Probabilistic Assessment of Concrete Structure Durability under Reinforcement Corrosion Attack

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Probabilistic Assessment of Concrete Structure Durabilityunder Reinforcement Corrosion Attack

Dita Vořechovská1; Břetislav Teplý2; and Markéta Chromá3

Abstract: In the context of performance-based approaches, sustainability and whole life costing, the concrete structure durability issuehas recently gained considerable attention. The present paper deals with service life assessment using durability limit states specialized forconcrete structures. Both initiation and propagation periods of reinforcement corrosion are considered and a comprehensive choice of limitstates is provided. The approach is based on degradation modeling and probabilistic assessment, enabling the evaluation of service life andthe relevant reliability level, serving thus to facilitate the effective decision making of designers and clients. For this purpose the selectedanalytical models for degradation assessment are randomized and appropriate software has been developed. Three numerical examples arepresented: a comparison of modeled carbonation depth with in situ measurements on a cooling tower, and analyses of crack initiation dueto corrosion and loss of reinforcement cross section.

DOI: 10.1061/�ASCE�CF.1943-5509.0000130

CE Database subject headings: Reliability; Durability; Service life; Concrete structures; Corrosion.

Author keywords: Reliability; Durability; Service life; Modeling; Concrete; Reinforcement corrosion.

Introduction

During the last 20 years the concrete structure service life issuehas been given considerable attention and a great number ofworks have been published. As it is not in the scope of this paperto review them, let us mention only a few: Siemes et al. �1985�,Sarja and Vesikari �1996�, DuraCrete �1999�, and Rostam �2005�.

In the context of performance-based approaches, sustainabilityand whole life costing, time is the decisive variable and the du-rability issues are pronounced—often within the framework ofperformance-based design, the advanced trend in structural engi-neering design. This approach deals with durability and reliabilityissues, which both rank among the most decisive structural per-formance characteristics. This is clearly demonstrated by numer-ous international events and is reflected also in recentstandardization activities: ISO �2008� and fib �2010�. Both thesedocuments advocate probabilistic approaches and enhance the de-sign of structures for durability—i.e., a time-dependent limit stateapproach with service life consideration. The prescriptive ap-proach of current standards does not directly allow for designfocused on a specific �target� service life and/or a specific level ofreliability. Advanced design for durability requires dealing withthe inherent uncertainties in material, technological, and environ-

1Dr., Faculty of Civil Engineering, Institute of Structural Mechanics,Brno Univ. of Technology, Veveří 95, 602 00 Brno, Czech Republic�corresponding author�. E-mail: [email protected]

2Dr., Faculty of Civil Engineering, CIDEAS Center, Brno Univ.ofTechnology, Žižkova 17, 602 00 Brno, Czech Republic.

3Faculty of Civil Engineering, CIDEAS Center, Brno Univ. of Tech-nology, Žižkova 17, 602 00 Brno, Czech Republic.

Note. This manuscript was submitted on September 21, 2009; ap-proved on February 18, 2010; published online on February 27, 2010.Discussion period open until May 1, 2011; separate discussions must besubmitted for individual papers. This paper is part of the Journal ofPerformance of Constructed Facilities, Vol. 24, No. 6, December 1,
2010. ©ASCE, ISSN 0887-3828/2010/6-571–579/$25.00.
JOURNAL OF PERFORMANCE OF CONST

J. Perform. Constr. Facil.

mental characteristics while assessing the service life of a struc-ture. It appears that a more consistent approach to the durabilitydesign of concrete structures is needed, i.e., fully probabilisticdurability design, which necessarily requires the utilization of sto-chastic approaches, analytical models of degradation effects basedon experimental evidence and relevant observations of structuresin real conditions, and also simulation techniques. The theoreticalapparatus for this approach has already been developed but itsutilization in practice is still rare and surprisingly even some re-cent research works are based on deterministic approaches only�see e.g., Wang and Liu 2010�.

