On Mathematical Models of Degradation Processes According to ISO16204 and fib Model Code

On Ma hema ical Models o Deg ada ion P ocesses Acco ding o ISO 16204 and ib Model Code Ma ina Šomodíko á1 | Jiří Doležel1 | Da id Lehký1 1 In oduc ion In he ib Model Code 2010 [1], se ice li e is de ined as “ he pe iod in which he equi ed pe o mance o a s uc- u e o s uc u al elemen is achie ed, when i is used o i s in ended pu pose and unde he expec ed condi ions o use”. I is a empo al (quan i a i e) alue ha is no a ma e ial p ope y bu is ela ed o he abili y o ma e ials, componen s and sys ems o main ain speci ic u ili y and o he p ope ies a he equi ed le el unde no mal main enance, o e a ce ain ime pe iod and unde gi en ope a ing and en i onmen al condi ions. The me hods o de e mining he esidual se ice li e o new and exis ing s uc u es may be as ollows: Based on knowledge o he se ice li e o a simila s uc u e (wi h simila cha ac e is ics) loca ed in sim- ila condi ions; On he basis o accele a ed es s; Using ma hema ical models; Using s ochas ic me hod – a me hod using a eliabili y model o a me hod using a combina ion o s a is ical and de e minis ic models may be used. In he case o he de e mina ion o se ice li e based on he knowledge o he se ice li e o a simila s uc u e, i is a he an es ima e due o he la ge a iabili y, especially in he p ope ies o he ma e ials used and he in luence o he a ious en i onmen s. Fo he de e mina ion based on accele a ed es s, i is necessa y ha he deg ada ion mechanisms o he ma e ial in accele a ed condi ions a e he same as in he eal en i onmen . E en i his condi ion is me , he lack o da a on he deg ada ion a e unde no - mal condi ions, based on long- e m moni o ing o he con- di ion o s uc u es and long- e m es ing, is a majo p ob- lem in he de e mina ion o se ice li e. The e o e, ma hema ical modelling and s ochas ic me hods, which a e he ocus o his pape , appea o be app op ia e me h- ods o he se ice li e assessmen . Rela ed o his issue he u iliza ion o a s anda dized me hodology implemen ed in he indus y wo ldwide plays an impo an ole. A he u n o he millennium, e o s we e ini ia ed o de elop a pla o m o du abili y design o conc e e s uc u es ha con ained he same elemen s and philosophy as ha o mode n s uc u al design [2]. In ollowing ew yea s he ib Bulle in No. 34 [3] was endo sed. The close coope a ion be ween he ib and ISO commi ees was es ablished and based on he p inciples o ISO 2394 [4] he ib Model Code 2010 [1] and he ISO 16204 [5] documen s we e inalized. These wo documen s a e oday close o being iden ical when assessing he se ice li e and du abili y o con e e s uc u es. 2 Ma hema ical modelling o deg ada ion p o- cesses In mos cases, he se ice li e o a s uc u e, S, ela ed o ORIGINAL ARTICLE Abs ac The main ac o s a ec ing he se ice li e o conc e e s uc u es a e he p esence o chlo ide ions and ca bon dioxide, al e na ing os ac ion and mechanical s esses on he s uc u e. In p ac ice, hese e ec s can be aken in o accoun by using ma h- ema ical modelling o hese phenomena. In his con ibu ion, he au ho s ocus on he p ocesses o chlo ide ions and ca bon dioxide (ca bona ion p ocess) di usion h ough conc e e and he associa ed subsequen co osion o ein o cemen . Ma h- ema ical models ecommended by he ib Model Code and ISO 16204 a e desc ibed and he e ec o simul aneous en i onmen al and mechanical load on he se ice li e o he s uc u e is discussed. The models a e used o s udy he esidual se ice li e o a simple ein o ced conc e e ame b idge based on he pe cen age loss o ein o cemen a ea analysis o e ime. Keywo ds ib Model Code, ISO 16204, Conc e e Ca bona ion, Chlo ide Ions Ing ess, Co osion o Rein o cemen Co espondence Ing. Ma ina Šomodíko á, Ph.D. B no Uni e si y o Technology Facul y o Ci il Enginee ing, Ins i u e o S uc u al Mechanics Ve eří 331/95 60200 B no Email: somodiko a.