr, S3 wr adopd o vsga h ffc of h vhcl spd o h paramr dfcao. N scaros of h laboraor prm wr cosdrd as show Tabl. Assumg a ral brdg whos frs aural frquc was.35hz ad spa lgh was 40.4m, h spds of h modl vhcls, S, S 35

5 Joural of JSCE, Vol., 3-43, 03 ad S3, corrspodd o.m/h, 7.6m/h ad 38.8m/h rspcvl accordg o h odmsoal spd paramr 3 of Eq. (8: v a (8 fbl whr L dcas h spa lgh (m; f b, h frs aural frquc of h brdg (Hz; ad v, vhcl spd (m/s. Th dmsolss paramr s mpora for h scalg of h prmal modl as s usd o maa a rlaoshp bw vhcl spd ad frquc ad spa lgh of h 5.4m bam, a rlaoshp smlar o ha for a 40.4m brdg subjc o ral raffc. Thr pos a /4, / ad 3/4 of h spa lgh wr obsrvao pos for acclrao rsposs as show Fg.. Th samplg ra was 00Hz. Eampl acclrao rsposs of h modl brdg bfor ad afr applg damag ar show Fg. 4 wh Fourr amplud spcra. Th doma frquc ar 3Hz, whch was assocad wh h hrd bdg mod of h modl brdg, was cd wh a vhcl al h o of h bumps o h road profl. Th road profl ad wavl rasform of acclrao rspos obsrvd a po I of h ac brdg udr scaro (SCN ar show Fg. 5, whch dmosras h doma frquc ar 3Hz wh a vhcl passs ovr h bumps. Th AIC cosss of wo rms. Th frs rm s a log-llhood fuco ad h scod rm s a pal fuco for h umbr of h AR ordr. Fg. 6 shows h AIC vrsus h AR ordr. A rsg obsrvao s ha, as show Fg. 6, h AIC valu a h opmal AR ordr was chagd apparl du o damag. Alhough ds furhr comprhsv vsgao, hr s a possbl ha h AIC could b ulzd as a damag-ssv faur. ( Ssv aalss Cosdrg radom vbraos such as raffc-ducd vbraos of brdg, s o as o dcd whch ordr of AR coffc s h mos ffcv o g a ssv DI j du o damag. Thrfor a ssv aalss was prformd o dcd h mos ssv DI j du o srucural damag 6. Afr sg svral dffr combaos wh h AR coffcs, was foud ha h frs AR coffc ormalzd b h squar roo of h sum of h squars of h frs hr AR coffcs gvs h mos ssv DI j du o damag, as show Fg. 7. Fg. 7 shows h avrag rsduals of DI j bw ac ad damagd brdgs wh rspc o j. Th avrag rsduals wr smad usg 35 obsrvaos from combaos of 9 scaros, 5 prms ad 3 obsrvao pos (9 5 3= STRUCTURAL DIAGNOSIS USING DAMAGE INDICATOR Ths sco focuss o dagoss of brdgs usg DI j dfd Eq. (7, whch was a from AR coffcs. Th opmal ordr of h AR modl of ach prm was slcd b mas of h Aa Iformao Crro (AIC 4, show Eq. (9: AIC N log( ˆ M ( M N (9 whr N dcas h umbr of daa; M, AR ordr; ad, ma squar of M h prdco rror. ˆ M Fg. 6 AIC w.r.. h modl ordr (p=: opmal AR ordr. a b c Fg. 7 Avrag rsduals of DI valus from 35 obsrvaos w.r.. o h ordr of AR coffcs domaor of Eq. (7: a Iac vs. D; b Iac vs. D; c Iac vs. D3. 36

6 Joural of JSCE, Vol., 3-43, 03 S S S3 a Po I b Po II c Po III Fg. 8 Varao DI accordg o dffr pars, spds ad obsrvao pos. Th dffrc DI valus bw ac ad damagd brdgs as a pa wh j as 3, as show Fg. 7. Thrfor hs sud adopd DI 3 as h damag-ssv faur. ( Damag dcaor accordg o svr of damag Ths sud amd h chag DI 3 accordg o damag, obsrvao pos ad vhcl spds as show Fg. 8. Th obsrvd DI 3 valus a po I ad po III udr all vhcl spds wr chagd du o damag. O h ohr had, for