Modelng of Cobned Deeroraon of Concree Srucures by Copeng Hazard Model Kyoyuk KAITO Assocae Professor Froner Research Cener Osaka Unv., Osaka, apan kao@ga.eng.osaka-u.ac.p Kyoyuk KAITO, born 97, receved hs cvl engneerng degree fro he Unv. of Tokyo. Koch SUGISAKI Senor Reseacher BMC Corp. Chba, apan sugsak@hashor.p Koch SUGISAKI, born 976, receved cvl engneerng degree fro Tokyo Insue of Technology. Kyosh KOBAYASHI Professor, Vce Dean Graduae School of Manageen, Kyoo Unv. Kyoo, apan kkoba@psa.box.eda.kyoou.ac.p Kyosh KOBAYASHI, born 95, receved cvl engneerng degree fro Kyoo Unv. Suary In asse anageen of nfrasrucures, predcng deeroraon of srucures s one of an essenal echnque. However, n order o esae her deeroraon process wh hgh accuracy, no only sngle cause deeroraon bu also cobned deeroraon appearng on for exaple concree srucures has o be consdered. Therefore hs paper addresses he copeng hazard odel whch s possble o odel coplex deeroraon by ulple causes, and especally focuses on ha deeroraon by neuralzaon and sal aack on concree brdges. Specally, separaely deeroraons by he are forulaed by he Webull hazard odel ha can express he edependen deeroraon phenoena. Eployng hs odel, he phenoena of whch deeroraon probably s ncreasng wh e can be descrbed. Furherore, he coplex deeroraon s forulaed by neuralzaon hazard rae s lnearly affecng sal aack one. Keywords: Copeng hazard ode; cobned deeroraon; corroson; sascal deeroraon predcon; asse anageen.. Inroducon As a ehod for ananng and anagng socal nfrasrucures effcenly, asse anageen s aracng aenon. A he frs sage of asse anageen, s exreely poran o predc how deeroraon proceeds n srucures fro a acro perspecve. In hs crcusance, as deeroraon predcon echnology, a sascal ehod, n whch a predcon odel s sascally developed based on nspecon daa, has been proposed. Especally, n recen years, a large aoun of daa have been accuulaed hrough he research no deeroraon predcon ehods ulzng hazard odels[], whch ndcaes ha her praccales are hgh. However, he convenonal deeroraon predcon odel arges he daage caused by a sngle facor and descrbes s deeroraon process. Meanwhle, under he suaon where srucural and aeral characerscs are dversfed and he usage and envronenal condons vary lke socal nfrasrucures, here are any evens ha have any facors n causng he sae daage sae. Moreover, such daagng evens can be classfed no he followng wo evens: ( deeroraon evens n whch each of several facors proceeds ndependenly and hen he os rapd facor becoes sgnfcan, and ( coplex deeroraon evens n whch soe of several facors are copounded durng he deeroraon process and accelerae he deeroraon process. As a represenave coplex deeroraon, he corroson of renforcng bars nsde concree srucures can be enueraed. The corroson of renforcng bars s consdered due o sal daage and neuralzaon, and here are experenal resuls ha he process of neuralzaon acceleraed he process of sal daage. In addon, he crackng due o he cobnaon of freezng daage & sal daage or alkal aggregae reacon & sal daage can be sad o be coplex deeroraon, because f one even proceeds, he oher even akes he caalyc role, accelerang he deeroraon process. Furherore, wh respec o seel aerals, here s no fague l n repeaed sress under he corroson condon, and so here s a possbly ha corroson wll ncrease he probably of fague crack no aer he value of repeaed sress. Therefore, he developen of a odel of he deeroraon process caused by ore han one facor ndependenly or only can be sad o be a useful echnology ha would conrbue o he advance of asse anageen.