[email p o ec ed].cz 1 B no Uni e si y o Technology, Facul y o Ci il Enginee ing, B no, Czech Republic P oceedings in ci il enginee ing This is an open access a icle unde he e ms o he C ea i e Commons A ibu ion-NonComme cial-NoDe i s License, which pe mi s use and dis ibu ion in any medium, p o ided he o iginal wo k is p ope ly ci ed, he use is non-comme cial and no modi ica ions o adap a ions a e made. h ps://doi.o g/10.1002/cepa.2135 wileyonlinelib a y.com/jou nal/cepa ce/pape s 6 (2023), No. 5 © 2023 E ns & Sohn GmbH. 1221 he du abili y limi s a e, is de ined as he ime ha elapses om he ime he s uc u e is pu in o se ice o he ime when he ein o cemen co osion p ocess is ini ia ed, i.e., S = i. In he case o pene a ion o chlo ide ions and ai CO2 h ough he conc e e, he co osion p ocess is ini ia ed by di usion o chlo ide ions in he conc e e o he dep h a which he s eel ein o cemen is loca ed, o by ca bona- ion, which lowe s he pH o he conc e e a i s con ac wi h he s eel ein o cemen . Taking a less conse a i e iew, he se ice li e can be de e mined as he sum o wo ime pe iods, i.e., he ini ia ion phase, i, and he p opa- ga ion phase, p, as S = i + p. Since he co osion o he ein o cemen educes i s e ec i e a ea, he sa e y c i e- ion can be de ined, o example, on he basis o eaching a limi alue o pe cen age loss o a ea o he ein o ce- men ba s. No e he e ha while he a ea o he ein o ce- men dec eases, he co osion p oduc s o med ha e a wo old o six old inc ease in olume, which leads o an inc ease in ensile s ess in he su ounding conc e e, and hus o he o ma ion o longi udinal con inuous c acks and consequen spalling o he conc e e co e . 2.1 Co osion o ein o cemen The cou se o ein o cemen co osion, which akes place du ing he p opaga ion s age o he assumed se ice li e o he s uc u e, i.e., a e he ini ia ion ime o ein o ce- men depassi a ion is eached, can be modelled acco ding o [6]. Fo he case o uni o m co osion, he ollowing equa ions a e used o es ima e he ein o cemen diame- e d [mm] o e ime: 𝑑(𝑡)={ 𝑑i 𝑑i−0.0116𝑖co 𝑅co (𝑡−𝑡i) 0 (1a) o he p esc ibed condi ions o : { 𝑡≤𝑡i 𝑡i<𝑡≤𝑡i+𝑑i 0.0116𝑖co 𝑅co 𝑡>𝑡i+𝑑i 0.0116𝑖co 𝑅co (1b) whe e di [mm] is he ini ial diame e o he ein o cemen , ico [μA/cm2] akes he co osion cu en densi y (co o- sion a e) in o accoun and Rco [-] e lec s he ype o co osion (uni o m o pi ing). Simila ly, he pi ing dep h p [mm] can be de e mined o e ime o pi ing co osion as [7]: 𝑝(𝑡)={ 0 o 𝑡≤𝑡i 0.0116𝑖co 𝑅co (𝑡−𝑡i) o 𝑡>𝑡i (2) Assuming a hemisphe ical shape o he pi he esidual (ne ) c oss-sec ional a ea o he co oded ba (A [mm2], g ey a ea in Fig. 1 le ) a ime > i can be calcula ed as [8]: 𝐴 (𝑡)= { π𝑑i2 4−𝐴1−𝐴2 o 𝑝(𝑡)≤√2 2𝑑i 𝐴1−𝐴2 o √2 2𝑑i<𝑝(𝑡)≤𝑑i 0 o 𝑝(𝑡)>𝑑i (3a) wi h 𝐴1=12[𝜃1(𝑑i 2)2−𝑎p|𝑑i 2−𝑝(𝑡)2 𝑑i|] 𝐴2=12[𝜃2𝑝(𝑡)2−𝑎p𝑝(𝑡)2 𝑑i] 𝑎p=2𝑝(𝑡)√1−(𝑝(𝑡) 𝑑i)2 𝜃1=2a csin(𝑎p 𝑑i) 𝜃2=2a csin(𝑎p 2𝑝(𝑡)) (3b) The c oss-sec ional a ea can be calcula ed wi hou signi i- can loss o accu acy using a simple equa ion ha as- sumes a ci cula pi shape and a ein o cemen a ea cal- cula ed based on he o e lap o wo ci cles o adius di (see Fig. 1 igh ), as in [9]: 𝐴 (𝑡)={π𝑑i2 4−𝐴p o 𝑝(𝑡)≤𝑑i 0p o 𝑝(𝑡)>𝑑i (4a) wi h 𝐴p=𝑑i2 4(𝛿−sin 𝛿) 𝛿=2a ccos(1−𝑝(𝑡) 𝑑i) (4b) Figu e 1 A ea o ein o cing ba wi h pi ing co osion – hemisphe