Broad application of probabilistic design is still prevented bythe insufficient dissemination of basic ideas or by the lack ofefficient and user friendly design instruments �software andother�. Reinforced concrete is supposed to be very durable and isa widely used construction material. Despite this fact there arestill a large number of cases with severe degradation and costlyreconstruction work needed due to reinforcement corrosion.When considering the limit states �LS� caused by the degradationof reinforced concrete structures, four kinds of attack may bedistinguished1. Mechanical �mechanical load—static or dynamic�;2. Chemical �carbonation, chloride and acid attack�;3. Electrochemical �corrosion of reinforcement�; and4. Physical �freeze-thaw, abrasion, fire and others�.One of the most frequent types of degradation of concrete struc-tures is corrosion of reinforcement, which has significant �nega-tive� implications for life cycle costing. Therefore, the presenttext focuses on Cases 2 and 3. Owing to either carbonation of theconcrete or the ingress of chlorides into the concrete, depassiva-tion of reinforcing steel occurs �the initiation period�; this may befollowed by a steel corrosion process �the propagation period��see e.g., Tuutti 1982�. With the focus on these periods, the mod-eling of relevant LS and degradation models is dealt with in thepresent paper. Types 1 and 4 of concrete degradation are not
encompassed in the present paper and are discussed elsewhere;
RUCTED FACILITIES © ASCE / NOVEMBER/DECEMBER 2010 / 571

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e.g., frost or fire attack on concrete �fib 2010; Matesová et al.2006�.

In the present paper the durability LS �DLS� focused on thecorrosion process in concrete structures are described. Some rel-evant analytical models �adopted from the literature� for rein-forced concrete degradation assessment are randomized and usedin a software tool developed by the team of writers: Novák et al.�2003� and Teplý et al. �2007�, thus allowing for the full proba-bilistic design/assessment of reinforced concrete structures interms of degradation. The possibilities of the software are shownin numerical examples.

Design for Durability

Let us note that the reliability approach �and hence the designprocess� has to cover, apart from safety and serviceability, alsodurability �see European Committee for Standardization 2002�.Durability is related to the design working life �or service life�.The most generally used design method—the partial safety factormethod—does not assess the reliability level directly, i.e., it doesnot arrive at a specified value of relevant reliability level and isnot suitable for service life assessment. This situation can beamended by utilization of the full probabilistic approach, which islegally applicable �as an alternative to the partial safety coeffi-cients approach� according to both the ISO and Eurocode docu-ments, but not commonly accepted in practice.

The ultimate LS �ULS� and the serviceability LS �SLS� areassessed while designing a concrete structure. The general condi-tion for the probability of failure Pf reads

Pf = P�A � B� � Pd �1�

where A=action effect; B=barrier; and Pd=design �acceptable,target� probability value. The index of reliability � is alternativelyused instead of the probability of failure in practice. Generally,both A and B �and hence Pf or �� are time dependent; this has notbeen considered for common cases of ULS or SLS in designpractice very frequently up to now.

Eq. �1� may be also expressed by means of service life formatas

Pf = P�tS � tD� � Pd �2�

where tD=design life and tS can be determined as the sum of twoservice-life predictors

tS = ti + tp �3�

where ti=time of the initiation of reinforcement corrosion andtp=service life after corrosion initiation �the propagation period�.

Considering the durability issue, a new category of LS hasbeen introduced which is intended to prevent the onset of dete-rioration or allow only a limited range of degradation. The newLS category represents such a type of LS which may be calledDLS �see ISO 2008; fib 2010; Sarja 2006�. DLS may be formallyregarded as belonging in the SLS category.

When considering the degradation of reinforced concretestructures, the corrosion of reinforcement is the dominating effect.In the context of the initiation period only one DLS can berecognized/defined—depassivation of reinforcement due to car-bonation or chloride penetration creating the possible startingpoint for reinforcement corrosion. The relevant values of vari-ables A and B used in Eq. �1�, which are random quantities, have
to be assessed by utilization of a suitable degradation model or by
572 / JOURNAL OF PERFORMANCE OF CONSTRUCTED FACILITIES © AS


field or laboratory investigations. For the purposes of the formercase effective probabilistic software tools are needed.

Initiation Period

The degradation phenomena affecting RC structures dealt with inthis paper are concrete carbonation, ingress of chloride ions andsteel corrosion models.

