Identification of structural changes using symbolic representations of modal quantities

Transcription

1 8th Europea Worshop O Structural Health Moitorig (EWSHM 2016), 5-8 July 2016, Spai, Bilbao Idetificatio of structural chages usig symbolic represetatios of modal quatities More ifo about this article: 1 João SANTOS 1, Jua MATA 2, Christia CRÉMONA 3, Paulo SILVEIRA 4 Dpt Structures, LNEC. Av. Brasil 101, Lisboa, (PORTUGAL) josatos@lec.pt 2 Dpt Dams, LNEC. Av. Brasil 101, Lisboa, (PORTUGAL) jmata@lec.pt 3 Techical Divisio, Bouygues Travaux Publics, 1 Aveue Eugèe Freyssiet, Guyacourt (FRANCE) c.cremoa@bouygues-costructio.com 4 Dpt Structures, LNEC. Av. Brasil 101, Lisboa, (PORTUGAL) paulosilveira@lec.pt Key words: Idetificatio, Operatioal Modal Aalysis, Stochastic subspace idetificatio, Cluster aalysis, Symbolic metrics, Suspeded bridge Abstract A structural health moitorig strategy based o the cotrol of structural frequecy data over time, obtaied from operatioal modal aalysis, is preseted i the preset paper. The strategy relies o modal estimatio based o Stochastic Subspace Idetificatio ad clusterig methods ad, ulie most methodologies foud i previous wors, it does ot require the tracig of each structural mode through time. Istead, it relies o extractig histograms of frequecy data ad i quatifyig the dissimilarities betwee sets of these histograms, over time. The strategy is tested ad validated o modal estimates obtaied from the moitorig system of the suspeded 25 de Abril bridge, located i Lisbo, Portugal. The obtaied results show that the proposed strategy is capable of highlightig small-magitude chages i a few umber of mode shapes, while cotrollig a large rage of structural frequecies (ad, cosequetly, a large umber of structural modes). Whe applied to smaller frequecy subrages, the strategy proves capable of idetifyig the frequecy values more susceptible to the damage beig observed, thus cotributig for the localizatio ad magitude assessmet of the chages moitored o site. 1 INTRODUCTION Structural health moitorig (SHM) ca be defied as the process of developig ad implemetig techiques capable of idetifyig damage i structures usig sesig systems [1], [2]. Ideally, these systems ad techiques should operate cotiuously ad automatically, ad be capable of providig, without false detectios ad i real-time, iformatio that ca be directly related to the structural coditio [2]. I this cotext, operatioal modal aalysis (OMA) has become oe of the most used ad importat approaches sice it allows obtaiig, i real-time, iformatio that is directly related with the stiffess of structural systems, which is assumed to vary whe these experiece chages [2], [3]. Strategies for idetifyig structural chages based o OMA have recetly bee the subject of umerous research wors [3] [7] ad are geerally composed of two distict steps: modal estimatio ad modal tracig. Modal estimatio is owadays coducted usig time domai