Consderng he above enoned ssues, he obecve of hs sudy s o esablsh a deeroraon predcon odel ha akes no accoun copounded deeroraon evens, by usng a copeng hazard odel. In he followng secons of hs paper, Chaper forulaes a Webull deeroraon hazard odel ha akes no accoun he copeve naure aong several facors, by usng wo odels ha were classfed based on wheher or no he copeve naure depends on e. Chaper 3 focuses on coplex deeroraon by he wo copeng facors: neuralzaon and sal daage, argeng concree srucures, and esaes he ng of he corroson of renforcng bars, usng acual nspecon daa, and eprcally analyzes he effecveness of he proposed odel. The followng chapers enon he renforceen corroson nsde concree srucures as a specfc arge for a copeve deeroraon hazard odel. Needless o say, s possble o apply hs odel o oher daage and deeroraon phenoena.. Copeng Webull hazard odel. Forulaon of a copeve hazard odel The auhors consruc a odel of he coplex deeroraon phenoenon relaed o he renforceen corroson nsde concree srucures, usng a hazard odel. The arge concree ebers are represened by ( =,..., K, and he elapsed e fro he sar of use of he eber o presen s denoed by. In addon, s assued ha here s ore han one facor ( =,..., n renforceen corroson a he concree eber. Here, s assued ha he elapsed e ζ (lfespan unl renforceen corroson occurs a he eber s a rando varable and s subec o he probably densy funcon f ( ζ and he dsrbuon funcon F ( ζ for each facor. Here, he doan of he lfe-span s [,. In addon, he probably F ( ζ ha renforceen corroson wll no occur for years can be expressed by he followng equaon: ob ζ = F( ζ = F ( ζ ( { } Pr The condonal probably ha renforceen corroson does no occur a he eber unl he arbrary ng and hen renforceen corroson occurs due o he facor durng he perod, + Δ can be expressed by he followng equaon: [ f Δ λ Δ = ( F( where λ represens he hazard funcon for each corroson facor, whch s dscrnaed fro he hazard funcon ha akes no accoun all corroson facors. Ths equaon ndcaes ha he hazard funcon s probably densy. Furherore, he paral survval dsrbuon funcon and paral densy funcon for each corroson facor can be expressed by he followng equaons, usng Equaons ( and (, respecvely []: F = exp { ( u du} { exp ( u du} λ (3 f = λ λ (4 Nex, he hazard funcon for all corroson facors s defned. Snce he survval dsrbuon funcon for all facors F s he probably ha any corroson facor does no rgger corroson, ha s, he on probably of he paral survval dsrbuon funcon for each facor F ( ( =,...,, F = = F By subsung Equaon (3 no Equaon (5, he equaon can be arranged as follows: { } F = exp λ ( u du = exp λ ( u du (6 = = By defnng he hazard funcon for all facors λ lke Equaon (3 and coparng wh ( (5
Equaon (6, he followng equaon s obaned. λ = λ (7 = Lkewse, he probably densy funcon of all facors can be expressed as follows, based on he relaon of Equaon (. f = λ F = L (8 f = ( λ + L+ λ F( = f( + + f =. Copeng Webull deeroraon hazard odel.. Hazard odel n whch he copeve naure does no depend on e A concree funconal for s provded for he hazard funcon, and he copeng hazard odel s specfed. In hs sudy, he Webull deeroraon hazard odel s appled, whle assung ha he probably of renforceen corroson ncreases wh e. Here, he exponenal deeroraon hazard odel n whch he probably of renforceen corroson does no depend on e s a specal for of he Webull deeroraon hazard odel. Here, as deeroraon facors, neuralzaon (deeroraon facor, sal daage (deeroraon facor, and coplex deeroraon are specfed. Suppose ha deeroraon proceeds nsde he concree srucure because of neuralzaon (deeroraon facor only. The hazard funcon for deeroraon facor a arbrary ng can be defned wh he followng equaon as a general Webull deeroraon hazard odel. = γ λ (9 where γ s he paraeer represenng he arrval rae of corroson, and s he acceleraon paraeer represenng he e dependency of he hazard