The durability-oriented limit state condition �Eq. �1�� special-ized for the case of carbonation may be written as

Pf�tD� = P�a − xc�tD� � 0� � Pd �4�

where a=concrete cover; xc=depth of carbonation at time; andtD=design service life. Note that the carbonation process is drivenby the diffusivity of ambient CO2 in concrete and the reactivity ofCO2 with concrete. The CO2 penetrating from the surface de-creases pH to a value of 8.3. When the carbonation depth equalsthe concrete cover, the steel is depassivated and corrosion maystart �when oxygen and moisture is present�. The rate of carbon-ation progress from the concrete surface to the reinforcement de-pends on many parameters, e.g., concrete cover thickness andpermeability, the ambient temperature, relative humidity and car-bon dioxide content, whereas the concrete cover permeability it-self depends on the concrete mix type and composition, theaggregate gradation and the processing and curing of the concretemix.

Considering chloride ingress �e.g., due to deicing salts�, Eq.�1� may be transformed into the formula

Pf�tD� = P�Ccr − Ca�tD� � 0� � Pd �5�

where Ccr=critical concentration of dissolved Cl− leading to steeldepassivation and Ca=total concentration of Cl− at the reinforce-ment at time tD. Generally, this quantity is the sum of the initialconcentration Ca,i �due to chloride-contaminated compounds ofconcrete� and Ca,e �the Cl− concentration resulting from externalsources, i.e., deicing salts or sea water effects�. Condition �5� isapplicable for new as well as existing structures. Both limit con-ditions �Eqs. �4� and �5�� represent the initiation period limits andare—in the sense of degradation—conservative limits; they de-scribe a situation where corrosion has not started yet. Typically,these conditions fall into the DLS category.

Details regarding the chloride diffusion process are describedelsewhere, e.g., in �fib 2010; Papadakis et al. 1996�. From thesteel corrosion point of view let us only mention that free chlo-rides �dissolved in pore solution� destroy the passive film of thesteel rebar and lead to possible corrosion as they reduce the pH ofthe pore water, decrease the solubility of Ca�OH�2, and increasethe electrical conductivity and the moisture content due to thehygroscopic properties of chloride salts �Papadakis et al. 1996;Hunkeler et al. 2005�.

The chloride threshold concentration may be presented bymeans of the total amount of chloride by weight of cement, theamount of free chloride, the concentration ratio of free chlorideions to hydroxyl ions or the ratio of acid-soluble chloride content,and the acid neutralization capacity �the content of acid neededto reduce the pH of concrete and cement paste suspended in waterto a particular value� �Ann and Song 2007�. In terms of currentlyused representations, the total chloride content related to the ce-ment weight is considered to be the best alternative. The valueof 0.4% for a building exposed to a European climate and0.2% for structures exposed to a more aggressive environment
are currently suggested as the chloride threshold concentration
CE / NOVEMBER/DECEMBER 2010

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�Glass and Buenfeld 1995; Duprat 2007�. In fib �2010� the lowerboundary of critical chloride content has been specified as 0.20and the mean value as 0.60 �wt.-%/cement�; the statistical char-acteristics of Ccr can be quantified as follows: Ccr=Beta distribu-tion �m=0.6; s=0.15; a=0.2; b=2.0�. According to EN 206-1�2000�, the maximal admissible value of the initial chloride con-tent Ca,i in concrete reinforced by steel rebars is 0.4% �with re-spect to the weight of cement in the concrete mix�, which is inreasonable correlation with the aforementioned threshold concen-tration values.

Propagation Period

Concentrating on reinforcement corrosion, the following LS maybe specified:1. The rate of steel corrosion is governed by �among other fac-

tors� the availability of water and oxygen. Also, the presenceof chlorides in the concrete surrounding the steel bars mayinfluence the corrosion. The volume expansion of rust prod-ucts develops tensile stresses in the surrounding concreteleading to concrete cracking �mainly affecting the concretecover�. The relevant limit condition for DLS may be con-structed either with the tensile stress limit or crack widthlimit

Pf�tD� = P��cr − ��tD� � 0� � Pd �6�

Pf�tD� = P�wcr − wa�tD� � 0� � Pd �7�

where �cr=critical tensile stress that initiates a crack in con-crete �on an interface with a reinforcing bar� and �=tensilestress in concrete at the time of design service life tD. The wcr

