Structural damage identification by sensitivity of modal strain energy based on optimization function

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1 Structural damage dentfcaton by senstvty of modal stran energy based on optmzaton functon Alreza Entezam 1 *, Hashem Sharatmadar 2 1. M.Sc. of Structural Engneerng, Department of Cvl Engneerng, Ferdows Unversty of Mashhad, Iran, entezam@stu-mal.um.ac.r 2. Assocate Professor, Department of Cvl Engneerng, Ferdows Unversty of Mashhad, Iran, sharatmadar@um.ac.r Abstract Many methods for the assessment of structural damage based on vbraton-based technque are surveyed for many decades. In ths paper, an effectve damage detecton methods based on numercal and expermental evaluaton of structural damage s presented by senstvty of modal stran energy and usng a new model updatng optmzaton functon. Frst, damage localzaton n the dynamc structures s formulated by fundamental modal stran energy (MSE) to precsely locate the eventual damage of a structure. Wth locaton determned, the correspondng damage extent can be obtaned by senstvty of modal stran energy based on changes n the stffness components. Subsequently, usng of senstvty matrx and dfference between mode shapes of healthy and damaged structures, a new optmzaton functon s defned to accurately estmate the damage severty. For verfcaton of the proposed methods, two llustratve examples as a numercal model of a cantlever beam and a 3-story laboratory frame as expermental model are used. In the expermental nvestgaton, modal analyss wth mpact hammer s carred out to laboratory frame and vbratonal modes are dentfed as real data. Eventually, results show that the modal stran energy ndcator can be exactly detected the locaton of damage. hen, the estmated damage extent ndcates the proposed method based on senstvty of modal stran energy and mnmzaton of optmzaton functon can provde the convenent results n the numercal and expermental evaluatons. Key words: Structural damage detecton, Modal stran energy ndctor, Senstvty of modal stran energy, Optmzaton functon 1. Introducton Vbraton-based methods have been developed and appled to assessment of structural damage n many exstng structures that related to cvl, mechancal and aerospace engneerng felds for many decades. Among the vbraton-based methods, those based on modal analyss are wdely used. Generally modal parameters are defned as natural frequency, mode shape and modal dampng rato. hese data depend on the physcal propertes of a 1

2 structure and not the exctaton appled. herefore, parameter dentfcaton of structures has become very mportant as researchers attempt to correlate changes n test data to the changes n the structural element propertes. Modfcaton of physcal parameters of structures such as mass, stffness or dampng propertes are lead to occur changes n the vbratonal response of structures or modal data. Hence, changes n the physcal propertes wth adversely performance of dynamc behavour of structures are descrbed as structural damage. ypcally, damage detecton algorthms are categorzed as three steps, namely detecton of present of damages, detecton of the structural damage locatons and estmaton of the damage extents [1]. herefore, the knowledge of vbraton-based methods can be used to determne the exstence as well as the locaton and the extent of damage. For the damage detecton problem, many researchers have worked n these felds for many decades. Doeblng et el. [2], stubbs et al [3] and Yan et al. [4] have been provded lterature revews n damage detecton process. Gudmundon [5] ntroduced a frst order perturbaton method to predct cracks, other geometrcal changes and mathematcally showed that the change n egenvalue was related to the change n stran energy of the system. Lee and Chung [6] appled Gudmundson s theory to dentfy the locaton and severty of an edge crack n a cantlever beam. He et al, [7] presented a computatonally attractve damage ndex method s proposed for structural damage detecton of cylndrcal shell. Accordng ths approach, the modal stran energy values computed for undamaged and damaged states were used n the correspondng damage ndex algorthms and two parameters as moment response power spectral densty and curvature response power spectral densty were dvded. For all measured mode shapes, the damage ndex was defned by usng the statstcal parameter of relatve root mean-square error of case before and after damage. Fan and Qao [8] ntroduced a new stran-based damage detecton for plate-type structures. hey proposed the concepts of a damage locaton factor (DLF) matrx and a damage severty correcton factor (DSCF) matrx, whch can be derved from the elemental modal stran energy. Hence, the damage dentfcaton method usng the DLF and DSCF was developed for damage localzaton and quantfcaton n plate-type structures. Usng fnte element model updatng for damage detecton method, Jash and Ren [9] proposed a new multobjectve optmzaton technque for damage localzaton and quantfcaton n the beam-lke structures. In that methods, egenfrequency resdual and modal stran energy resdual were used as two objectve functons of the multobjectve optmsaton. Also, Seyedpoor [10] provded a two-stage method to properly dentfy the ste and extent of multple damage cases n structural systems. In the frst stage, a modal stran energy based ndex (MSEBI) was presented to precsely locate the eventual damage of a structure. he modal stran energy was calculated usng the modal analyss nformaton extracted from a fnte element modellng. In the second stage, the extent of actual damage was determned va a partcle swarm optmzaton (PSO) usng the frst stage results. 2

