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A simplified numerical approach ofglobal behaviour of RC beams degradedby corrosionLucas Adelaide a , Benjamin Richard a b , Frédéric Ragueneau b &Christian Cremona ca Laboratoire Central des Ponts et Chaussées, Université de Paris-Est, Paris, Franceb Laboratoire de Mécanique et Technologie, UniverSud Paris,Cachan, Francec Direction de la Recherche et de l’Innovation, MEDDTL TourPascal, La Défense, France

Available online: 01 May 2012

To cite this article: Lucas Adelaide, Benjamin Richard, Frédéric Ragueneau & Christian Cremona(2012): A simplified numerical approach of global behaviour of RC beams degraded by corrosion,European Journal of Environmental and Civil Engineering, 16:3-4, 414-439

To link to this article: http://dx.doi.org/10.1080/19648189.2012.667990

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A simplified numerical approach of global behaviour of RC beamsdegraded by corrosion

Lucas Adelaidea*, Benjamin Richarda,b, Frédéric Ragueneaub and Christian Cremonac

aLaboratoire Central des Ponts et Chaussées, Université de Paris-Est, Paris, France;bLaboratoire de Mécanique et Technologie, UniverSud Paris, Cachan, France; cDirection de laRecherche et de l’Innovation, MEDDTL Tour Pascal, La Défense, France

One of major causes responsible for the performance loss of reinforced concretestructures is the corrosion phenomenon. Thereby, taking into account the localeffects of the steel/concrete interface is of primary importance to predict properly theresponse of corroded reinforced concrete structures. A multifiber-based model includ-ing the steel/concrete interface is proposed. This interface model allows taking intoconsideration the bond strength variation due to corrosion. Such an approach leadsto reasonable computational costs which a powerful feature of the model. A numeri-cal study of beams already studied in the French project « benchmark des poutres dela Rance » is proposed in order to show the efficiency and reliability of the proposedmodel:Une des causes majeures pouvant conduire à une perte de performance est due auphénomène de corrosion. De ce fait, considérer le comportement de l’interface acier/béton est de première importance pour prédire la réponse de structures en bétonarmé corrodées. Un modèle simplifié incluant les effets locaux de l’interface acier/béton est proposé. Ce modèle permet de prendre en compte la variation d’adhérencedue à la corrosion. Une telle approche conduit à des coûts de calcul raisonnables cequi est un des points forts du modèle. Pour ce faire, une étude numérique de poutresissues du projet de recherche « Benchmark des poutres de la Rance » est proposéedans le but de montrer l’efficacité et fiabilité de la méthode proposée.

Keywords: steel/concrete interface; modelling; multifiber; corrosion; reinforced con-crete

Mots-clés: interface acier/béton; modélisation; multifibre; corrosion; béton armé

1. Introduction

In this study, a special interest is focused on the rebar corrosion phenomenon becauseis one of the major causes of damage of the civil engineering structures and is able tobe considered as a key issue for reinforced concrete (RC) structures ageing. The corro-sion phenomenon is electrochemical by nature. It is a destructive attack of the steelrebar from the development of electrochemical reactions. To avoid this attack, the con-crete surrounding the rebar plays an important part in the physical protection againstthe corrosion. By considering a RC structure element, an interstitial solution is presentin the concrete pores with high alkalinity (pH value above 12.5). Under these condi-tions, the steel rebar is protected from corrosion by forming a passive film on its sur-face. Thus, this process is called the passivation of steel rebar.

*Corresponding author. Email: lucas.adelaide@lcpc.fr

European Journal of Environmental and Civil EngineeringVol. 16, Nos 3–4, March–April 2012, 414–439

ISSN 1964-8189 print/ISSN 2116-7214 online� 2012 Taylor & Francishttp://dx.doi.org/10.1080/19648189.2012.667990http://www.tandfonline.com

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Nevertheless, this passive film can be dissolved due to action of external aggres-sions such as carbonation or chlorides ingress.

According to the type of corrosion, it is possible to get either a uniform corrosionin the case of carbonation or a localised (pitting) corrosion in the case of chlorideinduced corrosion. A particular attention will be turned to chloride induced reinforcedcorrosion. This kind of corrosion can come from sea water, salt spray particles or de-icing salts used on road bridges.