h DI 3 obsrvd a po II, a rlavl clar par chag DI 3 du o damag was obsrvd udr vhcl spds S ad S3 comparso wh h DI 3 udr h vhcl spd S. 5. FAULT DETECTION BY MTS METHOD MTS s a par formao cholog, whch s usd dagosc applcaos o ma quaav dcsos b cosrucg a mulvara masurm scal calld Mahalaobs dsac 9 (hrafr MD. Th cocp of MD s show Fg. 9. Th ma objc of h MTS mhod s o ma accura prdcos of MD from uow daa b comparg wh MD obad from ow daa. Thrfor, hs sud, h MTS mhod was adopd for srucural faul dco of brdgs. Th MTS mhod s summarzd as follows. Th u spac, also calld rfrc spac, s obaabl from ow daa, as show Tabl. I Tabl, h colums dca varabls ad h rows, obsrvaos. Th ma valu ad sadard dvao of ach varabl ar appld ordr o ormalz h ow daa as X p p (0 whr dcas h umbr of valuad ms, ad p dcas h umbr of obsrvaos. ad ar dfabl b Eq.(: p p, p ( p Th corrlao coffc mar R MD s obaabl from ormalzd daa as show Eq. (. r r r r R ( MD r r whr h corrlao coffc s dfd as follows: 37

7 Joural of JSCE, Vol., 3-43, 03 r j p p X p X pj (3 X p p Th vrs mar of R MD s dfd as show Eq. (4: - A R MD (4 Obsrvao Obsrvao X pj Tabl U spac (ow daa. Evaluao m ( p Tabl 3 Sgal spac (uow daa. Evaluao m ( p Fg. 9 Cocp of Mahalaobs dsac (MD. Fg. 0 Schm of cross-valdao. Th MD of h ormalzd ow daa h u spac ca b smad from Eq. (4 usg mar A ad ormalzd ow daa: X p MD p X p X p A (5 X p N, h sgal spac s obad from uow daa as show Tabl 3. Uow daa ar ormalzd b ulzg ad, whch ar h ma valu ad sadard dvao of ow daa dscrbd Eq. ( rspcvl: Y p p (6 Th MD of h ormalzd uow daa h sgal spac MD p s dfabl b Eq. (7 from h ormalzd uow daa ad A, whch s obad from ow daa: Yp MD p Y p Yp A (7 Y p Th rqurd codos o ulz MTS ar as follows 9 : a h umbr of valuao ms of ow daa s quval o ha of uow daa; b h umbr of obsrvao daa s largr ha ha of valuao m ; ad c h sadard dvao of ow daa σ s o zro. ( Cross-valdao Cross-valdao was adopd o dcd o a hrshold of h obsrvd paramrs ad o df fauls brdgs. A schm of h cross-valdao s show Fg. 0. A frs, - daa h u spac ar slcd from ow daa ad o ow daa m whch s o slcd as ow daa s assumd o b uow. N, b usg h - ow daa, h MD for h assumd uow s smad b mas of h MTS mhod. Th, aohr daa m s slcd ad assumd o b uow, ad cross-valdao s prformd. Ths sps for cross-valdao ar rpad ms, ad fall MDs h sgal spac ar obad. I hs sud, h largs ad smalls valus of MD a from h cross-valdao wr rmovd, ad h rmmd ma valu was adopd as h hrshold usg (- MD dsacs o rduc h ffc of oulrs o h MDs. ( Faul dco Th MTS mhod was appld for srucural faul dco usg DI 3. Thr valuao ms wr usd: DI 3 obad from obsrvao po I; 38