3 he objectve of ths artcle s dentfcaton of structural damage by senstvty of modal stran energy wth fnte element model updatng optmzaton functon. For damage localzaton a modal stran energy ndcator s expanded at frst and then, changes of stffness components as damage ndex case can be detected. ypcally, the damage quantfcaton pertans to locaton of damage. herefore, the senstvty of modal stran energy s determned to use n the model updatng optmzaton functon. After provdng the optmzaton functon, the extent of damage s estmated by mnmzaton of proposed optmzaton functon based on MSE senstvty matrx and error vector contanng the dfferences n mode shapes before and after damage. For verfcaton of the proposed methods, two llustratve examples as a numercal model of a cantlever beam and a 3-story laboratory frame are used. In the expermental model, modal analyss wth mpact hammer s carred out to laboratory frame and modal parameters are dentfed as real data. Eventually, results show that the modal stran energy ndcator can be exactly detected the locaton of damage. hen, the estmated damage extent ndcates the proposed method based on senstvty of modal stran energy and mnmzaton of optmzaton functon can provde the convenent results n the numercal and expermental evaluatons. 2. heory 2-1- Modal stran energy ndex he equatons of moton of the free vbraton of a lnear undamped dscrete system wth N degrees of freedom can be gven by K M, 1,2,..., N (1) Where, M and K are the mass and stffness matrces, respectvely. λ and φ are the th egenvalue (square of natural frequency, λ =ω 2 ) and assocated egenvector (mode shape), respectvely. Also, N s the total degrees of freedom of the structure. he mode shapes are usually normalzed wth respect to the mass matrx. Hence, n ths paper consders the problem for undamped symmetrc systems wth dstnct egenvalues. As the mass matrx M s non-sngular, the egenvectors are usually normalzed as M 1 (2) Snce the mode shape vectors are equvalent to nodal dsplacements of a vbratng structure, therefore n each element of the structure stran energy s stored [10]. he stran energy of a structure due to mode shape vector are usually referred to as modal stran energy (MSE) and can be consdered as a valuable parameter for damage dentfcaton. he modal stran energy n th mode of the structure can be expressed as 1 MSE K (3) 2 Assume the global stffness matrx s assembled by m ndvdual element stffness matrces and that s 3

4 e K K (4) j1 he eth element MSE can be then gven by e 1 MSE Ke, 2 1,2,..., N (5) e1 e 2-1- Damage localzaton by modal stran energy method It s possble to use of Eq. (5) to detecton of damage locaton. For computatonal propose, t s approprate to normalzed the MSE of elements wth respect to the total MSE of the structure NMSE K (6) e e Ke where, NMSE s the normalzed MSE of eth element n th mode of the structure. Sometmes, achevement to complete modal data s mpossble. herefore, for only m dentfed modes the MSE can be rewrtten as follow MNMSE m e NMSE 1 e (7) m he modal stran energy ndex of healthy and damaged structures as form NMSE h and NMSE d must be defned to detect the damage locaton, respectvely. Generally, placng the mode shapes of healthy and damaged structures nto Eq. (6), the correspondng MSE ndex for m modes are obtaned. On the other hand, healthy modal stran energy ndex s smlar to typcal relatonshp for global modal stran energy method. he modal stran energy ndex for damaged structure s wrtten as follow 1 MNSE K, 1,2,..., m (8) d e e1 2 where, and α denote the damaged mode shape and stffness modfcaton factor for damaged states, respectvely. he damage occurrence s led to ncreasng the MSE and consequently the effcent parameter NMSE for m modes. As a result, an ndcator s termed as general modal stran energy ndex (λ MSE ), whch can be determned as MNMSEd MNMSEh MSE (9) MNMSE It should be noted that, as the damage locatons are unknown for the damaged structure wth respect to real data applcatons, therefore for ths case the element stffness matrx of the healthy structure s used for estmatng the parameter (MNMSE) d. Accordng to the Eq. (9), for a healthy element the ndex wll be equal to zero (λ MSE =0) and for a damaged element the ndex wll be greater than zero (λ MSE >0). h 4