Recently, both experimental (Almusallam, Al-Gahtani, Aziz, Dakhil, & Rash-eeduzzafar, 1996a; Almusallam, Al-Gahtani, Aziz, & Rasheeduzzafar, 1996b; Almusal-lam, Al-Gahtani, Maslehuddin, Khan, & Aziz, 1997; Almusallam, 2001; Cabrera, 1996;Castel, François, & Arliguie, 2000a, 2000b, 2002; Rodriguez, Ortega, & Garcia, 1994;Rodriguez, Ortega, & Casal, 1995; Rodriguez, Ortega, Casal, & Diez, 1996; Rodriguez,Ortega, & Casal, 1996; Rodriguez, Ortega, & Casal, 1997) and modeling (Lundgren,1999, 2005a, 2005b; Coronelli & Gambarova, 2004; Dekoster, Buyle-Bodin, Maurel, &Delmas, 2003; Lee, Noguchi, & Tomosawa, 2002; Spacone and Limkatanyu, 2000;Wang and Liu, 2004, 2006; Berto, Simioni, & Saetta, 2008; Sæther & Sand, 2009)approaches have been carried out to provide a better understanding of the corrosionphenomenon. But the time and spatial evolution of steel rebar corrosion in concrete isnot yet completely understood. However, it is known that once, corrosion is initiated, itprogressed which shortens the service life and consequently implies a performance loss(Tekeste Teshome Gebregziabhier, 2008). In other words, the more the corrosion rateincreases, the more the load-carrying capacity decreases. The corrosion affects not onlythe steel rebar and the steel/concrete interface but also the concrete. It manifests in:

• a reduction of the steel rebars cross-section which implies a brittleness of the steelrebars. Actually, this steel cross-section loss results from the amount of surfacesteel transforms in rust (iron oxide).

• a cracking of the concrete cover. Indeed, the corrosion products (rust) grow andexpand sometimes until about eight times the volume of the original material.This increase in volume leads to tensile stresses in the surrounding concretewhich can cause cracking and ultimately spalling of the concrete.

• a variation in steel/concrete bond properties. In fact, during the growth of corro-sion products, a decrease of confinement stresses occurs which results in a reduc-tion of the steel/concrete bond.

One can notice that the corrosion causes concrete cracking which shows the pointin considering a satisfying concrete model during the numerical simulations of corrodedRC structures.

Although the corrosion kinetics and its effects are not yet completely understood, itis essential to take into account an efficient and reliable constitutive model able to rep-resent the particular behaviour of the steel/concrete interface in presence of corrosionwith reasonable computational costs. The constitutive steel/concrete interface modelused has been developed by Richard, Ragueneau, Cremona, Adelaide, and Tailhan(2010a). This approach requires the use of zero-thickness joint elements included in thefinite element mesh of the RC structure. Note that the steel rebars must be meshedexplicitly leading high computational costs when dealing with large-scale RC structures.

In the field of seismic engineering, some researchers such as Spacone, Filippou, andTaucer (1996), Mazars, Ragueneau, Casaux, Colombo, and Kotronis (2004), Kotronisand Mazars (2005) Ragueneau, Nguyen, and Berthaud (2006) Ragueneau, Dominguez,

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and Ibrahimbegovic (2006) have resorted to the multifiber numerical framework inorder to decrease the computational cost. In our study, the strategy which has been setup is to consider the steel/concrete interface and concrete models within the multifiberframework (Richard et al., 2010a; Richard, Ragueneau, Cremona, & Adelaide, 2010b;Richard, 2010).

Our study has been based on the experimental results of the French research project« Benchmark des poutres de la Rance » (L’Hostis, 2007). This project studied 20 RCbeams exposed for 40 years in a marine environment. And its objective was to betterunderstand and know the effect of the main corrosion parameters on the structuralmechanical behaviour from experimental and numerical points of view and validate thenumerical model with respect to the experimental results.

The present article is outlined as follows. First, the multifiber framework is brieflyexposed especially, the way to consider the relative slip between concrete and steel.Second, steel/concrete interface constitutive law is presented in details and concrete andsteel laws are briefly showed. Third, a structural case study is presented. It is based onRC beams which are subjected to four-point bending tests. At last, a critical comparisonof the experimental and numerical results is performed. And thus, that aims at exploringthe possibilities offered by the proposed approach to consider the steel/concrete inter-face within the framework of multifiber approach.

2. Principle of the proposed multifiber approach

When it is not necessary to predict refined information, the use of multifiber approachcan be a satisfactory alternative. Indeed, this approach allows dealing with large-scalestructures with low computational cost but does not allow obtaining local informationsuch as crack openings or crack spacings. This strategy has been used in the field ofseismic engineering to perform complex analyses (Kotronis & Mazars, 2005; Kotronis,Ragueneau, & Mazars, 2005; Mazars, Kotronis, Ragueneau, & Casaux, 2006; Wanget al., 2007; Souid, Delaplace, Ragueneau, & Desmorat, 2009).

