Structural Damage Detection Using Optimal Sensor Placement Technique from Measured Acceleration during Earthquake

Transcription

1 Cover page Tle: Auhors: Srucural Damage Deecon Usng Opmal Sensor Placemen Technque from Measured Acceleraon durng Earhquake Graduae Suden Seung-Keun Park (Presener) School of Cvl, Urban & Geosysem Engneerng Seoul Naonal Unversy San 56-, Shllm-dong, Kwanak-gu Seoul, 5-74, Korea Phone: Fax: E-mal: Professor Soobong Shn School of Cvl and Envronmenal Engneerng Inha Unversy 53 Yonghyun-dong, Nam-gu Incheon 40-75, Korea Phone: Fax: E-mal: Professor Hae Sung Lee School of Cvl, Urban & Geosysem Engneerng Seoul Naonal Unversy San 56-, Shllm-dong, Kwanak-gu Seoul, 5-74, Korea Phone: Fax: E-mal:

2 Srucural Damage Deecon Usng Opmal Sensor Placemen Technque from Measured Acceleraon durng Earhquake Seung-Keun Park, Soobong Shn, Hae Sung Lee ABSTRACT Ths paper presens a sysem denfcaon (SI) scheme n me doman usng measured acceleraon daa. The error funcon s defned as he me negral of he leas square errors beween he measured acceleraon and he calculaed acceleraon by a mahemacal model. Dampng parameers as well as sffness properes of a srucure are consdered as sysem parameers. The srucural dampng s modeled by he Raylegh dampng. A regularzaon funcon defned by he L -norm of he frs dervave of sysem parameers wh respec o me s proposed o allevae he ll-posed characerscs of nverse problems and o accommodae dsconnues of sysem parameers n me. The me wndow concep s proposed o race varaon of sysem parameers n me. Fsher Informaon Marx s formulaed n erms of acceleraon sensvy wh respec o srucural sysem parameers. A scheme of an effecve ndependence dsrbuon vecor has been appled o deermne opmal locaons of acceleromeers. Numercal smulaon sudy s performed hrough a wo-span connuous russ. Key Words: Sysem Idenfcaon, Regularzaon, Tme Wndow, Fsher Informaon Marx, Effecve Independence dsrbuon vecor Seung-Keun Park, School of Cvl, Urban & Geosysem Engneerng, Seoul Naonal Unversy, Seoul, 5-74, Korea Soobong Shn, School of Cvl and Envronmenal Engneerng, Inha Unversy, Incheon, 40-75, Korea Hae Sung Lee, School of Cvl, Urban & Geosysem Engneerng, Seoul Naonal Unversy, Seoul, 5-74, Korea

