A Computationally Efficient Bayesian Framework for Structural Health Monitoring using Physics-Based Models

Transcription

1 Paper 27 Abstract A Bayesan nference framework for structural damage dentfcaton s presented. Sophstcated structural dentfcaton methods, combnng vbraton nformaton from the sensor network wth the theoretcal nformaton bult nto a hgh-fdelty fnte element model for smulatng structural behavour, are ncorporated nto the system n order to montor structural condton, track structural changes and dentfy the locaton, type and extent of the damage. The methodology for damage detecton combnes the nformaton contaned n a set of measurement modal data wth the nformaton provded by a famly of compettve, parameterzed, fnte element model classes smulatng plausble damage scenaros n the structure. The computatonal challenges encountered n Bayesan tools for structural damage dentfcaton are addressed. Smulated modal data from the etsovo Brdge are used to valdate the effectveness of the methodology. Keywords: Bayesan nference, structural health montorng, damage dentfcaton, model reducton, krgng, hgh performance computng. 1 Introducton Cvl-Comp Press, 2015 Proceedngs of the Fourth Internatonal Conference on Soft Computng Technology n Cvl, Structural and Envronmental Engneerng, Y. Tsompanaks, J. Krus and B.H.V. Toppng, (Edtors), Cvl-Comp Press, Strlngshre, Scotland A Computatonally Effcent Bayesan Framework for Structural Health ontorng usng Physcs-Based odels C. Papadmtrou 1, C. Argyrs 1 and P. Panetsos 2 1 Department of echancal Engneerng Unversty of Thessaly, Volos, Greece 2 Egnata Odos S.A., Captal antenance Department Therm, Greece Bayesan nference s used for quantfyng and calbratng uncertanty models n structural dynamcs based on vbraton measurements, as well as propagatng these uncertantes n smulatons for updatng robust predctons of system performance, relablty and safety [1-3]. The Bayesan tools are based on Laplace methods of asymptotc approxmaton [4,5] and samplng algorthms [6]. These tools nvolve solvng optmzaton problems, generatng samples for tracng and then populatng the mportant uncertanty regon n the parameter space, as well as evaluatng ntegrals over hgh-dmensonal spaces of the uncertan model and loadng parameters. They requre a moderate to very large number of system re-analyses to be performed over the space of uncertan parameters. For hgh-fdelty fnte element 1

2 (FE) models requred n model-based structural health montorng applcatons, the use of Bayesan technque may result n excessve computatons. The computatonal challenges encountered n Bayesan tools for structural damage dentfcaton are addressed. An effectve structural health montorng system requres the development of computatonally effcent technques and specalzed software that ntegrates nformaton from physcs-based mathematcal models of structural components wth the nformaton collected from vbraton measurements under varous operatonal condtons, ncludng normal operaton under the acton of everyday loads, wnd loads and envronmental effects (e.g. temperature), as well as sudden extreme load events such as moderate to strong earthquakes or strong wnds. The specfcatons of a complete montorng system of collectng and processng data for structural health montorng based on physcsbased FE models [7,8] are revewed and the mportance of FE model updatng technques s emphaszed. Novel methods to drastcally reduce computatons n SH systems are presented. Specfcally, hgh performance computng technques are ntegrated wth Bayesan technques for structural damage dentfcaton to effcently handle largeorder models of hundreds of thousands or mllons degrees of freedom, and nonlnear actons actvated durng system operaton. Fast and accurate component mode synthess technques [9] are proposed, consstent wth the fnte element model parameterzaton, to acheve drastc reductons n computatonal effort n a sngle fnte element model smulaton. Surrogate models are also used wthn mult-chan CC algorthms wth annealng propertes to substantally speed-up computatons [10], avodng full system re-analyses. Sgnfcant computatonal savngs are also acheved for hghly-parallelzed operatons manfested n system smulatons and samplng algorthms, by adoptng the Π4U software [11] to effcently dstrbute the computatons n avalable mult-core CPUs. Such technques allow one to handle detaled lnear and nonlnear models of structural components and thus mprove SH capabltes. The mportance of the proposed computatonal framework s demonstrated for applcatons on model-based structural damage dentfcaton of cvl nfrastructure. The developed structural health montorng methodology s llustrated usng smulated damage and correspondng modal data from a brdge of the Egnata Odos motorway. 2 Damage dentfcaton methodology The Bayesan nference methodology for model class selecton based on measured modal data s presented n order to detect the locaton and severty of damage n a structure. A substructure approach s followed where t s assumed that the structure s comprsed of a number of substructures and damage n the structure causes stffness reducton n one of the substructures. In order to dentfy whch substructure contans the damage and predct the level of damage, a famly of model classes,, s ntroduced, and the damage dentfcaton s 1 m accomplshed by assocatng each model class to damage contaned wthn a 2