in Eq. �7� is the critical crack width on the concrete surfaceand wa is the current crack width on the concrete surface attime tD. Eq. �6� is typically a DLS, while Eq. �7� is either aDLS or an SLS depending on the wcr value, which may havea considerable impact on durability. Note that the exceedingof wcr or �cr depends not only on structural degradationcaused by corrosion of reinforcement but also may coactwith a stress field developed due to other acting loads de-pending on the type of structure and its configuration. Also,note that the �cr and � mentioned above should demonstratethe tensile strength of cementitious material surrounding thereinforcement. Due to the thinness of this layer and due tothe fact that Eq. �6� should be viewed more generally inpractical situations �considering also the possible combina-tion with mechanical loading effects�, the tensile stress ofconcrete is used. It is commonly treated in this way by otherauthors too �see e.g., Liu and Weyers 1998; Li et al. 2006�.Note that Condition �7� is essential for the prediction of du-rability �serviceability� and is more important from the prac-tical point of view compared to Eq. �6�, which only reflectsthe beginning of the cracking process in concrete. In theexample presented in this paper only the time to crack initia-tion is calculated. However, the crack width may be assessede.g., by the model proposed by Li et al. �2006�, where thekey parameter is the stiffness reduction factor �Bažant andPlanas 1998�, taking into account concrete tensile strength,modulus of elasticity, fracture energy, and crack spacing.

2. When the progression of corrosion and consequently theopening of cracks continue, the network of cracks is propa-gated, possibly reaching the surface of the concrete cover.
Together with cracks due to mechanical loading, the crack


network may form separating concrete elements. The con-crete stress at the delaminating surface may be considered asa governing quantity. Delamination is a complex effect de-pending e.g., on the diameters of reinforcing bars, their loca-tion, concrete quality, cover, type and amount of loading, andthe configuration of the structure. Such a state is either anSLS or a ULS—depending on the location and the severityof this effect. To our knowledge, an analytical model existsfor the assessment of such damage derived on the basis offracture mechanics �Li and Lawanwisut 2004�. However, thisissue is not treated in the present paper.

3. A decrease in the effective reinforcement cross section due tothe corrosion, leading to excessive deformation, loss of bear-ing capacity and finally to collapse, may fall into the SLS�deformation capacity� or ULS �load bearing capacity� cat-egories. The limit condition reads

Pf�tD� = P�A�tD� − Amin � 0� � Pd �8�

where A=reinforcement cross-sectional area at time tD andAmin=minimum acceptable reinforcement cross-sectionalarea with regard to either the SLS or ULS. Both pittingand/or a uniform type of corrosion may be considered inEq. �8�.

4. Due to reinforcement corrosion bond capabilities may be sig-nificantly affected. A low level of corrosion �a diameter lossof up to about 1% �Chung et al. 2008�� causes an increase inbond capacity. When the corrosion exceeds a certain level,the bond stress decreases considerably. The key parametersare, apart from corrosion level, the confinement �provided bytransverse steel and concrete cover�, bar diameter and ap-plied current density �Ouglova et al. 2008; Saether 2009�. Itwould not be practical to construct a specific limit conditionin the form of Eq. �1� considering bond capacity. Instead, thegradual decrease of bond due to the actual degree of rebarcorrosion �time dependent� may be reflected in changes tothe bond stress-slip function and incorporated directly in thenonlinear finite-element code while assessing the SLS orULS of a structure in question—a discussion of such LS andtheir sensitivity to corrosion attack is provided by Zhanget al. �2009�. In this respect the “bond of corroded reinforce-ment” is not considered as a DLS and is mentioned here onlyfor the sake of completeness. The bond stress-slip functioninfluenced by corrosion has been shown and some modelsare proposed e.g., in �Wang and Liu 2004; Maaddawy et al.2005; Fisher et al. 2009�.

The key factor for the modeling of steel corrosion as a timedependent process is the corrosion rate, which is usually ex-pressed through current density, icorr. This variable is stronglyaffected by ambient conditions such as humidity �Živica 1994�and temperature, moisture and oxygen availability at the level ofthe steel, the degree of concrete carbonation, and the amount ofchlorides �Morinaga 1988; Escalante and Satoshi 1990�; thus de-pending on the diffusion characteristics of concrete �Matsushimaet al. 1996� that are affected e.g., by the water to cement ratio �Vuand Stewart 2000� or cracks in the concrete cover �Schiessel andRaupach 1997�. According to Alonso et al. �1988�, the rate ofcorrosion is inversely proportional to concrete resistivity. Thisapproach may be applied for carbonated concrete structures, butespecially for chloride-contaminated concrete it should be modi-fied by several factors �DuraCrete 1999� considering the influenceof chloride content, galvanic effects, the formation of rust prod-ucts, and availability of oxygen. However, sufficient data for fac-
tor evaluation is lacking. Escalante and Satoshi �1990� have

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investigated the effect of oxygen concentration, chloride concen-tration and the pH of anodic areas on corrosion propagation. Ac-cording to their studies, oxygen controls the rate of corrosion andthe chloride concentration affects the amount of areas where cor-rosion initiates. Also, decreasing the pH of anodic areas helps toreinitiate the corrosion in the moisture cycle following the dryingcycle.

