Final. Report. Finite Element Model Updating and Damage Detection for Bridges Using Vibration Measurements. December 2013

Transcription

1 University Transportation Research Center - Region 2 Final Report Finite Element Moel Upating an Damage Detection for Briges Using Vibration Measurements Performing Organization: Columbia University December 23 Sponsor: University Transportaton Research Center - Region 2

2 University Transportation Research Center - Region 2 The Region 2 University Transportation Research Center (UTRC) is one of ten original University Transportation Centers establishe in 987 by the U.S. Congress. These Centers were establishe with the recognition that transportation plays a key role in the nation's economy an the quality of life of its citizens. University faculty members provie a critical link in resolving our national an regional transportation problems while training the professionals who aress our transportation systems an their customers on a aily basis. The UTRC was establishe in orer to support research, eucation an the transfer of technology in the iel of transportation. The theme of the Center is "Planning an Managing Regional Transportation Systems in a Changing Worl." Presently, uner the irection of Dr. Camille Kamga, the UTRC represents USDOT Region II, incluing New York, New Jersey, Puerto Rico an the U.S. Virgin Islans. Functioning as a consortium of twelve major Universities throughout the region, UTRC is locate at the CUNY Institute for Transportation Systems at The City College of New York, the lea institution of the consortium. The Center, through its consortium, an Agency-Inustry Council an its Director an Staff, supports research, eucation, an technology transfer uner its theme. UTRC s three main goals are: Research The research program objectives are () to evelop a theme base transportation research program that is responsive to the nees of regional transportation organizations an stakeholers, an (2) to conuct that program in cooperation with the partners. The program inclues both stuies that are ienti ie with research partners of projects targete to the theme, an targete, short-term projects. The program evelops competitive proposals, which are evaluate to insure the mostresponsive UTRC team conucts the work. The research program is responsive to the UTRC theme: Planning an Managing Regional Transportation Systems in a Changing Worl. The complex transportation system of transit an infrastructure, an the rapily changing environment impacts the nation s largest city an metropolitan area. The New York/New Jersey Metropolitan has over 9 million people, 6, businesses an 9 million workers. The Region s intermoal an multimoal systems must serve all customers an stakeholers within the region an globally.uner the current grant, the new research projects an the ongoing research projects concentrate the program efforts on the categories of Transportation Systems Performance an Information Infrastructure to provie neee services to the New Jersey Department of Transportation, New York City Department of Transportation, New York Metropolitan Transportation Council, New York State Department of Transportation, an the New York State Energy an Research Development Authorityan others, all while enhancing the center s theme. UTRC-RF Project No: Project Date: December 23 Project Title: Finite Element Moel Upating an Damage Detection for Briges using Vibration Measurement Project s Website: inite-elementmoel-upating-an-amage-etection Principal Investigator: Dr. Raimono Betti Professor, Civil Engineering an Engineering Mechanics Columbia University Phone: (22) betti@civil.columbia.eu Performing Organization: Columbia University Sponsor: University Transportation Research Center - Region 2, A Regional University Transportation Center sponsore by the U.S. Department of Transportation s Research an Innovative Technology Aministration Eucation an Workforce Development The moern professional must combine the technical skills of engineering an planning with knowlege of economics, environmental science, management, inance, an law as well as negotiation skills, psychology an sociology. An, she/he must be computer literate, wire to the web, an knowlegeable about avances in information technology. UTRC s eucation an training efforts provie a multiisciplinary program of course work an experiential learning to train stuents an provie avance training or retraining of practitioners to plan an manage regional transportation systems. UTRC must meet the nee to eucate the unergrauate an grauate stuent with a founation of transportation funamentals that allows for solving complex problems in a worl much more ynamic than even a ecae ago. Simultaneously, the eman for continuing eucation is growing either because of professional license requirements or because the workplace emans it an provies the opportunity to combine State of Practice eucation with tailore ways of elivering content. Technology Transfer UTRC s Technology Transfer Program goes beyon what might be consiere traitional technology transfer activities. Its main objectives are () to increase the awareness an level of information concerning transportation issues facing Region 2; (2) to improve the knowlege base an approach to problem solving of the region s transportation workforce, from those operating the systems to those at the most senior level of managing the system; an by oing so, to improve the overall professional capability of the transportation workforce; (3) to stimulate iscussion an ebate concerning the integration of new technologies into our culture, our work an our transportation systems; (4) to provie the more traitional but extremely important job of isseminating research an project reports, stuies, analysis an use of tools to the eucation, research an practicing community both nationally an internationally; an (5) to provie unbiase information an testimony to ecision-makers concerning regional transportation issues consistent with the UTRC theme. To request a har copy of our inal reports, please sen us an at utrc@utrc2.org Mailing Aress: University Transportation Reserch Center The City College of New York Marshak Hall, Suite 9 6 Convent Avenue New York, NY 3 Tel: Fax: Web:

