International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

The assessment of the reliability of potentially deteriorated reinforced concrete elements with Bayesian… Hackl, Jürgen; Köhler, Jochen Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015The Assessment of the Reliability of Potentially DeterioratedReinforced Concrete Elements with Bayesian NetworksJürgen HacklResearch Associate, Dept. of Civil Engineering, ETH, Zürich, SwitzerlandJochen KöhlerAssociate Professor, Dept. of Structural Engineering, NTNU, Trondheim, NorwayABSTRACT: The concrete deterioration caused by corrosion is a complex physical, chemical and me-chanical process. The modeling of this process is subjected to significant uncertainties, which are basedon a simplistic model representation of the actual physical process and limited information on material,environmental and loading characteristics.The present work proposes a generic framework for the esti-mation of structural reliability of a potentially corroded reinforced concrete element. This frameworkcombines structural reliability analysis and Bayesian networks. Thereby uncertainties by model param-eters, but also additional information, provided by measurements, monitoring and inspection results, areconsidered.1. INTRODUCTIONOne of the major deterioration mechanisms in re-inforced concrete (RC) structures is corrosion ofthe reinforcing steel. This process causes effectssuch as cracking, spalling, or delamination of theconcrete and also leads to a reduction in the rein-forcement cross-section and a loss of bond strength(Bertolini et al., 2004). These changes are accom-panied by a decrease of the load bearing capacityand structural reliability of the corresponding struc-tural elements. Therefore RC structures may be in-spected and maintained in order to control the de-crease in structural reliability. However, it is an on-going challenge to direct these activities within thelarge portfolios of RC structures cost effectively.A vast amount of research did result in a res-onable understanding of the phenomena that causeRC deteriotation and in corresponding probabilisticrepresentation of the initiation of corrosion dam-age. The representation of corrosion progressionand how a system of partly corroded reinforcement-bars affect the loadbearing capacity, the structuralreliabilty and herewith the principal performancecriterion of a RC element represents still an inter-esting research challenge.In this paper a generic framework for stochasticmodeling of the structural reliability of deteriorat-ing RC elements is presented. Thereby a combi-nation of structural reliability analysis (SRA) andBayesian networks (BNs) is used to model the pro-cess from environmental exposure towards time-dependent component reliability.2. DEGRADATION AND RELIABILITYMODELSFor the degradation of concrete structures, severalmodels have been developed to provide methodsto estimate the duration of time during which RCstructures maintain a desired level of functional-ity. Service life models such as DuraCrete (2000),LIFECON (2003), or fib Bulletin 34 (2006) providevaluable information about the durability character-istics of concrete structures.The basic approach of such a service life modelis based on Tuutti (1982), where the service lifeis subdivided to two phases, initiation and propa-gation. While the models for the initiation phaseare well documented, there is a lack of informationfor the propagation phase. Additionally, modelsfor both phases are developed separately, such thatconnections from the initiation phase to the propa-gation phase cannot be made in terms of a unified112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015model. This is unsatisfactory in scope of a holis-tic view of the service life and the identification ofoptimal decisions during the management of dete-riorated RC structures.In the remainder of this paper the initiation andpropagation models proposed by DuraCrete (2000)and the model for effects of corrosion suggested byVal and Melchers (1997) and modified by Stewart(2004, 2009, 2012), are going to be utilized.2.1. Chlorde induced corrosionA frequent cause of reinforcement steel corrosionis contamination by