STRUCTURAL HEALTH MONITORING VIA STIFFNESS UPDATE

Transcription

1 ISE Journal of Earthquake echnology, Paer No. 509, Vol. 47, No. 1, March 2010, SUCUAL HEALH MONIOING VIA SIFFNESS UPDAE Prashant V. Pokharkar and Mansh Shrkhande Deartment of Earthquake Engneerng Indan Insttute of echnology oorkee oorkee ABSAC he erformance of an udated tme-doman least-squares dentfcaton method for dentfyng a reduced-order lnear system model n the case of lmted resonse measurements and ts use n structural health montorng s evaluated. It s shown that the ncororaton of a mass-nvarablty constrant enhances the robustness of the arametrc dentfcaton rocedure. he full structural stffness and mass matrces are dentfed from the dentfed reduced-order model by usng the condensed model dentfcaton and recovery method. he damage state s consdered to be reresented by an ncremental stffness degradaton model. he degradaton n stffness s estmated through the mnmzaton of an error functon defned n terms of aylegh quotents. he erformance of the roosed scheme s examned wth reference to the smulated damage due to earthquake exctaton n a 10-story buldng wth rgd floor dahragm. KEYWODS: Detecton, Dynamc Condensaton, Least-Squares Identfcaton, Structural Health Montorng, System Identfcaton INODUCION Structural system dentfcaton has ganed n mortance over the last coule of years as a dagnostc tool for the structural health assessment rmarly due to the requrements of enhanced functonalty and reduced downtme of buldngs and servces. he conventonal aroaches to structural health assessment requre hyscal access to the regons of nterest n the structural system and are also very tedous and tme consumng. he damage/degradaton n structures causes reducton of natural frequences, ncreased energy dssaton, and changes n the mode shaes. herefore, montorng of vbraton characterstcs of structural system should ermt the detecton of both the locaton and severty of damage. he vbratonbased system dentfcaton and health assessment s romsng because substantal nformaton can be gathered by only a few sensors dstrbuted across the structural system. he system dentfcaton aroaches can be classfed as ether arametrc or non-arametrc methods. In arametrc methods, the structural models to be dentfed are characterzed n terms of a fnte set of arameters, such as the coeffcents of the governng dfferental equatons of moton, or the coeffcents of the ratonal olynomal aroxmaton for transfer functon, etc. he non-arametrc methods, on the other hand, characterze the dynamc systems n terms of mulse resonse functons, or frequency resonse functons derved from the drect measurements of exctaton and resonse at varous locatons n the structural system. As the stffness characterstcs are most romnently nfluenced by the damage, f any, n the structural systems, several aroaches have been develoed to dentfy stffness, or related characterstcs of a structural system from the analyss of ts vbraton sgnatures. Udwada (2005) resented a method for the dentfcaton of the stffness matrces of a structural system from the nformaton about some of ts observed frequences and corresondng mode shaes of vbraton. Pandey and Bswas (1994) suggested the use of mode shae curvature n detectng damage. For large and/or comlex structures, however, the changes n mode shaes and ther curvatures may be so small that ther use for the detecton of damage mght not be ractcal. Stubbs et al. (1995) develoed a methodology based on the comarson of modal stran energy before and after damage to dentfy damage n structure. However, excet for very smle structures, moderate damage does not sgnfcantly affect the lower modes of vbraton, whch can be dentfed wth greater relablty. Baruch and Bar Itzhack (1978) roosed a matrx udate method, wheren a norm of the global arameter matrx erturbatons s mnmzed wth the alcaton of symmetry constrant on roerty matrx. Many other aroaches develoed n ths area are based on the

