A procedure for the identification of concentrated damages on beams by free vibration tests

Transcription

1 A procedure for the detfcato of cocetrated damages o beams by free vbrato tests Salvatore Caddem, Ivo Calò, Sadro Lse Dpartmeto d Igegera Cvle ed Ambetale, Uverstà d Cataa, Itala E-mal: scaddem@dca.uct.t, calo@dca.uct.t, lse@dca.uct.t Keywords: Idetfcato, cocetrated damages, dyamc tests. SUMMARY. A ovel procedure for the detfcato of multple cocetrated damages o a straght beam, based o the kowledge of the relevat ege-mode plct pressos, s preseted. The specal aalytcal structure of the drect problem soluto allows for the determato of plct pressos also for the verse dyamc damage detfcato problem. Namely the damage testy ad the posto of the cocetrated cracks may be detfed through the kowledge of the compoet of two vbrato modes the cross sectos betwee two cracks ad the correspodg frequeces. INTRODUCTION Structural health motorg techques are ofte based o dyamc respose-based damage detecto methods. Most of the dyamcs-based structural health motorg techques rely o free vbrato of beams wth cracks whch s a problem tesvely studed the last three decades [-6]. May crack models have bee proposed the lterature to smulate the effect o the dyamc behavour of beams [7]. The most wdely adopted model s that based o a local flblty duced by a trasverse edge crack whch s smulated by a teral hge edowed wth a rotatoal sprg at the locato of the crack []. Accordg to ths model, the beam s subjected to a slope dscotuty at the locato of the crack. A commo approach to formulate the drect problem for the aalyss of the free vbrato of a beam wth multple cracks s to dvde the beam to sub-beams wth dfferet modal dsplacemet fuctos for each sub-beam. Therefore, case of -cracks the beam, four boudary codtos ad 4 cotuty codtos have to be employed, ad the egevalue equato of the problem s pressed by a 4+) order determat equated to zero []. Recet studes amed at fdg more effcet approaches able to reduce the order of the determat dow to + []. However, the most terestg approach s that preseted by L [3] proposg a determat of order two, avodg the fulfllmet of the cotuty codtos at the crack locatos by meas of a recursve presso. I ths work a approach to study the free vbrato of damaged beams based o modellg the cracks by meas of dstrbutos geeralsed fuctos), such as the Drac s delta, s proposed. For the case of a beam wth cracks, the proposed approach leads to plct pressos for the egemodes depedet o the testes ad postos of the damages ad four tegrato costats, whle the egevalue equato s obtaed plct form by the evaluato of a fourth order determat. O the bass of the plct pressos provded for the egemodes a ovel multple damage detfcato procedure ca be set, oce at least the frst two ege-modes, together wth the

2 respectve egevalues, are gve by free vbrato permetal tests. By equatg the values of the frst permetal ege-mode to the gve aalytcal plct presso a o-lear set of equatos s obtaed. A sutable cremet of varables to be detfed leads to a tragular structure for the latter set of equatos. A coveet closed form soluto cascade for the damage testes uder the hypothess that the cocetrated damage locatos cocde wth the measuremet postos s obtaed. I case the cocetrated damages le betwee two measuremets, the secod permetal ege-mode allows ther localzato by meas of a umercal procedure whch s however decoupled for each damage. THE DIRECT PROBLEM The dfferetal equato goverg the free vbrato of a mult-cracked beam may be wrtte the followg form: EI o o γδ x xo) u xt, ) + mu xt, ) = 0 ) where EI x) = E oio γδ x xo)) descrbes a flural stffess wth cracks whch are represeted as Drac delta dstrbutos δ x x o ) the flural stffess cetred at crosssectos xo, =,,. Ths model, already appled statc [8,9] ad stablty [0], s equvalet to cosder a straght beam wth massless elastc rotatoal sprgs. By cosderg the o-dmesoal coordate ξ = x/l, the dfferetal Eq. ) takes the followg form: 4 ml γδ ξ ξo) u ξ, t) + u ξ, t) = 0 EI ) o o where the property δ x xo ) = / L) δ ξ ξo ) of the Drac s delta dstrbuto has bee ploted, ad the dmesoless damage parameters γ = γ / L have bee troduced. The soluto of Eq. ), wth the use of separato of varables, ca be gve the followg form: u ξ, t) = y t) φ ξ). 3) Substtuto of Eq. 3) Eq. ) yelds to the followg dfferetal equato for modal dsplacemets, that, after some smple algebrac mapulato, ca be wrtte the form: 4 γδ ξ ξo) φ ξ) α φ ξ) = 0 4) 4 4 where the frequecy parameter α = ω ml EoIo has bee troduced. Eq. 4), by performg double dfferetato wth respect to ξ of the frst term cotag the Drac s delta dstrbuto, ad after smple algebra, may be gve the followg form: v 4 φ ξ) α φ ξ) = B ξ) 5) where the fucto B ξ ) collects all the terms wth the Drac s deltas ad ther dervatves as follows: ) ν ) ) B ξ = γφ ξ δ ξ ξo) + γφ ξ δ ξ ξo) + γφ ξ) δ ξ ξo). 6)