Langford and Broomfield �1987� specified four ranges of re-sistivity for the orientation classification of corrosion rates �cor-responding current densities were identified in Broomfield et al.1993�: icorr�0.1 �A /cm2 is low; 0.1� icorr�0.5 �A /cm2 is lowto moderate; 0.5� icorr�1 �A /cm2 is moderate to high; andicorr�1 �A /cm2 is a high rate of corrosion. When icorr is not �orcannot be� measured in situ then it may be treated as a randomquantity to allow for the scatter expected in reality. This approachis followed in the present paper, also taking advantage of findingsin �Vořechovská et al. 2009�.

Software Tool and Numerical Examples

Software Tool

In the present work, modeling of degradation processes is basedon simple models �often semiempirical�. The input variables aretreated as random quantities; therefore, the outputs are also ca-pable of expressing statistical and probabilistic quality with re-spect to time evolution. Models selected on the basis of theauthors´ literature survey were primarily published as determinis-tic ones; for our purposes all of them were randomized, i.e., in-puts were treated as random variables described by theirprobability density function �PDF� with proper parameters. Theyare included in the probabilistic software FReET �Novák et al.2003� �www.freet.cz�—a combination of analytical models andsimulation techniques—creating a special degradation module,FReET-D �Teplý et al. 2007� for assessing the potential degrada-tion of newly designed as well as existing concrete structures. Theimplemented degradation models may serve directly in the dura-bility assessment of structures in tasks such as the assessment ofservice life and the level of relevant reliability. The user maycreate different limit conditions. For the statistical analysis of thefollowing examples the Latin Hypercube Sampling method wasapplied, although FReET also works with the crude Monte Carlomethod or FORM. Numerical and/or graphic forms of outputs areavailable, i.e., the results of statistical analyses of degradationmeasures �e.g., depth of carbonation�, sensitivity factors for indi-vidual inputs of the chosen model, and reliability measures �theprobability of failure of the index of reliability� considering thegiven limit condition. For the output quantities the best fit of PDFmay be found using the Kolmogorov-Smirnov goodness-of-fit test�KST�; also, Bayes updating is an option.

FReET-D serves for the probabilistic assessment of different

Table 1. Carbonation Depths in a Cooling Tower: Comparison of an Ana

Surface

Mean�mm�

In situ measurement�Keršner et al. 1996�

Outdoor �RH=70%� 14.9

Indoor �RH=93%� 8

DLSs encompassing together:



• Nine models for carbonation assessment in concretes fromPortland or blended cements and for concrete from Portlandcement with lime-cement mortar coating �outputs: carbonationdepth at time t or time to depassivation�;

• Four models for chloride ingress modeling �variants of output:depth of chloride penetration at time t, time to depassivation orconcentration of chlorides at depth x and time t�; and

• Seven models for effects of steel corrosion �variants of output:net cross-sectional area of rebar at time t, pit depth at time t,time to concrete cracking due to corrosion, width of cracks inconcrete due to corrosion, and the stress intensity factor at thepit tip at time t�.The main criteria in selecting the degradation model for each

specific use—apart from the relevant degradation mechanism andrequired accuracy of the model—are mostly concerned with theavailability of input data �statistical data�. Some of those modelsor their combinations are used in the following examples to illus-trate possible practical applications. The selection of input vari-ables, their PDFs and the values of their statistical characteristicsin the following illustrative examples are partially based on theauthors’ experience and/or on literature sources.