funcon. When he acceleraon paraeer s equal o, he hazard funcon corresponds o he exponenal deeroraon hazard odel ha does no depend on e. When he Webull deeroraon hazard funcon s used, he survval dsrbuon funcon can be expressed by he followng equaon: F { λ ( u du} = exp( γ = exp ( Nex, le us defne he hazard funcon a he ng n he case where deeroraon proceeds because of sal daage (deeroraon facor. In hs case, he coplex deeroraon phenoenon beween sal daage and neuralzaon (deeroraon facor s also aken no accoun, and so he followng equaon s defned: { γ + δ αγ } λ = ( In he above equaon, ha s, he hazard funcon for sal daage (deeroraon facor, s assued ha when coplex deeroraon s observed, he arrval rae paraeer of he renforceen corroson due o deeroraon facor γ s lnearly nfluenced by he arrval rae paraeer due o deeroraon facor γ. Naely, hs forulaon s ade so ha he hazard funcons for neuralzaon and sal daage do no nfluence each oher, bu neuralzaon unlaerally nfluences sal daage. Here, δ s he duy varable: when coplex deeroraon occurs δ = ( oherwse As enoned laer, s possble o oban he nforaon on wheher or no coplex deeroraon has occurred, hrough vsual nspecon or he lke. In addon, α s he paraeer represenng he degree of he nfluence of neuralzaon on he hazard funcon for sal daage. When he copeve naure s defned based on Equaon (, he copeve naure becoes consan regardless of e. Ths basc odel s herenafer called Model I. In addon, he survval
dsrbuon funcon for sal daage (deeroraon facor s expressed by he followng equaon: { λ } { } ( u du = exp ( γ + αδ γ F exp = (3 Accordngly, he unknown paraeers, whch are esaed by he copeve hazard odel for renforceen corroson, are he followng fve: he corroson arrval rae paraeers: γ and γ, he acceleraon paraeers: and, and he nfluence degree paraeer: α... Hazard odel n whch he copeve naure depends on e I s assued ha he speed of he concenraon of chlorde on, whch s he deernan of renforceen corroson, changes (ncreases wh e. Here, s possble o defne he hazard odel for sal daage (deeroraon facor akng no accoun he e dependency of he copeve naure as follows: { γ + δ αλ } λ = (4 The hazard odel n whch he e dependency of he copeve naure s descrbed s called Model II. Whle Model I s consruced under he assupon ha he nfluence of neuralzaon on sal daage s represened by he corroson arrval rae paraeer γ, Model II assues ha hs nfluence s represened by he hazard funcon λ (. In hs case, he survval dsrbuon funcon for sal daage s expressed by he followng equaon: F = exp + { λ ( u du} = exp{ { γ + αδ λ ( u } u du} = exp γ αδ γ.3 Esaon ehod + (5.3. Conens of nspecon daa Suppose ha K vsual nspecon daa regardng concree srucures have been obaned. The nspecon daa s coposed of anly nspecon resuls and characersc nforaon. Inspecon resuls have he nforaon on wheher or no renforceen corroson has occurred and he facors n corroson (neuralzaon, sal daage and coplex deeroraon for each nspecon daa saple ( =,..., K. On he oher hand, for characersc nforaon, he srucural, usage, and envronenal condons ha would nfluence renforceen corroson are recorded. Frsly, nspecon resuls are descrbed. Ou of K nspecon daa saples, he saples n whch renforceen corroson has occurred are classfed as follows, for each facor: neuralzaon and sal daage. (Snce coplex deeroraon s expressed n he hazard funcon for sal daage, hs s ncluded n he sal daage saples n hs dscusson. The populaon of he saples n whch renforceen corroson has occurred due o he facor s represened by Φ, and he oal nuber of such saples s represened by K(, and he k-h saple of Φ s denoed by (, k. In he above dscussons, saples are dsngushed by usng bu n he followng dscussons, saples are dscrnaed by usng (, k. Accordngly, he populaon of he saples n whch renforceen corroson has occurred due o neuralzaon (deeroraon facor, = s represened by Φ, he oal nuber of such saples