2 48 Structural Health Montorng va Stffness Udate mnmzaton of the rank of erturbed matrx wth connectvty and sarsty constrant on the orgnal matrx. In large scale comlex structural systems the natural frequences and mode shaes towards the hgher end of the sectrum can rarely be dentfed wth suffcent accuracy, whch n turn affect the relablty of damage detecton on the bass of mode shae nformaton. Agbaban et al. (1990) and Smyth et al. (2000) roosed a least-squares method for the arametrc dentfcaton of a lnear system for estmatng the coeffcents of the governng dfferental equaton. For a lmted number of sensors to record the vbraton resonse, the tme-doman least-squares dentfcaton rocedure yelds the mnmum norm estmates for the coeffcents of the reduced order system, whch may be sgnfcantly dfferent from the true coeffcents (Choudhury, 2007). hs roblem of non-unqueness of the results of tme-doman least-squares dentfcaton rocedure s addressed n ths study. educed-order equvalent lnear models are dentfed for dfferent tme wndows. he full system matrces are recovered from the estmated reduced order models and an attemt s made to detect the resence of damage by trackng the changes n the stffness coeffcents of the recovered full stffness matrx of the structure. he erformance of ths scheme s examned wth the hel of a smle analytcal model for a steel structural frame wth rgd floor dahragms and subjected to earthquake exctaton. IME-DOMAIN LEAS-SQUAES IDENIFICAION Let us consder the governng equatons of moton for a mult-degree-of-freedom (MDOF) system subjected to the external forces { f } : [ M ]{ y} + [ C]{ y } + [ K]{ y} { f } (1) where, [ M ], [ C ] and [ K ] are the mass, damng and stffness matrces of the system and { y } reresents the vector of deformatons at each degree of freedom (DOF). In ractce, the system resonse s recorded only at a few select DOFs corresondng to the sensor locatons, and we consder these DOFs as rmary (or master) DOFs. he least-squares arametrc dentfcaton would allow us to estmate the coeffcents of the reduced-order model consstng only of the rmary DOFs. By arttonng the vector { } of deformatons as { y} { ys},{ y }, where { } y and { } y denote the rmary and secondary DOFs, resectvely, the system consdered for dentfcaton s M { y } + { C y} + K { y} { f } (2) where, M, C and K denote the nerta, damng and stffness characterstcs of the structural system after condensng out the secondary DOFs. he vector { f } reresents the equvalent forces on the rmary DOFs and deends on the transformaton of the full analytcal model of Equaton (1) to the reduced-order model of Equaton (2). Assumng that a total of n number of resonse measurements are avalable at dfferent sensor locatons, a vector of resonses, say { r } can be constructed as s 13 n { r} { 1( ) ( ), 1( ) t yn t ( ), 1( ) ( )} t yn t y t yn t Consderng the resonse at all tme nstants ( t 1 through n defned as { r1 } { 2} [ ] r { rn } ( ), at the th tme nstant y y (3) t ), a resonse matrx [ ] he coeffcents of Equaton (2) corresondng to each rmary DOF may be arranged as n 3n ( ) can be (4)

3 ISE Journal of Earthquake echnology, March { j} { mj1 mjn c 1 1 } j cjn k j kjn α,,,,,,,, ; j 12,,, n (5) where m j1, c j1 and k j1, resectvely, denote the coeffcents of the condensed system matrces M, C and 13 n K, and { α } ( j ) s the arrangement of these coeffcents of the j th row n a row vector. Equaton (2) can then be rearranged n the form: ˆ { ˆ α } { b ˆ } (6) 2 3 where ˆ nn ( n ) s a block dagonal matrx wth the resonse matrx [ ] on ts dagonal, { ˆ α } ( n 1 ;,, ˆ nn b ( ) s the vector of corresondng exctaton measurements gven by {{ α j} { α n } } ), and { } wth { j} { j( 1), j( 2),, j( n) } ˆ b { b1} { b } 2 { bn } b f t f t f t, 12,,,. f j t, j 12,,, n denotes the equvalent forces on the rmary degrees of freedom of the condensed system at tme t and are descrbed n the followng secton. he least-squares soluton of Equaton (6) may be then obtaned as { ˆ α } ˆ { ˆ b} (8) where ˆ s the Moore-Penrose seudo-nverse (Golub and Van Loan, 1996) of ˆ. he leastsquares soluton comuted n Equaton (8) corresonds to the soluton of assocated normal equatons ( ˆ ˆ { } ˆ ˆ α { b ˆ } ). he soluton vector { ˆ α } rovdes the mnmum-norm least-squares estmates of the desred system arameters. It s often requred to mrove the numercal condtonng of the system of equatons and also to mose the constrant of symmetry of coeffcent matrces to elmnate hyscally nconsstent results of system dentfcaton. j n Here, ( ) ANALYICAL EDUCION OF SYSEM MAICES Snce the above-mentoned tme-doman least-squares arametrc dentfcaton rocedure can only dentfy a reduced-order model corresondng to the rmary DOFs (Smyth et al., 2000), t s desrable to have a set of benchmark values for assessng the qualty of arameter estmates before usng the results of ths dentfcaton rocedure to draw further nferences. A comarable reduced-order system model can be obtaned by the elmnaton of secondary DOFs from the system of equatons by usng the dynamc condensaton method (Paz, 1984, 2004). Let us wrte the equatons of moton for free vbraton n arttoned matrx form as 2 2 [ K ] [ ] ss ω Mss Ks ω Ms { ys} 2 2 { y} {} 0 {} 0 Ks ω M s K ω M from whch the secondary varables can be elmnated as ( 2 1 ) ( 2 ){ } [ ]{ } { } [ ] [ ] ys Kss ω Mss Ks ω Ms y y (10) (7) (9)