3 Boar of Directors The UTRC Boar of Directors consists of one or two members from each Consortium school (each school receives two votes regarless of the number of representatives on the boar). The Center Director is an ex-of icio member of the Boar an The Center management team serves as staff to the Boar. City University of New York Dr. Hongmian Gong - Geography Dr. Neville A. Parker - Civil Engineering Clarkson University Dr. Kerop D. Janoyan - Civil Engineering Columbia University Dr. Raimono Betti - Civil Engineering Dr. Elliott Sclar - Urban an Regional Planning Cornell University Dr. Huaizhu (Oliver) Gao - Civil Engineering Dr. Mark A. Turnquist - Civil Engineering Hofstra University Dr. Jean-Paul Rorigue - Global Stuies an Geography Manhattan College Dr. Anirban De - Civil & Environmental Engineering Dominic Esposito - Research Aministration New Jersey Institute of Technology Dr. Steven Chien - Civil Engineering Dr. Joyoung Lee - Civil & Environmental Engineering New York Institute of Technology Dr. Naa Marie Ani - Engineering & Computing Sciences Dr. Marta Panero - Engineering & Computing Sciences New York University Dr. Mitchell L. Moss - Urban Policy an Planning Dr. Rae Zimmerman - Planning an Public Aministration Polytechnic Institute of NYU Dr. John C. Falcocchio - Civil Engineering Dr. Kaan Ozbay - Civil Engineering Rensselaer Polytechnic Institute Dr. José Holguín-Veras - Civil Engineering Dr. William "Al" Wallace - Systems Engineering Rochester Institute of Technology Dr. J. Scott Hawker - Software Engineering Dr. James Winebrake -Science, Technology, & Society/Public Policy Rowan University Dr. Yusuf Mehta - Civil Engineering Dr. Beena Sukumaran - Civil Engineering Rutgers University Dr. Robert Nolan - Planning an Public Policy UTRC Consortium Universities The following universities/colleges are members of the UTRC consortium. City University of New York (CUNY) Clarkson University (Clarkson) Columbia University (Columbia) Cornell University (Cornell) Hofstra University (Hofstra) Manhattan College New Jersey Institute of Technology (NJIT) New York Institute of Technology (NYIT) New York University (NYU) Polytechnic Institute of NYU (Poly) Rensselaer Polytechnic Institute (RPI) Rochester Institute of Technology (RIT) Rowan University (Rowan) Rutgers University (Rutgers)* State University of New York (SUNY) Stevens Institute of Technology (Stevens) Syracuse University (SU) The College of New Jersey (TCNJ) University of Puerto Rico - Mayagüez (UPRM) * Member uner SAFETEA-LU Legislation UTRC Key Staff Dr. Camille Kamga: Director, UTRC Assistant Professor of Civil Engineering, CCNY Dr. Robert E. Paaswell: Director Emeritus of UTRC an Distinguishe Professor of Civil Engineering, The City College of New York Herbert Levinson: UTRC Icon Mentor, Transportation Consultant an Professor Emeritus of Transportation Dr. Ellen Thorson: Senior Research Fellow, University Transportation Research Center Penny Eickemeyer: Associate Director for Research, UTRC Dr. Alison Conway: Associate Director for New Initiatives an Assistant Professor of Civil Engineering Naia Aslam: Assistant Director for Technology Transfer Dr. Anil Yazici: Post-oc/ Senior Researcher Nathalie Martinez: Research Associate/Buget Analyst State University of New York Michael M. Fancher - Nanoscience Dr. Catherine T. Lawson - City & Regional Planning Dr. Ael W. Saek - Transportation Systems Engineering Dr. Shmuel Yahalom - Economics Stevens Institute of Technology Dr. Sophia Hassiotis - Civil Engineering Dr. Thomas H. Wakeman III - Civil Engineering Syracuse University Dr. Riya S. Aboutaha - Civil Engineering Dr. O. Sam Salem - Construction Engineering an Management The College of New Jersey Dr. Thomas M. Brennan Jr. - Civil Engineering University of Puerto Rico - Mayagüez Dr. Ismael Pagán-Trinia - Civil Engineering Dr. Diier M. Valés-Díaz - Civil Engineering Membership as of January 24

4 TECHNICAL REPORT STANDARD TITLE PAGE. Report No. 2.Government Accession No. 3. Recipient s Catalog No. 4. Title an Subtitle 5. Report Date Finite Element Moel Upating an Damage Detection for Briges using Vibration Measurement December Performing Organization Coe 7. Author(s) 8. Performing Organization Report No. Dr. Raimono Betti, Columbia University 9. Performing Organization Name an Aress. Work Unit No. Columbia University Department of Civil Engineering an Engineering Mechanics 5 W. 2th St., 6 Mu, New York, NY 27. Contract or Grant No Sponsoring Agency Name an Aress 3. Type of Report an Perio Covere University Transportation Research Center City College of New York-Marshak 9 6 Convent Avenue NewYork, NY 3 Final Report 4. Sponsoring Agency Coe 5. Supplementary Notes 6. Abstract In this report, the results of a stuy on eveloping a amage etection methoology base on Statistical Pattern Recognition are presente. This methoology uses a new amage sensitive feature evelope in this stuy that relies entirely on moal characteristics, i.e. natural frequencies an moe shapes, irectly ientifie from measurements of the structural response. A proceure for training the amage etection methoology to account for variability inuce by operational conitions, i.e. temperature, traffic, win, etc., has been propose to etermine bounaries of variability of the cumulative istribution functions of the various features. Two ifferent test setups have been consiere: ) a full set of sensors, an 2) a limite number of sensors. The results show that this methoology has been proven successful in etecting the occurrence of amage an, in the case of full sensor setup, also in accurately locating the element where amage has occurre an the amount of element stiffness reuction. In the case of a limite instrumentation setup, the propose methoology is successful in ientifying the occurrence of amage, an even though it loses accuracy in pinpointing the exact amage location, it still successfully ientifies the region containing the amage element. In conclusion, the results obtaine using the propose SPDSF are promising an allow for amage ientification, localization an estimation. The next step will be to use the same amage etection algorithm but relying only on measurement ata of the structural response (output-only). Preliminary results conucte as a part of this stuy show that using non mass-normalize moe shapes, as usually obtaine from output-only ientification algorithm, can generate misleaing results. The key point will be the etermination of the mass normalizing factors of a moe shape matrix ientifie via output-only system ientification algorithms. Also, the evelopment an application of moe shape expansion techniques to obtain the complete mass normalize moe shape matrix in the situation of limite instrumentation will constitute a part of future stuy. 7. Key Wors 8. Distribution Statement Damage, Detection, Sensors, Structural, Algorithms, Localization 9. Security Classif (of this report) 2. Security Classif. (of this page) 2. No of Pages 22. Price Unclassifie Form DOT F 7.7 (8-69) Unclassifie