5 2-3- Damage quantfcaton by modal stran energy senstvty analyss Desgn senstvty analyss s used to quantfy the relatonshp between parameters used to defne an optmum desgn and calculate outputs used to measure ther performance. Desgn senstvty analyss of structural and mechancal systems wth respect to structural desgn parameters plays a crtcal role n nverse and dentfcaton problems n engneerng applcatons [11]. Generally senstvty analyss descrbes the rates of change of some of key propertes of the dynamc model such as natural frequences and mode shapes wth small changes n some of the physcal propertes consst of ndvdual mass and stffness matrces [12]. herefore, dervatves of dynamc response of structures toward to physcal propertes are usually descrbed the senstvty analyss n the dynamc structures. MSEe 1 Ke K e (10) p p 2 p he dervatves of egenvalues wth respect to the desgn varable p can be easly obtaned by dfferentaton of the undamped egenvalue, but the dervatves of mode shapes cannot be found drectly due to t needs overcome the sngular problem [11]. For dealng wth these lmtaton to calculaton of modal stran energy senstvty, Yan and Ren [13] derved a compact analytcal expresson of the element MSE senstvty based on the algebrac method. hs method computes the senstvty of element MSE usng the followng as MSEe * K (11) p where K M 1 * K M M p p 1 Ke Ke 0.. M 0 1 M 2 p 2 K As t can be noted, ths method s an accurate method, whch only requres the egenvector of nterest. And t can be found the desgn senstvty of element MSE n a very smple and straghtforward manner. In ths study, damage s assumed to be drectly related to a decrease n stffness. herefore, damage can be located usng the senstvty of the modal stran energy wth respect to the stffness parameters. Hence, wth neglectng of the mass matrx modfcaton, the Eq. (12) s rewrtten to form 1 K * K M M 1 Ke K Ke 0.. p M 0 (13) 2 p 0 Once the senstvty of modal stran energy are computed, the senstvty matrx s buld and the change n the stffness parameters s estmated by mnmzng of the optmzaton functon, p (12) 5

6 J S k W S k k W k (14) MSE MSE kk S MSE s the modal stran energy senstvty matrx, whch can be descrbed as follow 1 K K M M 1 Ke SMSE Ke 0.. p M 0 2 p (15) 0 Δφ s the error vector contanng the dfferences n mode shapes before and after damage. W εε, W kk are postve defnte weghtng matrces. W εε s a dagonal matrx whose elements are gven by the recprocals of the varance of the correspondng measurements. W kk =αi s a dagonal matrx whose elements are equal to the regularzaton parameter α (konov regularzaton). A detaled explanaton of the dervaton of ths equaton s found n the book of Frswell and Mottershead [14]. he soluton of equaton (14) s obtaned through least squares as follow, 1 MSE MSE kk MSE k S W S W S W (16) where, Δk s the damage quantfcaton based on modal stran energy senstvty method. In the numercal evaluaton, the damage detecton process s usually carred out to nduce the damage ndex and accordng to proposed method, predcated damage ndex can be estmated as stffness reducton. 3. Applcaton 3-1- A cantlever beam In ths secton, the damage detecton and damage severty models descrbes n the precedng secton used to dentfy the locaton and determne the magntude of reducton of stffness on a cantlever beam. he beam has been shown n Fg. 1. he length, thckness and wdth of the beam are 1.20, 0.05 and 0.1 m, respectvely. he mass densty s 7850 kg/m3 and the elastcty modulus s 210 GPa. In ths example, the frst 5 vbratng modes are used for dentfyng the damage. herefore, consder the ncomplete modal data are avalable. Fgure 1. A cantlever beam he fnte element analyss s carred out to smulate the modal data, usng two-node beam elements [15]. Here, four damage cases are assumed to nvestgate the capabltes of the proposed methods n detecton of the occurred damage of a flexural structure. In the frst damage case, the stffness of element 2 was decreased by 30%. In damage case number two, the stffness of elements 5 reduced by 40%. In the thrd damage case, the stffness of element 6

7 2 and 5 decreased va 20% and 30%, respectvely. Fnally, n the damage case number four, the stffness of element 8 reduced by 25%. Based on the proposed damage assessment algorthms, locaton of nduced damages s detected by modal stran energy ndex from Eq. (9). Correspondng to damage quantfcaton, senstvty of MSE s frstly determned to specfy the changes of dynamc behavour. Subsequently, dfference of egenvectors (mode shapes) between healthy and damaged structures are computed. Eventually, usng of Eq. (16) the vectors of damage parameters wll be estmated. Fgure 2. Damage localzaton of the cantlever beam n scenaro 1, a) For 3 dentfed modes, b) For 10 dentfed modes Fgure 3. Damage localzaton of the cantlever beam n scenaro 2, c) For 3 dentfed modes, d) For 10 dentfed modes 7