ical pi (le ), ci cula pi ( igh ) 2.2 Conc e e ca bona ion and chlo ide ions in- g ess As al eady men ioned abo e, he depassi a ion o ein- o cemen can be caused mainly by ca bona ion p ocess and/o by he chlo ide ions ing ess, whe e he ini ia ion ime is de e mined on he basis o he assump ion ha he ca bona ion on eaches he dep h a which he s eel membe s a e loca ed o he chlo ide ion concen a ion a he dep h o he s eel ein o cemen eaches a c i ical alue. The ma hema ical models o ca bona ion and chlo- ide pene a ion a e based on Fick's second law o di u- sion: 𝜕𝐶 𝜕𝑡=𝐷𝜕2𝐶 𝜕𝑥2 (5) whe e C is he concen a ion o he pe mean , x is he dis- ance om he su ace, is he di usion ime and D is gene ally he di usion coe icien , which depends on he p ope ies o he conc e e and he en i onmen . The solu- ion o his di e en ial equa ion is ound by he C ank p o- cedu e using he Gaussian e o unc ion “e (.)”. Acco ding o ISO 16204 [5], he ca bona ion dep h xc [mm] a ime [yea s] can be calcula ed as: 𝑥c(𝑡)=𝑊𝑘√𝑡 (6) 1222 25097075, 2023, 5, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.1002/cepa.2135 by Technical Uni e si y In B no, Wiley Online Lib a y on [23/10/2023]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License whe e he pa ame e W [-] akes changing clima ic condi- ions (such as humidi y and empe a u e) in o accoun , k [-] is a ac o desc ibing he basic esis ance o he con- c e e mix u e o he ca bona ion p ocess. When designing new s uc u es, he pa ame e s W and k can be de e - mined based on da a measu ed on simila s uc u es in simila clima ic condi ions; when e i ying exis ing s uc- u es, hey mus be de e mined om measu emen s. Fo modelling, he sophis ica ed model ecommended in he ib Bulle in No. 34 [3] can be used, whe e he ca bona ion dep h is de e mined based on he o mula: 𝑥𝑐(𝑡)=√2𝑘e𝑘c(𝑘 𝑅ACC,0 −1 +𝜀 )𝐶CO2∙𝑊(𝑡)∙√𝑡 (7) The en i onmen al unc ion pa ame e , ke, includes he in- luence o he mois u e con en o he conc e e su ace (in luence o he ela i e humidi y RH [%]) on he alue o he di usion coe icien , he pa ame e kc akes he in lu- ence o he cu ing ime o he esh mix (pa ame e s c [days] and bc [-]) on he e ec i e esis ance o he con- c e e o ca bona ion in o accoun . The pa ame e 𝑅ACC,0 −1 [(m2/s)/(kg/m3)] is he in e se o he e ec i e esis ance o he d y mix o ca bona ion, he alue o which is de e - mined by es ing conc e e specimens exposed o condi- ions in which he ca bona ion p ocess is accele a ed ( he ACC es ). The pa ame e s k [-] and ε [(m2/s)/(kg/m3)] co e he di e ences be ween samples es ed unde ac- cele a ed condi ions and he s uc u e es ed unde na u- al condi ions. The CCO2 pa ame e includes he e ec o he CO2 concen a ion and he ime-dependen wea he unc ion W( ) [-] accoun s o he wea he condi ions due o ain (pa ame e s pSR [-], w [days] and bw [-]). We e e he eade o [3] o de ailed ecommenda ions on he in- di idual model pa ame e s. In he case o chlo ide ions ing ess, he chlo ide concen- a ion, C(x, ) [w . %/cemen ] ( ela i e o he weigh o cemen ), a dep h x [mm] (mos o en a he dep h o he conc e e co e o he s eel ein o cemen ) and ime [yea s] can be exp essed o 1D cases by a simple analy - ical o mula: 𝐶(𝑥,𝑡)=𝐶0+(𝐶S−𝐶0)[1−e (𝑥 2√𝐷c𝑡)] (8) whe e CS [w . %/cemen ] is he chlo ide concen a ion on he conc e e su ace, Dc [mm2/yea ] is he di usion coe - icien o chlo ide pene a ion h ough he conc e e and C0 [w . %/cemen ] is he ini ial chlo ide concen a ion in he conc e e. The solu ion o Eq. (8) is alid assuming homo- geneous ma e ial ully sa u a ed wi h wa e and cons an CS and Dc alues o e