3 substructure. For ths, each model class s parameterzed by a number of structural model parameters q controllng the stffness dstrbuton n the substructure, whle all other substructures are taken to have fxed stffness dstrbutons equal to those correspondng to the undamaged structure. Damage n the substructure wll cause stffness reducton whch wll alter the measured modal characterstcs of the structure. The model class that contans the damaged substructure wll be the most lkely model class to observe the modal data snce the parameter values q can adjust to the modfed stffness dstrbuton of the substructure, whle the other model classes that do not contan the substructure wll provde a poor ft to the modal data. Thus, the model class can predct damage that occurs n the substructure and provde the best ft to the data. Bayesan nference s used to dentfy the most probable damaged substructure gven the modal data [1,12]. 2.1 Bayesan model class selecton 0 Let D {ˆ, ˆ N = w f Î R, r = 1,, m } be the avalable measured data consstng of r r modal frequences ˆr and modeshape components f ˆr at N 0 measured DOFs, where m s the number of observed modes. Let P ( ; ) = N0 q Μ { w ( q; Μ ), f( q; Μ ) Î R, r = 1,, m} be the predctons of the r r modal frequences and modeshapes from a partcular model n the model class Μ correspondng to a partcular value of the parameter set q. Before the selecton of data, each model class Μ s assgned a probablty P( Μ ) of beng the approprate class of models for modelng the structural behavor. Usng Bayes theorem, the posteror probabltes P( Μ D) of the varous model classes gven the data D s pd ( Μ ) P( Μ ) P( Μ D) = m å pd ( Μ ) P( Μ ) = 1 (1) where pd ( Μ ) s the probablty of observng the data from the model class Μ, gven by pd ( Μ ) = (,, ) (, ) ò pd q s Μ p q s Μ ds dq (2) Q where { : u q 0 q q } s the doman of ntegraton n Equaton (2) that u depends on the range of varaton of the parameter set q, and q are the values of 3

4 q at the undamaged condton of the structure. In Equaton (2), pd ( q, s, ) s the lkelhood of observng the data from a gven model n the model class Μ. For each model class Μ ths lkelhood s obtaned usng predctons P( q ; Μ ) and the () () () assocated probablty models for the vector of predcton errors e = [ e,, e ] 1 m defned as the dfference between the measured modal propertes nvolved n D for all modes r = 1,, m and the correspondng modal propertes predcted by a model () () () n the model class Μ. The model error e = [ e ] r w e r f for the model class Μ are r assumed to be gven separately for the modal frequences and mode shapes from the predcton error equatons: () wˆ = w ( q ; Μ ) + wˆ e r = 1,, m (3) r r r wr ˆ () ˆ () f = b f( q; Μ ) + f e r = 1,, m (4) r r r r () () () () where ˆ T T () b = ff / f f, wth f º f( q ; Μ ), s a normalzaton constant r r r r r r r that accounts for the dfferent scalng between the measured and the predcted modeshape. The model predcton errors are due to modelng error and measurement nose. Assumng that they are modeled as ndependent Gaussan zero-mean random varables wth varance s, the lkelhood s gven by [13] where 2 f r é 1 N ù J pd ( q, s, Μ ) exp J( ˆ ;, D) N /2 - q Μ 2 (5) J s 2s j êë j úû ˆ 1 1 a J( q ; Μ, D) = + (6) m m 2 m é m w ( ; ) wˆ ù ( ; )- r r r r r q Μ - å 2 r= 1 wˆ å f q Μ f êë r úû r= 1 fˆ r 2 represents the measure of ft between the measured modal data and the modal data predcted by a partcular model n the class, s the usual Eucldan norm, and N = m( N + 1). J 0 Also, gven the model class Μ, the pror probablty dstrbuton p( q, s Μ ), nvolved n Equaton (2), of the model and the predcton error parameters [ q, s ] of the model class Μ are assumed to be ndependent and of the form p ( q, s Μ ) = p ( q ) p ( s ). q s The optmal model class s selected as the model class that maxmzes the best probablty P( D) gven by Equaton (5). The probablty dstrbuton 4