4 50 Structural Health Montorng va Stffness Udate 2 ω s an aroxmaton for the th egenvalue of the structural system, and [ ] [ ] 2 1 K ω [ M ] K ω 2 M reresents the transformaton matrx relatng the rmary where ( ss ss ) ( s s ) (master) DOFs to the secondary (slave) DOFs for the current aroxmaton of the th egenvalue obtaned from the reduced-order system contanng only rmary DOFs. he full DOFs-vector {{ s},{ } } y y may be then exressed n terms of the rmary DOFs as where [ ] { y } [ ] s { y } [ ] { y } { } y I I denotes an dentty matrx and s the matrx relatng the full-dofs vector to the rmary DOFs. he reduced mass and stffness matrces are then obtaned as wth [ ] M M and (11) 2 K D + ω M (12) ( 2 2 ) ( s s )[ ] D K ω M + K ω M (13) hese reduced-order system matrces are used to calculate an mroved estmate of the th egenvalue, whch s then substtuted n Equaton (13) and the teratve rocess s reeated untl the egenvalue s close enough to that for the full model. hs rocess may be reeated for estmatng the next egenvalue. Snce the reduced mass and stffness matrces are nfluenced by the choce of natural frequency, an teratve scheme s necessary to converge to a set of reduced matrces whch have the same natural frequences n the lower half of the sectrum as those calculated for the full-order system. he fnal reduced-order matrces can be used for the modelng of the forced vbraton roblem, for whch the resonse s only montored at the rmary DOFs. he modfed force vector for the reduced-order model n the case of exctaton by base moton may be exressed as [ ] Mss Ms { f } {} 1 ug( t) (14) M s M where s the transformaton relatng the DOFs of the chosen reduced-order analytcal model to the DOFs of the full analytcal model as defned n Equaton (11). he damng matrx s not consdered n the condensaton scheme as the level of damng forces s, generally, very small n structural systems, artcularly so n the case of steel structures. he examle structural system consdered n ths study s an ntermedate frame of a ten-story steel buldng wth two bays as shown n Fgure 1. he Indan standard rolled beam secton ISMB-225 (BIS, ) wth yeld strength f y of 250 MPa and Young s modulus E of 2 10 MPa are used for modelng the frame n the SAP2000 structural analyss software. A erfect elasto-lastc consttutve behavour s assumed. As a frst-order aroxmaton to the structural behavour and also for reducng the roblem sze, the rgd floor dahragm assumton s made and the vertcal and rotatonal DOFs are also restraned. hs reduces the total number of unrestraned DOFs to 10. he numerals on the rght sde of the frame n Fgure 1 ndcate the DOF numbers assocated wth dfferent floors (assgned nternally n SAP2000 after mosng the rgd floor dahragm constrant). A modal damng rato of ζ 0.05 s assumed. he undamed natural frequences of ths system are found to be ω 4.049, , , , , , , , , rad/s. he buldng resonse data s assumed to be recorded at the 1st, 4th, 8th and 10th floors, whch corresond to the otmal locatons for a 10-story buldng wth rgd floor dahragms (Hereda-Zavon and Esteva, 1998; Datta et al., 2002). he DOFs assocated wth these floors,.e., 2, 3, 9, and 6, resectvely, are desgnated as the rmary DOFs and the remanng DOFs (.e., secondary DOFs) are condensed out by the dynamc condensaton rocedure. he reduced system matrces are obtaned as