ime. Due o i s simplici y, his so- lu ion has been used o se e al decades and is also ec- ommended in a ious modi ica ions by a numbe o na- ional and in e na ional no ma i e documen s. C i ical poin s o he use o models o chlo ide pene a ion h ough conc e e a e summa ized in e.g. [10], whe e, among o he hings, he high a iabili y o modelling esul s caused by o en conside able unce ain y in he alues o model pa- ame e s is men ioned. Ano he widely discussed opic is he simul aneous ac ion o mul iple deg ada ion p ocesses and he ime dependence o di usion coe icien and su - ace chlo ide concen a ion The ime dependence o he di usion coe icien Dc can be modelled acco ding o he ISO 16204 [5] using he o - mula: 𝐷c(𝑡)=𝐷app(𝑡)=𝐷app(𝑡0)(𝑡0 𝑡)𝛼 (9) whe e Dapp( 0) [mm2/yea ] is he ac ual alue o he di u- sion coe icien de e mined a he e e ence ime, 0 [yea s] ( 0 = 28 days = 0.0767 yea s) and α [-] is he ageing ac o o he conc e e, which akes he inc ease in he esis ance o he conc e e o he pene a ion o ag- g essi e subs ances due o i s ageing in o accoun , i.e., i akes in o accoun he dec ease in he Dapp alue o e ime due o he hyd a ion o he cemen componen s (changes in he po e s uc u e). I should be no ed ha , bo h in he design o new s uc u es and in he assessmen o he e- sidual li e o exis ing s uc u es, he ageing ac o α should be ob ained om obse a ions o s uc u es in si u, whe e he conc e e composi ion, pe o mance and condi ions o exposu e o agg essi e subs ances a e simila o hose o he ac ual s uc u e. To calcula e he ageing ac o , obse - a ions du ing a leas wo pe iods o exposu e (wi h a su icien in e al be ween obse a ions) a e necessa y. The model acco ding o he ib Bulle in No. 34 [3] accoun s o he ime dependence o he di usion coe icien as ol- lows: 𝐷c(𝑡)=𝐷app(𝑡)=𝑘en ∙𝐷RCM(𝑡0)∙𝑘u∙(𝑡0 𝑡)𝛼 (10) whe e ken [-] is he en i onmen al pa ame e ha akes he e ec o empe a u e on he di usion coe icien alue (pa ame e s T [°C] and be [°C]) in o accoun , DRCM( 0) [m2/s] is he chlo ide mig a ion coe icien , he alue o which can be de e mined om he Rapid Chlo ide Mig a- ion es (RCM), and k [-] is he con e sion ac o o he di usion coe icien uni (ku = 3.1536∙106; 1 m2/s = 3.1536∙106 mm2/yea ). The concen a ion o chlo ide ions is hen de e mined based on Eq. (8). A he su ace o conc e e exposed o ex e nal chlo ide, a laye whe e di usion is no he main p ocess and he chlo- ide ions pene a ion p ocess di e s om he Fick's second law o di usion due o exposu e o equen we ing and subsequen e apo a ion, he so-called con ec ion zone, Δx, usually o ms. Taking in o accoun his ac , he Eq. (8) can be u he modi ied o he o m o [11]: 𝐶(𝑥,𝑡)=𝐶0+(𝐶S,∆𝑥−𝐶0)[1−e (𝑥−∆𝑥 2√𝐷app(𝑡)∙𝑡)] (11) The maximum concen a ion o chlo ide ions he e is no a he ou e su ace (x = 0), bu a ises p og essi ely a he posi ion x = ∆x, wi h his hickness gi en in he ange o 6–11 mm. In he con ec ion zone, chlo ide concen a ions can de ia e conside ably om no mal alues, so he model acco ding o Eq. (11) neglec s hese alues and wo ks wi h he so-called su oga e chlo ide concen a ion CS,∆x [w . %/cemen ], which is applied only om a dep h g ea e han he dep h o he con ec ion zone. This modi- ica ion in de e mina ion o he chlo ide concen a ion in ime is only meaning ul o long exposu e imes when he con ec ion zone has o med and emained mo e o less cons an o a long ime. 1223 25097075, 2023, 5, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.1002/cepa.2135 by Technical Uni e si y In B no, Wiley Online Lib a y on [23/10/2023]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License