8 Fgure 4. Damage localzaton of the cantlever beam n scenaro 3, e) For 3 dentfed modes, f) For 10 dentfed modes Fgure 5. Damage localzaton of the cantlever beam n scenaro 4, g) For 3 dentfed modes, h) For 10 dentfed modes It can be observed that the modal stran energy ndex acheves to true locaton of nduced damage cases, even for multple damage case and lmtaton of dentfed modes. Accordng to Fgs. 2-5, the locaton of nduced damage s precsely detected, whle 10 dentfed modes have better results than 3 dentfed modes. he error functon for undamaged element n each damage cases s nconsderable values and accordng to all Fgs. 2-5, the peaks of damaged element are clearly demonstrated to damage localzaton. Also, Fgs. 6-9, llustrate the extent of damage cases based on senstvty modal stran energy and optmzaton functon. 8

9 Fgure 6. Damage quantfcaton of the cantlever beam n scenaro 1, a) For 3 dentfed modes, b) For 10 dentfed modes Fgure 7. Damage quantfcaton of the cantlever beam n scenaro 2, a) For 3 dentfed modes, b) For 10 dentfed modes Fgure 8. Damage quantfcaton of the cantlever beam n scenaro 3, a) For 3 dentfed modes, b) For 10 dentfed modes 9

10 Fgure 9. Damage quantfcaton of the cantlever beam n scenaro 4, a) For 3 dentfed modes, b) For 10 dentfed modes As can be seen, extent of damage cases were estmated for 3 and 10 dentfed modes, respectvely. he fgures belong to three dentfed modes, error functon of predcted damage than nduce damage has consderable values, and hence the damage quantfcatons do not valdate based on numercal results. Whereas, for ten dentfed modes, error functon of predcted have better results and close to nduced damage. As a result, whatever the number of modes ncreases, more accurate results can be acheved A 3-story expermental frame For expermental evaluaton, the 3-story laboratory frame used n ths study was approxmately 2.1 m hgh and constructed from equal angel alumnum column sectons and steel floor plates bolted together wth alumnum brackets as shown n Fg. 10. he steel floor plates were 4 mm thck and 650 mm 650 mm square. Hence, each story has kg weght. he column sectons at each story were 30 mm 30 mm equal angles. wo secton thcknesses were used for the columns, ether 4.5 mm or 3 mm, for the undamaged and damaged states, respectvely. Each column was made of m hgh segments, rather than one long angle, n order to make them easly replaceable for smulaton of localzed damage at dfference stores. Based on actual propertes of components of each story, stuffness values ncludng of N/m, 9891 N/m and 9078 N/m for the frst, second and thrd stores, respectvely. he proposed damage localzaton approach was accomplshed on three-story laboratory frame by expermental modal analyss. A modal analyss wth mpact hammer s performed n the undamaged state. able (1) and (2) ndcate the extracted modal parameters of healthy frame. 10

11 Fgure 10. he 3-story laboratory frame able 1. Expermental mode shapes of 3-story laboratory frame [φ] Mode 1 Mode 2 Mode 3 Story Story Story able 2. Expermental natural frequency of 3-story laboratory frame (Hz) Degree of freedoms Mode 1 Mode 2 Mode 3 Natural frequency Accordng to dentfyng of modal parameters of healthy structures, two damage cases were ntroduced and expermental modal analyss was separately carred out for all cases of damage n the three-story laboratory frame. In the frst damage case, the alumnum angels wth 30 mm 30 mm and thckness of 3mm were replaced rather than the columns of frst story. In thrd damage case, the columns of thrd story as well as the frst story were replaced wth columns smlar to damage case number one. herefore, changes of dynamc behavour of laboratory frame can be llustrated based on able 3. able 3. Natural frequences evaluaton n the damage cases for 3-story laboratory frame (Hz) Damage cases Mode 1 Mode 2 Mode 3 11

12 Case Case In the frst stage of dentfyng the damage nduced, the damage localzaton for complete modal data by ndcator modal stran energy ndex s detected. Fgs. 11a-b show the values of MSE for damage localzaton. In can be seen, n the expermental evaluaton, the locaton of damage s also predcted based on dentfed modal data. In the other words, the hghest columns of charts show the damage locaton. Fgure 11. Damage localzaton of expermental frame, a) Damage case 1, b) Damage case 2 he damage parameter Δk, based on Eq. (16) and dentfed expermental modal data has been estmated. As mentoned before, the nduced damage n the expermental evaluaton, mposed to laboratory frame by replacng the columns of selected stores. herefore, for representaton of damage severty the changed stffness matrx of damaged structures wll be compared to ntal stffness quanttes of healthy frame. Generally, stffness matrx of damaged frame s calculated by summaton of stffness damage parameters as well as stffness matrx of healthy frame as K d =K h - Δk. able 4. Damage severty assessment by comparson of stffness components Stffness of the 3 story laboratory frame (N/m) Damage cases 1st 2nd 3rd Healthy Damage case Damage case As can be seen, usng of degraded materal and sectons n the laboratory frame s led to reducton of stffness values as well as adversely dynamc behavour of structure. Also, the 12