8 Joural of JSCE, Vol., 3-43, 03 DI 3 obad from obsrvao po II; ad 3 DI 3 obad from obsrvao po III. Th oal umbr of obsrvaos was 5. Rsuls of MTS accordg o vhcl spds ar show Fg., whr h horzoal rd l dcas h hrshold, ad prcags do h probabl of MDs crossg h hrshold. I hs sud, boh prcag ad h ma valu of MDs crossg h hrshold wr cosdrd. Th prcag ad ma valu of MDs crossg h hrshold ar show Tabl 4 ad Tabl 5 rspcvl. I cosdrg h prcag of h MD crossg h hrshold summarzd Tabl 4 ad Fg. (b for D udr vhcl spd S, s hard o rad a clar par chag h MD du o h damag bcaus h prcag of D s smallr ha ha of h cross-valdao. O h ohr had, h MD of D udr vhcl spds S ad S3 (s Fg. a, Fg. c ad Tabl 4 shows appar chags DI 3 du o damag as h prcag s grar ha ha of h cross-valdao. For D ad D3 udr all vhcl spds, bcoms as o dc a faul sc all MDs cd h hrshold. I cosdrg h ma valu of h MDs crossg h hrshold summarzd Tabl 5, h ma valu of D udr vhcl spd S s grar ha ha of h cross-valdao v hough h prcag of crossg h hrshold s lowr ha ha of h cross-valdao as prvousl mod. For D udr vhcl spd S, o h ohr had, h ma valu s smallr ha ha of h cross-valdao v hough h prcag of crossg h hrshold s hghr ha ha of h cross-valdao, as plad Tabl 4. Th obsrvaos dmosra ha damag ca b dcd b h proposd mhod: boh prcag ad ma valu of MDs crossg h hrshold ar ulzd o ma a dcso o h halh codo of brdgs. For svr damag (D ad D3, cosdrg h prcag of MDs crossg h hrshold would b usful for dcso mag. O h ohr had, for lgh damag (D, cosdrg boh prcag ad ma valu of h MDs crossg h hrshold would b usful. Th Mahalaobs dsacs of ssm frqucs ad dampg cosas dfd b mas of h AR modl 4, 6 ar summarzd Appd B for formao. Sc, applg MTS, h umbr of valuao ms has o b ufd across all obsrvaos, frqucs ad dampg cosas hav o b dfd a all hr obsrvao pos whou fal. Howvr, h wr o alwas dfd a vr obsrvao m b mas of h AR modl. Th umbr of smad MDs was, hus, dffr ach brdg codo (Iac, D, D ad D3. Ths plas wh h hrshold could o b drmd Fgs. Bc ad Bc: followg sco 5(, h umbr of obsrvaos cross-valdao bcoms 3, whl ha umbr for h ac cas s 4, whch volas h rqurd codo o ulz h MTS. a b c Fg. Mahalaobs dsac of DI wh hrshold: a S; b S; c S3. Tabl 4 Prcag of MD crossg h hrshold. S S S3 Iac 7% 7% 7% Cross-valdao 33% 47% 47% D 53% 40% 80% D 00% 00% 00% D3 00% 00% 00% Tabl 5 Ma valu of MD crossg h hrshold. S S S3 Iac Cross-valdao D D D

9 Joural of JSCE, Vol., 3-43, 03 I s oworh ha hr s o formao abou h probabl of cdg h hrshold Fg. Bc ad Fg. Bc bcaus of h falur o dcd h hrshold. Th MDs of h dfd ssm frqucs ad dampg cosas dmosra ha h frqucs ad dampg cosas for h frs mod ar ssv o damag (s Fgs. B ad B compard o hos for h hrd mod (s Fgs. B3 ad B4. A oworh po s ha h probabl of succssfull dcg damag dpds o h vhcl spd v h cas of h hrd mod: h oal umbr of succssfull dfd frqucs ad dampg cosas for h hrd bdg mod of damag scaro D3 udr vhcl spd S3 (.63m/s was lss ha h umbr for ohr scaros, whch lads o a smallr umbr of smad MDs for damag scaro D3 apparg Fg. B3c ad Fg. B4c. Obsrvaos from frqucs ad dampg cosas dmosra ha h classcal modal paramrs basd faul dco rqurs dcdg damag-ssv faurs whch should b cosdrd halh moorg, such as mod, frquc ad dampg cosa. I comparso, ulzg DI 3 lads o succssful faul dco. Thrfor h approach combg DI ad MTS mhods provds a ffcv wa of moorg h halh codo of brdgs. 