5 p( q D, ) quantfyng the uncertanty n the parameters q of a model class gven the data s obtaned by applyng Bayes theorem [1], as follows pd ( q, s, Μ ) p( q, s Μ ) p( q, s D, Μ ) = (8) pd ( Μ ) Where pdμ ( ) s the evdence of the model class Μ. The most probable value of the parameter set that corresponds to the most probable model class Μ s denoted best by q ˆbest. Usng the Bayesan model selecton framework n Secton 2, the model classes are ranked accordng to the posteror probabltes based on the modal data. The most probable model class that maxmzes p( D) n Equaton (5), through best ts assocaton wth a damage scenaro on a specfc substructure, wll be ndcatve of the substructure that s damaged, whle the most probable value q ˆbest of the model parameters of the correspondng most probable model class, compared to the best parameter values of the undamaged structure, wll be ndcatve of the severty of damage n the dentfed damaged substructure. The percentage change Dq between the best estmates of the model parameters q ˆ of each model class and the values q ˆ of the reference (undamaged) structure und, s used as a measure of the severty (magntude) of damage computed by each model class, = 1,, m. The selecton of the compettve model classes, = 1,, m depends on the type and number of alternatve damage scenaros that are expected to occur or desred to be montored n the structure. The m model classes can be ntroduced by a user experenced wth the type of structure montored. The pror dstrbuton P( ) n Equaton (5) of each model class or assocated damage scenaro s selected based on the prevous experence for the type of brdge that s studed. For the case where no pror nformaton s avalable, the pror probabltes are assumed to be equal, P( ) = 1/ m, for all ntroduced damage scenaros. 3 Computatonally Effcent Technques The Bayesan tools for FE model selecton and parameter estmaton used for structural damage dentfcaton are Laplace methods of asymptotc approxmaton and stochastc smulaton algorthms. These tools requre a moderate to very large number of repeated system analyses to be performed over the space of uncertan parameters. Consequently, the computatonal demands depend hghly on the number of system analyses and the tme requred for performng a system analyss. The 5

6 Transtonal CC (TCC) algorthm [14] s employed n ths work to carry out the computatons. Hgh performance computng technques are ntegrated wthn the TCC tool to effcently handle large number of DOF n FE models. Specfcally, fast and accurate component mode synthess technques [9] are used, consstent wth the FE model parameterzaton, to acheve drastc reductons n computatonal effort. Surrogate models [10] are also used to replace full system smulatons by fast approxmatons. Fnally computatonal savngs are acheved by adoptng parallel computng algorthms to effcently dstrbute the computatons n avalable multcore CPU [15]. Detals of such HPC technques are presented n [11] where the software Π4U s ntroduced to perform a model non-ntrusve Bayesan analyss n a parallel envronment. The model reducton technques and the surrogate modelng based on krgng algorthm are brefly descrbed n the followng. 3.1 odel Reducton At the model level, model reductons technques have been proposed to consderably reduce the sze of the stffness and mass matrces by several orders of magntude. In partcular, computatonal effcent model reducton technques based on component mode synthess have been developed to handle certan parameterzaton schemes for whch the mass and stffness matrces of a component depend ether lnearly or nonlnearly on only one of the free model parameters to be updated, often encountered n FE model updatng formulatons. In such schemes, t has been shown that the repeated solutons of the component egen-problems are completely avoded, reducng substantally the computatonal demands, wthout compromsng the soluton accuracy. For the case of lnear dependence of the stffness matrx of a structural component on a model parameter, the methodology s presented n [9]. The method can readly be extended to treat the case of nonlnear dependence of the stffness matrx of a structural component on a model parameter. 3.2 Surrogate models At the level of the TCC algorthm, an adaptve krgng model has been ntroduced n [10,15] to reduce the computatonal tme by avodng the full model runs at a large number of samplng pont n the parameters space. Ths s done by explotng the functon evaluatons that are avalable at the neghbour ponts from prevous full model runs n order to generate an estmate at a new samplng pont n the parameter space. Surrogate models are well-suted to be used wthn the TCC algorthm, resultng to the X-TCC algorthm proposed n [10]. In X- TCC, the krgng technque s used to approxmate the functon evaluaton at a samplng pont usng the functon evaluatons at neghbour ponts n the parameter space. To ensure a hgh qualty approxmaton, certan condtons are mposed n order a surrogate estmate be accepted. Detals can be found n [10]. In contrast to the model reducton technque n whch several orders of magntude reducton n computatonal effort may be achevable, for surrogate models wthn TCC only an order of magntude reducton n the number of full model runs nvolved n 6