chloride. To initiate the cor-rosion process, the chloride content at the surfaceof the reinforcement has to reach a certain thresh-old value (Bertolini et al., 2004). The transport ofchloride in concrete towards the reinforcement canbe represented in a simplified way by use of Fick’ssecond law of diffusion :∂Ccl∂ t = Dcl∂ 2Ccl∂x2cl(1)where Ccl is the concentration of chloride ions atdistance xcl from the concrete surface after time tof exposure to chlorides and Dcl the chloride diffu-sion coefficient. The obtained solution of the partialdifferential equation is:Ccl(xcl, t) =C(ee)s,cl(1− erf(xcl2√Dcl · t))(2)where Cs,cl is the surface concentration of chloridesand erf(·) denotes the error function. According toDuraCrete (2000) Dcl can be calculated as:Dcl = k(ee)e,cl · kt,cl · k(tcur)c,cl ·D(w/c)o ·(to,clt)n(ee)cl(3)where ke,cl is the environmental parameter, kt,cl isa test method parameter, kc,cl is the executions pa-rameter, Do is the empirical diffusion coefficient,to is the reference time and ncl the age factor; de-pending on the exposure events: the exposure en-vironment ee, the curing time tcur and the water-cement ratio w/c. Each parameter can be expressedas a random variable, the corresponding distribu-tions and parameters that are used in this study aredocumented in the Appendix.For the onset of corrosion, the limit state can beassumed as the probability that the critical chlorideconcentration Ccrit is reached at the depth of the re-inforcement denoted by dc.p f ,cl = pi[Ccrit−Ccl(xcl = dc, t)≤ 0] (4)2.2. Propagation of corrosionAfter the depassivation of the reinforcement has oc-curred and the passive layer broke down, the so-called propagation phase starts and the reinforce-ment steel starts to corrode.The corrosion rate Vcorr is usually expressed asthe penetration rate and is measured in [mm/yr].A simplified propagation model based on Nilssonand Gehlen (1998) is used in the DuraCrete (2000)model. Hereby, the basic assumption is to representthe corrosion rate as a product of material parame-ters and local influencing factors:Vcorr =moρ ·F(Pcl)Cl ·FGalv ·FO2 (5)where mo is a constant for corrosion rate versus re-sistivity based on Faraday’s law, ρ is the concreteresistivity, FCl is the chloride corrosion rate fac-tor considering the chloride induced corrosion Pcl,FGalv the galvanic effect factor, and FO2 the oxygenavailability factor.In the model of Nilsson and Gehlen (1998), theresistivity ρ is a function of the concrete properties,the temperature and the moisture conditions in theconcrete cover (DuraCrete, 2000):ρ = ρo(tHydrto,r)nrkt,r · kc,r · k(T )T,r · k(RH)RH,r · k(p f ,cl)Cl,r (6)where ρo is a potential concrete resistivity for a ref-erence environment, tHydr the time of hydration, toa reference time, nr the age factor, kt,r a test methodparameter, kc,r the execution parameter, kT,r thetemperature factor depending on the temperature T ,kRH,r the humidity factor depending on the relativehumidity RH and kCl,r the chloride factor dependingif chloride induced corrosion Pcl occur; the corre-sponding distributions together with the parameterscan be found in the Appendix.2.3. Effects of corrosionIf corrosion is initiated, the consequences are a re-duction in the cross section of the load carrying re-inforcement steel, increase in bar diameter result-ing from the volumetric expansion of the corrosion212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015products, and a change in the mechanical prop-erties of the reinforcement steel and the concrete.These effects do not only lead to serviceability is-sues of corresponding structural components, butmay also affect its structural reliability and there-fore the safety of the component. Correspondinglythe corrosion affects the reinforcement itself andthe surrounding concrete. In the remainder of thispaper only the effects of corrosion on the reinforce-ment steel will be considered.Local or pitting corrosion is only associated withchloride induced corrosion. According to Gonzalezet al. (1995) the corrosion rate for pitting corrosionis four to eight times higher than the average pen-etration pav on the surface of a reinforcement bar.This ratio between maximum and average corrosionpenetration