6. CONCLUSIONS Ths papr vsgad h fasbl of srucural faul dco usg raffc-ducd vbrao masurms o a modl brdg hrough a laboraor prm. Ths sud cosdrd h damag dcaor (DI obad from AR coffcs as a damag-ssv faur. Th MTS mhod was appld o suppor dcso mag o brdg halh codos. Th summarzd rsuls ar as follows: AR coffcs ar drcl assocad wh damc characrscs of h srucural ssm. Accordg o h ssv aalss, cosdrg AR coffcs up o h hrd ordr ld o h mos ssv DI for srucural damag h modl brdg cosdrd hs sud. 3 Th DI j would provd a ffcv wa of moorg halh codos of shor-spa brdgs. 4 Th MTS mhod succssfull dcd aomals h modl brdg. 5 Th proposd mhod succssfull dcd svr damag. For lgh damag, howvr, cosdrg boh probabl ad ma valus of h MD crossg h hrshold would b usful mag a dcso o h halh codo of brdgs. 6 I mag a dcso for brdgs halh codo usg h proposd approach, a hghr vhcl spd lads o a br rsul. Ths ma b bcaus h fasr h spd, h bggr h damc rspos, gral, of a brdg, ad bggr rsposs of brdgs could provd mor formao abou brdg codos ha smallr rsposs. Howvr, hs also ds furhr vsgao. 7 As a rsg obsrvao, h AIC could b ulzd as a damag-ssv faur. Th sp for hs sud s o vsga h fasbl of h proposd approach for ral-world applcaos. Th auhors ar ow cosdrg, as a ral-world applcao, faul dco of a ral sl russ brdg b applg arfcal damag. Aohr problm rmag o b solvd s how o dcd h opmal obsrvao pos ad valuao ms, whch ds furhr vsgao. ACKNOWLEDGEMENT: Ths sud s parl sposord b JSPS, Gra--Ad for Scfc Rsarch (B udr projc No Such facal ads ar grafull acowldgd. APPENDIX A Damc quaos of moo for a ssm ca b wr as Eq. (A ad Eq. (A: m c ( c c ( c ( 0 (A ( h0 ( 0 ( 0 (A whr h dos h dampg cosa of h ssm, ad ω 0 s crcular frquc. A gral soluo s ( G s( (A3 0 h (A4 0 h (A5 whr G s a uow cosa ad dcas a uow phas agl. Th m hsor of a damc ssm show Eq. (A ca b modld b h AR procss as a a a (A6 m Usg Eq. (A3, h damc rspos ca b rwr as G s( (A7 G G s{ ( } {s( s( } (A8 40

10 To smplf h problm, w cosdr h AR procss up o h scod ordr: } s( {s( G (A9 s( ( s s( G } s( {s( G (A0 From Eq. (A7, w oba h followg: s( G (A B subsug Eq. (A o Eq. (A9, w ca oba h followg quao: s( s( (A Subsug Eq. (A o Eq. (A0 gvs s( s( ( s ( s ( cos ( s (A3 (A4 Eq. (A4 ca b rwr gral formao as (A5 Comparg Eq. (A5 wh Eq. (A6 of up o h d ordr, a, a (A6 From Eq. (A6, w ca oba followg rlaoshp: a l( (A7 cos a a (A8 Ths also shows ha AR coffcs ar drcl assocad wh damc characrscs of ssms. APPENDIX B a b c Fg. B Mahalaobs dsac of frquc of frs bdg mod wh hrshold: a S; b S; c S3. a b c Fg. B Mahalaobs dsac of dampg cosa of frs bdg mod wh hrshold: a S; b S; c S3. Iac D D D3 Po II Frquc (Hz Iac D D D3 Po II Frquc (Hz Joural of JSCE, Vol., 3-43, 03 4