Carbonation Depth Prognosis: Cooling Tower

Using a model based on Papadakis et al. �1992�, the carbonationdepth on an RC cooling tower with a height of 206 m was ana-lyzed. The tower was investigated at the age of 19.1 years and thedepth of carbonation was measured at 75 locations on both theindoor and outdoor surfaces �Keršner et al. 1996�, thus providingstatistical data. Table 1 provides a comparison of the analyticalresults with the findings of tests, namely the mean and coefficientof variation �COV�. The agreement is satisfactory. For computa-tions the input values described in Table 2 were used. The com-putational analysis also provides a prognosis for future decadeswhich is effectively corrected by Bayess updating while usingthe real �measured� data for the age of 19.1 years. After doing thisone arrives at a lower dispersion of carbonation depth fort�19.1 with COV=20% �outdoor tower surface� or 22% �indoorsurface� and a more “realistic” mean value with the consequenceof more reliable residual service life prediction. The comparisonof the result calculated using the model and Bayess updating isplotted in Fig. 1 for the indoor surface in the range of 0 to 50years.

Crack Initiation due to Corrosion

In this illustrative example the calculations of both initiation andpropagation periods are presented. First, the time to reinforcementdepassivation, ti due to carbonation and/or chloride ingress depen-dent on the concrete cover thickness �this quantity is assumed

Model with Measurements on a Real Structure at the Age of 19.1 Years

COV�%�

eET-DIn situ measurement�Keršner et al. 1996� FReET-D

12.4 56 21

11.5 29 30

lytical

FR

here as being in the range from 25 to 60 mm� is calculated. Next,


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the crack initiation due to corrosion is assessed using the resultsfrom depassivation calculations. The deterministic models wereadopted from Papadakis et al. �1992� for carbonation, Papadakiset al. �1996� for chloride ingress and Liu and Weyers �1998� forcrack initiation.

A full description of all input values for the assessment of timeto depassivation is given in Table 3. The results of the statisticalanalysis are shown in Fig. 2, where mean values together withstandard deviations �std� are plotted. Let us focus on a concretecover of 45 mm and apply conditions �Eqs. �4� and �5�� where thetarget design life tD is equal to 50 years. We obtain �=2.85,�Pf =2�10−3�, and �=0.7 �Pf =2�10−1, not an acceptable value�for depassivation due to carbonation and chloride ingress, respec-tively. The well known fact that the rate of chloride ingress isgreater compared to the carbonation rate with respect to time todepassivation is also evident from this example.

Tower

OV PDF Reference

— Deterministic

.12 Normal fib �2010�

.07 Beta�bounds a=0, b=100� fib �2010�.03



.03 Normal EN 206-1 �2000�

.03 Normal EN 206-1 �2000�

.03 Normal EN 206-1 �2000�

.02 Normal

.02 Normal

.05 Normal

.05 Normal

.15Lognormal

�2 par�Joint Committee onStructural Safety �JCSS� 2006

sivation

COV PDF Reference

0.15

Lognormal Joint Committee onStructural Safety �JCSS� 2006�2 par�

0.12 Normal fib �2010�

0.07

Beta

fib �2010��a=0, b=100�

0.03 Normal EN 206-1 �2000�

0.03 Normal EN 206-1 �2000�

0.03 Normal EN 206-1 �2000�

0.03 Normal EN 206-1 �2000�

0.03 Normal EN 206-1 �2000�

0.02 Normal

0.02 Normal

0.02 Normal

0.02 Normal

— Deterministic

— Deterministic Papadakis et al. �1996�

— Deterministic

— Deterministic Papadakis et al. �1996�9 — Deterministic Papadakis et al. �1996�

Table 2. Input Parameters for Calculation of Carbonation Depths in a Cooling

Input parameter UnitMeanvalue C

Time of exposure Years 19.1

CO2 content in the atmosphere mg /m3 800 0

Relative humidity: outdoor; indoor %

70 0

93 0

Unit content of cement in concrete kg /m3 342 0

Unit content of water in concrete kg /m3 188 0

Unit content of aggregate �0–4 mm� kg /m3 834 0



Specific gravity of cement in concrete kg /m3 3,100 0

Specific gravity of aggregate �0–4 mm� kg /m3 2,590 0



Uncertainty factor of model — 1 0

Table 3. Input Parameters for the Calculation of Time to Reinforcement Depas

Variable UnitMeanvalue

Uncertainty factor of model — 1

CO2 content in the atmosphere mg /m3 820

Relative humidity % 70

Unit content of OPC cement kg /m3 313

Unit content of water kg /m3 185

Unit content of aggregate �0–4 mm� kg /m3 847



Specific gravity of cement kg /m3 3,100

Specific gravity of aggregate �0–4 mm� kg /m3 2,590



Concrete cover mm 25–60

Concentration of Cl− on nearest concrete surface mol /m3 50

Saturation concentration of Cl− in solid phase mol /m3 140

Threshold concentration of Cl− in liquid phase mol /m3 13.4

Diffusion coefficient of Cl− in infinite solution m2 /s 1.6�10−

0

5

10

15

20

25

30

Time [years]