is called pitting factor and denoted byRpit = pmax/pav. Stewart (2004) proposes, that thepitting factor Rpit can be treated as a random vari-able modeled by a Gumbel distribution.For simplicity, a hemispherical form of pits is as-sumed. The radius of the pit, pmax, at time tcorr, canbe estimated as: pmax(tcorr) = Vcorr · tcorr ·Rpit. Thepit configuration is used to predict the cross sec-tional area of the pit, denoted by Apit.2.4. Structural Reliability AnalysisStructural reliability analysis (SRA) is generallyused to estimate the probability of adverse eventsrelated with the performance of structures that ishere defined with the event g(Z) ≤ 0. Here, Z de-scribe a set of random variables Z1, . . .Zn which in-fluence the performance of a structure.The probability of failure p f is equal to the prob-ability that an undesired performance will occurand is expressed thought an integral of the form:p f = pi(g(Z)≤ 0) =∫· · ·∫g(Z)≤0fZ(z)dz (7)where fZ(z) describe the complete joint probabilitydensity function, for the set of random variables Z.3. BAYESIAN NETWORK APPROACH3.1. TransformationIn order to provide a generic framework forstochastic modeling of the structural reliability ofdeteriorating RC elements, each physical model,explained in Section 2 is transformed in a BN. Ev-ery parameter of the model is represented as a vari-able or node in the single model (SM) network.Also the exposure events are represented as nodes.The edges denote the certain causal relationshipsbetween different variables corresponding to thephysical models.Based on the Eqs. (2) to (4), the BN for chlorideinduced corrosion can be represented as shown inFigure 1.PclCcl Ccritke,clkt,clkc,cl ncl Cs,cl Dow/ctcur eedcShared parametersPerformance parametersProbability of failureFigure 1: Bayesian network for corrosion.A coupled model (CM) over the whole phys-ical process is constructed. Therefore, the SMshave to be connected; shared and dependent ran-dom variables from the different BNs must be sin-gled out and replaced with the corresponding nodesor BNs. In certain circumstances some additionalnodes must be added to the CM.• Shared parameters are denoted parameters thatoccur in multiple places across one or morenetworks. Where the notation parameter de-scribes a node in which the values are not de-termined by others.• Dependent parameters are similar to sharedparameters for a SM, but with the differencethat the parameter itself depends on an otherBN. In a CM these parameters are part of thecoupled BN and link the different models to anoverall consistent model representation, start-ing by the edification of the RC structure andending by reaching a critical limit state.• In some cases it is beneficial to add extra nodesto the existing BNs, to extract some extra in-formation from the network and on the otherhand to simplify the existent model.312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015ChlorideCorrosionPropagationEffectsResistanceActionProbability of failurePclCclCcritke,clnclCs,clDokt,clkc,cleew/cdctcurPcorr ρonrkt,rkc,rkCl,rkRH,rkT,rTRHρFClmoFGaFO2Vcorrtcorrds,opmaxRpitLUαyfy,oApitQcorrAnomAsfydsR. . .. . .. . .SPfekc,clDoobserved parametershared parameterdependent parameterlogical parameterFigure 2: Coupled model for degradation of concrete, caused by corrosion.Figure 2 shows the CM for the structural reliabil-ity of a deteriorating RC element. This model com-bines the probabilistic models for: chloride inducedcorrosion, propagation of corrosion, effects of cor-rosion and structural reliability. Constant variablesfrom the physical model are modeled as uniformdistributed nodes and set evidence at the initialvalue, here illustrated as shaded nodes. Nodes de-noted by “. . .” represent additional nodes that arenecessary to compute the structural reliability.3.2. SimplificationIn order to describe the physical process of corro-sion, it is not necessary to represent every randomvariable in the model as explicit as a node in theBN. Instead parts of the BN can be reduced to asingle random variable, containing the informationof the other ones. For the information transforma-tion from several nodes to one specific node SRA isused.Consider a Bayesian network B