To exp ess he ime dependence o he chlo ide concen- a ion on he su ace o conc e e, CS, a linea o oo - mean-squa e dependence on ime wi h cons an k acco d- ing o [12] can be used: 𝐶S(𝑡)=𝐶0+𝑘𝑡 o 𝐶S(𝑡)=𝐶0+𝑘√𝑡 (12) No e he e ha he unc ions in Eq. (12) a e no clea ly sui able o desc ibing he chlo ide ions ing ess om he sp eading sal s, which is ela ed o he change o seasons, he ac ion o ain (washing he sal s om he conc e e su - ace) and possibly o he clima ic o o he in luences. The ime dependence o CS alue is also no included in ISO 16204 due o he complexi y o he whole issue and he e- o e he use o hese models should always be ca e ully conside ed in iew o possible e o s in he de e mina ion o he se ice li e o s uc u es. As an example, he e ec o seasonal applica ion o hawing sal s has been ad- d essed o 2D cases by he cellula au oma a echnique, see e.g. [13]. 2.3 Simul aneous e ec s o en i onmen al ac- ions and mechanical load In ui i ely, i can be assumed ha deg ada ion p ocesses will be as e in s uc u es wi h c acks. Howe e , nei he he ib no he ISO commi ees ha e been able o come up wi h any gene al model ha akes his e ec in o accoun . The e o e, i was ag eed o use a simpli ied app oach ha assumes ha ein o cemen co osion is no a ec ed up o a ce ain c ack wid h. Depending on he se e i y o he en i onmen al in luence and he sensi i i y o he s uc- u e, he limi ing c ack wid h is usually gi en by a cha ac- e is ic alue (i.e., 5% uppe quan ile) in he ange o 0.2 o 0.4 mm [2]. Acco ding o s udies epo ed in e.g. [14], he in luence o s ess on he a e o ca bona ion and chlo ide di usion can be easily aken in o accoun by means o co ec ion ac- o s. The dep h o ca bona ion can be p edic ed wi h e- spec o he s ess s a e acco ding o he o mula: 𝑥c(𝑡)=𝑘𝜎𝐴√𝑡 (13) whe e he cons an A can be calcula ed using any sui able ca bona ion model depending on he composi ion and cu - ing o conc e e, ype o cemen , humidi y o he en i on- men , CO2 con en , o o he pa ame e s (see e.g. he abo e men ioned model acco ding o [3]). The co ec ion ac o kσ [-] is de ined sepa a ely o elemen s unde en- sion (s ess σ ) and comp ession (s ess σc) as: 𝑘𝜎(𝜎 /𝜎u, )=1+1.41(𝜎 /𝜎u, )+0.82(𝜎 /𝜎u, )2 𝑘𝜎(𝜎c/𝜎u,c)=1−2.27(𝜎c 𝜎u,c)+4.86(𝜎c/𝜎u,c)2 (14) whe e σu, and σu,c ep esen he ul ima e ensile and com- p essi e s ess o he conc e e. The dependence o kσ on he ensile/comp essi e s ess a io and i s limi ing alue is shown in Fig. 2. I is e iden ha a modes comp essi e load, i.e., alues o (σc/σu,c) in he ange o 0 o app oxi- ma ely 0.5, dec eases he a e o conc e e ca bona ion. This e ec is due o pa ial closing o mic o-c acks unde an applied comp essi e s ess. Howe e , i he load is in- c eased abo e app oxima ely 50 % o he ul ima e com- p essi e s eng h, he on-going closing mic o-c acks is o e compensa ed by o ma ion o new c acks h ough which ai CO2 can pene a e he conc e e. This accele a es he ca bona ion p ocess. As he ensile load inc eases, he ca bona ion p ocess is only accele a ed, since e en a mod- es ensile s ess leads o he o ma ion and opening o mic o-c acks, which se e as new pa hways o he pene- a ion and mig a ion o CO2 in o he conc e e. Figu e 2 Dependence o he kσ ac o on he a io o ensile/comp es- si e s ess and i s limi alues As in he case o ca bona ion, he models o chlo ide ions ing ess can be ex ended by he e ec o mechanical load on he s uc u e. Wi h espec o c acks in he conc e e, o ma ed due o s ess om loading, he di usion o chlo- ide ions is accele a ed. When he e ec o c acks is con- side ed, he alue o he di usion coe icien (gene