5 Disclaimer The contents of this report reflect the views of the authors, who are responsible for the facts an the accuracy of the information presente herein. The contents o not necessarily reflect the official views or policies of the UTRC [, (other project sponsors),] or the Feeral Highway Aministration. This report oes not constitute a stanar, specification or regulation. This ocument is isseminate uner the sponsorship of the Department of Transportation, University Transportation Centers Program, in the interest of information exchange. The U.S. Government an other project sponsors assume no liability for the contents or use thereof.

6 CONTENTS Section Page. Introuction 2. Selection of Damage Sensitive Features 7 3. Damage Detection Base on Empirical Cumulative Distribution of Damage Sensitive Features 4. Numerical Results 6 4. Full Set of Sensors Reuce Number of Sensors Conclusions an Future Stuies 37 References 45

7 Finite Element Moel Upating An Damage Detection For Briges Using Vibration Measurements Introuction The term Structural Health Monitoring (SHM) is use to inicate the task of analyzing the current conition of a given structural system, e.g. a builing, a brige, an airplane, a rotating machine, etc., in orer to establish whether or not amage has occurre. With our nation s infrastructure that is rapily ecaying because of lack of resources an/or proper maintenance, a successful SHM strategy is of obvious interest to engineers an government authorities that are responsible for the up-keeping of complex structural systems such as briges. Nowaays, briges have become vital links in the urban lanscape (nearly 9 percent of the national foo chain moves on state briges) an nee to be kept constantly operational in orer to prevent isruption to large sectors of our society. However, accoring to the latest report of the American Society of Civil Engineers, about 25 percent of the entire national brige inventory comprises briges that are either structurally eficient or

8 functionally obsolete. In about 5 years, 5 percent of this nation s briges will be over 5 years ol an this will imply an unpreceente commitment of both financial an human resources. Consequently, there will be the nee for more reliable, economic an easy to conuct inspections that can take avantage of the many avances in other areas, like computer technology an material science. It is in this framework that the concept of SHM has to be frame: thanks to the innovations in computer an sensor technology, it is now possible to collect large amounts of ata that represent the response of the brige to the environment excitation an, through the analysis of such ata, to provie an instantaneous assessment of the brige conitions. For its great potential, SHM has been the object of extensive stuies for the past thirty years, starting from the amage assessment in rotating machine to the analysis of the integrity of win turbines an, now recently, of briges. Because of such a broa spectrum of applications, various approaches to SHM have been propose an stuie, some of more general breath an applicability (e.g. vibration base SHM) an some relate to well efine applications (e.g. groun penetrating raar to assess elamination in brige ecks). Looking at the more general (global) approaches that look at the overall behavior of the structure, the majority of such SHM approaches focus on vibration base techniques, which employ the measurement of the ynamic response of the structure, measure at some locations on the system of interest, to perform the amage etection assignment. This is becoming the preominant tren in civil engineering applications for briges, where now monitoring sensor systems are installe permanently, either at the time of initial construction or uring the service life, an recor, in real time, quantities 2

9 like isplacements an accelerations. The broa set of vibration base SHM techniques can be ivie into two major categories: ) moel base methos an 2) ata base methos. As for the first group, moel base methos require the selection of a moel of the structure that has to be ientifie from the measure structural response. The basic iea behin such methos is that a moel is ientifie, either irectly or iteratively, whose response mimics very closely the measure response from the real structure. Sometimes these moels have physical meaning, e.g. a mass-amping-stiffness moel, sometimes they o not, e.g. black-box moel. Usually, to etect amage, these algorithms often employ the moal characteristics of the moel, such as natural frequencies an/or moe shapes, ientifie from the structural response. As state by one of the axioms of amage etection [], the amage etection problem requires the knowlege of the unamage conitions in orer to be solve. In moel base techniques exploiting moal analysis, the aforementione requirement is satisfie by first ientifying the moal properties of the structural moel that represents the structure in what can be referre to as unamage conition or healthy state, an then by comparing such properties with those corresponing to a newly ientifie moel obtaine from the analysis of the response of the system uner unknown conitions. If the novel properties result to be sensibly ifferent from the ones observe on the healthy structure, i.e. ifferent values of corresponing frequencies or ifferent types of moes (translational vs. rotational), then the system can be eclare amage. Of course, how sensibly ifferent is something that varies from system to system an it requires some engineering jugment. In aition, a substantial boy of literature has been prouce on the analysis of the raw- 3