The multifiber approach used has been implemented into Cast3M, finite elementsoftware developed by the CEA (Commisariat à l’Energie Atomique) (Combescure,1999, 2000). This approach allows including nonlinear constitutive laws in a finite ele-ment model built from Timoshenko’s beam elements. Two discretisation levels havebeen considered: the beam level (Timoshenko’s element) and the cross-section one(classical 2D finite elements). The link between both levels is ensured by the classicalforce/stress relationships coming from the beam theory. Moreover, one-dimensional con-stitutive models can be used at the cross-section level, allowing representing nonlineari-ties. Each material (steel, concrete…) is represented by fibers, at the cross-section scaleand the interface between the different fibers is perfect.

Due to kinematic constraints related to the beam theory, it is not easy to considerexplicitly the bond/slip relationship between the steel and the concrete. Nevertheless,an interesting approach has recently been developed and used in the field of seismicengineering (Combescure & Wang, 2007; Wang et al., 2007).The non perfect steel/concrete interface can be taken into account in an implicit way in the steel fiber. Theprinciple lies in assuming that the total strain in the steel fiber exx can be split intotwo contributions, the proper strain in the steel rebar eAxx and the sliding strain occur-ring at the steel/concrete interface eIxx. This partition can be expressed in the followingform:

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exx ¼ gexx þ ð1� gÞexx ¼ eAxx þ eIxx ð1Þ

where g is the so-called partition factor.The value of this factor g varies between 0 and 1. To determine this factor, the equilib-rium between the tension force in the steel rebar and the shear force acting at the steel/concrete interface is assumed (Figure 1).

And this equilibrium condition can be formulated as follows:

ZSðxÞ

rðsÞdS � la

Z@SðxÞ

sðsÞd� ¼ 0 ð2Þ

where SðxÞ is the current cross-section surface of the steel fiber at the point abscissa x,@SðxÞ and la are respectively the boundary and the anchorage length of SðxÞ, r and sare the normal stress and the shear stress associated respectively with the real steelstrain and the sliding strain at the steel/concrete interface. By considering the behaviouroperators of the steel and of the steel/concrete interface denoted respectively =Að:Þ and=I ð:Þ, Equation (2) can be rewritten in the following way:

=Aðexx; gÞSðxÞ � la=Iðexx; gÞPðxÞ ¼ 0 ð3Þ

where SðxÞ and PðxÞ are the area and the perimeter of the current cross-section of thesteel fiber. Moreover, exx and g are supposed constant in the element.Due to the nonlinearity of the behaviour operators, Equation (3) can be seen as a non-linear equation with one variable g. The resolution of the resulting equilibrium is car-ried out by a modified Newton-based scheme.

3. Constitutive models: concrete, steel and steel/concrete interface

In this section, a detailed description of the steel/concrete interface constitutive model isgiven, a less detailed one for the concrete constitutive model and a brief one of the steelmodel.

3.1. Steel/concrete interface model

The steel/concrete interface constitutive model is expressed within the framework of theirreversible processes thermodynamics, which ensures consistency with respect to wellknown physics principles. It has been developed by Richard et al. (2010a) and has beenreduced to a one-dimension because of the multifiber framework. Therefore, only thebond strength variations due to corrosion are taken into account. Swelling cannot beconsidered due to the particular kinematics prescribed by the beam theory.

Figure 1. Local stress state at the steel/concrete interface due to a tension force to the steelrebar.

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In the first place, in the case where there is no corrosion at the steel/concrete inter-face, the different steps of the degradation process of the steel/concrete interface can bedescribed into three different stages (Lutz & Gergely, 1967). The first stage occurswhen the chemical adhesion ends. Furthermore, this adhesion being very weak oftenimplies a modelling of the latter by elastic models. During the second stage, it is possi-ble to observe the first signs of the concrete shear cracking between the rebar and theconcrete and small displacements at the interface. And the third stage concerns thepost-peak domain. Sliding becomes high and the steel/concrete interface is entirelydegraded. Degradation mode II (shearing) is entirely characterised by these three stages.Due to the Timoshenko’s kinematic hypotheses, the degradation mode I (opening) can-not be considered explicitly. Moreover, the structural failure mode (splitting or pull-outfailure) will be conditioned by the ability of the concrete constitutive model used torepresent accurately the stress state.

The state potential can be written in the form of the Helmholtz free energy expres-sion denoted W as follows:

qw ¼ ð1� dÞleIxx:eIxx þ ldðeIxx � ep;Ixx Þ:ðeIxx � ep;Ixx Þ þ1

2ca : aþ HðzÞ ð4Þ

where q is the material density, l the second Lamé’s coefficient (shear modulus), ep;Ixxthe inelastic sliding strain, a the kinematic hardening variable, z the isotropic hardeningvariable, H the consolidation function and c a material parameter which needs to beidentified.