with the prob-ability measure piB over the outcome space of aset of random variables X = {Y,Z,F}, where Y ={Y1, . . . ,Y j} denote a set of shared parameters withthe set of children Z = {Z1, . . . ,Zk}. Accordingto Section 2.4, the set of random variables Z in-fluences the performance of a structure. An unde-sired performance will occur if g(Z) ≤ 0, which isequal to the probability of failure F and it impliesan unsafe structure. The joint probability measurefor this network is given as:piB(X) = pi(Y1, . . . ,Yj,Z1, . . . ,Zk,F) (8)=j∏i=1pi(Yi)k∏i=1pi(Zi|pa(Zi)) pi(F |pa(F))Where pi(F |pa(F)) can be expressed as the indica-tor function 1[g(Z)≤0]. Hence it follows:piB(X) = pi(Y1, . . . ,Yj,F(Z)) (9)=j∏i=1pi(Yi)∫g(Z)≤0k∏i=1pi(Zi|pa(Zi)) dz1 . . .dzkThe integral in Eq. (9) can be expressed as:pi(F |Y) = pi(g(Z)≤ 0 |Y) (10)Consequently, Eq. (8) can be rewritten as:piB(X) =j∏i=1pi(Yi) pi(g(Z)≤ 0 |Y) (11)Which describes the simplified form of theBayesian network B, where the second part ofEq. (11) can be solved by using SRA.In case of the SM for chloride induced corrosion,the random variable of interest is the probability of412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015chloride induced corrosion Pcl. All other randomvariables are not explicitly necessary to describe forthe structural reliability computation. So the infor-mation of the material, environmental, execution,and test variables are collected and treated in thenode for the probability of failure. Only the inputparameters have to be modeled beside Pcl.The disadvantage of this simplification is that in-formation of the model values, such as the age fac-tor ncl, are no longer available. The huge advantageis that those values need no longer to be modeledand so a discretization of those continuous randomvariables is omitted. Beyond this, SRA can be usedto calculate the probability of failure, which leadsto more accurate values for the node Pcl as any dis-cretization of the unsimplified BN. Also the prop-erty that the node Pcl only can reach two states, al-lows to transmit those outcome values without anyloss of information to an other model (or node).The same principle as discussed previously canbe used to simplify the SM for chloride inducedcorrosion. However, Vcorr, As and fy are also onlyintermediate results for the structural reliability.Using the idea of simplification as before, these val-ues can be included in R, the random variable forthe resistance of a RC element or even included inPf , the probability of failure of the structure. Thisleads to a two-phase model, where the first phaseis the failure of corrosion and the second phase thefailure of the system according to Tuutti (1982), butnow coupled through a probabilistic model basedon a BN.3.3. Temporal probabilistic modelThe last step to provide a sophisticated model forthe deterioration process of RC structures evolvingin time is to “unroll” the previous discussed CMover the service life of the structure.A dynamic Bayesian network (DBN) is just an-other way to represent stochastic processes usinga DAG. To model domains that evolve over time,the system state represents the system at time tand is an assignment of some set of random vari-ables V . Thereby the random variable Xi itselfis instantiated at different points in time t, repre-sented by X ti and called template variable. To sim-plify the problem, the timeline is discretized intoa set of time slices with a predetermined time in-terval δ . This leads to a set of random variables inform of V 0,V 1, . . . ,V t , . . . ,V T with a joint proba-bility distribution pi(V 0,V 1, . . . ,V t , . . . ,V T ) overthe time T , abbreviated by pi(V 0:T ). This distri-bution can be reparameterized by using the chainrule for probabilities.To unroll the CM over the service life of thestructure, the properties of DBNs are used. Thatlead to a so-called dynamic coupled model (DCM).Key of this procedure is to model the DBN effi-ciently, which includes the simplification and opti-mization methodologies introduced in Section 3.2,to decrease the size of the network.After the simplification of the CM and the subdi-vision of the shared parameters into initial and tem-poral values, the BN can