3 The geeral plct soluto of Eq. 5) has bee derved [] by makg use of geeralsed fuctos ad may be wrtte as follows: φ ξ) = C λμ sα ξ ξo ) sh α ξ ξo ) U ξ ξo ) sαξ C λυ sα ξ ξo ) sh α ξ ξo ) U ξ ξo ) cosαξ ) + C3 λζ sα ξ ξo ) sh α ξ ξo ) U ξ ξo ) shαξ C4 λη sα ξ ξo ) + sh α ξ ξo ) U ξ ξo ) + coshαξ where: U ξ ξo) represets the Heavsde ut step fucto, λ = γ/ Aγ), wth A a arbtrary costat, are the damage testy parameters related to γ, ad the terms μ, ν, ζ, η are gve by the followg pressos: j α μj = λμ sα ξoj ξo) shα ξoj ξ ) s + o α αξoj j α υj = λυ sα ξoj ξo) shα ξoj ξo) cos oj + α αξ 8) j α ζ j = λζ sα ξoj ξo) shα ξoj ξo) α shαξoj + + j α ηj = λη sα ξoj ξo) + shα ξoj ξ o ) + α cosh αξoj. The soluto gve by Eq. 7) ca also be pressed the followg more compact form: φ ξ) = C λμ S α, ξ) U ξ) + s αξ + C λυ S α, ξ) U ξ) + cosαξ + 9) + C3 λζ S α, ξ) U ξ) + sh αξ + C4 λη S α, ξ) U ξ) + coshαξ where the compact otato U ξ ) = U ξ ξo) has be adopted ad, for smplcty, the followg fucto S αξ, ) = sα ξ ξo) + shα ξ ξo) has bee defed. It s worth otg that the soluto pressed by the Eqs. 7) or 9) s vald for the overall beam ad for ay umber ad postos of cracks, furthermore, t preserves the same aalytcal structure of the udamaged beam. Eqs. 7) or 9) ca also be used for aalysg the o-lear dyamc behavour of mult-damaged beams wth closg cracks [] as well as for dervg the dyamc stffess matrx of the mult-cracked beam wth ope cracks. 3 THE INVERSE CRACK IDENTIFICATION PROCEDURE It s worth otg that, due to the aalytcal structure of Eq. 7), the values of the m-th mode shape φm ξ ), cept for the tegrato costats C, C, C3, C 4, deped oly o the damages that are located at postos ξ < ξ. Cosderg the + beam segmets dvduated by the cracked cross-sectos, as reported Fgure, the soluto for each segmet, pressed for the overall beam by Eqs. 7) or 9), may be specalsed, for each segmet, as follows:

4 0) ) C C C C ) φ ξ = sαξ + cosαξ + shαξ + coshαξ = φ ξ 0 < ξ ξ 3 4 o 0) λ φ ξ) = φ ξ) + S α, ξ)[ Cμ+ Cυ+ C3ζ+ C4η] ξo < ξ ξo 0) λ φ ξ) = φ ξ) + S α, ξ)[ Cμ+ Cυ+ C3ζ+ C4η] +... λk... + S αξ, )[ C μ + C υ + C ζ + Cη ] ξ < ξ ξ k k k 3 k 4 k o k ) ok 0) Fgure : The beam segmets betwee the eds ad the cracked cross-sectos. From the observato of the aalytcal structure of the plct soluto emerges that t possesses a tragular structure wth referece to the testy ad postos of the damages, cept for the tegrato costats whose values are flueced by the boudary codtos ad by the damage testes ad postos as t has bee show []. For the m-th mode shape φm ξ ), the tragular structure of the Eq. 7), also mataed by ts dervatves, ca be hghlghted as follows: φ ξ = φ C, C, C, C 0< ξ ξ ) ) ) C, C, C, C,, ) ) C, C, C, C,,,, ) ) C, C, C, C,,...,,,..., ) Accordg to Eq. ), the geerc mode shape m o) m o m 3 4 o o φ ξ = φ ξ λ ξ < ξ ξ m o m 3 4 o o o o φ ξ = φ ξ ξ λ λ ξ < ξ ξ m o m 3 4 o o o o o3 φ ξ = φ ξ ξ λ λ ξ < ξ ξ + m ok m 3 4 o ok k ok ok ok φ ξ evaluated at abscssa o ξ located betwee the abscssa 0 ad the frst cracked cross secto 0 < ξo ξo ) depeds oly o the tegrato costats C, C, C3, C 4. The value of the mode shape φ m ξ ok) evaluated at the geerc abscssa ok ξ stuated betwee the cracks k ad k+ ξ ok < ξok ξ ok + ), besdes the tegrato costats, depeds oly o the prevous damages,.e. the damages collocated at postos ξ,..., ξ < ξ. o ok ok Ths partcular aalytcal structure of the soluto suggests two procedures for solvg the verse problem based o dyamc tests whch are descrbed ad dscussed the followg. The frst approach accouts for posto of the sesors cocdet wth the cracked cross-sectos ad, measurg oe vbrato mode, leads to plct pressos of the damage testes to be detfed. The secod procedure s able to provde both the postos ad the testes of the damages, based o the measuremets of two vbrato modes, by meas of a umercal procedure. I partcular the case of a mult-cracked free-free beam s treated ad the relablty of the proposed procedures wll be verfed by meas of fte elemet umercal smulatos o damaged beams modelled by meas of two-dmesoal shell elemets. )

5 3. Evaluato of the damage testy wth measuremets at the cracked cross-sectos The prevously descrbed aalytcal structure of the soluto leads to a detfcato procedure whch provdes plct pressos of the damage testes as a fucto of the values of a mode shape or ts dervatves) at the damaged cross-sectos, assumg that the measuremets are take at the damage postos. Accordg to Eq. ), the values of the m-th mode shape φm ξ ok) at the abscssa ξ ok of the k-th crack depeds learly o the tegrato costats C, C, C3, C 4 ad o the prevous damages oly λ,..., λk located at the abscssae ξo, ξo,..., ξok as follows: φm ξok) = φm C, C, C3, C4, λ,..., λk ). ) If the beam s restraed by meas of perfect costrats oly two costats are eeded to represet the correspodg mode shape ad Eq. ) ca be wrtte as: φm ξok) = φm C, C, λ,..., λk ). 3) Therefore order to evaluate the + ukows damage testes ad tegrato costats) the correspodg codtos ca be obtaed by equatg the permetal ad the theoretcal results at + cross sectos. I order to take advatage of the tragular scheme of the soluto, frstly t s coveet to detfy the tegrato costats C, C, ad afterwards the successve measuremets wll provde plct pressos for each damage testy as a fucto of the prevously evaluated ukows. The values of the tegrato costats ca be easly detfed cosderg oe measuremet of the frst vbrato mode at a posto that precedes the frst damage cross secto o ξ < ξ o, ad a further measuremet correspodg to the frst cracked cross-secto ξ o. By equatg the permetal values wth the theoretcal presso, correspodg to the measured frst frequecy parameter α, the followg lear system of two equatos s obtaed: th th φ ξo) = φ α, C, C), φ ξo ) = φ α, C, C) 4) I Eq. 4) the oly ukows are the tegrato costats C, C whch ca be evaluated closed form. Oce the tegrato costats have bee evaluated, the measuremet at the secod cracked cross-secto, accordg to Eq. 3) wrtte at abscssa ξ o, leads to the evaluato of the frst damage parameter λ, ad so o. For the evaluato of the last damage testy λ a further measuremet located at a posto that follows the fal damaged cross-secto s eeded. 3. Evaluato of both the posto ad the testy of damages wth measuremets betwee cracked cross-sectos If the damage postos are ot kow the beam may be strumeted wth several sesors to verfy the presece of damage betwee two measuremet pots. Assumg perfect costrat codtos, the value of the m-th mode shape φ m ξ ok) at the geerc abscssa ok ξ, where the sesor s located, depeds learly o two tegrato costats C, C ) ad o the prevous damage testes λ,..., λ k ) ad olearly o the correspodg locatos represeted by the abscssae ξo, ξo,..., ξ ok. Oce the tegrato costats have bee detfed as the procedure outled secto 3., the damage testes ad the correspodg postos ca be obtaed by equatg the permetal measuremets ad the theoretcal pressos of two vbrato modes m,s, for each segmet of the beam betwee two subsequet cracks, as follows:

6 th ) = C, C,,...,,,..., ) th ) = C, C,,...,,,..., ) φm ξok φm ξo ξok λ λk ξ ok < ξok ξ ok + 5) φs ξok φs ξo ξok λ λk ad solvg wth respect to λ, ξ the olear sstem 5) by meas of a sutable umercal k ok procedure. Also ths case, whch the damage postos are ot kow a pror, order to take advatage of the tragular scheme of the soluto, t s coveet to evaluate, frst, the tegrato costats ad tha proceed to the evaluato of the damage testy ad locato, crack by crack, startg from the frst crack. 4 IDENTIFICATION OF MULTIPLE CRACKS ALONG A FREE-FREE BEAM I ths secto the closed-form soluto preseted Eq. 9) s adopted to treat the case of a Free-Free Euler-Beroull beam ad the verse problem wll be formulated. For a Free-Free beam, the boudary codtos at the left ad rght eds may be wrtte as φ 0) = 0; φ 0) = 0; φ ) = 0; φ ) = 0; 6) by meas of the codtos 6), the followg pressos of the four tegrato costats for the k-th vbrato mode ca be derved: C = C; C = ϑkc; C3 = C; C4 = ϑkc. 7) Therefore the plct presso of the geerc vbrato mode of a mult-cracked free-free beam may be wrtte as. φk ξ) = C λ μ + ζ) S αk, ξ) U ξ) + sαξ k + shαξ k k 8) + ϑk λ υ + η) S αk, ξ) U ξ) + cosαkξ + cosh αkξ. k It s worth otg that, due to the aalytcal structure of Eq. 8), the values φk ξ ) deped oly o the damages that are located at postos ξ < ξ, amely: φk ξ) = C sαkξ + shαkξ) + ϑk cosαkξ + coshαkξ) 0 < ξ ξo φk ξ) = C λ μ+ ζ) S αk, ξ) + sαkξ + shαkξ k + ϑk λ υ+ η) S αk, ξ) + cosαξ k + coshαξ k ξo < ξ ξ 9) o k j φk ξ) = C λ μ + ζ) S αk, ξ) + sαkξ + shαkξ k j + ϑk λ υ + η) S αk, ξ) + cosαkξ + cosh αkξ ξo j < ξ ξoj. k Accordg to the procedure descrbed secto 3., t s possble to detfy the presece ad quatfy the damage at the cross-sectos where the measuremet sesors are located. The procedure s based o the kowledge from permetal tests of at least a frequecy parameter α m ad the correspodg mode φ m ξ). I the followg, referece to the frst frequecy α ad the correspodg vbrato mode φ ξ ) s made. It s assumed that at the