10 backs of using such approaches in real applications, as moal properties, especially moal frequencies, are highly sensitive to environmental variability ue, for instance, to temperature or humiity changes. For example, measurements of the brige response collecte at ifferent times uring the year have shown that there is a change in the magnitue of the natural frequencies of about percent between summer an winter, making it ifficult for the engineer to separate the amage-inuce changes from the seasonal changes. Furthermore, all system ientification algorithms are sensitive to measurement noise, making the risk of false alarms error very likely to occur uring the amage etection performance. In orer to tackle such problems, it has been propose to cast the structural amage etection problem into a probabilistic framework, which is naturally tailore to cope with variability in the ata. With regar to the secon set of methos, ata-base methos, as the name suggests, rely entirely on information extracte from the ata an o not require the aoption of a physics-base moel of the structure. As an example, in this class, we can group all the Autoregressive (AR), moving-average (ARMA), with exogenous terms (ARX) moels that are commonly least-square solutions of sets of equations that relate response an input measurements. Because of the nature of such equations that map current outputs to previous outputs an/or input measurements, such ata-base moels can be successfully use to preict the response of the system (e.g. a brige) to a future excitation (e.g. preicte groun motion) but cannot be irectly applie in etecting structural amage. However, such moels become quite useful in amage ientification when a statistical pattern recognition approach is followe. 4

11 In amage etection, the basic iea behin a statistical pattern recognition approach is the following: first, training ata from the unamage an all possible amage states is collecte an analyze an a statistical moel of the ata, for example the probability ensity function, is create. Once this moel is built, then, when the new monitoring ata set becomes available, it is possible to see whether its moel fits within the statistical moel or not. In case it oes not, this is an inication that structural amage has occurre an means have to be foun to relate such iscrepancy to the presence, amount an location of amage. In the fiel of SHM, the statistical pattern recognition approach is mainly use to solve unsupervise verification problems. The term unsupervise verification problem is use to represent those problems in which only ata from the unamage structure is available to create the statistical moel. The general outline of a unsupervise statistical pattern recognition technique comprises of two phases: ) the training phase an 2) the testing phase. During the training phase, a large amount of ata is collecte from the structure in a state that is labele as the healthy conition, then a set of information, calle amage sensitive features, is extracte from such ata an the features probability istribution is etermine. This istribution represents the statistical moel of the ata from the healthy structure against which the ata from any future monitoring test will be compare. During the testing phase, the amage sensitive features are extracte from the response of the system in an unknown state (either still unamage or amage), an the probability of the new features of being realizations of the traine istribution is 5

12 evaluate. If such probability is lower than a prescribe threshol, the structure is eclare amage. The challenges in this secon approach stem mainly from the selection of the most appropriate istribution to moel the amage sensitive features statistical properties, an from the metric to employ to measure the istance of the novel features from the traine istribution. Normal istribution an square Mahalanobis istance are two popular choices to aress the aforementione issues, as well known results are available for both statistical tools, making the treatment of the overall problem much easier. Nevertheless, the assumption of normally istribute features can be erroneous, especially when the number of available observations is not large. If a set of observations is not normally istribute, mean an covariance matrix are no longer sufficient to completely escribe the statistics of the set of observations an consequently, square Mahalanobis istance oes not represent a robust tool to measure the istance of a new observation from the traine istribution. However, pattern recognition remains a viable approach able to account for the physiological changes in structural properties ue to external factor effects, even when there are uncertainties, as for the type an istribution of the amage sensitive features. It must be note, though, that ue to the choice of the amage sensitive features, which are often selecte as abstract information ifficultly relatable to the structural properties, the amage etection algorithm evelope within the statistical pattern recognition framework can selom locate an quantify amage. On the contrary, ue to their intuitive relationship with the structure topology an characteristics, moal properties, especially when in the form of moe shapes, can solve the problem of amage location an quantification. In view of the aforementione reasons, a mixe approach has been explore recently an 6

13 presente in this stuy, where the amage sensitive features are efine using the moal properties ientifie from the response of a brige system, while the amage etection, location an quantification is evelope accoring to the statistical pattern recognition paraigm. In this approach, the finite element moel upating was not necessary because it was possible to irectly obtain the iagonal elements of the mass an stiffness matrices from the ientifie moe shapes an frequencies. In this way, lengthy an computationally cumbersome calculations for the finite element moel upating are avoie. Aitionally, while finite element moel upating requires the knowlege of the moel parameters of an existing finite element moel, an the reliability of this knowlege in turn affects the upate moel parameters, in the current approach no such a priori knowlege of the moel parameters are necessary, thereby also proviing great avantages in terms of accuracy of the results. The results presente in this report are those relate to numerical simulations of the brige superstructure. 2 Selection of Damage Sensitive Features One of the most ifficult tasks in performing SHM is to select a set of parameters, to be ientifie from the measurement ata, that are sensitive to structural amage an so that can provie early warning. In the past, natural frequencies were use but they were foun too insensitive to amage an too much affecte by environmental effects. Here, a new set of features, erive from the ientifie moal characteristics, is propose. 7

14 Consier that the brige structure can be represente as a iscrete mass, amping an stiffness moel with N egrees of freeom. The equations escribing the motion of such a system can be expresse as: Mẍ(t) + Cẋ(t) + Kx(t) = F () where M, C, an K represent the N N mass, amping an stiffness matrices, respectively, while x, ẋ, an ẍ inicates the N vectors of the noal isplacements, velocities an accelerations, respectively. The vector F represents the N vector containing the noal input forces representing the external excitation. If the assumption of lumpe mass moel is use, as it is in this stuy, the mass matrix can be consiere iagonal. Let {M; K} an {M ; K } enote the mass an stiffness matrices of two moels of the same structure in two ifferent states, for example the amage an unamage states. In amage etection analysis, amage is usually characterize as a change (reuction) of some members stiffness (e.g. inuce by cracking) while the mass is consiere unchange. Hence, it is esirable to erive features that somehow reflect such stiffness reuction. The amage sensitive feature use in the present work is tailore to give a measure of the ifference between the iagonal elements of K an K. Equation (2) expresses the relation between the amage sensitive feature evaluate at the jth DOF an the jth iagonal elements of K an K : DSF j = K j,j K j,j K j,j = K j,j K j,j, for j =,..., N. (2) 8