5 ISE Journal of Earthquake echnology, March M kg and (15) K N/m hese matrces wll now be used as benchmarks for comarng the system matrces estmated by the leastsquares dentfcaton rocess. Fg. 1 Structural frame used for analyss

6 52 Structural Health Montorng va Stffness Udate DAASE FO SYSEM IDENIFICAION In the tme-doman least-squares system dentfcaton aroach, t s necessary to have the exctaton and the resonse data of the system. he buldng resonse data s generated through a non-lnear dynamc analyss of an analytcal model n the SAP2000 envronment. he dataset conssts of relatve dslacement, relatve velocty and relatve acceleraton resonses at each of the desgnated floors assumed to have vbraton sensors, namely, the 1st, 4th, 8th and 10th (roof) floors. A scaled tme hstory (wth the scale factor of 2) of the Northrdge, Calforna earthquake ground moton recorded at Pacoma Dam Uer Left Abutment (wth the closest dstance of 8 km to the fault ruture) on January 17, 1994 for the comonent 104 (as n the PEE database 1 ) and samlng nterval of 0.02 s s consdered as the base exctaton. hs ground moton has the eak ground acceleraton (PGA) of 1.58g. Fgure 2 shows the (unscaled) tme hstory of ths ground moton. Fg. 2 Northrdge earthquake ground acceleraton tme hstory he recorded tme hstory s scaled (by a factor of 2) to enforce the develoment of lastc hnges n the structural frame durng the shakng. A zero-mean Gaussan random nose s added to smulate the effect of measurement nose such that σ n 0.05 σ s, where σ n denotes the standard devaton of nose and σ s reresents the standard devaton of the actual tme-doman sgnal. A reresentatve lot of the comuted relatve acceleraton, velocty and dslacement tme hstores at the 10th floor s shown n Fgure 3. All tme hstores are samled at 0.02 s nterval. In the case of sesmc exctatons, resonse data s generally acqured by usng accelerometers, whch record absolute acceleratons at the base of the transducers. he relatve acceleraton resonse s then obtaned by subtractng the base acceleraton from the recorded floor acceleratons. he velocty and dslacement tme hstores are obtaned by ntegratng the acceleraton tme hstory fltered to correct for baselne errors. hese tme hstores are then used for the least-squares system dentfcaton. However, consderng the full-duraton data at once smears away the tme-varyng nformaton and one only gets gross tme-averaged nformaton to draw nferences. Snce the damage of buldng frame durng an earthquake s a gradual rocess, t s exected that ths effect should be notceable n the arameter estmates obtaned from the data segments from small tme wndows. he length of the data wndow s an mortant consderaton for any such movng wndow analyss and s decded by examnng the temoral evoluton of the frequency content n the structural resonse tme hstory. he temoral evoluton s dected by the sectrogram,.e., the lot of the squared amltude of the short tme Fourer transform (SF) of the resonse tme hstory. he SF of a sgnal may be defned as wu Sf ( t, ω ) f ( u) g( u t) e du (16) 1 Webste of PEE Strong Moton Database, htt://eer.berkeley.edu/smcat/ (last accessed on January 2, 2010)