Dep

tho

fca

rbo

nat

ion

[mm

]

model - meanmodel - mean ±std

in situ measurement - mean±std

0 10 20 30 40 50

Bayess updating - meanBayess updating mean- ±std

Fig. 1. Comparison of carbonation model results and Bayess updat-ing for the indoor surface


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Let us note that for concretes with supplementary cementitiousmaterials �SCM� or with blended cements the results wouldbe different, as the effect of partial replacement of Portland ce-ment by SCM on the resistance to both deteriorative effects isknown and reported elsewhere �Thomas and Bamforth 1999;Khunthongkeaw et al. 2006; Sisomphon and Franke 2007�. Ap-propriate models are also encompassed in FReET-D; however,such cases are not treated in the present text due to the greatvariety of SCM types, their dosing and other conditions, all ofwhich would deserve extensive study. Analyses of such a kindand a comparison of the results to experimental findings �Jianget al. 2000; Khunthongkeaw et al. 2006� were published byChromá et al. �2005, 2007�.

In addition to the previous calculation, let us assume a timeto corrosion initiation ti due to chloride ingress for a cover of40 mm. The best fit for the resulting ti gained by KST is thetwo-parametric lognormal distribution function: ti=LN�47.4;11.5� years. This result can now be used in combination with themodel for the time to crack initiation due to reinforcement corro-sion, tc. A decisive input quantity is the concrete tensile strengthfct, which is considered in this study as being in the range from3 to 10 MPa. A full description of the input parameters is given inTable 4. The resulting time of crack initiation tc= ti+ tp, dependenton the tensile strength of concrete, is plotted in Fig. 3. Using KSTit follows that for fct=3 and 4 MPa a lognormal two-parametric

Table 4. Input Parameters for Calculation of Time to Crack Initiation du

Input parameter UnitMeanvalue

Initial bar diameter mm 30

Porous zone thickness mm 0.01

Concrete cover mm 40

Time to corrosion initiation Years 47.4

Current density �A /cm2 1

Specific gravity of rust kg /m3 3,600

Specific gravity of steel kg /m3 7,850

Ratio of steel to rust molecular weight — 0.57

Tensile strength of concrete MPa 3–10

Modulus of elasticity of concrete GPa 27

Poisson’s ratio of concrete — 0.18

Creep coefficient — 2

Uncertainty factor of model — 1a

0

100

200

300

400

500

25 30 35 40 45 50 55 60

Tim

eto

dep

assi

vat

ion,

[yea

rs]

t i

Concrete cover [mm]

carbonation-mean

chloride ingress-meanchloride ingress-std

carbonation-std

Fig. 2. Time to depassivation �meanstd� due to carbonation andchloride ingress versus concrete cover

Only type of PDF, not COV.



PDF and for fct=5 to 10 MPa a lognormal three-parametric PDFare the best fits of the output histograms of tc. It is illustrated inFig. 3.

If we apply the limit state condition given by Eq. �2� fortS= tc and tD=50 years, we obtain the reliability indices plottedin Fig. 4. Assuming a design value of the reliability index of�d=1.3 �the recommendation of the fib 2010 for a DLS� it followsfrom the figure that concrete with approximately fct�7 MPawould satisfy these reliability recommendations.

Loss of Reinforcement Cross Sectiondue to Corrosion

Let us assume that an initial reinforcement diameter is log-normally distributed �2 par�: di=LN�30;0.75� mm, and that thecritical loss of the reinforcement area is assessed to be 10% �sucha loss may e.g., lead to the exceeding of the reliability levelfor the ULS or SLS—depending on the structure and loadingconfiguration�. This limit corresponds to the critical net rebardiameter of 28.46 mm. The uniform type of corrosion isconsidered. The main input parameters of the chosen model�based on Rodriguez et al. 1996� are current density: icorr