be expanded over time,which is shown in Figure 3.PftcorrPcorr Pclee dctcur w/cTRHdofy,o . . .TFigure 3: Dynamic Bayesian network for degradationof concrete caused by corrosion, given as plate model.This proposed DCM framework, based on SRAand BNs, allows to couple the probabilistic mod-els for the initiation and propagation of corrosion.Thereby uncertainties by model parameters, butalso additional information, provided by measure-ments, monitoring and inspection results, can beconsidered.4. APPLICATION: BENDING BEAM4.1. Structural configurationThe structural configuration is assumed to be a sim-ply supported RC beam with a rectangular cross-section. Therefore, the ultimate flexural capacityMu of the RC beam can be used to describe the re-sistance R of the structure. The resistance R can beexpressed in terms of ultimate flexural capacity Mu512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015for a simple supported RC beam:R = Mu = As fy(d− As fy1.6 fcb)Xm (12)where As is the cross-sectional area of the reinforce-ment, fy the yield strength, d the effective depth, fcthe compressive strength of concrete, b the beamwidth, and Xm describes the model uncertainties.The nominal resistance in RC design is obtainedfrom the design condition (Eurocode, 2012):1γmRk = αγGGk +(1−α)γQQk (13)where γm, γG and γQ are the corresponding partialsafety factors for the resistance and for the load, andα describes the ratio between permanent loads Gkand variable loads Qk and is defined in the rangefrom 0 to 1.The time period is taken as 50 years. Accord-ing to Eurocode (2012), a target reliability indexof βt = 3.8 should be reached, assuming no de-terioration of the RC structure. This can be en-sured by using a suitable set of partial safety factors(γm = 1.4,γG = 1.35,γQ = 1.5) and a permanent tovariable load ratio of α = 0.5.4.2. ResultsThe following sections, show some results and ca-pabilities of the proposed DCM framework. Due tothe complexity and vast amount of data, only fewsimplified results will be presented here.Therefore, it is assumed, that no evidence isgiven for the considered system, meaning all ex-posure parameters treated as unknown and equallylikely. Hence, the quantitative values are indicativeand not necessarily realistic.The proposed framework allows an analysis forthe whole process of corrosion, i.e. the initiationprocess, the propagation process and the mechani-cal performance of a RC structure that depend oneach other. This leads to the result that the quali-tative service life model proposed by Tuutti (1982)actually can be represented as quantitative model.Figure 4 shows a schematic of the typical servicelife modelling approach based on Tuutti (1982),represented by the solid line. The DCM frameworkis shown by the dashed line. Here, the end of ser-vice life depends on the defined limit state; for ex-ample, the reliability index β falling below an ac-ceptable reliability index βacc.Figure 4: Comparison of service life models.In the DCM framework, a conventional initialphase is no longer outlined, which describes theperiod during the depassivation of reinforcement,because of the fact that even in the first weeks theprobability of corrosion onset is considered. Thisassumption has been confirmed under experimentalconditions and field conditions.Instead, the initiation phase in the DCM de-scribes the period of time where no significant lossof structural performance can be expected. This cri-teria is related to the general requirements concern-ing the safety of the RC structure, in terms of theprobability of failure, as shown in Figure 5. Thisperiod of time depends not only on the model forchloride induced corrosion but rather on the wholeDCM, which includes also the propagation and theeffects of corrosion.The probabilistic models for the case of corro-sion initiation and propagation, caused by chloridepenetration, are functions of a number of randomvariables, discussed in Section 2. The simplifiedmodels in Section 3.2 have reduced the amount ofrandom variables to a set of shared parameters.These parameters can also be called initial con-dition indicators and are used in the probabilisticmodel as prior estimations for the probability thatthe RC