ally deno ed by D) changes. This can be di ided in o wo pa s, D0 [m2/s] and D [m2/s], as shown in Fig. 3. The di usion coe icien D [m2/s] is de e mined wi h espec o he wid h o he c acks o med, w [mm], and hei maximum dis ance s ,max [mm] as [15]: 𝐷=(1− 𝑤 𝑠 ,max)𝐷0+𝑤 𝑠 ,max𝐷 (15) whe e D0 is he alue o he di usion coe icien o in ac conc e e and D is he alue o he di usion coe icien in- side he c ack, he pa ame e s w and s ,max can be ob ained e.g. by measu emen on a eal s uc u e o by calcula ion acco ding o e.g. [16]. The alues o D can be de e mined wi h espec o he c ack wid h as ollows [17]: 𝐷 ={0 m2/s (0,16𝑤−3)∙10−10 m2/s 13∙10−10 m2/s (16a) o he c ack wid h o : {𝑤<30 μm 30 μm ≤ 𝑤≤100 μm 𝑤>100 μm (16b) Figu e 3 Di usion coe icien D o chlo ide pene a ion h ough c acked conc e e 1224 25097075, 2023, 5, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.1002/cepa.2135 by Technical Uni e si y In B no, Wiley Online Lib a y on [23/10/2023]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License The applica ion o hese ela ions o modelling he simul- aneous e ec o mechanical and en i onmen al load and he e i ica ion o he abili y o hese models o achie e esul s close o eali y is p esen ed in [18]. 3 S udy on ma hema ical models wi h o wi h- ou he e ec o mechanical load A simple ein o ced conc e e ame b idge was selec ed o he esidual se ice li e s udy. The b idge was pu in o op- e a ion in 2019, and a e wo yea s o use (2021), a con- inuous c ack g ea e han 0.1 mm wide was obse ed in he middle o he span o he ame gi de (see Fig. 4). Figu e 4 Con inuous c ack g ea e han 0.1 mm wide in he middle o he ame span, eco ded a e wo yea s o b idge use – ups eam side (le ), subs uc u e bo om iew (middle) and downs eam side ( igh ) The se ice li e o he b idge was analysed using ma he- ma ical modelling o deg ada ion p ocesses. The inpu ma- e ial pa ame e s conside ed we e as ollows: C30/37 con- c e e class (exposu e class XD1 – medium we , humid en i onmen and conc e e su ace exposed o chlo ides dispe sed in ai ; wa e o cemen a io w/c = 0.55), B500B,A ein o cemen p o ile o 32 mm diame e , con- c e e co e 50 mm, s i ups o 12 mm diame e (i.e., main ein o cemen co e a = 62 mm). Fo modelling o deg ada ion p ocesses, he a o emen- ioned models acco ding o he ib Bulle in No. 34 [3] and ISO 16204 [5], espec i ely, we e used. The inpu pa am- e e s we e de ined acco ding o Tab. 1. No e he e ha a combina ion o de e minis ic and s ochas ic models was used o de e mine he esidual se ice li e. De e minis ic alues o ca bona ion dep h, chlo ide concen a ion and loss o ein o cemen a ea o e ime we e calcula ed based on he mean alues o he inpu pa ame e s (column “Mean” in Tab. 1). To ake he a iabili y in ma e ial p op- e ies and en i onmen al cha ac e is ics in o accoun , he same ou pu s we e calcula ed using s ochas ic modelling based on a sui ably chosen s a is ical model o he inpu a iables (columns “COV” and “PDF” in Tab. 1, meaning he coe icien o a ia ion and p obabili y densi y unc ion, espec i ely). The esul s a e summa ized in Fig. 5, which shows he mean alues (de e minis ic model) ± s anda d de ia ions (s ochas ic model) o ca bona ion dep h and chlo ide con- cen a ion o e ime. FReET-D so wa e was used o s o- chas ic modelling (see e.g. [19]), s a is ical cha ac e is ics we e calcula ed based on one hund ed o andom simula- ions. No he de e minis ic no he s ochas ic calcula ion does no assume ha he ein o cemen will be depassi a ed due o ca bona ion o chlo ide a ack o e he design li e o he s uc u e (solid lines in Fig. 