15 Let us enote as {ω, ψ} an {ω, ψ } the full sets of moal frequencies, ω s, an moe shape vectors, ψ s, ientifie through any system ientification proceure from the states of the system referre to as {M; K} an {M ; K }, respectively. For convenience in the following erivations, the natural frequencies are store in an N N iagonal matrix Λ, whose elements are: Λ = iag{ω 2, ω 2 2, ω 2 N} (3) while the moe shape vectors are the columns of the eigenvector matrix Φ, Φ = [ψ, ψ 2, ψ N ] (4) Equivalently, for the moel corresponing to the amage case, Λ = iag{ω 2, ω 2 2, ω 2 N } (5) Φ = [ψ, ψ 2, ψ N] (6) If only the response time histories of the system are employe to ientify the moal characteristics of the structure (e.g. this coul be the case when ambient vibrations of the brige are monitore), the ientifie moe shape matrices Φ an Φ are not mass normalize, but rather satisfy the normalizing conitions (7) an (8): Φ T MΦ = α = iag{α, α 2, α N } (7) Φ T KΦ = Λα (8) 9

16 where α i, for i =,..., N, are scalars. Similar relationships apply for the state of the system characterize by {M ; K } an {Λ, Φ }. In the case that both input an output histories of the system are measure (e.g. force vibration tests on the brige), the moe shapes normalizing α i factors are all equal to, as the following conitions on mass an stiffness apply: Φ T MΦ = I (9) Φ T KΦ = Λ () where I inicates an ientity matrix of orer N. In this stuy, only the time-histories of the structural response will be available for the ientification of the moal characteristics of the brige (output-only ientifications), simulating the case that is most commonly occurring in real practice when ambient vibrations of the brige are recore. In this case, exploiting conitions (7) an (8) an the mass iagonality assumption, it is possible to express the {j, j}th elements of the mass matrix, M, an of the stiffness matrix, K, only in terms of the moal properties an normalizing scalars α i : N K j,j = Mj,j 2 ( N M j,j = i= i= φ 2 j,i α i φ 2 j,iλ i () α i ) (2)

17 Substituting Equations () an (2) into the initial efinition of the propose amage sensitive feature, Equation (2), it is possible to express the propose amage sensitive feature only in terms of the ientifie moal characteristics as: DSF j = ( N φ 2 j,i i= ( N φ 2 j,i i= α i ) 2 ( N α i i= ) 2 ( N i= ) φ 2 j,i λ i α i ). (3) φ 2 j,i λ i This is a quite interesting result because it allows us to look at changes of the stiffness elements irectly affecte by amage through the ientification of moal characteristics as natural frequencies an moe shapes. Because of this link with the stiffness coefficients, we can efine this feature as Stiffness Proportional Damage Sensitive Feature (SPDSF). 3 Damage Detection Base on Empirical Cumulative Distribution of Damage Sensitive Features Inherent in the efinition of the Stiffness Proportional Damage Sensitive Feature (SPDSF) presente in the previous section, there is a comparison between two states of the system, so that the feature can actually be thought of as a measure of the variability of the stiffness properties of the system subjecte to ifferent conitions. For example, a change in the stiffness value of the element connecting noe i an j shoul be reflecte in a change in the values of the ith an jth elements of the main iagonal of the structure s stiffness matrix.

18 Inee, it is customary to moel amage as a ecrease in the stiffness of a part (an element or a joint) of a structure. Nonetheless, as alreay mentione, environmental an operational conitions may as well inuce changes in the stiffness values of a structure. Therefore, it is extremely important that a amage etection algorithm be able to istinguish between the fluctuations of the amage sensitive features (in this case, the propose SPDSF) ue to the influence of external factors an those ue to amage occurrence. Because of this variability that coul lea to false alarms or false safety, it is possible, as propose in [2], to cast the problem in a probabilistic context by introucing the probability of amage assigne to each iagonal element of the stiffness matrix. Such a probability can be efine as the probability that the {j, j}th element, Kj,j, of the stiffness matrix associate to a possibly amage state be less than a prescribe fraction of the same element, K j,j, associate to a known, unamage, state: P amage () = P (K j,j < ( )K j,j ) for [, ) (4) where is a parameter inicating the percentage of amage an can range from to. By using the previously erive expressions for the SPDSF (Equation (3)), it is possible 2

19 to rewrite Equation (4) in terms of the SPDSF efine in Section 2: P amage () = P (Kj,j < ( )K j,j ) = (5) ( Kj,j K ) j,j = P > = K j,j { Kj,j K } j,j = CDF K j,j where CDF represents the Cumulative Distribution Function. Then, the objective of the training phase will be to efine bounaries for the fluctuations of the SPDSF that can be consiere normal, using ata from what can be consiere as the unamage state, in orer to set a reference against which new realizations of features extracte from the system uner unknown conitions can be compare. The general proceure of the training phase can be briefly summarize as follows. Let us enote as n tr the number of measurements that have been conucte on the system uner various healthy conitions. These measurement sets represent the measurements obtaine at ifferent times uring the year, with ifferent traffic conitions, etc. They encompass all the ifferent operating conitions of the brige uring which the brige is consiere to be in a healthy state. Let us further enote with m the number of sensors available for each measurement campaign, an with N the number of egrees of freeom of the Finite Element moel of the structure in question. From these measurements, a set Y = {λ (i), Φ (i) }, for i =, 2,, n tr, of moal properties can be ientifie, where λ (i) represents the N vector containing the square circular moal frequencies ientifie 3