2 methods, from which the most used is the stochastic subspace idetificatio (SSI) [3], [5], [7] which assumes that structural systems are excited by radom-lie actios ad allows obtaiig accurate estimates of modal quatities. More recet wors describe the associatio of the SSI with clusterig aalysis [4], [6], [8] with the objective of icreasig accuracy ad separatig structural mode shapes from spurious oes related to oise ad other exteral factors. Modal tracig cosists of associatig each of the estimated structural modes with those obtaied i the past, so as to cotrol the structural respose ad detect chages. This tas is geerally coducted usig predetermied baselie sets of modal quatities, obtaied a priori i a o-automated maer, to which ew estimates are compared ad allocated based o their values of frequecy, MAC (modal assurace criterio) ad dampig [3], [4], [6], [8]. While modal SSI estimates ca be accurately ad automatically obtaied from the esemble of SSI ad clusterig, their automatic associatio with the differet atural mode shapes ca be difficult to carry out o: (i) large flexible structural systems composed of umerous atural mode shapes with idetical frequecy values, (ii) structural systems excited by o-radom loadigs of differet atures ad whe (iii) the umber of atural mode shapes to be estimated ad cotrolled over time is larger tha the umber of sesors istalled o site. The preset wor addresses these difficulties ad presets a SHM strategy that allows cotrollig modal quatities over time without the eed to coduct modal tracig, i.e., without the eed to compare modal estimates ad to allocate observed structural modes to those preset i a previously established baselie. Istead, it relies o SSI ad clusterig to estimate the frequecies of atural modal shapes ad o calculatig symbolic metrics betwee these, over time. The testig ad validatio of the proposed strategy is coducted usig data permaetly acquired from a log-spa multi-modal suspeded bridge. Followig this itroductio, the case study is preseted, after which a brief descriptio of the adopted modal estimatio procedure is give. The, the strategy for cotrollig modal quatities over time is preseted ad tested usig a simulated sceario of a structural chage. Fially, the mai coclusios are draw from the preseted wor. 2 CASE STUDY The case study used i the preset paper is the suspeded 25 de Abril bridge, located i Lisbo, Portugal. The bridge has a total legth of 2177m, with a 1013m log mai spa, two 483m lateral spas ad two 190m high pylos (Figure 1a). The bridge dec (see Figure 1b) cosists of a steel truss carryig 6 roadway laes ad two railway lies (Figure. 1b). The bridge suspesio system is composed of 4 suspesio cables ad 1344 vertical hagers suspedig 168 trasversal steel trusses coected by four mai logitudial beams that spa the etire legth of the structural system. The structural health moitorig system istalled o the 25 de Abril bridge acquires data from 200 sesors at a rate of 500 samples/secod. This leads to a total of 8.6x10 9 values per day. This rate was chose to tae advatage of the hardware s aalog low pass filters. Data is the digitally filtered to a value of 20Hz ad oly 50 samples per secod, per sesor, are ept for aalysis, thus leadig to a total of 8.6x10 8 values processed each day. The eed to have such a large set of data is imposed by the eed to characterize ad quatify fast effects iduced by traffic ad by the eed to coduct OMA. For the preset case study, fiftee accelerometers istalled o five cross-sectios across the bridge dec are cosidered. The cross-sectios coicide with the suspeded trasversal 2

3 trusses (Figure 1a) amed as 0, 22S ( S stads for South, with referece to the ceter of the mai spa), 66S, 22N ( N stads for North) ad 66N ad their locatios were chose as the ceter ad quarters of the mai spa ad as the ceter of the lateral suspeded spas. (a) (b) (c) Figure 1: The 25 de Abril bridge, (a) side view, (b) cross-sectio with istalled accelerometers ad (c) acceleratios cosidered for modal aalysis. The accelerometers were istalled, i each sectio, o the top of the upper logitudial beams. Two of them were istalled to acquire vertical acceleratio (a2v ad a3v i Figure 1b) while the third oe (a1h) is acquirig horizotal acceleratio perpedicular to the dec s axis (as show i Figure 1b). Based o the accelerometers istalled i each sectio, three structural global acceleratio compoets are calculated ad cosidered i modal aalysis. These cosist of the horizotal (ah), vertical (av) ad rotatio (ar) acceleratios (Figure 1c), calculated as follows: ah a1h (1) av a2v a3v 2 (2) ar a2v a3v 2 (3) Modal quatities are obtaied hourly from time-series with a legth equal to the same time iterval, thus comprisig 1.8 x10 5 values, per accelerometer. The frequecy cotet of these series, for acceleratios acquired i the ceter of the mai spa durig oe hour of data acquisitio, is show i Figure 2. 3 MODAL ESTIMATION 3.1 Stochastic subspace idetificatio Operatioal modal aalysis geerally relies o time based methods such as the SSI, either i its DATA or COV versio, or as the p-lscf, istead of frequecy based methods such as the several variats of the FDD (frequecy domai decompositio). Detailed descriptio of these ca be foud i [3], [8]. I the preset paper, the most commoly used SSI-COV is 3