In Equation 4, the continuous damage variable d, ranging from 0 (virgin material)to 1 (ruined material) allows considering cracking effects.

The fact that this damage variable is chosen scalar is based on the point that crackshave a single and fixed orientation at the steel/concrete interface. Moreover, it is possi-ble to notice that the amount of friction between cracked surfaces and the level of dam-age d are linked (Ragueneau, Dominguez, & Ibrahimbegovic, 2006b). And, storedenergies due to both kinematic and isotropic hardenings are introduced in the statepotential so that cyclic behaviours are described suitably.

3.1.1. State equations

From the state potential expressed by Equation 4, the following state equations areobtained.

The Cauchy’s shear stress is given by:

s ¼ q@ w@ eIxx

¼ 2lð1� dÞeIxx þ 2 ldðeIxx � ep;Ixx Þ ð5Þ

The friction shear stress sp is expressed as follows:

sp ¼ �q@ w@ ep;Ixx

¼ 2ldðeIxx � ep; Ixx Þ ð6Þ

It can be noticed that a coupling between the damage variable and the inelastic slidingstrain in the friction shear stress expression.

The energy released rate due to damage Y is:

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Y ¼ �q@ w@ d

¼ leIxx:eIxx � lðeIxx � ep;Ixx Þ:ðeIxx � ep;Ixx Þ ð7Þ

The previous relation 7 is composed of two terms associated with the damage. Thefirst and second terms represent the energy rate released due to mode II (shear) and dueto inelastic frictional sliding respectively.

The thermodynamic force associated to the kinematic hardening, X called backstress is defined as follows:

X ¼ q@ w@ a

¼ ca ð8Þ

And, the thermodynamic force Z associated to the isotropic hardening is:

Z ¼ q@ w@z

¼ dHðzÞdz

ð9Þ

3.1.2. Damage and isotropic hardening flow rules coupled with corrosion

On the assumptions that the corrosion effect is not considered, the dissipative mecha-nisms, damage and isotropic hardening are supposed to follow an associated evolution.And the expression of the associated yield function is:

fdðY ; Z; Y0Þ ¼ Y � ðZ þ Y0Þ ð10Þ

where Y0 is the initial threshold to activate damage mechanism and Y the part of theenergy released due to damage.

In the case where the corrosion is considered, it is important to know the corro-sion evolution. This evolution can be described by three different stages. The first oneis called the free expansion. The development of corrosion products will not createany stress on the surrounding concrete. The second one represents the stress initiation.During the development of corrosion products and once the porous zone around thesteel/concrete interface is filled by rust, the pressure on the surrounding concrete willkeep on increasing with the expansion of the corrosion products. The third onedescribes the concrete cracking. When assuming the expansion of corrosion productsgoes on and the amount of corrosion products reaches a critical level, consequently,the internal stress will exceed the tensile strength of concrete which leads to concretecracking.

To take into account the corrosion phenomenon, some simplifications due to Timo-shenko’s hypotheses have to be considered. Thereby, this implies that swelling cannotbe considered. Nevertheless, the bond strength variation can be it. Other corrosioneffects exist and influence the steel rebar behaviour such as the steel cross-sectionreduction and the steel brittleness decrease (Ouglova, 2004). This means that the latterhave not been included in the steel/concrete interface law. In order to take into accountthe bond strength variation due to the corrosion phenomenon, an additional term (afunction) W is added to the damage yield criterion capable of producing a delayed acti-vation of damage mechanism (increase of bond strength) or to generate an initial dam-age level (decrease of bond strength). The damage yield surface becomes:

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fdðY ; Z; Y0; YRÞ ¼ �Y � ðZ þ Y0 þW ðYRðTcÞÞÞ ð11Þ

where W is a given function which has to be identified from experimental pull-out testsperformed on corroded specimens such as the ones performed by (Al Sulaimani, Kale-emullah, Basunbul, & Rashiduzzafar, 1990), YR the part of strain energy released duringthe evolution of corrosion products (Richard et al., 2010b) and Tc a corrosion degreeresulting from either steel mass loss or steel cross-section reduction. In the work per-formed, it is an empirical formulation for the W function which is used. It would beinteresting to get a reliable and predictive analytical expression for the W function. Butuntil now, there are very few experimental tests based on local observations of the cor-rosion behaviour in the literature and besides, there are no clear conclusions on thelocal mechanisms explaining the bond variations due to corrosion.