s K(; hose for sal daage (deeroraon facor, = are Φ and K(, respecvely. On he oher hand, he sound condon saples n whch renforcng bars dd no corrode are defned as =. Obvously, here s he followng relaon: K ( + K ( + K ( = K (6 In hs sudy, s assued ha afer he occurrence of renforceen corroson s confred, corroson facors are denfed hrough nspecon. In hs assupon, ordnary vsual nspecon s consdered. However, s noeworhy ha corroson facors can be denfed before he occurrence of renforceen corroson hrough he core-saplng es or he lke. Based on he above enoned defnons and assupons, s possble o descrbe he nforaon on he occurrence of renforceen corroson and he facors n corroson (neuralzaon and sal daage, regardng
concree srucures (, k. In addon, he elapsed e of he saple (, k s represened by k. Wh regard o coplex deeroraon, he duy varable n Equaon ( s redefned as δ k, and n he sal daage saple (, k, s possble o check wheher or no coplex deeroraon s he corroson facor. Nex, characersc nforaon s descrbed. I s assued ha he sar ng of he corroson of renforcng bars depends on he aeral and envronenal characerscs of concree srucures. Accordngly, he srucural, usage, and envronenal characerscs ha would nfluence he corroson facors: neuralzaon and sal daage are specfed. For each facor, he characersc k k k vecor s represened by x = ( x, x. Here, he subscrps and denoe neuralzaon and sal k k k k k k daage, respecvely. In addon, x = ( x,..., xe, x = ( x,..., xe, and e and e represen he oal nubers of characersc nforaon ha would nfluence respecve facors. Accordngly, he nforaon on concree srucures (, k ha can be obaned hrough vsual nspecon can be k k expressed by ξ =, δ, x. k k.3. Esaon of unknown paraeers wh he axu-lkelhood ehod In order o esae he copeng Webull deeroraon hazard odel based on he nspecon daa populaon ξ, s assued ha he dfferences n corroson occurrence due o he srucural, usage, and envronenal condons can be expressed by he arrval rae of corroson: γ k, whch s represened by a characersc vecor as follows: k γ = exp( β x =, (7 k e where β = ( β, β, L, β s he row vecor coposed of unknown paraeers. The sybol n he above equaon denoes ransposon. Accordngly, he characersc nforaon on each k concree srucure: x, whch can be obaned hrough nspecon, s refleced n he arrval rae paraeer. Therefore, he laer-enoned esaon of unknown paraeers can be consdered as he esaon of β raher han he drec esaon of he arrval rae paraeer γ k. Then, he unknown paraeers ha should be esaed are β, β,,, and α. These are represened by θ = ( β,, α. Nex, when he k-h saple (, k n whch corroson facor ( =, nduces renforceen corroson s focused on, he lkelhood funcon of he saple can be expressed by he followng equaon, based on Equaons ( and (5. L k ( θ = f = λ F = λ F L λ (8 k k k k k F k = k Fh k h= On he oher hand, he lkelhood funcon of he sound condon saple (eher facor does no nduce daage; =, n whch renforceen corroson dd no occur, s expressed as follows: L (θ (9 k = F( k = Fh k h= Furherore, he su of log lkelhoods for all saples s expressed as follows: K ( ln L( θ = ln ( θ ( = k = L k When he axu-lkelhood ehod s used, he axu lkelhood esaor of he paraeer θ ha axzes he log lkelhood funcon ( can be obaned as θˆ ha sasfes he followng relaon: ln L( ˆ θ = θ (
Table Overvew of Used Daa (oal nuber of saples: 45 Average Sandard Devaon Age 46. 