einforcement Corrosion

COV PDF Reference

— Deterministic

— Deterministic

0.19Lognormal

�2 par� Engelund and Faber �1999�a

0.24Lognormal

�2 par�

0.2 Normal Vořechovská et al. �2009�

0.02 Normal

0.01 Normal

— Deterministic

— Deterministic

0.08Lognormal

�2 par�

— Deterministic

— Deterministic

0.15Lognormal

�2 par�Joint Committee onStructural Safety �JCSS� 2006

40

50

60

70

80

90

100

110

3 4 5 6 7 8 9 10

Tim

eto

crac

kin

itia

tion,

[yea

rs]

t c

Concrete tensile strength, [MPa]fct

std mean

LN 3par PDF

std

Fig. 3. Time to crack initiation �meanstd� versus tensile strength ofconcrete

e to R

25


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=N�1;0.2� �A /cm2 �Vořechovská et al. 2009�, and time to cor-rosion initiation: ti=LN�47.4;11.5� years, which was gained bythe model for the chloride ingress used in the previous example.The decrease in rebar diameter over time is plotted in Fig. 5. Thetimes ti and td,crit �the time of a critical drop in rebar diameter� aremarked in this figure.

The best-fitted PDFs for the output net rebar diameters are LN�2 par� for 0, 10, and 20 years, Student t for 30 years, Laplace for40 and 50 years, LN �2 par� for 60, 70, 80, 90, and 100 years, LN�3 par� for 110 and 120 years and LN �2 par� for 130, 140, and150 years. Chosen histograms of output parameters together with

0

0.5

1

1.5

2

3 4 5 6 7 8 9 10

Rel

iabil

ity

index

,[-

]�

Concrete tensile strength, [MPa]fct

�d = 1.3

Fig. 4. Reliability indices for tD=50 years versus tensile strength ofconcrete

22

24

26

28

30

32

34

0 25 50 75 100 125 150

Net

reb

ard

iam

eter

[mm

]

Time [years]

meanstd

ti td,crit

Fig. 5. Net rebar diameter �meanstd� versus time

meanstd std

0.60.50.40.30.20.1

27 28 29 30 31 32 33

LN (2 par)

10 years

0.60.50.40.30.20.1

270.60.50.40.30.20.1

27 28 29

mstd

20 21 22 23 24 25 26

Fig. 6. Histograms of output net rebar diameter plotted in Fig. 5 �desteps



fitted PDFs are depicted in Fig. 6, showing the complexity of thestatistical description of the problem solved. Note that lognormalPDFs appear for time intervals of 0–20 and 60–150 years. In thefirst interval the steel is not yet depassivated, while in the secondtime interval the steel is already depassivated in the majority ofstochastic realizations. Therefore, the std of the output net rebardiameter in the time interval of 0 to 20 years is influenced by thestd of the input initial bar diameter only while the std in the timeinterval of 60–150 years is affected by the scatter of all inputvariables. In the time interval between these �i.e., 20–60 years�the std gradually increases �see Fig. 5�.

If we apply the limit state condition given by Eq. �8�, replacingthe cross sections by diameters, and assume a design service lifeof 50 years, we obtain a reliability index of �=0.38 �a nonaccept-able level of reliability�. If the reliability limit �=1.3 is consid-ered, the corresponding service life would be �30 years only. Thecoherency of reliability level and service life is clearly demon-strated in this example.

Conclusions

• Concrete is the premier construction material and design fordurability is a decisive issue in sustainability-based buildingstrategy.

• For the effective decision making of designers and clients, theassessment of both service life and reliability level play adominant role; such activities may be enhanced by the pre-sented approach.

• An effective probabilistic software tool is briefly described andits utilization in numerical examples is shown, demonstratingthe features of durability design of reinforced concrete struc-tures.

• The advanced durability design approach—a probabilisticperformance-based approach—is shown in the paper. DLSspecialized for the initiation period and propagation period arepresented, restricted to reinforcement corrosion effects, al-though a comprehensive choice of LS is offered and the meth-odology is general enough to serve for other degradation typesas well.

Acknowledgments

This outcome has been achieved with the financial support of theCzech Ministry of Education, Project No. 1M0579, within theactivities of the CIDEAS research center.

29 30 31 32 33

meanstd std

Student t

30 years

1 32 33 34 35 36 37 38 39 40

std

Laplace

50 years

due to corrosion� together with the best-fitted PDFs in chosen time

28

30 3

ean

creased


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Probabilistic Assessment of Concrete Structure Durability under Reinforcement Corrosion Attack