3 INTRODUCTION Immedae safey assessmen srucures afer an earhquake s exremely mporan n evaluang servceably and funconaly of socal nfrasrucures. Nowadays, no only ground acceleraon bu also acceleraon of mporan socal nfrasrucures s monored durng earhquakes. I would be very helpful for quck resoraon of socal acves f srucural damage caused by an earhquake s accessed wh he measured acceleraon durng an earhquake n real me or near real me. Varous damage assessmen schemes based on sysem denfcaon (SI) have been exensvely nvesgaed for socal nfrasrucures durng he las few decades. The modal analyss approaches have been wdely adoped o deec srucural damage usng measured acceleraon of srucures. The modal analyss approaches, however, suffer from drawbacks caused by nsensveness of modal daa o changes of srucural properes. To overcome he drawbacks of he modal analyss approaches, hs paper presens a sysem denfcaon scheme n me doman usng measured acceleraon daa. The error funcon s defned as he me negral of he leas square errors beween he measured acceleraon and he calculaed acceleraon by a mahemacal model. The srucural dampng s modeled by he Raylegh dampng. A regularzaon echnque s employed o overcome he ll-posedness of nverse problems. A regularzaon funcon defned by he L -norm of frs me dervaves of sffness parameers s proposed o accommodae abrup changes of sysem parameers n me. The L -runcaed sngular value decomposon (TSVD) s adoped o opmze he error funcon wh he L -regularzaon funcon. To race he varaon of sffness parameers n me, a me wndowng echnque s nroduced. In he me wndowng echnque, SI s performed sequenally whn a fne me nerval, whch s called a me wndow. The me wndow advances forward a each me sep o denfy changes of sysem parameers n me. Desgn of sensor layou s an mporan ask o provde useful measuremen nformaon on srucural monorng bu s mporance has no been serously acknowledged n he feld applcaon ye. Alhough dverse ypes of sensors are appled n he acual applcaons he man focus of any sensor s o provde nformaon useful for guaraneeng he srucural safey as effcen and economcal as possble. Snce a cvl srucure s huge and complex requrng many degrees of freedom (DOF) n s analycal model, such SI resuls are nfluenced by he measuremen locaons and measured DOFs. Through mahmacal manpulaons wh assumpons on measuremen nose, a Fsh nformaon marx (FIM) s obaned by he Cramer-Rao nequaly as he nverse of he lower bound of he esmaon error. Each column of he formulaon FIM s a vecor of acceleraon sensvy wh respec o srucural parameers. Snce s generally known ha opmal locaons of acceleromeers can be deermned by maxmzng characersc properes of FIM wh a possble mnmum covarance of he esmaon error, a scheme of an effecve ndependence dsrbuon vecor has been appled o deermne opmal locaons of acceleromeers. Numercal smulaon sudy s performed hrough a wo-span connuous russ. PARAMETER ESTIMATION SCHEME IN TIME DOMAIN The dscrezed equaon of moon of a srucure subjeced o ground acceleraon a g caused by an earhquake s expressed as follows. Ma + C( x ) v + K( x ) u Ma () c s g

4 where M, C and K represen he mass, dampng and sffness marx of he srucure, respecvely, and a, v and u are he relave acceleraon, velocy and dsplacemen of he srucure o ground moon, respecvely. The dampng parameers and he sffness parameers of he srucure are denoed by x c and x s n (), respecvely. Newmark β-mehod s used o negrae he equaon of moon. Snce he operaonal vbraons of a srucure are neglgble compared o hose nduced by an earhquake, he nal condon of () s se o zero. In case ground acceleraon as well as acceleraons of a gven srucure a some dscree observaon pons are measured, he unknown sysem parameers of a srucure ncludng sffness and dampng properes are denfed hrough mnmzng leas squared errors beween compued and measured acceleraon. In case he sysem parameers are nvaran n me, he parameer esmaon procedure s represened by he followng opmzaon problem. ~ Mn Π E ( ) ( ) subjec o ( ) 0 a x a d R x () x 0 where ~ a, a, x and R are he calculaed acceleraon and he measured acceleraon a observaon pons relave o ground acceleraon, sysem parameer vecor and consran vecor, respecvely, wh represenng he -norm of a vecor. Lnear consrans are used o se physcally sgnfcan upper and lower bounds of he sysem parameers. The mnmzaon problem defned n () s a consraned nonlnear opmzaon problem because he acceleraon vecor ~ a s a nonlnear mplc funcon of he sysem parameers. In case he sysem parameers vary wh me, he me wndow echnque s proposed. Fg. llusraes he me wndow concep. In hs echnque he mnmzaon problem for he esmaon of he sysem parameers s defned n a fne me nerval, whch s referred o as a me wndow. Mn Π x E + d w ~ a( x( )) a d subjec o R( x( )) 0 (3) Here, and d w s he nal me and he wndow sze of a gven me wndow. I s assumed ha sysem parameers are consan n a me wndow, and ha sysem parameers esmaed by (3) represen he sysem parameers a me. As he me wndow advances forward sequenally n me, he varaons of sysem parameers n me are denfed. Acual sysem parameer Sysem parameer by SI Acceleraon D : Damage occurs Tme Sysem Parameer D : Damage occurs Tme