7 ISE Journal of Earthquake echnology, March where gu ( t) denotes a real, symmetrc wndow n tme doman and serves to localze the Fourer ntegral n the neghbourhood of u t. he sectrogram s defned as the energy densty of the SF as 2 St (, ω) Sf( t, ω ) (17) he length of wndow functon s an mortant factor n SF-based tme-frequency analyses too short a wndow enhances the resoluton n tme at the cost of oor resoluton n frequency doman, whereas too long a wndow rovdes a shar sectral resoluton wth low resoluton n tme doman. In ths study a 256-samle Hann wndow wth 0.9 overla factor s used for calculatng sectrograms. he varaton of the energy of varous harmoncs n a sgnal wth resect to tme s color coded wth large amltudes shown n red to very small amltudes shown n volet colour. Fgure 4 shows the sectrograms of the ground moton and the relatve acceleraton resonse at the roof level of the examle buldng. Fg. 3 Comuted relatve acceleraton, velocty and dslacement tme hstores at the 10th floor

8 54 Structural Health Montorng va Stffness Udate (a) Ground moton (b) elatve acceleraton at roof level Fg. 4 Sectrograms of the acceleraton tme hstores From the examnaton of the above sectrograms and the tme hstores (as shown n Fgures 2 and 3), t may be seen that the erod of strong shakng, marked by the S-wave arrval, commences after about 4 s from the onset of shakng. hs tme san of 4 s s also marked by an almost unform dstrbuton of energy wth resect to frequences. A tme wndow of 4-s length s therefore consdered to be adequate to cature the tme-varyng characterstcs n the dataset. Moreover, the data after 20 s can be neglected from the movng wndow analyss as the amltude of shakng s very small and s therefore nconsequental for the urose of damage detecton. hs reduces the total tme frame for the dataset to 20 s wth 5 tme wndows of 4 s each. LEAS-SQUAES IDENIFICAION ESULS Snce the least-squares soluton rocedure yelds a mnmum-norm soluton for the unknown system arameters, the estmated coeffcents for the reduced-order model can be greatly underestmated (Choudhury, 2007) and a sutable constrant on the ossble solutons s desrable. As the rocess of damage n a structural system durng an earthquake does not lead to any changes n the mass/nerta characterstcs of the structure, ths constrant of the mass nvarance can be mosed on the ossble solutons of the least-squares dentfcaton. hs mass-nvarance constrant s mosed by ncludng a requste number (.e., as many as the number of mass terms n the arameter vector { ˆ α }) of the denttes of the tye β m β m (18) to the system of equatons gven by Equaton (6). Here, j j m j denote the analytcally comuted mass coeffcents of the reduced-order model as n Equatons (15) and (18). hs rocess s smlar to the stff srng aroach for mosng rescrbed values for some varables n a system of lnear algebrac equatons. he coeffcent β should be chosen large enough so that the mass-nvarance dentty has substantal weght n the soluton of equatons. A good choce s to consder β to be tmes the largest coeffcent n [ ]. he augmented system of equatons for the least-squares soluton after the ncluson of mass-nvarance constrant may be exressed as ˆ { ˆ} [ ] { ˆ } b α [ ]{ ˆ * mc α } mc where [ mc ] s a matrx of β s and 0 s such that the mass-nvarant constrant can be ncororated. Another hyscal constrant of the symmetry of mass and stffness matrces s mosed by consderng coeffcents from the uer trangular regons only n the arameter vector { ˆ α } n Equaton (19) and by rearrangng the elements of ˆ so as to assocate the multle of an element of lower trangular art wth ts symmetrc counterart n the uer trangular art (Smyth et al., 2000). For the examle buldng (19)

9 ISE Journal of Earthquake echnology, March and the dataset for the frst tme wndow of 4 s, the dentfed system arameters for the reduced-order model wthout the mass-nvarant constrant are obtaned as and M dn K dn kg N/m whereas these arameters after alyng the mass-nvarant constrant are estmated as and M dn K dn kg N/m On comarng these estmated values wth the benchmark values shown n Equaton (15), t may be seen that the arameter estmates obtaned after mosng the mass-nvarant constrant are n good agreement wth the analytcal results. he error n the dentfed stffness matrx wth resect to the analytcally condensed stffness matrx, evaluated n terms of the Frobenus norm, s 55% when the mass-nvarant constrant s not used and 19% wth ths constrant n lace. It may be mentoned here that some good results for the modal frequences and damng of large-scale structures have also been obtaned wth the least-squares method wthout the use of mass-nvarant constrant (Smyth et al., 2003). he use of massnvarant constrant s amed at mrovng the robustness of the least-squares dentfcaton of the system roerty matrces themselves and not just the modal characterstcs. he natural frequences of the (analytcal and dentfed) reduced-order models are shown n able 1 along wth the frst four natural frequences of the full analytcal model. able 1: Natural Frequences of Analytcal and Identfed Models Model Natural Frequences (rad/s) Mode 1 Mode 2 Mode 3 Mode 4 Analytcal Model Analytcal educed Model Identfed educed Model It may be seen that the frst two natural frequences of the dentfed reduced-order model are n good agreement wth the frst two frequences of the analytcal model. herefore, the frst two egenvalues of the dentfed reduced-order model wll be used for the further analyss for damage dentfcaton as descrbed next.