7 TCC algorthm has been reported whch results n addtonal computatonal savngs. 4 Applcaton The methodology s demonstrate usng a smulated damage scenaro and smulated modal data from the etsovo brdge, shown n Fgure 1. The brdge s the hghest renforced concrete brdge of Egnata Odos motorway located n Greece, wth the heght of the taller per P2 equal to 110m. The total length of the brdge s 537m. The commercal software package COSOL ultphyscs s used for developng the FE model of the brdge based on the desgn plans, the geometrc detals and the materal propertes of the structure. The followng nomnal values of the materal propertes of the concrete deck, pers and foundatons are consdered: Young s 3 modulus E= 37Gpa, Poson s rato n = 0.2 and densty r = 2548 kg m. For the pers and the foundaton the nomnal value of the Young s modulus s E = 34GPa. A detaled FE model s created usng three-dmensonal tetrahedron quadratc Lagrange fnte elements. The selected model has 97,636 fnte elements and 562,101 DOF. Fgure 1: etsovo brdge. 7

8 [2] [4] [5] [8] [10] Fgure 2: Parameterzed FE models correspondng to sngle damage scenaros. To demonstrate the methodology, the etsovo brdge s dvded nto 15 [] [, j] substructures. A number of compettve model classes and are ntroduced to montor varous probable damage scenaros for the brdge correspondng to sngle [] and multple damages at dfferent substructures. The model class contans one parameter related to the stffness of substructure. Representatve models correspondng to a sngle damage scenaro at dfferent substructures are shown n Fgure 2. These models can montor damage assocated wth the stffness reducton [, j] n the substructure. The model class contans two parameters related to the 8

9 stffness of substructures and j. Representatve models are shown n Fgure 3 and they can be used to montor damage assocated wth the stffness reducton n ether substructures and j or smultaneously at both substructures. A fve-parameter model shown n Fgure 3 s also ncluded n the famly of model classes to montor smultaneous damages at fve dfferent substructures. Ths fve-parameter model [5-par ] class s denoted by. All model classes are generated from the updated FE model of the undamaged structure. For each model class, CS technques are used to allevate the computatonal burden assocated wth the model updatng problems that needs to be solved. For ths, two dfferent cases of reduced-order FE models are consdered. The frst case corresponds to models obtaned by reducng the nternal DOF, whle the second case corresponds to models obtaned by reducng both the nternal and nterface DOF. [7,10] [8,10] [2-par] Fgure 3: Parameterzed FE models correspondng to multple damage scenaros. The number of components ntroduced for each model class depends on the [] parameterzaton. Specfcally, the model class s dvded nto two, three or four components. One component s selected to be the substructure shown n Fgure 2, whle the remanng components are selected to be the parts of the remanng [1] structure that connect to the nterfaces of component. The model classes, [10], [14] and [15] have one nterface, the model classes [2], [5] to [8], [11] 9

10 [12] [3] [9] [13] and have two nterfaces, whle the model classes, and have three nterfaces wth the remanng structure. A smlar dvson nto components s [, j] [10,8] ntroduced for the famly of model classes. For example, model class s dvded nto four components, the frst two components concde wth the physcal substructures 10 and 8, the thrd ncludes the physcal substructures 9, 11 to 15 and [5-par ] the fourth ncludes the substructures 1 to 7. The components n the model class are kept the same as the ones shown n Fgure 3. odel Evdence Dq DOF Class (log) (%) (NFES) [2] ,724 (8,000) [4] (8,000) [5] ,747 (9,000) [8] ,824 (9,000) [10] ,393 (12,000) [10,7] , (14,000) [10,8] , (14,000) [5 -par ] , (19,000) CE (n) Total 2155 Table 1: Damage dentfcaton results, model DOF, number of FE smulatons (NFES) and computatonal effort (CE) n mnutes for each model classes A. The effectveness of the proposed methodology, n terms of computatonal effcency and accuracy, s nvestgated by ntroducng a smulated damage at the hghest per. The nflcted damage corresponds to a stffness reducton of 30% the nomnal stffness value. Smulated, nose contamnated, measured modal frequences and mode shapes are generated for the damaged structure. Among all [10] [10, ] [5-par ] models n Fgure 2 and 3,, and contan the actual damage. The model class selecton and the model updatng s performed usng the TCC algorthm wth 1000 samples per TCC stage. The results for the log evdence for representatve model classes and the correspondng magntude of 10