4 used, which is based o the classical discrete state-space model describig a liear N-DOF (degree of freedom) time ivariat systems uder white oise excitatio: x1 Ax w, (4) y Cx v, (5) where idetifies the samplig istat, A (x, =2N) is the state matrix, C (rx) is the output matrix, built usig r measured sigals, x is the state vector, y the measuremets vector, while w ad v are idepedet zero mea stochastic processes which represet uow effects, oise, etc. Give these equatios ad assumptios, it ca be show that the modal quatities ca be obtaied usig oly the structural resposes by cosiderig that their covariace matrix describes the free dyamic behavior of the moitored structural system [4], [8], [9]. Figure 2: Spectra of the acceleratio series acquired durig oe hour i the ceter of the mai suspeded spa. I practical terms, the SSI-COV relies o buildig a bloc Hael matrix (H) from the sigals covariace matrices, calculated at each time istat. Afterwards, the observability matrix (O) is obtaied by extractig the sigular values of H. By solvig a liear least-squares problem o a subset of O, a estimate of the state matrix A is obtaied. Its eigevalues are associated with the modal frequecies ad dampig ratios, while its eigevectors cosist of the structural mode shapes [7] [9]. The method requires the iput of a model order to allow estimatig the state matrix A, whose optimal value is ot ow i advace, eve if it ca be estimated with more or less accuracy [3]. As a result, the practical approach for OMA based o SSI-COV is to cosider a wide rage of model orders, most of which are larger tha the umber of mode shapes cosidered i the aalyzed frequecy rage, ad to plot eige frequecies Vs. model orders (this plot is also amed stabilizatio diagram). I this type of plot, the physical mode shapes are show as vertical lies, i which eige frequecies repeat themselves across umerous model orders, while spurious modes associated with oise ad other effects, do ot [3]. The procedure described i the previous paragraph was applied to the data acquired o the 25 de Abril bridge, at each hour, i the frequecy rage of 0-4Hz ad up to order 50. For greater accuracy, the frequecy rage was divided ito subrages of 0.5Hz, thus leadig to eight SSI-COV aalyses per hour, geeratig stabilizatio diagrams as the oe show i Figure 3, where it ca be observed that umerous poles (modes i stabilizatio diagram) are vertically aliged while others are ot. The first set is expected to be related with structural 4

5 modes while the others are supposed to be spurious. To distiguish these two sets, the followig validatio costraits were imposed to the poles obtaied i each stabilizatio diagram: 1. Dampig ratios must cosist of values betwee 0% ad 10%. 2. Each pole s frequecy must have a differece of less tha 1% with at least 10 other poles frequecies foud i the same stabilizatio diagram. 3. Each pole s mode shape must exhibit a MAC larger tha 99% with those of 10 other mode shapes. After imposig the three validatio costraits, the stabilizatio diagram show i Figure 3 allowed obtaiig the stabilizatio diagram show i red color, Figure 4. Figure 3: Stabilizatio diagram obtaied from 15 time-series of acceleratios acquired o the 25 de Abril bridge. 3.2 Automatic mode shape selectio usig clusterig methods The challege of extractig the structural modes observed i situ, from the stabilizatio diagrams obtaied from the SSI-COV is tacled herei as i other recet wors [3], [4], [6] by usig clusterig methods. These cosist of usupervised statistical learig algorithms capable of allocatig poles as belogig to specific structural modes (clusters of poles) such that those allocated to each structural mode are more similar to oe aother tha to those assiged to differet oes. The aim of a clusterig method ca be mathematically posed as [10] the attempt to miimize the dissimilarity betwee poles assiged to the same structural mode (withi-cluster dissimilarity) ad, cosequetly, maximize the dissimilarity betwee poles assiged to differet structural modes (betwee-cluster dissimilarity). The most well-ow clusterig methods are the iterative ad the hierarchical oes [10]. The first type addresses the problem of fidig the structural modes iteratively, while the secod fulfills the same objective by creatig a hierarchy i which more similar poles are merge before more dissimilar oes. This secod type of clusterig methods was used herei, with the Ward mergig criterio [10], for estimatig the structural modes i each stabilizatio 5

6 diagram extracted from the 25 de Abril data. The dissimilarity used as iput i this method is the oe foud i other OMA wors [3], [4], ad it depeds o frequecy, dampig ad mode shapes, as follows: fi f j di d j dij 1 MAC i, j, (6) max f, f max d, d i j where i ad j are two poles of the stabilizatio diagram, f i is the eige frequecy of pole i, d i its dampig ratio ad i the vector of its mode shape coordiates. As for the SSI-COV, oe cluster aalysis was performed o each 0.5Hz rage, thus leadig to a total of 8 cluster aalysis o each stabilizatio diagram. The umber of structural modes (clusters) cosidered i each of these aalyses was purposely fixed over time, ad chose as equal to the umber of sesors istalled o site, i.e., 15, thus leadig to a total of 120 structural modes estimated i the rage 0-4Hz, at each hour. This umber is higher tha the true umber of structural modes, which is approximately 65 for this frequecy rage. However, as it will be see i the ext sectio, if modal tracig is ot required, the accurate estimatio of this umber is ot eeded for health moitorig based o OMA. The 120 modes obtaied from cluster aalysis are show i Figure 4 as dashed blac colored vertical lies. i j Figure 4: Stabilizatio diagram obtaied from 15 time-series of acceleratios acquired o the 25 de Abril bridge ad after imposig the validatio criteria (red crosses) alog with the output obtaied from cluster aalysis, represeted as blac dashed vertical lies. For the preset paper, the procedure described so far, cosistig of SSI-COV followed by cluster aalysis, was repeated for each hour of data acquisitio i the 25 de Abril bridge, for a period of eleve moths, from December 2014 to November The correspodig set of eige frequecies obtaied is show i Figure 5, where those associated with structural modes ca be easily observed as deser horizotal bads. 6