7 abscssae o ξ ad ξ +, where the frst ad the last sesors are located, o damage occurs, whle the other strumets are located at the cracked cross-sectos, ξ,..., ξ, as depcted Fgure. o o Fgure : Crack ad sesor postos alog the beam spa. Employg the frst two measuremets of the frst ege-mode φ ξ o ), ξ o ), uder the assumpto that o ξ < ξ o, ad by equatg the permetal ad the theoretcal results, accordg to Eq. 7), the followg system of equatos s obtaed: ) ) φ ξ o = C ϑ sα ξo + shα ξo + cosα ξo + coshα ξ ) o 0) φ ξo ) = C ϑ sα ξo + shα ξo ) + cosα ξo + coshα ξo ) from whch the followg values of the costats C ad ϑ are derved plct form: φ ξ o)[s α ξo ) + sh α ξo )] φ ξo )[s α ξo) + sh α ξo)] ϑ = φ ξ ) o cos α ξo) cosh α ξo) φ ξo) cos α ξo ) cosh α ξo ) + + ) φ ξo ) C =. s α ξo) + sh α ξo) + ϑ cos α ξo) + cosh α ξo) Equatg the measuremet at the secod cracked cross-secto φ ξ o ) at ξ to the theoretcal presso φ th ξ o ) of the frst mode leads to the followg equato: φ ξo) = C ϑ λ μ+ ζ) S α, ξo) + sα ξo + shα ξo ) λ υ+ η) S α, ξo) + cosα ξo + coshα ξo whch the oly ukow s the tet of frst damage λ preset at ξ o. Oce the frst damage has bee detfed, the secod damage testy λ ca be obtaed by meas of the further measuremet φ ξ o 3) at ξ o3 ad so o. The testy λ of the geerc damage ca be wrtte plctly as follows: [ φ ξo+ )] C[s α ξo+ ) + sh α ξo+ ) + ϑcos α ξo+ ) + cosh α ξo+ ))] λ = + C[ μ + ς + ϑ ν + η)][s α ξo+ ξo)) + sh α ξo+ ξo))] 3) λjc[ μ + ς + ϑ ν + η)][s α ξo+ ξo)) + sh α ξo+ ξo))] j=. C[ μ + ς + ϑ ν + η )][s α ξ ξ )) + sh α ξ ξ ))] o+ o o+ o If there s o crack at the cross-secto ξ o, the detfed damage parameter λ gve by Eq. 3) wll be zero, dcatg the absece of damage. O the other had, the case the measuremet postos are ot cocdet wth the damage locatos ξ o.e. the sesor are placed at abscssae o ξ betwee damages ξ o < ξo < ξ o + ), both testy ad damage posto are ukow. The measuremets of the secod frequecy ad the correspodg ege-mode provde the suffcet addtoal data, ad, accordg to the procedure φ

8 secto 3., Eq. 5) for the case uder study, where the frst ad the secod vbrato modes are gve by permetal data, ca be wrtte as follows: j ) ) ) φ ξo = C, s sh λ μ + ζ S α ξo + α ξo + α ξo j ), ) cos + ϑ o o cosh λ υ + η S α ξ + α ξ + α ξo ξ o < ξo ξ o +. 4) j ) ) φ ξ, ) s o = C λ μ + ζ S α ξo + α ξ o + shα ξo j ), ) cos ϑ + k λ υ + η S α ξo + α ξo + coshα ξo The olear sstem of equatos 4) has to be solved wth respect to ξ o, appearg the terms μ, ς, υ, η, ad to the testy λ, by meas of a umercal procedure, for each damage cascade startg from the frst damage. 5 APPLICATION The damage detfcato procedure proposed has bee tested agast a fte elemet model of a cracked Free-Free beam ad the results are brefly summarsed ths secto. A steel beam, free at ts both eds, of leght L=. m wth a rectagular cross-secto b=9.8 mm, h=. mm) has bee cosdered for FEM smulato by makg use of the code Sap000. The beam s subjected to 3 damages cocetrated at the abscssae ξ o = 0., ξ o = 0.5, ξ o3 = 0.7, whose testes λ = 0.075, λ = , λ 3 = 0.586, are related, as descrbed [0], to the ratos of the crack depth to the cross-secto hegth d / h= 0.375, d / h= 0.5, d3 / h = 0.75, respectvely. For ths smulato, measuremets placed as Fgure 3 has bee cosdered avalable ad 3 of them are cocdet wth the damage postos accordg to the hypothess secto 3.. The beam has bee modelled by meas of shell elemets ad the results of the FEM modal aalyss atural frequeces ad vbrato modes) has bee cosdered for the detfcato procedure. Fgure 3: Crack ad sesor postos alog the beam spa. Accordg to the plct soluto proposed Eq. 3), by makg use of the frst atural frequecy ad vbrato mode, the damage testy parameters λ ca be evaluated for each measuremet posto ad the results are reported Fgure 4 terms of the rato d / h showg a maxmum error of 7.8%. Eq. 3) provdes zero values for the damage testes for those measuremets where the damage s abset. The same steel beam has bee the cosdered subjected to damages at ξ o = 0.5, ξ o = 0.45 whose testes λ = , λ = 0.586, correspod to d / h= 0.5, d / h = 0.75, respectvely. The measuremets are placed as Fgure 5 ad they are ot cocdet wth the damage postos accordg to the hypothess secto 3.. The frst ad the secod atural