From the maximum dissipation principle (Lemaître & Chaboche, 1985), the evolu-tion laws associated with the dissipative mechanisms for the damage and isotropic hard-ening can be expressed as follows:

_d ¼ _kd@fd@Y

¼ _kd and _z ¼ _kd@fd@Z

¼ � _kd ð12Þ

where _kd is the Lagrange’s multiplier.To consider the damage mechanism entirely, the consolidation function chosen is

based on the proposal made by La Borderie (1991). And its expression is written in thefollowing way:

HðzÞ ¼ 1

Adð�zþ lnð1þ zÞÞ ð13Þ

where Ad is a material parameter which can be interpreted as a brittleness degree.So, the thermodynamic force associated with the isotropic hardening becomes:

Z ¼ dHðzÞdz

¼ �z

Adð1þ z0Þ ð14Þ

From the consistency conditions _fd ¼ 0 and also Equations 11 and 14, the damage vari-able expression is:

d ¼ 1� 1

1þ AdðY � Y0 �W ðTcðt0ÞÞÞð15Þ

In the proposed approach, the corrosion effects related to the bond strength variation atthe steel/concrete interface are considered. For that, a W function has been used whichallows representing the global behaviour from a local mechanism like the corrosion andalso a scalar variable d allows considering the degradation of the interface thanks to theone of the elastic modulus of the steel/concrete interface. Indeed, the three stages of thesteel/concrete interface degradation are not considered explicitly in the proposed consti-tutive model due to Timoshenko’s hypotheses. Thus, this enables to show off a simpli-fied model in order to estimate the mechanical behaviour of corroded structures whichis the aim of this article.

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3.1.3. Sliding and kinematic hardening flow rules

We now consider sliding and kinematic hardening mechanisms. Contrary to the pre-vious case, the evolution of these mechanisms is not associated which allows consid-ering the nonlinear effects capable of appearing when the kinematic hardening isactivated (Ragueneau, 1999; Ragueneau, Dominguez, & Ibrahimbegovic, 2006b).Thus, the sliding mechanism is supposed follow an associated evolution which isnot the case of the kinematic hardening. Firstly, a yield function fp associated withsliding is expressed as a classical Drucker-Prager criterion:

fpðsp;X Þ ¼ jsp � X j ð16Þ

In order to obtain a nonlinear kinematic hardening rule, the pseudo-potential function ischosen to be different than the yield function fp. To consider a nonlinear kinematichardening law, the model initially introduced by Armstrong and Frederick (1966) andfurther developed by (Lemaître & Chaboche, 1985) has been selected. This pseudopotential is described by a plasticity-like behaviour from the nonlinear kinematic hard-ening rule. The generalised normality rules of thermodynamics can be consistent andsatisfied in the case where the following form of the pseudo potential up is defined asfollows:

upðsp;X Þ ¼ jsp � X j þ a

2XX ð17Þ

where a is a material parameter. We can note that the last term on the right-hand sidein the pseudo potential has been added in order to overcome the main inconvenient ofthe classical form of the Drucker-Prager criterion (linearity of the state law). It is thereason why the nonlinear kinematic hardening rule follows a non associated evolution.Moreover, the sliding and kinematic stress tensors are purely deviatoric. Hence, thepseudo potential up is also purely deviatoric.

Consequently, the flow rules can be expressed by the following equations:

_eI ;pxx ¼ _kp@fp@sp

and _a ¼ � _kp@up

@Xð18Þ

where _kp is the Lagrange’s multiplier linked to sliding and kinematic hardening.Lagrange’s multiplier is determined numerically by using a return mapping algo-

rithm proposed by (Ortiz and Simo, 1986) from the consistency condition:

fp ¼ 0 and _fp ¼ 0 ð19Þ

During the sliding mechanism, two cases can occur. The first one represents the slid-ing with no friction. It is activated through the mode I degradation mechanism. Onlysliding is involved. The second case occurs when both sliding and friction willappear. That is possible under confinement conditions. So, this frictional sliding isresponsible for the strain irreversibility. In other words, the inelastic part mainlyresults from the frictional sliding. Therefore, a frictional sliding dissipative mecha-nism is introduced in order to consider these two mechanisms. This is manageddefining two material parameters a and c which affect flow rules and state lawsrespectively.