7 Figure 5: Eige frequecies obtaied usig SSI-COV ad clusterig betwee Dec ad Nov CONTROL OF MODAL QUANTITIES OVER TIME The SHM strategy proposed relies o cotrollig frequecy values without the eed to trac specific structural modes through time. Istead, the strategy relies o computig symbolic dissimilarities betwee the mode s frequecies (estimated usig the strategy proposed i the previous sectio) ad i statistically testig these dissimilarities values. Symbolic dissimilarities cosist of distace metrics betwee data objects described by oe or more statistical quatities, from which iterquartile rages ad histograms are the most used [2], [11], [12]. I the preset paper, choice was made to use histograms to quatify the dissimilarity of all frequecies estimated over a pre specified rage, at each hour, i the 25 de Abril bridge. Cosiderig a histogram with classes describig all frequecies estimated durig time period u (f u1,,f u ), ad a idetical oe for time period v (f v1,,f v ), the dissimilarity, D uv, betwee these two objects describig the frequecies i a cosidered rage is equal to the stadardized categorical distace, defied as follows [11], [12]: D uv j1 uj uj j1 j1 uj uj vj vj j1 j1 vj vj 2, (7) where uj ad vj are the umber of structural mode frequecies estimated i the class j, durig periods u ad v, respectively. The umber of histogram classes,, as well as their width must be chose a priori accordig to the target applicatio, ad ept uchaged durig the etire aalysed period. For the the 25 de Abril bridge case study, histograms with 200 classes of 0.02Hz each were hourly built usig the frequecies of the preceedig 14 days. From these, the categrocial distace betwee each histogram ad the first oe was calculated ad it is displayed i blac color i Figure 6. Sice these histograms comprise the etire frequecy rage acquired i situ, this sigle time-series of categorical distaces show i the figure cosists of a idicator describig all the frequecy cotet show i Figure 5. 7

8 Figure 6: Categorical distace obtaied from histograms computed with a rage of 0 to 4Hz, ad 200 classes with a rage of 0.02Hz each. Alog with the categorical distace show i blac color, i Figure 6, a additioal oe is show i blue color. This was obtaied usig the same strategy ad histogram parameters, but from a set of frequecies which was previously chaged so as to simulate a small-magitude damage occurrece. This simulatio cosisted i decreasig by 1% the values of frequecy estimates (show i Figure 5) ow to correspod to local mode shapes i sectio 66S, i.e, to modes exhibitig high values of modal displacemets i sectio 66S (see Figure 1) while havig the remaiig oes ull (or ear ull). Amog the 65 structural modes idetified i the frequecy rage 0-4Hz, oly six were foud to have local character i sectio 66S ad their average frequecies are approximately of 1.2, 2.2, 2.7, 2.9, 3.7 ad 3.9Hz. As it ca be observed, i the blue plot show i Figure 6, the categorical distace obtaied from the etire frequecy 0-4Hz rage appears to be highly represetative of the structural behavior, sice it represets the variatios of a large umber of structural modes. However, it is also highly sesitive to chages, sice a 1% chage i a small umber of mode frequecies geerated the importat variatio i the distace s magitude, show i Figure 6. The dashed red colored lie show i the same figure cosists of a 99% cofidece limit obtaied usig oly the distaces computed before the 1 st of September, date i which the damage was simulated, ad assumig that chages i the quatity follow a Normal distributio. As it ca be observed i Figure 6, the blac lie does ot exceed this limit, while the blue lie clearly overpasses it after the damage simulatio. The time-series of categorical distaces obtaied from histograms describig the etire frequecy rage acquired o site allow geeratig a sigle-valued idicator with high represetativeess ad sesitivity to detect structural chages. If, however, there is the eed to ow which frequecy bad is the most affected by a observed chage, ot oly for aomaly localizatio based o the modal displacemets, but also for quatifyig the magitude of damage 1, several categorical distaces ca be computed from histograms represetig frequecy subrages. For the case study cosidered, the frequecy rage 0-4Hz was divided ito 20 subrages of 0.2Hz, each with te categories of 0.02Hz. The correspodig categorical distaces alog with their cofidece limits are show i Figure 7, where it ca be observed that i five of the twety categorical distace s series the cofidece limit is clearly exceeded. A simple assessmet of the most importat modal 1 Based o the premise that lower frequecy mode shapes are affected by more importat damage. 8