5 Fgure. Tme wndow concep L -REGULARIZATION SCHEME The parameer esmaon defned by he mnmzaon problems s a ype of ll-posed nverse problems. Ill-posed problems suffer from hree nsables: nonexsence of soluon, non-unqueness of soluon and dsconnuy of soluon when measured daa are pollued by nose. Because of he nsables, he opmzaon problem gven n () and (3) may yeld meanngless soluons or dverge n opmzaon process. Aemps have been made o overcome nsables of nverse problems merely by mposng upper and lower lms on he sysem parameers. However, has been demonsraed by several researchers ha he consrans on he sysem parameers are no suffcen o guaranee physcally meanngful and numercally sable soluons of nverse problems. The regularzaon echnque proposed by Tkhonov s wdely employed o overcome he ll-posedness of nverse problems. In he Tkhonov regularzaon echnque, a posve defne regularzaon funcon s added o he orgnal opmzaon problem. Mn x Π( ) Π E + λ Π R subjec o R( 0 (4) where Π R and λ are a regularzaon funcon and a regularzaon facor, respecvely. Varous regularzaon funcons are used for dfferen ypes of nverse problems. Kang e al proposed he followng regularzaon funcon defned by he L -norm for he SI n me doman. Π R 0 dx d d (5) The regularzaon funcon defned n (5) s able o represen connuously varyng sysem parameers n me. Snce, however, he sysem parameers may vary abruply (Fg.) wh me durng earhquakes due o damage, a regularzaon funcon ha can accommodae pecewse connuous funcons n me s requred o access damage ha occurs durng an earhquake. To represen dsconnuy of sysem parameers n me, hs paper proposes an L -regularzaon funcon of he frs dervave of sysem parameers wh respec o me. Π R + d w dx d d (6) where represenng he -norm of a vecor. Snce he error funcon s nonlnear wh respec o sffness parameers, a Newon-ype opmzaon algorhm, whch requres graden nformaon of an objecve funcon, s usually employed n SI. As he L -regularzaon funcon s non-dfferenable, he objecve funcon n he Tkhonov regularzaon scheme defned n (4) conans a non-dfferenable funcon, and hus a Newon-ype opmzaon algorhm canno be appled. To avod hs dffculy, hs paper employs he L -TSVD o mpose he L -regularzaon funcon n he opmzaon of he error funcon. In he proposed mehod, he ncremenal soluon of he error funcon s obaned by solvng he quadrac sub-problems whou he consrans. The nose-pollued soluon componens are runcaed from

6 he ncremenal soluon. Fnally, he regularzaon funcon s mposed o resore he runcaed soluon componens and he consrans. The above procedure s defned as follows. Pecewse-connuous funcon Sysem Parameers Connuous funcon Tme Mn Π x R Fgure. Connuous and pecewse-connuous funcon + dw + d w dx d d subjec o R( 0 and Mn Π x E ~ a( a d (7) The error funcon and he regularzaon funcon are easly dscrezed n me doman usng smple numercal mehods. The runcaed soluon of he mnmzaon problem of he error funcon s obaned by he runcaed sngular value decomposon, whle he smplex mehod s employed o solve he mnmzaon problem of he L -regularzaon funcon wh consrans. Dealed soluon procedures are presened n References. FISHER INFORMATION MATRIX To esmae unknown srucural parameers x by mnmzng he error funcon Π(x,) of (), he measured nformaon by an expermenal should be maxmzed. In oher words, opmal parameers can be obaned by mnmzng he esmaon errors. By he Cramer-Rao nequaly, he esmaon error has a lower bound of F -, where F s he Fsher nformaon marx expressed n erms of he probably densy funcon by (8). T log f ( a log f ( a F( a, Ea x x x (8) + d w m / T f ( a [( π ) de C] exp ( ~ a( a) C( ~ a( a) d (9) C s covarance marx of he measuremens of acceleraons. By subsung he probably densy funcon of (9) no (8), he FIM can be formulaed by (0).