5). The ein o cemen depassi a ion is no assumed e en i he e ec o c acks in conc e e is aken in o accoun (dashed lines in Fig. 5). The esul s o he s ochas ic modelling hen show some a iabili y in he modelling esul s. Howe e , using he ib and ISO model, he dep h o conc e e co e and he c i ical alue o chlo ide ions concen a ion a he dep h o he conc e e co e is no eached e en when conside ing he a iabili y o he inpu a iable, and i is he e o e clea ha he design o he s uc u e wi h espec o he limi s a e o du abili y is pe ec ly ine wi h espec o a su i- cien conc e e co e . Table 1 De ini ion o inpu pa ame e s Inpu pa ame e [Uni ] Mean COV [-] PDF* No e CCO2 [mg/m3] 820 0.12 N RH [%] 70 0.07 B Limi s: 0–100 c [days] 7 - De . bc [-] –0.567 0.04 N 𝑅ACC,0 −1 [(m2/s)/(kg/m3)] 9.8×10–11 0.42 N Limi s: 1×10–12–1×10–9 k [-] 1.25 0.28 N ε [(m2/s)/(kg/m3)] 1×10–11 0.15 N pSR [-] 1 - De . w [days] 60 - De . bw [-] 0.446 0.37 N x [mm] 62 0.12 LN σ /σu, [-] 1 - De . Assuming eaching he ensile s eng h o conc e e CS [w . %/cemen ] 0.465 0.75 N Lowe limi : 0 C0 [w . %/cemen ] 0.0 - De . DRCM [m2/s] 1.97×10–11 0.20 N be [-] 4526.85 0.15 N T [°C] 10 0.05 N α [-] 0.3 0.40 B Limi s : 0–1 Cc [w . %/cemen ] 0.6 0.15 B Limi s : 0.2–2.0 w [mm] 0.3 - De . Based on measu emen s (e - ec o empe a u e changes and conc e e sh inkage) s ,max [mm] 600 - De Calcula ed acco ding o [16] di [mm] 32 0.02 N ico [-] 3 0.5 R Rco [-] 6 - De Mean = 2 o uni o m co osion No e: *De . = de e minis ic alue; B = be a, LN = logno mal, N = no mal, R = ec an- gula p obabili y densi y unc ion Al hough he ma hema ical modelling does no assume de- passi a ion o he ein o cemen and subsequen co osion o he ein o cemen ba s, a heo e ical analysis o he loss o ein o cemen a ea was pe o med; see Fig. 6. He e, 1225 25097075, 2023, 5, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.1002/cepa.2135 by Technical Uni e si y In B no, Wiley Online Lib a y on [23/10/2023]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License di e en o mulas we e used o ein o cemen a ea calcu- la ions in o de o compa e he po en ial pe cen age loss o ein o cemen a ea o e co osion p opaga ion pe iod. Uni o m as well as pi ing co osion was compa ed o his pu pose. Figu e 5 Resul s o a s udy o he esidual se ice li e assessmen – modelling o conc e e ca bona ion and chlo ide ions ing ess Figu e 6 Resul s o a s udy o he esidual se ice li e assessmen – modelling o ein o cemen co osion The sa e y c i e ion was de e mined acco ding o he na- ional s anda d ČSN 73 6221 [19] o e y poo (VI) o eme gency condi ion (VII) as eaching he limi ing alue o he co oded ein o cemen a ea a he le el o 15 %. This alue would be achie ed a e app oxima ely 40 yea s o p opaga ing co osion (Fig. 6). Based on he heo e ical ma hema ical models a co osion loss o ein o cemen a ea a he le el o 15 % was eached a e 40 yea s o co osion p opaga ion wi h a p obabili y o 47.6 o 56.4 % (acco ding o he o mula used o calcula e he esidual a ea o co oded ein o cemen , A ). A no mal dis ibu ion was assumed in his case. No e ha he p og ess o he ein o cemen a ea is e y simila o uni o m and pi ing co osion du ing he i s 50 yea s o p opaga ion. In he ollowing yea s, pi ing co osion p og esses much as e . I no addi ional measu es a e aken o ake in o accoun he iden i ied de ec s in he o m o a localized c ack, he s uc u e will be classi ied as being in s uc u al condi ion VI o VII a e 40 yea s o use wi h his p obabili y. The e- a e , es ic i e measu es will ha e o be aken wi hin he amewo k o i s use, in he ex eme case esul ing in i s closu e. I should be no ed he e ha i he loss o he e- in o cemen a ea due o co osion up o 1 % is de ec ed, i is ecommended o ecalcula e he load-bea ing capaci y in ela ion o he classi ica ion o he s uc u e in s uc u al condi ion IV (sa is ac o y). Based on ma hema ical mod- elling, such a loss o ein o cemen a ea is achie ed wi h a 50% p obabili y in 2 yea s o 11 yea s a e he ini ia ion o co osion p ocess. 