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3.2. Concrete model

In this section, the constitutive law for concrete is presented within the multifiberframework briefly. This law is an adapted version of the local concrete behaviour lawresulting from (Richard et al., 2010b). The constitutive equations of the proposed con-crete model are formulated within the framework of the irreversible processes thermo-dynamics. Different mechanisms are considered, among them, there are elasticity,damage and frictional sliding. With regard to the damage mechanism, it is coupled withelasticity and allows representing the stiffness reduction satisfactorily. This stiffnessreduction arises from the degradation phenomenon associated with the concrete crack-ing. We notice that when the material is subject to a tension/compression loading path,the damage mechanism appears. This mechanism has an effect on the mode I (opening).During the compression stage, a gain of stiffness is observed. This is due to the crackclosure (unilateral effect). In the proposed model, due to the fact that the damage vari-able is a single scalar, the unilateral effect is considered partially. Regarding the othermechanism, the frictional sliding, it is essential to consider it because generally, thesliding takes place between the crack lips and can even cause friction. That allows rep-resenting the hysteretic behaviour and so, improves the prediction properties of themodel. Moreover, it is obvious that the intensity of the energy released by friction alongcracked surfaces depends on the damage level. Nevertheless, a part of strain elasticenergy is stored because of friction and damage mechanisms during a loading path andto consider this energy, kinematic and isotropic hardenings are introduced. Besides, aBAEL (Limit state reinforced concrete) model is used as a reference. It is a model com-ing from of the French code for reinforced concrete (BAEL91) and considering a plasticmodel for the compression and a unilateral model for tension with a tensile strengthequal to zero.

3.2.1. State equations

The state potential is expressed thanks to Helmholtz free energy as follows:

qw ¼ E2ðð1� dÞexxþ2exxþ2

�Þ þ lð1� dÞe:eþ ldðe� epÞ:ðe� epÞþ12ca : aþ HðzÞ : ð20Þ

where exx represents the axial strain, e the shear strain, ep the sliding strain, a the kine-matic hardening variable. So, the state laws can be deduced.

Cauchy’s stresses are written as:

rxx ¼ q@w@exx ¼ Eðð1� dÞ\exx >þ þ�Þs ¼ q@w

@e ¼ 2lð1� dÞeþ 2ldðe� e/Þ�

ð21Þ

where s is the tangential stress.The frictional sliding stress associated to the frictional sliding between the crack lips

is defined in the following way:

sp ¼ �q@w@ep

¼ 2ldðe� epÞ ð22Þ

The damage energy released rate is written as:

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Y ¼ �q@w@d

¼ E

2

2

þþ le:eþ lðe� epÞ:ðe� epÞ ð23Þ

The back stress which is the thermodynamic force associated to kinematic hardeningand the thermodynamic force linked with the isotropic hardening can be expressedby Equations 24 and 25 respectively:

X ¼ q@ w@a

¼ ca ð24Þ

Z ¼ q@ w@z

¼ dHðzÞdz

ð25Þ

3.2.2. Damage and isotropic hardening flow rules

The damage mechanism and the isotropic hardening follow an associated evolutiondefined by a yield surface fd :

fdðY ; Z; Y0Þ ¼ Y � ðZ þ Y0Þ ð26Þ

where �Y ¼ E2exxexx.

By using the same process as the steel/concrete interface case, the damage variableis obtained explicitly as follows:

d ¼ 1� 1

1þ ADir@ðexxÞð�Y � Y0Þ þ AIndð1� @ðexxÞÞð�Y � Y0Þ ð27Þ

3.2.3. Sliding and kinematic hardening flow rules

The internal sliding and the kinematic hardening are handled by a non associative way.The yield surface proposed is the following one:

fpðsp;X Þ ¼ ksp � Xk2 ð28Þ

where k:k2 is Euclidien norm.The sliding flow rule follows then the normality rule deduced from the maximum

dissipation principle:

_ep ¼ _kp@fp@sp

ð29Þ

The flow rule related to the kinematic hardening needs a pseudo potential up:

upðsp;X Þ ¼ ksp � Xk2 þa

2X :X ð30Þ

And the flow rule linked to the kinematic hardening is written as:

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_a ¼ � _kp@up

@Xð31Þ

In this section, a simplified concrete model has been shown deriving from a 3Dlocal concrete model developed by (Richard et al., 2010a, 2010b). Since in some casesit is not necessary to obtain local information, the 3D local concrete model is notrequired. In other words, this simplified concrete model leads to a saving of computa-tional costs for complex structures.

3.3. Steel model

In this section, the steel model proposed by (Ouglova, 2004) is described. This shortpresentation aims to show how this constitutive model allows dealing with corrodedbehaviour of the steel. Additional and detailed explanations can be found in Ouglova(2004).