11 Joural of JSCE, Vol., 3-43, 03 a b c Fg. B3 Mahalaobs dsac of frquc of hrd bdg mod wh hrshold: a S; b S; c S3. a b c Fg. B4 Mahalaobs dsac of dampg cosa of hrd bdg mod wh hrshold: a S; b S; c S3. REFERENCES Doblg, S. W., Farrar, C. R., Prm, M. B. ad Shvz, D. W.: Damag dfcao ad halh moorg of srucural ad mchacal ssms from chags hr vbrao characrscs: A lraur rvw, Los Alamos Naoal Laboraor Rpor LA-3070-MS, 996. Prs, B. ad D Roc, G.: O-ar moorg of h Z4-Brdg: vromal ffcs vrsus damag vs, Earhqua Egrg ad Srucural Damcs, Vol. 30, No., Dramar, A., Rbdrs, E., D Roc, G. ad Kullaa, J.: Vbrao-basd srucural halh moorg usg oupu-ol masurms udr chagg vrom, Mchacal Ssms ad Sgal Procssg, Vol., No., pp , Zhag, Q. W.: Sascal damag dfcao for brdgs usg amb vbrao daa, Compurs ad Srucurs, Vol. 85, No. 7-8, pp , Dla, M. ad Morass, A.: Damc sg of damagd brdg, Mchacal Ssms ad Sgal Procssg, Vol. 5, pp , 0. 6 Km, C. W., Kawaa, M. ad Km, K. B.: Thr-dmsoal damc aalss for brdg-vhcl raco wh roadwa roughss, Compurs ad Srucurs, Vol. 83, No. 9-0, pp , Km, C. W. ad Kawaa, M.: Psudo-sac approach for damag dfcao of brdgs basd o couplg vbrao wh a movg vhcl, Srucur ad Ifrasrucur Egrg, Vol. 4, No. 5, pp , Grsch, W., Nls, N. N. ad Aa, H.: Mamum llhood smao of srucural paramrs from radom vbrao daa, J. of Soud ad Vbrao, Vol. 3, No. 3, pp , Shozua, M., Yu, C. B. ad Ima, H.: Idfcao of lar srucural damc ssms, J. Egg. Mch. Dv., ASCE, Vol. 08, No. 6, pp , Hosha, M. ad Sao, E.: Srucural dfcao b dd Kalma flr, J. Egg. Mch., ASCE, Vol. 0, No., pp , 984. Wag, Z. ad Fag, T.: A m-doma mhod for dfg modl paramrs, J. of Appl. Mch., ASME, Vol. 53, No. 3, pp. 8-3, 986. H, X. ad D Roc, G.: Ssm dfcao of mchacal srucurs b a hgh-ordr mulvara auorgrssv modl, Compurs ad Srucurs, Vol. 64, No. -4, pp , Oabaash, T., Oumasu, T. ad Naama, Y.: Eprmal sud o srucural damag dco usg h hgh accura srucural vbrao-smao ssm, J. of Srucural Egrg, JSCE, Vol. 5A, pp , 005 ( Japas. 4 Km, C. W., Kawaa, M. ad Hao, J.: Modl paramr dfcao of shor spa brdgs udr a movg vhcl b mas of mulvara AR modl, Srucur ad Ifrasrucur Egrg, Vol. 8, No. 5, pp , 0. 5 Yoshoa, T., Iou, S., Yamaguch, H. ad Masumoo, Y.: Srucural halh moorg of russ brdgs basd o dampg chag dagoal mmbr-coupld mod, J. of JSCE, Srs A, Vol. 66, pp , 00 ( Japas. 6 Km, C. W., Ismoo, R., Kawaa, M. ad Sugura, K.: Srucural dagoss of brdg usg oupu-ol vbrao movg vhcl laboraor prm, J. of Appld Mchacs, JSCE, Vol. 67, No., pp. I_833-I_84, 0 ( Japas. 7 Furuawa, A., Osua, H. ad Umbaash, F.: Damag dfcao of sl russ brdg b vbrao sg usg mcro shar, J. of Appld Mchacs, JSCE, Vol. 9, pp. 03-0, 006 ( Japas. 8 Nar, K. K., Krmdja, A. S. ad Law, K. H.: Tm srs-basd damag dco ad localzao algorhm wh applcao o h ASCE bchmar srucur, J. of Soud ad Vbrao, Vol. 9, No. -, pp , Taguch, G. ad Jugulum, R.: Nw rds mulvara dagoss, Ida Joural of Sascs, Vol. 6, Srs B, pp , Ljug, L.: Ssm Idfcao-Thor for h Usr, d d, PTR Prc Hall, Uppr Saddl Rvr, M.J,

12 Joural of JSCE, Vol., 3-43, 03 Bold, B.: Famous Problms of Gomr ad How o Solv Thm, Nw Yor, Dovr, p.56, 98. Kawaa, M., Nshama, S. ad Yamada, Y.: Damc rspos aalss of hghwa grdr brdg udr movg vhcls, Tcholog Rpors of h Osaa Uvrs, Vol. 43, No. 37, pp.09-8, Yag, Y. B., Lao, S. S. ad Lm, B. H.: Impac formulas for vhcls movg ovr smpl ad couous bams, J. of Sruc. Eg., ASCE, Vol., No., pp , Bshop, C. M.: Par Rcogo ad Mach Larg, pp. 3-33, Sprgr, 006. (Rcvd Ocobr, 0 43