9.4 Neuralzaon Deph [] 38.3 9. Chlorde Ion Concenraon a Renforcng Bars [kg/ 3 ]..4 Table Reference Values for Esang Renforceen Corroson Used daa Reference Value Neuralzaon Neuralzaon resdue 5 Sal daage Chlorde on concenraon (whole sal conen kg/ 3 Coplex deeroraon Inal chlorde on aoun kg/ 3 3. Eprcal Sudy 3. Overvew of nspecon daa In hs sudy, he auhors could use he dagnosc daa of he core exracon es, whch s dealed nvesgaon, n addon o vsual nspecon. Table shows he averages and varances of age, neuralzaon deph, and chlorde on concenraon a renforcng bars, whch are consdered o nfluence renforceen corroson, for all of 45 saples. The saples were aken fro concree srucures near he shorelne, and so chlorde on concenraon s hgh as a whole. Wh regard o age, varaon s relavely large. The necessary explaned varables for hs odel are he wo values represenng wheher or no renforceen corroson has occurred. However, n order o esae he copeng deeroraon hazard odel, s necessary o clarfy whch s he facor n renforceen corroson; neuralzaon, sal daage, or coplex deeroraon. In hs sudy, n order o nfer he facor n renforceen corroson, he auhors focused on neuralzaon resdue for neuralzaon, chlorde on concenraon for sal daage, and nal chlorde on aoun for coplex deeroraon, and hen obaned acual easureen values hrough he core exracon ehod argeed a corroson-seen saples. The facor n renforceen corroson was denfed by coparng he acual easureen values wh he reference values (hresholds shown n Table. Praccally, here are soe cases n whch s possble o conduc he core exracon es for all saples. In such a case, he facor n renforceen corroson s deerned based on he subecve udgen by nspecors or engneers. Wh regard o neuralzaon, he reference value of neuralzaon resdue n he specfcaon s, bu s known ha he acual easureen value of coverng deph vares consderably. Therefore, wh regard o neuralzaon, consderng safey and assung ha he specfed value of coverng deph vares fro 5 o 4, was recognzed ha when neuralzaon resdue exceeds 5, renforceen corroson occurs. Wh regard o sal daage, was recognzed ha renforceen corroson occurs when he sal conen a renforcng bars exceeds kg/ 3 ; hs value was specfed consderng ha he saples were aken n he vcny of he shorelne and ephaszng safey whle he reference value n he specfcaon s. kg/ 3. Wh regard o coplex deeroraon, he auhors udged by checkng wheher or no nal chlorde on concenraon s above he reference value by referrng o he prevous sudy. Wh regard o coplex deeroraon, s accepable o udge by seeng he dsrbuon for of chlorde on concenraon, bu has been poned ou ha coplex deeroraon s hghly correlaed wh nal chlorde on concenraon, and so he auhors decded o adop he forer one n hs sudy. In soe cases, here were saples n whch he values of neuralzaon and sal daage exceed he respecve reference values. Such saples were classfed no coplex deeroraon.
Table 3 Esaed Paraeers β β β Model I -6.4 3.3-3.48..93 63.49 Model II -33.55 7.36-7.84.7.6.45 α 3. Esaon of copeng hazard odel Ulzng nspecon daa, he auhors conduc he esaon of he copeng Webull deeroraon hazard odel, whch was forulaed n Secon. Fro he nforaon ha can be obaned hrough nspecon, he explanaory varables of he hazard odels for neuralzaon and sal daage were seleced hrough ral and error. As a resul, here were no daa ha can be used as explanaory varables for neuralzaon, and he surface chlorde on aoun was adoped for sal daage. Accordngly, he concree hazard funcons for neuralzaon γ and sal daage γ k can be defned as follows: k γ = exp(, γ = exp( β + β x ( (3 β k k where x s he surface chlorde on concenraon of he saple (, k ha was noralzed so ha s axu s equal o. The neuralzaon hazard funcon: γ s consan regardless of he saple (, k, because he explanaory varable s coposed of a consan er only. I s noeworhy ha γ does no have he subscrp k. In addon, as enoned n Secon.., when he copeve deeroraon hazard odel n whch he copeve naure does no depend on e: Model I s adoped, he log lkelhood for all saples becoes as follows: ln L( θ : ξ = = + K ( k = D( { γ ( γ + αδ γ } + { ln[ γ ] γ ( γ + αδ γ } [ ] D( { ln { γ k + αδ kγ } k γ k ( γ k + αδ kγ k } k = K ( = k = k ln L k = k K ( k = ln L k k + K ( k = k ln L k k = + K ( k = ln L k k On he oher hand, when he copeng deeroraon hazard odel n whch he copeve naure depends on e: Model II s used, he log lkelhood for all saples can be forulaed