The steel model is based on a coupling between isotropic damage and plasticity.The work carried out by Ouglova (2004) is based on the initial contribution made by(Lemaître & Chaboche, 1985). The main state equation is expressed according to:

r ¼ ð1� dÞEðe� epÞ ð32Þ

where E is the steel Young’s modulus, d a scalar damage variable, e the axial strain andep the plastic strain. The damage variable is used to describe the loss of ductility of thesteel with respect to the corrosion degree variations. The plastic strain allows modelingthe permanent strains when the material is subjected to an unloading. The damage vari-able is linked to the axial strain and is activated only if the plastic strain has been acti-vated beforehand. It is expressed according to:

d ¼ dce� ed

ef � edð33Þ

where dc is the damage maximum value, ed is the strain when the damage is activatedand ef is the strain at failure. The effect of the corrosion is considered as a relationbetween a macroscopic corrosion degree (expressed in terms of steel cross-section loss)and the strain ed . This relation is given in Ouglova (2004) and can be identified fromexperimental tension tests carried out on steel bars corroded at various levels.

3.4. Material parameter identification and discussion

The number of material parameters associated to the steel/concrete interface is equal to7. One parameter called l is related to the elasticity, two parameters Ad and Y0 areaccounted for the damage and the isotropic hardening mechanisms respectively, onefunction named W for the corrosion phenomenon, two parameters c and a for the kine-matic hardening mechanism, one parameter lIa (called the anchorage length) is linked tothe finite element mesh and one corrosion parameter which is the corrosion degree Tc:

Indeed, the elastic parameter l can be chosen equal to the concrete shear modulus,as recommended by (Ragueneau, Nguyen, & Berthaud, 2006a). The damage and isotro-pic material parameters ðAd and Y0Þ are identified from pull-out tests. To identify thesetwo parameters, the shear stress peak at the steel/concrete interface can be used. The

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parameter Ad named the bond strength parameter drives the brittleness of the bond-slipcurve. It is identified by means of pull-out tests without corrosion. It is able to representthe steel/concrete interface behaviour according to the type of the steel, either smoothor notched and also the bond strength level. In the case where the response of the inter-face model is rather ductile, a smooth steel rebar is involved, on the other hand, in thecase of notched steel rebar, the response of the steel/concrete interface is rather brittle.Consequently, to have the bond strength varied, a modification of the parameter Ad issufficient.

The W function is identified by performing several experimental pull-out tests oncorroded specimens with different corrosion levels. For that, an inverse analysis basedon experimental data is carried out. The identification criterion can be the maximumshear stress. Let us note that proposing an analytical expression of such a function isquite delicate and furthermore, it is necessary to perform an important experimental pro-gram on reinforced concrete specimens with various corrosion level, concretes andsteels. But, the proposed approach stays robust and simple numerically for modelingthe steel/concrete interface including corrosion. The parameters c and a correspondingto the frictional sliding are enough sensitive. And they have to be chosen according tothe sensitivity analysis performed by Ragueneau, La Borderie, and Mazars (2000). Theanchorage length lIa allows putting more weight behind the steel model or steel/concreteinterface model. It corresponds generally to the finite element length. The corrosiondegree Tc results from either steel mass loss or steel cross-section reduction obtainedfrom experimental tests.The identification of the W function has been performed fromthe experimental campaign established by Al Sulaimani et al. (1990). Figure 2 showsthe form of the W function and Figure 3 represents experimental and numerical resultsof the model response using the W function deriving from Figure 2. A very goodaccordance between the experimental data and the numerical calculation is noticed(Figure 3).

The identification of material parameters associated with the concrete constitutivelaw is detailed briefly in this article. And the recommendations related to this modelleads to seven material parameters. One parameter is related to the elasticity, the shearmodulus l. Regarding the damage mechanism, three parameters are necessary Y0, ADir

and AInd . The latter correspond to an initial threshold, the brittleness due to direct (ten-sion) and induced (Poisson effects in compression) extensions respectively. The parame-ters c and a are related to the frictional sliding and to the kinematic hardening. The

2.4 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3x 109

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YR (J)

W(Y

R) (

J)

Figure 2. W function shape from experimental results.

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anchorage length lca is linked to the finite element mesh of the concrete.The elastic parameters are deduced from the Young and the shear modulus resultingfrom classical experimental tests. The initial threshold is determined from the averagetensile strength through the relation Y0 ¼ f 2t =2E. To get the brittleness ADir and AInd , itneeds to perform tensile and compressive mechanical tests respectively. These parame-ters can be determined without particular difficulty. The parameters related to frictionare identified in the same way as those linked to the interface model.

The identification of material parameters associated to the steel constitutive law is notdetailed in this article. Ouglova (2004) performed an accurate sensitivity analysis of thesteel model and her conclusions and recommendations have been used in this article.