structure is in a condition state at some de-fined period of time during its service life.To make existing service life models more accu-rate and realistic, the proposed framework allows,on the one hand, to deal with uncertainties by theinput parameter of the model and on the other handto consider dependency in time.For example, the bar diameter ds can be used toprovide a prior estimated of the structural perfor-612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Figure 5: Probability of (a) corrosion and (b) failure over the service life.mance. However, additional information about thebar diameter affects only the probability of failure,as shown in Figure 5. In contrast, the effects of theconcrete cover dc or the surrounding environmentinfluences the probability of corrosion and proba-bility of failure. The exact estimation of those pa-rameters for an RC structure is not always possible.Hence, using the DCM framework allows to takethose uncertainties into account.During service life several parameters maychange and/or the RC structure evolves differentlyin time as expected. Hence, new information aboutthe condition of the RC structure have to be col-lected during the service life. One way is to moni-tor the RC structure and continuously collect dataabout the condition states. Another option is todo selective inspections of the condition of interestduring the service life.For instance, half-cell potential measurement(HCPM) is a widely recognized and standardizednon-destructive method for assessing the corrosionstate of the reinforcement in RC structure. Thequality of the HCPM is affected by different factors(e.g. moisture, cracks, etc.). Hence, the interpre-tation of a HCPM should be combined with othermeasurements and information.Based on these assumptions, Faber et al. (2006)proposed that whether the probability of corro-sion given by HCPM, is observed or not, is statedpi(Ihc|p f ,cl,y) = 0.9 and pi(Ihc|p¯ f ,cl,y) = 0.24, wherethe inspection is expressed through the probabilityof an indication of corrosion initiation Ihc, giventhat corrosion occurs p f ,cl,y, or given that corrosiondoes not occur p¯ f ,cl,y.Figure 5 shows the influence of HCPM on a RCstructure, performed after 10 and 30years in servicewith corresponding positive and negative test indi-cations. If no HCPM is performed the prediction isstill based on the prior estimations for the models.After a inspection the test result can be consideredin the DCM. Hence, the prediction of the structuralsafety can be updated.5. CONCLUSIONSThe present paper proposes a DCM framework,based on SRA and BNs, which enables the couplingof probabilistic models for the initiation and prop-agation of corrosion. Within this framework uncer-tainties of model parameters and additional infor-mation, provided by measurements, monitoring andinspection results, can be considered.The basic features of the framework are illus-trated along an indicative example. Another possi-ble feature of the proposed framework is the studyof the impacts of parameters, models and decisionson the results by sensitivity analysis. Such a studymight deliver important insights for the further de-velopment of the models in particular and might di-rect future research needs in the area in general.Using the presented simplification of the SMs,the critical part of discretizing a continuous ran-dom variable for BNs can be reduced to a minimumamount or even eliminated. Hence, the DCM can beexpanded over a period of time and updated whennew information becomes available.Keeping it feasible for the beginning, only oneelement of a RC structure that is exposed to the pro-cess of corrosion is considered. However, the dete-rioration of concrete caused by corrosion is stronglyrelated to spatial and temporal variability. Thisproperty can be modeled by different approachesand will be necessary for a holistic contemplationof the system.712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 20156. APPENDIXTable 1: Probabilistic parametersSym. Description Type Mean CoV Unit Ref.Do diffusion coefficient [1]w/c = 0.4 Norm 220.9 0.12 mm2/yrw/c = 0.45 Norm 315.6 0.1 mm2/yrw/c = 0.5 Norm 473.4 0.09 mm2/yrke,cl environmental parameter [1]Submerged