as follows: ln L( θ : ξ = + K ( + k γ k γ kk αδ kγ k = + K ( k = [ ln + k k k k (4 { ln[ γ ] γ ( γ + αδ γ } + k {( γ + αδ γ } γ γ αδ γ ] k k k k k K ( k = k k k k k + (5 The unknown paraeers ha axze he above equaons (4 and (5 are esaed wh he drec search ehod. The esaon resuls are shown n Table 3. Boh of he acceleraon paraeers are above. Ths ndcaes he characerscs of he Webull deeroraon hazard ype raher han he exponenal deeroraon hazard ype. Especally, s obvous ha he nfluence of he acceleraon paraeer for neuralzaon becoes sronger wh e. In addon, α n Model I s larger han ha n Model II. I can be consdered ha he e dependency of he copeve naure s aken no accoun n Model II, and so he coplex deeroraon process s descrbed even f α s sall. 3.3 Analycal resuls Wh he hazard odel, s possble o forulae survval probably. If he probably ha k k k
Survval Probably..8.6.4 Neuralzaon. Sal Daage Coplex Deeroraon. 4 6 8 Elapesd Te [Years] Fg. Survval Probably: Model I Survval Probably..8.6.4 Neuralzaon. Sal Daage Coplex Deeroraon. 4 6 8 Elapsed Te [Years] Fg. Survval Probably: Model II renforceen corroson occurs due o he corroson facor a arbrary ng τ s called survval probably, can be calculaed by solvng he dsrbuon funcon for each facor. Fgs. shows he survval probables n Model I, whch was calculaed usng he esaon resuls enoned n he prevous secon. Fg. shows he survval curve calculaed usng he average of surface chlorde on concenraon. For he fgure, he hgh facors n renforceen corroson are coplex deeroraon, sal daage, and neuralzaon n hs order. Therefore, can be concluded ha he coplex deeroraon proceeds ore rapdly han he deeroraon caused by a sngle facor of neuralzaon or sal daage. In deal, he perod unl he survval probably agans renforceen corroson due o coplex deeroraon, sal daage, and neuralzaon reaches.5 s abou 7 years, abou 83 years, and abou 3 years, respecvely n Fg.. I s noeworhy ha he reason why neuralzaon s slow s ha he saples used for hs eprcal analyss were aken fro concree srucures near he shorelne and so here were few saples of neuralzaon. Acually, he nuber of saples of neuralzaon was only 4 ou of a oal of 45 saples. On he oher hand, Fgs. shows Model II n whch he copeve naure depends on e. Wh regard o he survval curves calculaed fro he average of surface chlorde on concenraon, he coplex deeroraon proceeds lke he deeroraon due o sal daage a he early sage, bu he probably of corroson seeply ncreases fro a ceran age. In addon, as obvous fro he resuls n Table 3, n he presen odel ha akes no accoun he e dependency of he copeve naure, he corroson due o neuralzaon occurs earler and s nfluence on coplex deeroraon s larger. These are he reason why he corroson due o neuralzaon s earler and he corroson due o sal daage s slower n hs fgure han Fg.. 4. Conclusons In hs sudy, he auhors proposed a copeng hazard odel akng no accoun he copeve naure aong several facors, argeng he coplex deeroraon phenoenon a nfrasrucures. Focusng on he esaon of he ng of he sar of renforceen corroson nsde concree, he auhors confred ha he copeve naure beween neuralzaon and sal daage acceleraes he sar of renforceen corroson and suded when he effec of coplex deeroraon becoes sgnfcan, hrough he eprcal analyss usng acual nspecon daa. Accordngly, s expeced ha he proposed odel wll enable he descrpon of he coplex deeroraon phenoenon a socal nfrasrucures and conrbue o he advance of asse anageen. Par of hs sudy was conduced a Froner Research Base for Global Young Researchers, Graduae School of Engneerng, Osaka Unversy, suppored by Specal Coordnaon Funds for Proong Scence and Technology, Mnsry of Educaon, Culure, Spors, Scence and Technology. Reference []TSUDA, Y., KAITO, K., AOKI, K. and KOBAYASHI, K.: Esang Markovan Transon Probables for Brdge Deeroraon Forecasng, ournal of Srucural Eng./Earhquake Eng., SCE, Vol.3, No., pp.4s-56s, 6.