4 Conclusions When he b idge owne /manage o s uc u al enginee is aced wi h he need o assess he esidual se ice li e o conc e e s uc u es in ela ion o ongoing deg ada ion p o- cesses, ein o cemen co osion and he cu en s uc u al condi ion, he use o app op ia e ma hema ical models in combina ion wi h s ochas ic me hods appea s o be a e y as and e icien ool compa ed o esidual se ice li e as- sessmen p ocedu es based on knowledge o he se ice li e o a simila s uc u e loca ed in simila condi ions o based on accele a ed es ing. The use o ad anced me h- ods o s ochas ic se ice li e analysis allows a g ea e in- sigh in o he p og ess o he deg ada ion p ocesses aking place, wi h he abili y o make p edic ions o he damage o e ime. The alues o he heo e ical p obabili y o ail- u e when limi ing alues a e exceeded, such as loss o e- in o cemen a ea due o co osion, can also be calcula ed. Finally, we no e ha ma hema ical and s ochas ic models o conc e e deg ada ion and ein o cemen co osion e- qui e knowledge o he inpu pa ame e s and hei p oba- bilis ic models, which in some cases may appea o be a disad an age. Howe e , he ma hema ical models can be calib a ed based on he esul s o eal in-si u measu e- men s and diagnos ic su eys ca ied ou . Sui able p oba- bili y dis ibu ion unc ions and s a is ical cha ac e is ics can also be ob ained om he a ailable li e a u e and some no ma i e documen s. 5 Acknowledgemen s The au ho s would like o g a e ully acknowledge he i- nancial suppo o he Czech Science Founda ion p ojec No. 22-00774S. Re e ences [1] ib Bulle ins Nos. 65 and 66 (2012) Model Code 2010 1226 25097075, 2023, 5, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.1002/cepa.2135 by Technical Uni e si y In B no, Wiley Online Lib a y on [23/10/2023]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License – Final d a , Volume 1 and 2. In e na ional Fede a ion o S uc u al Conc e e ( ib), Lausanne, Swi ze land. [2] Helland, S. (2013) Design o se ice li e: implemen- a ion o ib Model Code 2010 ules in he ope a ional code ISO 16204. S uc u al Conc e e 14, No. 1, pp. 10–18. 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Cemen and Conc e e Resea ch 38, No. 6, pp. 877–883. [18] Vořecho ská, D.; Šomodíko á, M.; Pod oužek, J.; Lehký, D.; Teplý, B. (2017) Conc e e s uc u es unde combined mechanical and en i onmen al ac ions: Modelling o du abili y and eliabili y. Compu e s and Conc e e 20, pp. 99–110. [19] No ák, D.; Vořecho ský, M.; Teplý, B. (2014) FReET: So wa e o he s a is ical and eliabili y anal- ysis o enginee ing p oblems and FReET-D: Deg ada- ion module. Ad ances in Enginee ing So wa e 72, pp. 179–192. [20] ČSN 73 6221 P ohlídky mos ů pozemních komu- nikací. Úřad p o echnickou no malizaci, me ologii a s á ní zkušebnic í, 2018. [in Czech] 1227 25097075, 2023, 5, Downloaded om h ps://onlinelib a y.wiley.com/doi/10.1002/cepa.2135 by Technical Uni e si y In B no, Wiley Online Lib a y on [23/10/2023]. See he Te ms and Condi ions (h ps://onlinelib a y.wiley.com/ e ms-and-condi ions) on Wiley Online Lib a y o ules o use; OA a icles a e go e ned by he applicable C ea i e Commons License