20 from the ith measurement set an Φ (i) the corresponing N N ientifie moe shapes. The set Y is then ivie into two subsets Y H an Y V such that: Y H Y V = Y Y H Y V = Y H, Y V Y H = n H Y V = n V The moal properties (natural frequencies an moe shapes) containe in the set Y H are taken as representative of the reference state an use to evaluate the elements K (i) j,j, for j =, 2,, m an i =, 2,, n H, of the stiffness matrix of the entire structure in its reference state. The remaining sets of moal properties containe in Y V are instea use to estimate the elements K (i) j,j, for j =, 2,, m an i =, 2,, n V. These n V sets of moal properties might be slightly ifferent from the reference ones because of the possible ifferent operational conitions but are still referring to the healthy state of the structure. By comparing these ientifie K (i) j,j with the other ientifie K (i) j,j, it is possible to see a range of variability of such stiffness parameters (an consequently of the propose DSF) inuce by operational conitions. At this point, each K (i) j,j in Y V is compare to all the K (i) j,j in Y H using Equation (3) an, at the en of this proceure, the Empirical Cumulative Distribution Function (ECDF) of each of the n V sets containing n H values of 4

21 DSF j can be estimate as: P () = n H n H i= U( DSF (i) j ) (6) where U(x) is the Heavisie function: x < U(x) =.5 x = (7) x >. In this way, a set of acceptable values of an the probability associate to such values to occur are estimate. This set will be inicative of the variability of the operational conitions of the brige an so a conition that will follow within this range will be consiere relate to regular operation. At the time of testing the brige to assess its conition, a new set of measurement ata is collecte from the structure in unknown conitions, an a single set of moal properties is ientifie from such ata. The corresponing K j,j elements estimate from such moal ata are then compare to the K (i) j,j values previously ientifie, for i =, 2,, n H, obtaine from Y H uring the training operation, using again Equation (3). The empirical cumulative istribution function of this new set of ata can now be compare to the one obtaine in the training phase: a shift towars right of the testing ECDF, outsie the bounaries inicative of the operational conitions, inicates the amage occurrence in that specific egree-of-freeom, with a probability P () given by Equation (6). 5

22 4 Numerical Results In orer to test the valiity of the propose approach, the amage etection algorithm escribe in the previous section was use to etect an locate amage in a simple brige moel. The brige was moele using two interconnecte spring-mass chains: each chain consists of alternately place flexural links an lumpe masses, with the corresponing lumpe masses of the two chains being aitionally connecte by flexural links. A total of 8 masses an 4 flexural links were assemble in such a moel, as shown in Figure (). The baseline conition refers to the case where all the 8 masses, m i, for i =,, 8, are taken equal to.25 5 kg an the flexural stiffnesses of all the spring elements, ki, for i =,, 4, are set equal to N/m. This simple moel can be use to represent simple overpasses or single spans of supporte girer briges, where the spring chains represent the various segments of the girers an the concentrate masses the mass of the eck. Because of the structure of the moel, only global bening an global torsional moes have been consiere. Such moes, an the corresponing frequencies, are shown in Figure (2) an Figure (3). Seven ifferent amage scenarios have been consiere, as liste in Table (). The first five scenarios are those that simulate the operational conitions of the brige: the first case represents the so calle reference state, in which the initial values of the lumpe masses an of the flexural stiffnesses are use, while amage scenarios 2 an 4, with a reuction of the flexural stiffness of percent of the original value for only the elements of one chain, are inicative of a conition where only one part of the brige is subjecte to an increase in temperature with respect to the baseline conition. Analogously, amage scenarios 3 6

23 Figure : Brige moel consiere for numerical case stuy. Table : Damage an unamage states consiere. State Conition Description Unamage (U) baseline 2 Unamage (U2) k i =.99ki, for i =,..., 5 3 Unamage (U3) k i =.ki, for i =,..., 5 4 Unamage (U4) k i =.99ki, for i = 6,..., 5 Unamage (U5) k i =.ki, for i = 6,..., 6 Damage (D) k 3 =.85k3 7 Damage (D2) k 2 =.85k2 an 5 represent a conition where only half of the brige is subjecte to a ecrease in temperature. The real amage scenarios are represente by amage cases 6 an 7. Here, amage is moele by a 5 percent reuction of stiffness in only one element: even if the reuction might appear substantial, when looking at the overall stiffness at the joint, the reuction is not so evient an so it will represent a goo test for the algorithm. In this stuy, only acceleration time-histories will be use in the ientification of the moal parameters. Accelerations are the most common type of ata obtaine uring the moni- 7

24 - 5-5 Moe : Hz Moe 2: Hz With of brige With of brige Length of brige Moe 4: Hz Moe 6: Hz - 25 With of brige Length of brige With of brige Figure 2: Bening moes of baseline moel. 2 Length of brige 2 Length of brige 3 3 8

25 - 5-5 Moe 3: Hz Moe 5: Hz With of brige Length of brige With of brige Moe 7: Hz Moe 8: 9.5 Hz With of brige Length of brige With of brige Figure 3: Torsional moes of baseline moel. 2 Length of brige 2 Length of brige 3 3 9