9 displacemets preset i these frequecy rages would provide a estimate of the aomalies locatios, while the fact that the smaller frequecy rages are uaffected by the simulated chage suggests that it is ot of sigificat magitude. Figure 7: Categorical distaces obtaied from histograms computed with a rage of 0 to 4Hz, ad 200 classes with a rage of 0.02Hz each. 4 CONCLUSIONS The preset wor presets ad describes a SHM strategy based o operatioal modal aalysis that requires the estimatio of modal quatities such as frequecies, but which avoids the eed to cotrol these quatities for each structural mode, over time. It cosists i usig the time-domai method SSI-COV associated with clusterig methods to obtai modal estimates, ad i extractig the correspodig histograms, whose dissimilarity over time is quatified by calculatig symbolic categorical distaces. 9

10 The strategy was tested ad validated o modal estimates obtaied from the moitorig system of the suspeded 25 de Abril bridge, located i Lisbo, Portugal, ad allowed cocludig that the categorical distace obtaied from modal frequecies is capable of describig the variatios observed i a large umber of structural modes, while beig sesitive to highlight small chages observed i oly a few. The categorical distace was obtaied ot oly for the etire frequecy rage uder aalysis, but also for subrages ad allowed cocludig that, without loss of sesitivity, it is possible to idetify the frequecy rages most affected by a give structural chage, without the eed to trac ay specific mode shapes, thus providig a straightforward ad precise way of localizig ad quatifyig the magitude of structural chages. REFERENCES [1] S. D. Glaser ad A. Tolma, Sese of Sesig: From Data to Iformed Decisios for the Built Eviromet, J. Ifrastruct. Syst., vol. 14, o. 1, pp. 4 14, [2] J. P. Satos, Smart Structural Health Moitorig Techiques for Novelty Idetificatio i Civil Egieerig Structures, PhD Thesis. Istituto Superior Técico - Uiversity of Lisbo, [3] E. Reyders, J. Houbrechts, ad G. De Roec, Fully automated (operatioal) modal aalysis, Mech. Syst. Sigal Process., vol. 29, pp , May [4] A. Cabboi, F. Magalhães, C. Getile, ad Á. Cuha, Automated modal idetificatio ad tracig: Applicatio to a iro arch bridge, Struct. Cotrol Heal. Moit., o. 2016, p. /a /a, [5] F. Magalhães, E. Caetao, ad Á. Cuha, Operatioal modal aalysis ad fiite elemet model correlatio of the Braga Stadium suspeded roof, Eg. Struct., vol. 30, o. 6, pp , [6] F. Ubertii, C. Getile, ad A. L. Materazzi, Automated modal idetificatio i operatioal coditios ad its applicatio to bridges, Eg. Struct., vol. 46, pp , [7] M. Döhler ad L. Mevel, Fast multi-order computatio of system matrices i subspace-based system idetificatio, Cotrol Eg. Pract., vol. 20, o. 9, pp , [8] F. Magalhães, Operatioal Modal aalysis for testig ad Moitorig of Bridges ad Special structures, Uiversity of Porto, Faculty of Egieerig, [9] B. Peeters, System idetificatio ad damage detectio i civil egieerig, Katholiee Uiversiteit, Belgium, [10] T. Hastie, The Elemets of Statistical Learig, Data Miig, Iferece ad Predictio, 2d ed. Staford, USA: Spriger, [11] L. Billard ad E. Diday, Symbolic Data Aalysis, vol. 52, o. 2. Chichester, UK: Joh Wiley ad Sos, [12] E. Diday ad Noirhomme-Fraiture, Symbolic Data Aalysis ad the SODAS Software. Chicester, UK: Joh Wiley ad Sos,