10 56 Structural Health Montorng va Stffness Udate ECOVEY OF FULL SUCUAL MAICES he dentfed stffness coeffcents corresond to the mathematcally contrved reduced-order system and t s not ossble to ascertan the health of hyscal structure by examnng these coeffcents. For the dentfcaton of damage, t s necessary to reconstruct the full-system matrces from the dentfed reduced-order matrces. A matrx udatng based formulaton to dentfy the ncremental changes n the elements of stffness matrx s roosed based on the condensed model dentfcaton and recovery method (CMI). Koh et al. (2006) used an egensystem realzaton algorthm (EA) to dentfy a condensed model utlzng comlete tme-doman records. he roosed method dffers only n the use of the least-squares tme-doman dentfcaton method wth the mass-nvarant constrant for the dentfcaton of condensed models n dfferent tme wndows, such that a rogressve montorng of the changes n the stffness coeffcents wth reference to the undeformed confguraton s allowed. M a denote the full stffness and mass matrces of the vrgn, undamaged system and K and M be the full stffness and mass matrces of damaged system. Further, let [ δ K ] reresent the ncremental changes n the stffness matrx due to damage. It s ossble to relate the system matrces for the undamaged and damaged states as Let [ ] a K and [ ] K [ Ka] + [ δ K] and M [ M a] (20) where the stffness ncrements [ δ K ] are to be determned teratvely so as to mnmze the quadratc error functon gven by 2 j ε 1 (21) j 1 j Only two terms corresondng to the frst two modes of vbraton are consdered n constructng the error functon because for a reduced-order system of sze N, aroxmately frst N/2 egenvalues correlate well wth the egenvalues of the orgnal, full system. In Equaton (21) j and j resectvely denote the aylegh quotents comuted from the j th mode shae vectors for the analytcally condensed system matrces and the dentfed reduced-order matrces and are comuted as where { φ j } and { j } j { φ } K { } j φj { φ } M { } j φj and j 2 { φ } K { } j φj { φ } M { } j φj φ are the j th mode shaes of the dentfed and analytcally reduced matrces, resectvely, and K and M resectvely denote the reduced-order stffness and mass matrces dentfed by the tme-doman least-squares dentfcaton rocedure. Smlarly, K and M resectvely denote the analytcally reduced stffness and mass matrces as obtaned from the udated structural matrces K and M. he teratve rocedure to determne the ncremental changes n the stffness matrx, [ δ K ], requres the comutaton of the error functon for a confguraton of ncremental stffness matrces and s arranged n the followng order: 1. Consderng the current values of the vector of desgn varables corresondng to the ncremental stffness matrx [ δ K ], the udated structural stffness K s determned as n Equaton (20). 2. he matrces K and M are dynamcally condensed to retan the terms corresondng to the rmary DOFs only. Let K and M be the analytcally reduced udated matrces. 3. he aylegh quotents based on the dentfed and reduced udated matrces are calculated by usng Equaton (22). (22)