26 toring of a brige an so the algorithm will be teste using only acceleration ata. In this version of the algorithm, input-output ata will be consiere in the ientification. However, in the future extension, cases when only time-histories of the structural response are available will be consiere. The general formulation of the amage etection algorithm will not change. For the simulations analyze in this stuy, the input is represente by a white Gaussian noise applie at some noal points. In this stuy, the moal parameters of the various moels are ientifie using the inputoutput system ientification algorithm OKID/ERA (Observer Kalman filter IDentification/Eigensystem Realization Algorithm) [3]. This algorithm provies a first-orer realization of the system using any arbitrary number of time-histories of the structural response an of the input, from which it is possible to extract moal frequencies, moal amping ratios an moe shapes. However, the moe shapes can only be etermine at the locations where the time-histories of the structural response were measure (e.g. sensor locations) an this introuces a ifferentiation in the analysis between ) the case of a full set of sensors (e.g. sensors at every egree of freeom), an 2) the case of a limite set of sensors (e.g. sensors at only few egrees of freeom). This istinction will be highlighte in the analysis of the results that follows. 4. Full Set of Sensors The training ata set is obtaine by simulating the response of the system uner states from to 5 at each egree of freeom, using white Gaussian noise as input. For each 2

27 unamage state, 2 ifferent sets of input have been use, so that n tr is equal to in this example. In each simulation, the mass an stiffness values given in Table () are perturbe by ± percent, so as to simulate operational conition variability. Following the proceure etaile in the previous section, one hunre sets of moal properties have been then ientifie: these sets have been istribute to create the two sets Y H an Y V accoring to the criterion that the even realizations have been taken to construct the set Y H, i.e. Y H = {λ (i), Φ (i) } for i = 2, 4,, 5, while the o realizations have been use to form the set Y V, Y V = {λ (i), Φ (i) } for i =, 3,, 49. With this choice of sets Y H an Y V, each set contains realizations of each of the 5 unamage scenarios. For testing, only a single set of input has been use to simulate the response of the system in all the 7 states of Table (): the iea of using also the response of the system in operational conitions was ictate by the nee to test the ability of the algorithm not only to ientify one of the amage states but also to ientify as unamage a healthy state. Figures (4) to (7) shows the results for a test performe using ata simulate from state, not use to construct the training ECDF. The re curves represent the ECDFs of the SPDSF values consiere to be acceptable: if the new ientifie ECDF obtaine from the testing set of ata is within these two lines, the system can be consiere unamage. While the mean of the lowest values assume by the SPDSFs in healthy conitions are close to, the mean of the maximum values assume by the features can also excee 2 percent. This can be explaine consiering the probabilistic framework in which this analysis is conucte. In fact, within a eterministic framework, a value of the SPDSF higher than woul have to be consiere inicative of amage, as only one reference structure 2

28 woul be consiere, i.e. only one value for each K j,j woul be estimate, an all instances whose SPDSF woul epart from woul raise a amage alarm. On the contrary, the initial training phase performe in the currently propose approach enables us to set a reasonable range of values within which the eparture of Kj,j from K j,j can be consiere as ue to the influence of external factors, e.g. temperature, traffic, win, etc. Figures (8) to () shows the ECDF results for a test performe using ata simulate from amage state 6, in which the flexural stiffness of the thir element ha been reuce by 5 percent. This change shoul have affecte irectly the stiffness components K 2,2 an K 3,3. In this case, it is interesting to see that the ECDF estimates for K 2,2, K 3,3, K 4,4, K 6,6, K 7,7 an K 8,8 fall outsie the bounaries efining the amissible range of the SPDSF values. Nonetheless, the mean of DSF 2 an DSF 3 are about 7 percent, while those of the remaining features are aroun percent. This clearly inicates that the amage is between egrees of freeom 2 an 3, i.e. in the element k 3. Moreover, by exploiting the ECDFs of DSF 2 an DSF 3, it is possible to state that the variation of K 2,2 an K 3,3 from the healthy states excees 6 percent with a probability of.97. It can be conclue that the propose approach is then able to locate amage an also to give an estimate of the amage severity. For example, for the current amage scenario, amage is locate between egrees of freeom 2 an 3. Degree of freeom 2 is share between elements 2, 3 an 2; therefore, assuming that all the elements have the same stiffness, k, an that only one element stiffness has been ecrease to an amount equal to x, the following equation can be solve to estimate which fraction of the original stiffness characterize the stiffness 22

29 DSF DSF P(DSF >).6.4 P(DSF 2 >).6.4 P(DSF 3 >) DSF 5 DSF 6 Figure 4: Test using ata simulate from state : ECDFs for DSF an DSF (DSF 5 >).6.4 (DSF 6 >).6.4 (DSF 7 >).6.4

30 F 2 DSF 3 DSF P(DSF 3 >).6.4 P(DSF 4 >) F 6 DSF 7 DSF 8 Figure 5: Test using ata simulate from state : ECDFs for DSF 3 an DSF (DSF 7 >).6.4 (DSF 8 >).6.4

31 P(D.4 P(D.4 P(D DSF 5 DSF P(DSF 5 >).6.4 P(DSF 6 >).6.4 P(DSF 7 >) Figure 6: Test using ata simulate from state : ECDFs for DSF 5 an DSF 6. 25

32 P(D.4 P(D F 6 DSF 7 DSF P(DSF 7 >).6.4 P(DSF 8 >) Figure 7: Test using ata simulate from state : ECDFs for DSF 7 an DSF 8. 26