13 effect of reducton n damage case 2 s larger than case 1. Furthermore, n the all damage cases, stffness of second story (healthy story) has been slowly decreased. As a result, connecton n each story has great nfluence to dstrbute the damage effects. 4. Concluson A method to detect and locate damage based on senstvty of modal stran energy has been mplemented. he proposed method for damage localzaton utlzes the modal stran energy ndctor (MSE) of healthy and damaged structures. For damage quantfcaton, the senstvty of modal stran energy was determned at frst and then, the model updatng optmzaton functon as well as the error vectors contanng of dfferences of mode shapes were used to estmate changes of stffness components as damage ndex. he approaches were verfed wth two examples as numercal nvestgaton on the cantlever beam and expermental evaluaton on the 3-story laboratory frame. In the numercal model, results were compared wth those obtaned wth the numercal nduced damage ndex, when ncomplete modal data were present. In contrast, n the expermental model, replacng of column of laboratory frame was ntroduced the damage ndex. herefore, n ths state, results were determned based on comparson of stffness components of damaged frame wth correspondng healthy frame. Consequently, numercal and expermental results show that the proposed methods can accurately detect the damage locaton and severty. References [1] N. Hu, Wang, X., Fukunaga, H., Yao, Z.H., Zhang, H.X., Wu, Z.S., "Damage Assessment Of Structures Usng Modal est Data," Internatonal Journal of solds and structures, vol. 38, pp , [2] S. W. Doeblng, Farrar, C.R., Prme, M.B. and Shevtz, D.W., "Damage dentfcaton and health montorng of structural andmechancal systems from changes n ther vbraton characterstcs: a lterature revew," Research Rep. No. LA MS, ESA-EA, Los Alamos Natonal Laboratory NM, USA, [3] N. Stubbs, Broome,.H. and Osegueda, R., "Nondestructve constructon error detecton n large space structures," AIAA Journal, vol. 28, pp , [4] Y. J. Yan, Cheng, L., Wu, Z.Y., Yam, L.H., "Development n vbraton-based structural damage detecton technque," Mechancal Systems and Sgnal Processng, vol. 21, pp , [5] P. Gudmundson, "Egenfrequency changes of structures due to cracks, notches or other geometrcal changes," Journal of Mechnacal and Physcal Solds, vol. 30, pp , [6] Y. S. Lee, Chung, M.J., "A study on crack detecton usng egenfrequency test data," Computaton of Structures, vol. 77, pp ,

14 [7] M. H. Hu, u, S.., Xuan, F.Z., X, C.M., Shao, H.H. "Stran energy numercal technque for structural damage detecton," Appled Mathematcs and Computaton vol. 219, pp , [8] W. Fan, Qao, Pzhong., "A stran energy-based damage severty correcton factor method for damage dentfcaton n plate-type structures," Mechancal Systems and Sgnal Processng, vol. 28, pp , [9] B. Jash, Ren, W.X., "Fnte element model updatng based on egenvalue and stran energy resduals usng multobjectve optmsaton technque," Mechancal Systems and Sgnal Processng vol. 21, pp , [10] S. M. Seyedpoor, "A two stage method for structural damage detecton usng a modal stran energy based ndex and partcle swarm optmzaton," Internatonal Journal of Non-Lnear Mechancs, vol. 47, pp. 1-8, [11] L. L, Hu, Y., Wang, X., "Numercal methods for evaluatng the senstvty of element modal stran energy," Fnte Elements n Analyss and Desgn, vol. 64, pp , [12] D. J. Ewns, Modal estng: heory and Practce and Applcaton, Second Edton ed.: John Wley & Sons, Inc., [13] W. J. Yan, Ren, W.X., "A drect algebrac method to calculate the senstvty of element modal stran energy," Internatonal Journal for Numercal Methods, vol. 27, pp , [14] M. Frswell, Mottershead, J.E, Fnte element model updatng n structural dynamcs. Boston: Kluwer Academc Publshers, [15] S. S. Rao, he Fnte Element Method n Engneerng. London, New York, Pars, San Dego, Sydney, okyo: Elsever Butterworth Henemann,