11 ISE Journal of Earthquake echnology, March he error functon for the current estmate of the ncremental matrx [ δ K ] s calculated by usng Equaton (21). Due to the symmetry of the stffness matrx, only the uer trangular elements of [ δ K ] are consdered as the desgn varables a total of 19 elements for a 10-story buldng wth rgd floor dahragms for the mnmzaton roblem. he desgn varables are numbered row-wse for the uer trangular art of [ δ K ]. he sequental quadratc rogrammng s used to mnmze the error functon defned n Equaton (21) wth resect to the desgn varables,.e., stffness ncrements. he uer and lower bounds on the stffness coeffcents are chosen to be 0 and 30% of the orgnal stffness coeffcents, wth negatve sgn added to ndcate the nature of stffness degradaton wth damage. he startng vector to begn the otmzaton rocess s cked by randomly choosng the desgn varables from the gven range. o guard aganst the ossblty of convergng on a local mnmum, the mnmzaton rocess s reeated 40 tmes wth randomly selected ntal desgn vectors. he set of desgn varables corresondng to the mnmum ε (n 40 trals) s assumed to gve the desred ncremental matrx. hs rocedure s carred out for all the fve tme wndows. he values of varables obtaned for these wndows after otmzaton are shown n Fgure 5 and able 2. (a) Frst wndow (0-4 s) (b) Second wndow (4-8 s) (c) hrd wndow (8-12 s) (d) Fourth wndow (12-16 s) (e) Ffth wndow (16-20 s) Fg. 5 Fnal values of desgn varables for dfferent tme wndows

12 58 Structural Health Montorng va Stffness Udate able 2: Changes n Stffness Coeffcents ( 10 5 N/m) n Dfferent Data Wndows Varable 1st Wndow 2nd Wndow 3rd Wndow 4th Wndow 5th Wndow 1( δ k 1,1 ) ( δ k 1,3 ) ( δ k 1,10 ) ( δ k 2,2 ) ( δ k 2,4 ) ( δ k 3,3 ) ( δ k 3,5 ) ( δ k 4,4 ) ( δ k 4,5 ) ( δ k 5,5 ) ( δ k 6,6 ) ( δ k 6,8 ) ( δ k 7,7 ) ( δ k 7,9 ) ( δ k 7,10 ) ( δ k 8,8 ) ( δ k 8,9 ) ( δ k 9,9 ) ( δ k 10,10 ) he changes n the frst tme wndow (from 0 to 4 s) are neglgble n comarson to the orgnal stffness coeffcents. hs ndcates that the structure has not suffered any damage durng the frst tme wndow. For all the subsequent tme wndows, the desgn varables 4 and 18 consstently show a sgnfcantly large stffness decrement n comarson to the other varables. hese two varables corresond to the stffness coeffcents corresondng to the DOFs 2 and 9, resectvely. hs s n agreement wth the actual damage state of the structural system after the earthquake exctaton n the SAP2000 smulaton run wth the formaton of lastc hnges at the ground and eghth floors as shown n Fgure 6. However, there are a number of nconsstences as well, e.g., there are lttle or no changes n the coulng terms for the DOF 9 to the adjacent DOFs. In addton, there are some false alarms, whch are more romnent n the thrd data wndow. For the smle analytcal model consdered n ths study, t s ossble to segregate these anomales as outlers and dscard those but t may be dffcult to take a defntve call n the case of a more comlex and realstc analytcal model. CONCLUSIONS A matrx udate method based on the reduced-order model dentfed by the tme doman leastsquares dentfcaton rocedure s roosed. he effectveness of the roosed dentfcaton rocedure for damage detecton s demonstrated wth the hel of an examle roblem of a structural steel frame damaged by an earthquake ground moton. he 10-story frame s modeled n SAP2000 to smulate the