7 n T [ S CS + ] F ( a, T (0) where, n, S and T are number of mesep, senvy marx and race marx defned by (), respecvely. S ~ a x C C, T r C C x x () If we can assume he varance of measuremen covarance marx can be expressed as C σ n s he same for all measurng locaons, he σ I so ha / x 0. In oher words, s assumed n ha all sensors are exposed o he same bu uncorrelaed nose so ha covarance of nose s ndependen of he parameers. n T [ S ] C F ( a, S () EFFECTIVE INDEPENDENCE DISTRIBUTION VECTOR A usual approach o deermne opmal locaons of acceleromeers s o maxmze characersc properes of FIM of (). One of he mehods proved as effcen s he scheme of effecve ndependence dsrbuon vecor (EIDV). The orgnal dea of EIDV was o deermne sensor locaons wh he nformaon of mode shapes. The curren approach s o deermne locaon of acceleromeers wh he compued acceleraon sensves. The each value of EIDV represens he effcency of each DOF. ~ ~ T ~ ~ T q d dag( S[ S S] S ) (3) where, ~ S T T T T [ S S L S n ], q d s effecve ndependence dsrbuon vecor. DAMPING MODEL I s a dffcul ask o model dampng properes of real srucures. In fac, exsng dampng models canno descrbe acual dampng characerscs exacly, and are approxmaons of real dampng phenomena o some exens. Snce he dampng has an mporan effec on dynamc responses of a srucure, he dampng properes should be consdered properly n he parameer esmaon scheme. In mos of prevous sudes on he parameer esmaon, he dampng properes of a srucure are assumed as known properes, and only sffness properes are denfed. However, he dampng properes are no known a pror and should be ncluded n sysem parameers n he SI. Among varous classcal dampng models, he modal dampng and he Raylegh dampng are he mos frequenly adoped model. In he modal dampng, a dampng marx s consruced by usng

8 generalzed modal masses and mode shapes. In Raylegh dampng, a dampng marx s defned as a lnear combnaon of he mass marx and sffness marx as follows. C 0 a a M + K (4) The dampng coeffcens of he Raylegh dampng can be deermned when any wo modal dampng raos and he correspondng modal frequences are specfed. In case he modal dampng s employed n he parameer esmaon, he number of he sysem parameers assocaed wh he dampng s equal o ha of he oal number of DOFs, whch ncreases he oal number of unknowns n he opmzaon problem gven n (4). Snce neher modal dampng nor Raylegh dampng can descrbe acual dampng exacly, and he modal dampng requres more unknowns han he Raylegh dampng n he parameer esmaon, hs sudy employs he Raylegh dampng for he SI. The Raylegh dampng yelds a lnear f o he exac dampng of a srucure. To approxmae acual dampng of a srucure more accuraely, Caughey dampng, whch s he general form of he raylegh dampng, may be adoped. J M a ( M K), ndof 0 C J (5) where ndof s he oal number of degrees of freedom of he gven srucure. For J, he Caughey dampng becomes dencal o he Raylegh dampng. In case eher he regularzaon scheme or dampng esmaon s no ncluded n he SI, he opmzaon procedure does no converge or converges o meanngless soluons. Therefore, only he resuls wh he regularzaon scheme and dampng esmaon are presened here. EXAMPLE Two numercal smulaon sudes are presened o llusrae valdy of he opmal sensor placemen echnque. Numercal smulaon sudy s performed hrough a wo-span connuous russ subjeced o earhquake-nduced ground moon. Newmark negraon s used for obanng he acceleraon. The negraon consans of he Newmark β-mehod, β/, γ/4, are used for all cases. The valdy of he proposed opmal sensor placemen echnque s examned hrough a smulaon sudy wh a wo-span connuous russ shown n Fg. 3. Typcal maeral properes of seel (Young s modulus 0 GPa, Specfc mass Kg/m 3 ) are used for all members. The cross seconal areas of op, boom, vercal and dagonal members are 50 cm, 300 cm, 00 cm and 0 cm, respecvely. The naural frequences of he russ range from 6.6 Hz o 4.7 Hz. Damage of he russ s smulaed wh 40% and 50% reducons n he seconal areas of member 7 and 6, respecvely. The damaged members are depced by doed lnes n Fg. 3. I s assumed ha he damage suddenly occurs a 0.5 sec. Acceleraons of he russ are measured from a vbraon nduced by a ground moon (Fg. 3). The measuremen errors are smulaed by addng 3% random nose generaed from a unform probably funcon o acceleraons calculaed by he fne elemen model. The observaon pons and drecons are deermned by opmal sensor placemen echnque. Deermned observaon pons and drecons are represened n fgure 4. The oal numbers of seleced DOF are.