9 frequeces ad vbrato modes of the beam, obtaed by the FEM modal aalyss, have bee cosdered for the detfcato procedure. Fgure 4: Idetfed damage testy for each measuremet posto Fgure 5: Crack ad sesor postos alog the beam spa. The system of Eqs. 4) has bee solved umercally for each segmet of the beam betwee two measuremets order to detfy the damage testy ad posto of each damage separately. I partcular, the solutos of Eq. 4a) ad 4b) terms of damage testy λ are plotted agast the damage posto ξ o for each terval; the tersecto pot betwee the two solutos provdes the correct damage testy ad posto. The results are show Fgure 6 terms of λ ad of the rato d / h showg a maxmum error of.6 % for the detfed posto ad 7.0% for the detfed testy. For those tervals where the damage s abset, there s o tersecto pot betwee the solutos of Eqs. 4a) ad 4b). 6 CONCLUSIONS I ths work a model for the Euler-Beroull beam wth multple cocetrated cracks based o the geeralsed fuctos dstrbutos) have bee adopted. Closed form solutos for the vbrato modes have bee preseted terms of testes ad postos of the damages ad depedet o four tegrato costats to be determed by the stadard boudary codtos. The latter solutos ca be effcacously employed to set a detfcato procedure for beams wth multple damages based o measuremets of the mode shapes by vbrato tests. I partcular the advatage of the proposed procedure cossts a sequetal detfcato of sgle damages ether cocdet wth the posto of the permetal measuremets or lyg the beam tervals betwee two permetal measuremets.

10 Fgure 6: Idetfed damage testes betwee each measuremet posto Refereces [] P.F. Rzos, N. Aspragatos ad A.D. Dmarogoas, Idetfcato of crack locato ad magtude a catlever beam from vbrato modes, J. Soud ad Vbrato, 38, , 990) [] E.I. Shfr ad R. Ruotolo, Natural frequeces of a beam wth a arbtrary umber of cracks, J. Soud ad Vbrato, 3, , 999). [3] Q.S. L, Free vbrato aalyss of o-uform beams wth a arbtrary umber of cracks ad cocetrated masses, J. Soud ad Vbrato, 5, , 00). [4] Y. Narks, Idetfcato of crack locato vbratg smply supported beams, J. Soud ad Vbrato, 7, , 994). [5] A. Morass, Idetfcato of a crack a rod based o chages a par of atural frequeces, J. Soud ad Vbrato, 4, , 00). [6] M. Dlea ad A. Morass, The use of atresoaces for crack detecto beams, J. Soud ad Vbrato, 76, 95 4, 004). [7] A.D. Dmarogoas, Vbrato of cracked structure: a state of the art revew, Eg. Fract. Mech., 55, , 996). [8] B. Bod, S. Caddem, Closed form solutos of Euler-Beroull beams wth sgulartes, Iteratoal Joural of Solds ad Structures, 4, , 005). [9] B. Bod, S. Caddem, Euler-Beroull beams wth multple sgulartes the flural stffess Europea, Joural of Mechacs A/Solds, 65), , 007). [0]S. Caddem, I. Calò, Exact soluto of the mult-cracked Euler-Beroull colum, Iteratoal Joural of Solds ad Structures, 456), 33-35, 008). []S. Caddem, I. Calò, Exact soluto of the mult-cracked Euler-Beroull colum, Joural of Soud ad Vbratos, press, 009). []S. Caddem, I. Calò, M.Marletta, Exact soluto of the mult-cracked Euler-Beroull colum, Iteratoal Joural of Nolear Mechacs, submtted to, 009).