11 damages D q predcted by each model class are reported n Tables 1 and 2 for the two cases A and B of reduced-order models. Heren, for demonstraton purposes, the percentage change D q between the mean estmates of the model parameters of each model class and the correspondng values of the reference (undamaged) structure measures the severty (magntude) of damage computed by each model class. odel Evdence Dq DOF Class (log) (%) (NFES) [2] (8,000) [4] (8,000) [5] (9,000) [8] (9,000) [10] (12,000) [10,7] (13,000) [10,8] (13,000) [5 -par ] (19,000) CE (n) Total 40.1 Table 2: Damage dentfcaton results, model DOF, number of FE smulatons (NFES) and computatonal effort (CE) n mnutes for model classes B. Comparng the log evdence of each model class and also the correspondng magntude of damages D q predcted by each model class n Table 1 t s evdent that the proposed methodology correctly predcts the locaton and magntude of damage usng the reduced-order model classes. Specfcally, based on the reducedorder models A, the most probable model class s whch predcts a mean [10] 29.2% reducton n stffness whch s very close to the nflcted 30%. Among all [10] [10,7] [10,8] [5-par ] alternatve model classes,, and that contan the actual damage, the proposed methodology favors the model class [10] wth the least [5-par ] number of parameters and t predcts the fve parameter model class as the least probable model. Ths s consstent wth theoretcal results for model class penalzaton for over parameterzaton, avalable for Bayesan model class selecton 11

12 [5]. The model classes that do not contan the damage are not favored by the proposed methodology. Based on the reduced-order models B n Table 2, the predctons of the locaton and severty of damage are very close to the ones obtaned from the reduced-order models A for most model classes ncluded n Table 1. In partcular, the most probable model class for models B s also predcted to be [10], whle the mean damage severty s predcted to correspond to 29.2% reducton n stffness, exactly the same as the one predcted wth the reduced-order models A. In addton the use of krgng wthn TCC (algorthm K-TCC) also provdes accurate estmates of the evdence that lead to damage dentfcaton results that are dentcal to the ones obtaned wthout the krgng estmates. The resultng number of FE model re-analyses and the computatonal demands n mnutes for each model class are also s shown n Tables 1 and 2. The number of FE model runs for each model class depends on the number of TCC stages whch vary for each model class from 8 for the one-parameter model class to 19 for the fve-parameter model class. The resultng varable number of stages per model class was automatcally obtaned from the TCC algorthm by keepng constant the value tolcov of the TCC parameter to tolcov= 1.0. The parallelzaton features of TCC [11] were also exploted, takng advantage of the avalable four-core mult-threaded computer unt to smultaneously run eght TCC samples n parallel. For comparson purposes, the computatonal effort for solvng the egenvalue problem of the orgnal unreduced FE model s approxmately 139 seconds. ultplyng ths by the number of TCC samples shown n Tables 1 and 2 and consderng parallel mplementaton n a four-core mult-threaded computer unt, the total computatonal effort for each model class s expected to be of the order of 3 to 7 days for 8,000 to 19,000 samples, respectvely. For all eght model classes consdered n Tables 1 and 2, the total computatonal effort usng the unreduced models s estmated to be approxmately one month and seven days. In contrast, for the reduced-order models A, the computatonal demands for runnng all model classes are reduced to 30 hours (2155 mnutes as shown n the last row of Table 1), whle for the reduced-order models B these computatonal demands are drastcally reduced to 40 mnutes (see Table 2). The use of surrogate models such as K-TCC [10] reduces the computatonal effort by an addtonal 85% makng the total computatonal effort equal to approxmately 6 mnutes. It s thus evdent that a drastc reducton n computatonal effort for performng the structural dentfcaton based on a set of montorng data s acheved from approxmately 37 days for the unreduced model classes to 40 mnutes for the reduced model classes B, wthout compromsng the predctve capabltes of the proposed damage dentfcaton methodology. Ths results n a drastc reducton n the computatonal effort of more than three orders of magntude. Addtonal substantal reductons n computatonal effort are possble f one has avalable more computer workers to effcently dstrbute the CC samples n each TCC stage, as well as to run the dfferent model classes, correspondng to dfferent possble damage scenaros, n parallel. The avalablty of such computer workers s expected to yeld the damage dentfcaton results n a few seconds, provded that a database of reduced parameterzed fnte element models that cover all damage scenaros have been ntroduced. 12