13 ISE Journal of Earthquake echnology, March damage durng the earthquake exctaton. A mass-nvarant constrant s roosed for use wth the tmedoman least-squares dentfcaton rocedure and s found to be effectve n mrovng the robustness of the dentfcaton rocedure. Fg. 6 Damage state of frame after exctaton A movng wndow analyss s erformed to track temoral changes n the stffness roertes of the structural system. he model dentfcaton and recovery method s used to recover the full-system matrces from the dentfed reduced-order matrces for each of the fve tme wndows consdered n the analyss. An otmzaton roblem s formulated to dentfy the requred ncrements n stffness coeffcents n each tme wndow. It s observed that the desgn varables 4 and 18 (assocated wth the DOFs 2 and 9 of the structural frame of Fgure 1) are consstently large n all tme wndows, excet for the frst one, as comared to the other varables. In addton, the magntudes of these two varables are aroxmately same across all tme wndows (excet for the frst wndow), whle other varables exhbt relatvely more varatons over tme. he consstent ndcatons of reducton n stffness corresondng to these desgn varables suggest damage n the ground and eghth floors of the structural frame. Nevertheless, there are also a number of nconsstences n the dentfcaton results, whch are easly recognzed as anomales and dscarded n the smle examle system consdered n ths study. However, the results of arametrc dentfcaton of a chosen reduced-order model from dfferent tme wndows and ther extraolaton to a full model may not work well n more realstc and comlex systems. In ths regard, t would be better to use the tme-doman least-squares dentfcaton rocedure for estmatng modal arameters as those are better constraned than the stffness (and/or mass) coeffcents. EFEENCES 1. Agbaban, M.S., Masr, S.F. and Mller,.K. (1990). System Identfcaton Aroach to Detecton of Structural Changes, Journal of Engneerng Mechancs, ASCE, Vol. 117, No. 2, Baruch, M. and Bar Itzhack, I.Y. (1978). Otmum Weghted Orthogonalzaton of Measured Modes, AIAA Journal, Vol. 16, No. 4, BIS (1989). IS 808: 1989 Indan Standard Dmensons for Hot olled Steel Beam, Column, Channel and Angle Sectons (hrd evson), Bureau of Indan Standards, New Delh. 4. Choudhury, K. (2007). On the elablty of a Class of System Identfcaton Procedures, M.ech. hess, Deartment of Earthquake Engneerng, Indan Insttute of echnology oorkee, oorkee. 5. Datta, A.K., Shrkhande, M. and Paul, D.K. (2002). On the Otmal Locaton of Sensors n Multstoreyed Buldng, Journal of Earthquake Engneerng, Vol. 6, No. 1, Golub, G.H. and Van Loan, C.F. (1996). Matrx Comutatons, he Johns Hokns Unversty Press, Baltmore, U.S.A.

14 60 Structural Health Montorng va Stffness Udate 7. Hereda-Zavon, E. and Esteva, L. (1998). Otmal Instrumentaton of Uncertan Structural Systems Subjected to Earthquake Ground Motons, Earthquake Engneerng & Structural Dynamcs, Vol. 27, No. 4, Koh, C.G., ee, K.F. and Quek, S.. (2006). Condensed Model Identfcaton and ecovery for Structural Damage Assessment, Journal of Structural Engneerng, ASCE, Vol. 132, No. 12, Pandey, A.K. and Bswas, M. (1994). Damage Detecton n Structures Usng Changes n Flexblty, Journal of Sound and Vbraton, Vol. 169, No. 1, Paz, M. (1984). Dynamc Condensaton, AIAA Journal, Vol. 22, No. 5, Paz, M. and Legh, W.E. (2004). Structural Dynamcs: heory and Comutaton, Kluwer Academc Publshers, Dordrecht, he Netherlands. 12. Smyth, A.W., Masr, S.F., Caughey,.K. and Hunter, N.F. (2000). Survelance of Mechancal System on Bass of Vbraton Sgnature Analyss, Journal of Aled Mechancs, ASME, Vol. 67, No. 3, Smyth, A.W., Pe, J.S. and Masr, S.F. (2003). System Identfcaton of the Vncent homas Susenson Brdge Usng Earthquake ecords, Earthquake Engneerng & Structural Dynamcs, Vol. 32, No. 3, Stubbs, N., Km, J.. and Farrar, C.. (1995). Feld Verfcaton of a Nondestructve Damage Localzaton and Severty Estmaton Algorthm, Proceedngs of the 13th Internatonal Modal Analyss Conference, Nashvlle, U.S.A., Udwada, F.E. (2005). Structural Identfcaton and Damage Detecton from Nosy Modal Data, Journal of Aerosace Engneerng, ASCE, Vol. 18, No. 3,