Gam 1.325 0.17 −Tidal Gam 0.924 0.17 −Splash Gam 0.265 0.17 −Atmospheric Gam 0.676 0.17 −Cs,cl surface concentration∗ [1]Submerged Norm 10.35 (0.07,0.06) %wbTidal Norm 7.76 (0.18,0.14) %wbSplash Norm 7.76 (0.18,0.14) %wbAtmospheric Norm 2.57 (0.14,0.16) %wbncl age factor [1]Submerged Beta 0.30 0.17 −Tidal Beta 0.37 0.19 −Splash Beta 0.37 0.19 −Atmospheric Beta 0.65 0.11 −kc,cl execution parameter [1]tcur = 1 d Beta 2.4 0.29 −tcur = 3 d Beta 1.5 −tcur = 7 d Det 1 − −tcur = 28 d Beta 0.8 0.13 −kt,cl test method parameter Norm 0.832 0.03 − [1]to,cl reference time Det 0.077 − yr [1]Ccrit critical concentration [1]w/c = 0.3 Norm 0.9 0.17 %wbw/c = 0.4 Norm 0.8 0.13 %wbw/c = 0.5 Norm 0.5 0.2 %wbmo rate / resistivity Det 822 − µmΩ/yr [1]FCl corrosion rate factor [1]corrosion is true sLN 5.71 1.84 −corrosion is false Det 1 − −FGalv galvanic effect factor Det 1 − − [1]FO2 oxygen factor Det 1 − − [1]ρo pot. concrete resistivity Norm 77 0.16 Ωm [1]nr age factor Norm 0.23 0.17 − [1]kT,r temperature factor [1]T > 20 oC Norm 0.073 0.21 −T = 20 oC Det 1 − −T < 20 oC Norm 0.025 0.04 −kRH,r humidity factor Norm [1]RH = 50 % sLN 1669 1.79 −RH = 65 % sLN 15.8 1.63 −RH = 80 % sLN 5.42 0.82 −RH = 95 % LN 2.94 0.14 −RH = 100 % Det 1 − −kCl,r chloride factor Norm 0.72 0.15 − [1]kc,r execution parameter Det 1 − − [1]kt,r test method parameter Det 1 − − [1]to,r reference time Det 1 − yr [1]Rpit pitting factor [2]ds = 10 mm Gum 5.67 0.23 −ds = 16 mm Gum 6.23 0.24 −ds = 27 mm Gum 7.17 0.19 −ds bar diameter Norm(10,16,27) 0.01 mm [3]h beam height Norm 550 0.02 mmL beam length Det 5000 − mmb beam width Norm 350 0.03 mmfc compressive strength LN [25,45] 0.17 Mpa [3]dc concrete cover LN [15,85] 0.3 mm [3]Gk permanent loads Norm 1 0.1 kN/m2Qk variable loads Gum 1 0.4 kN/m2fy yield strength LN [450,600] 0.05 Mpa [3]Xm model uncertainty LN 1.1 0.07 − [3][1] DuraCrete (2000), [2] Stewart (2012), [3] JCSS (2008)∗ with (CoVA,CoVε ), Cs,cl = A · (w/c)+ ε , and ε ∼ N(1,σε )7. REFERENCESBertolini, L., Elsener, B., Pedeferri, P., and Polder, R. B.(2004). Corrosion of steel in concrete : prevention,diagnosis, repair. Wiley-VCH, Weinheim.DuraCrete (2000). “Statistical quantification of thevariables in the limit state functions.” Project BE95-1347/R9, Brite EuRam III.Eurocode (2012). Basic of structural design. ComiteEuropeen de Normalisation, Brussels. EN 1990.Faber, M. H., Straub, D., and Maes, M. A. (2006).“A computational framework for risk assessment ofrc structures using indicators.” Computer-Aided Civiland Infrastructure Engineering, 21(3), 216–230.fib Bulletin 34 (2006). “Model code for service life de-sign.” Report 34, Int. Fed. for Structural Concrete.Gonzalez, J., Andrade, C., Alonso, C., and Feliu, S.(1995). “Comparison of rates of general corrosionand maximum pitting penetration on concrete embed-ded steel reinforcement.” Cement and Concrete Re-search, 25(2), 257–264.JCSS (2008). “Risk assessment in engineering: Princi-ples, system representation and risk criteria. The JointCommittee on Structural Safety.LIFECON (2003). “Life cycle management of concreteinfrastructures for improved sustainability.” Report D3.2, VTT.Nilsson, L.-O. and Gehlen, C. (1998). “Statistical quan-tification of the environmental parameters in the cor-rosion rate model.” Report 4.1.8, Brite EuRam III.Stewart, M. G. (2004). “Spatial variability of pittingcorrosion and its influence on structural fragility andreliability of rc beams in flexure.” Structural Safety,26(4), 453–470.Stewart, M. G. (2009). “Mechanical behaviour of pittingcorrosion of flexural and shear reinforcement and itseffect on structural reliability of corroding rc beams.”Structural Safety, 31(1), 19–30.Stewart, M. G. (2012). “Spatial and time-dependent re-liability modelling of corrosion damage, safety andmaintenance for reinforced concrete structures.” Jour-nal of Structure and Infrastructure Engineering, 8(6),607–619.Tuutti, K. (1982). Corrosion of steel in concrete. CBIforskning. Swedish Cement and Concrete ResearchInstitute, Stockholm.Val, D. V. and Melchers, R. E. (1997). “Reliability ofdeteriorating rc slab bridges.” Journal of StructuralEngineering, 123(12), 1638–1644.8

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