13 6 Conclusons A structural dentfcaton framework based on vbraton measurements was outlned n ths work. The framework ntegrates Bayesan computng tools wth hgh fdelty fnte element models of the montorng structure. It requres a large number of fnte element model analyses that can result n excessve computatonal effort. HPC technques were proposed to drastcally reduce the computatonal burden by several orders of magntude. The effectveness of the algorthms, n terms of computatonal effcency and accuracy was demonstrated usng smulated modal data from the etsovo Brdge of the Egnata Odos motorway n Greece. The proposed framework can be used by managng authortes as part of an ntellgent structural management system to provde a useful tool for structural montorng, structural ntegrty assessment and desgn cost-effectve mantenance strateges. Acknowledgements Ths research has been mplemented under the ARISTEIA Acton of the Operatonal Programme Educaton and Lfelong Learnng and was co-funded by the European Socal Fund (ESF) and Greek Natonal Resources. References [1] J.L. Beck, L.S. Katafygots, Updatng models and ther uncertantes. I: Bayesan statstcal framework, Journal of Engneerng echancs (ASCE), 124 (4), , [2] C. Papadmtrou, J.L. Beck, L.S. Katafygots, Updatng robust relablty usng structural test data, Probablstc Engneerng echancs, 16(2), , [3] H. Sohn, K.H. Law, Bayesan probablstc approach for structural damage detecton, Earthquake Engneerng and Structural Dynamcs, 26, , [4] C. Papadmtrou, J.L. Beck, L.S. Katafygots, Asymptotc Expansons for Relablty and oments of Uncertan Systems, Journal of Engneerng echancs (ASCE), 123(12), , [5] J.L. Beck, K.V. Yuen, odel selecton usng response measurements: Bayesan probablstc approach, Journal of Engneerng echancs (ASCE), 130(2), , [6] J.L. Beck, S.K. Au, Bayesan Updatng of Structural odels and Relablty usng arkov Chan onte Carlo Smulaton, Journal of Engneerng echancs (ASCE), 128 (4), , [7] E. Ntotsos, Ch. Karakostas, V. Lekds, P. Panetsos, G. Nkolaou, C. Papadmtrou, T. Salonkos, "Structural Identfcaton of Egnata Odos Brdges based on Ambent and Earthquake Induced Vbratons", Bulletn of Earthquake Engneerng, 7(2), ,

14 [8].W. Vank, J.L. Beck, S.K. Au, Bayesan probablstc approach to structural health montorng, Journal of Engneerng echancs (ASCE), 126 (7), , [9] C. Papadmtrou, D.C. Papadot, "Component ode Synthess Technques for Fnte Element odel Updatng", Computers and Structures, 126, 15-28, [10] P. Angelkopoulos, C. Papadmtrou, P. Koumoutsakos, "X-TCC: Adaptve krgng for Bayesan Inverse odelng", Computer ethods n Appled echancs and Engneerng, 289, , [11] P.E. Hadjdoukas, P. Angelkopoulos, C. Papadmtrou, P. Koumoutsakos, "Π4U: A Hgh Performance Computng Framework for Bayesan Uncertanty Quantfcaton of Complex odels", Journal of Computatonal Physcs, 284(1), 1 21, [12] L.S. Katafygots, C. Papadmtrou, H.F. Lam, A probablstc approach to structural model updatng, Internatonal Journal of Sol Dynamcs and Earthquake Engneerng, 17 (7-8), , [13] C. Papadmtrou, L.S. Katafygots, Bayesan modelng and updatng, In Engneerng Desgn Relablty Handbook, N. Nkolads, D.. Ghocel, S. Snghal (Eds), CRC Press, [14] J.Y. Chng, Y.C. Chen, Transtonal markov chan monte carlo method for Bayesan model updatng, model class selecton, and model averagng, J. Eng. ech.-asce, 133, , [15] P. Angelkopoulos, C. Papadmtrou, P. Koumoutsakos, "Bayesan Uncertanty Quantfcaton and Propagaton n olecular Dynamcs Smulatons: A Hgh Performance Computng Framework", The Journal of Chemcal Physcs, 137(14), ,