9 Boh x- and y- drecon acceleraons are measured n he me perod from 0 sec o sec wh he nerval of /00 sec. To fler hgh frequency mode, he nerval of nverse analyss s /00 sec. The runcaon number of he TSVD s seleced as 7. The sze of me wndow s 0. sec. Fgure 5 represens a ground acceleraon nduced by he Kobe earhquake. The varaons of axal rgdes of he wo damaged members and one undamaged member wh me are drawn n Fgure 6. From he fgure, s clearly seen ha he damage occurs a 0.5 sec, and ha he esmaed sffness parameers of damaged members 7 and 6 converge o he acual values as me seps proceed. Fgure 7 shows he axal rgdy of each member denfed a he fnal me.0 sec. The vercal axes of boh Fgure 6 and Fgure 7 represen he normalzed axal rgdy wh respec o he nal value of each member. The denfed axal rgdes oscllae moderaely whn he range of ±0 % for 50 undamaged members ou of 5, whle he oscllaon magnudes of he oher undamaged members are a lle hgher han 0%. Snce, however, axal rgdes of he damaged members are reduced promnenly compared wh hose of he oher members, he damaged members are clearly dsngushed from undamaged members. m a m Fgure 3. -span connuous russ : seleced DOF : unseleced DOF Fgure 4. Observaon pons and drecons by opmal sensor placemen echnque

10 Acceleraon (m/sec ) Tme (sec) Fgure 5. Ground acceleraon nduced by he Kobe earhquake..0 Normalzed axal rgdy Exac axal rgdy of member 7 0. Exac axal rgdy of member 6 Member 7 (damaged) Member 6 (damaged) Member 8 (undamaged) Tme (sec) Fgure 6. Varaon of axal rgdes of wo damaged members and one undamaged member

11 .4..0 Normalzed axal rgdy : Damaged member Member Fgure 7. Idenfed axal rgdes a he fnal me sep CONCLUSION The me wndow echnque and opmal sensor placemen echnque are proposed for SI n me doman usng measured acceleraon daa s proposed. The sysem parameers nclude he dampng parameers as well as he sffness parameers of a srucure. The Raylegh dampng s used o esmae he dampng characerscs of a srucure. The leas square errors of he dfference beween calculaed acceleraon and measured acceleraon s adoped as an error funcon. The regularzaon echnque s employed o allevae he ll-posedness of he nverse problem n SI. The L -TSVD s ulzed o opmze a non-dfferenal objec funcon. The proposed mehod exhbs very compromsng characerscs n deecng damage, and s able o esmae he sffness properes accuraely even hough he dampng characerscs are approxmaed by he Raylegh dampng. The example presened n hs paper shows capables of he me wndow echnque for he denfcaon of damage caused by earhquakes. REFERENCES I.H. Yeo, S.B. Shn, H.S. Lee and S.P. Chang, Sascal damage assessmen of framed srucures from sac responses, Journal of Engneerng Mechancs, ASCE, Vol. 6, No. 4, pp. 44-4, 000 Sh, Z.Y., Law, S.S. and Zhang, L.M., Damage localzaon by drecly usng ncomplee mode shapes, Journal of Engneerng Mechancs, ASCE, Vol. 6, No. 6, pp , 000 Vesoun, F. and Capecch, D., Damage deecon n beam srucures based on frequency measuremens, Journal of Engneerng Mechancs, ASCE, Vol. 6, No. 7, pp , 000

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