33 value of the amage element: 3k (2 + x)k 3k =.6 x =.82 (8) Therefore, from the observation of Figures (8) an (9), it is possible to conclue that the stiffness of the flexural element 3 has ecrease to 8 percent, very close to the actual value of 5 percent reuction. Finally, Figures (2) to (5) show the results for a test performe using ata simulate from amage state 7. Following the consierations mae for the previous amage scenario, in this case it can be conclue that amage has occurre between egrees of freeom 2 an 6, an that the stiffness value of the flexural element 2 has ecrease at least to 85 percent its original value. This represents the actual reuction of stiffness introuce in the moel, confirming the great potential of the propose approach in locating an quantifying structural amage. 4.2 Reuce Number of Sensors This case is consiere to represent situations that usually occur in real life, when the number of available sensors, e.g. accelerometers, is less than the number of egrees-offreeom use in the structure moel. Here, the same 7 states consiere for the previous case have been consiere, but, in this secon case stuy, only the response of the system at egrees of freeom, 3, 6 an 8 are use to ientify the moal properties. The 27

34 DSF DSF P(DSF >).6.4 P(DSF 2 >).6.4 P(DSF 3 >) DSF 5 DSF 6 Figure 8: Test using ata simulate from state 6: ECDFs for DSF an DSF (DSF 5 >).6.4 (DSF 6 >).6.4 (DSF 7 >).6.4

35 F 2 DSF 3 DSF P(DSF 3 >).6.4 P(DSF 4 >) F 6 DSF 7 DSF 8 Figure 9: Test using ata simulate from state 6: ECDFs for DSF 3 an DSF P(DSF 7 >).6.4 P(DSF 8 >).6.4

36 P( P( P( DSF 5 DSF P(DSF 5 >).6.4 P(DSF 6 >).6.4 P(DSF 7 >) Figure : Test using ata simulate from state 6: ECDFs for DSF 5 an DSF 6. 3

37 P(D.4 P(D F 6 DSF 7 DSF P(DSF 7 >).6.4 P(DSF 8 >) Figure : Test using ata simulate from state 6: ECDFs for DSF 7 an DSF 8. 3

38 DSF DSF P(DSF >).6.4 P(DSF 2 >).6.4 P(DSF 3 >) DSF 5 DSF 6 Figure 2: Test using ata simulate from state 7: ECDFs for DSF an DSF (DSF 5 >).6.4 (DSF 6 >).6.4 (DSF 7 >).6.4

39 F 2 DSF 3 DSF P(DSF 3 >).6.4 P(DSF 4 >) F 6 DSF 7 DSF 8 Figure 3: Test using ata simulate from state 7: ECDFs for DSF 3 an DSF P(DSF 7 >).6.4 P(DSF 8 >).6.4

40 P(.4 P(.4 P( DSF 5 DSF P(DSF 5 >).6.4 P(DSF 6 >).6.4 P(DSF 7 >) Figure 4: Test using ata simulate from state 7: ECDFs for DSF 5 an DSF 6. 34

41 P(D.4 P(D F 6 DSF 7 DSF P(DSF 7 >).6.4 P(DSF 8 >) Figure 5: Test using ata simulate from state 7: ECDFs for DSF 7 an DSF 8. 35

42 OKID/ERA algorithm is again use for the ientification: the algorithm is able to ientify all 8 moes, but only the rows corresponing to the instrumente egrees of freeom of the mass normalize moe shape matrix may be estimate. This ientification can be carrie out using input-output balance equations in the time omain [4]. The results for the case of a limite number of sensors are presente in Figure (6) to Figure (2). Also for this case, the amage ientification is successfully complete, eclaring the structure healthy for the first test, an amage in the last two tests. Nonetheless, using this limite sensor setup, the probable area of amage is now less confine than in the first case: for the first amage case (state 6 in Table ()) where the structure is eclare amage, it is possible to state that amage has occurre aroun egree of freeom 3, i.e. either element 3, 4 or 3 coul be amage. However, in this case, the extent of such amage is uncertain owing to the absence of measurements at egrees of freeom 2, 4 an 7: it coul range from mil an extene, in the case where all three elements are amage (in such scenario, each element woul have ecrease its stiffness to 96 percent of its original value), to substantial but localize, in the case where only one element has its stiffness ecrease to 82 percent of its original value. Similar consierations apply for the case where the response of the system uner amage state 7 is consiere: amage may be present in elements 7, 8 an 2, an, in such case, either each element woul have been subjecte to a ecrease to 96 percent of its original value, or only one of such element coul have been amage leaing to a reuction in stiffness of at least 5 percent of its original value. Nevertheless, even though the results in this secon case are less accurate than the case when the full set of sensors is available, the propose approach guarantees a 36

43 reasonable inication of amage presence an location. The amage etection in the situation of limite instrumentation may be further improve by applying moe shape expansion to expan the mass normalize moe shapes, ientifie at the instrumente egrees of freeom using OKID/ERA an input-output balance, from these instrumente to the non-instrumente egrees of freeom. In this way, the complete mass normalize moe shape matrix, neee for the calculation of all the DSF j s, can be estimate, an will hence improve the accuracy of amage location an severity estimation. To this en, the moe shape expansion equations of [4], base on the eigenvalue equations an on equations from the structural topology, may be consiere in conjunction with the amage etection metho propose here. This approach has the avantage of obtaining reliable estimates of the moel s complete mass normalize moe shapes without any a priori information on the structural properties, of being robust in the presence of significant measurement noise an of being flexible with respect to the type of measurement ata necessary, e.g. accelerations an/or velocities an/or isplacements. 5 Conclusions an Future Stuies In this report, the results of a stuy on eveloping a amage etection methoology base on Statistical Pattern Recognition are presente. This methoology uses a new amage sensitive feature evelope in this stuy that relies entirely on moal characteristics, i.e. natural frequencies an moe shapes, irectly ientifie from measurements of the struc- 37