The concrete cracking drives the structural failure consequently a relevant concretemodel is capable to highlight this effect because it conditions the stress redistribution.Consequently, the relevancy of the concrete model influences the steel/concrete interfacebehaviour. The one dimensional steel/concrete interface model does not consider explic-itly the confinement (coming from the stirrups, an external loading or the concrete coverthickness). Given that pull-out tests with and without corrosion are necessary in orderto calibrate the interface model including corrosion. That means that the influence ofthe interaction between the concrete and the steel is considered therefore the concreteconfinement effect. In the case where high corrosion degrees are considered spallingcan be observed experimentally. By using the 3D interface model, the cover spallingcan be noticed implicitly through the damage parameter d of the concrete model.Because when this parameter is equal to one, that means that the concrete is totallydeteriorated. Nevertheless, in the multifiber framework, the cover spalling cannot beseen.

Besides, a change of failure modes is observed due to the corrosion growth. Thisphenomenon is not explicitly taken into account by the one dimensional steel/concreteinterface model.

From the previous discussion, it results that the change of structural failure is man-aged by the set of used constitutive models.

3.5. Local results

In this section, the case of the simplified concrete model is not dealt with. For furtherdetails on this concrete model, see Richard et al. (2010b). With the aim of obtaininglocal results at the Gauss’ point level, one steel fiber is considered. Thus, the anchoragelength lIa has been chosen equal to 1.0cm and all other material parameters previouslyseen have to be identified. First, one assumes that mechanical tests are performed with-

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Experimental dataModel response after identification

Figure 3. Response of the model using the W function from Figure 2.

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out considering the corrosion effect. These tests are carried out under tensile load andare depicted in Figure 4.

Figure 4 shows the local response of the steel and the steel/concrete interface aswell as the partition factor evolution. From Figure 4.1 exhibiting the local response ofthe steel/concrete interface in terms of shear stress versus sliding strain, we can noticethe three degradation stages published by (Lutz and Gergely, 1967). Figure 4.2 showsan elastoplastic response which represents the local response of the steel rebar in termsof normal stress as a function of total strain. An unloading has been noticed althoughthe loading went on increasing because when the steel/concrete interface becomes toodamaged therefore, to ensure the equilibrium with the steel rebar the only one way isan unloading of the steel stress. Figure 4.3 represents the evolution of the partition fac-tor versus the number of load steps. Three stages can be distinguished. The first one isa decrease of the partition factor traducing the predominance of the steel/concrete inter-face. The second one shows an increase which reflects the predominance of the steelbehavior. Finally, for the last stage, the partition factor decreases drastically, highlight-ing the steel rebar unloading. That clearly shows that the coupling between the behav-iors of the steel and of the steel/concrete interface.

Considering the corrosion, the same simulation as above has been performed with acorrosion degree equal to 5.8 % (Figure 5). It can be notice that the shear stress peakassociated to the steel/concrete interface clearly decreases (Figure 5.1) with respect tothe results from Figure 4.1. This leads us to deduce that the corrosion influences on thebond strength at the interface. From Figure 5.2, the steel remains linear and thereforedoes not reach the plastic stage because the steel/concrete interface is too damaged sothat a stress transfer occurs. Figure 5.3 presents a decrease of the partition factor up tozero which highlights a strong predominance of the steel/concrete interface.

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Partition factor evolution

Figure 4. Local results obtained for Tc = 0 %.

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4. Numerical simulations of tests and comparison to experimental results

Our numerical study is based on the experimental test carried out on two pre-stressedand RC beams, P621 and P412. These beams are exposed for 40 years in a marineenvironment (L’Hostis, 2007). The aim of this structural cases study is to show the rele-vance of the use of the steel/concrete interface including corrosion for modelling thebehaviour of corroded RC structural components.

4.1. Experimental testing set up

This project aims at evaluating the possibilities offered by numerical approaches to pre-dict properly the mechanical behaviour of corroded structural components. Severalmechanical tests have been carried out. In our study, the interest is focused on four-point bending tests. Two beams have been selected P621 and P412. The beam P621was tested under monotonic loading and the beam P412 under non reverse cyclic load-ing. The beam dimensions are the following ones: 2500mm long, 200mm wide and200mm high. The ends of the beams are protected by asphalt on 250mm long from thebeam end. The configuration of the steel rebars is different according to the beam. Thepassive rebars are 6mm diameter and the 6mm diameter stirrups are all disposed at250mm. These beams are also pre-stressed with 7mm diameter wires located into aplastic sheath of 12mm in diameter and anchored at the beam ends. The concrete coveris equal to 16mm (Figure 6).

We can notice that the pre-stress wires are located in the upper part of the beams inother words the pre-stressing effects can be negligible. The support and loading spanswere 2000mm and 800mm, respectively.

The mechanical properties of the constitutive materials have been measured for steeland concrete respectively by mechanical classical tests on concrete and steel specimensfrom the beams exposed to chloride-induced corrosion during 40 years.

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Number of loading steps

Parti

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Figure 5. Local results obtained for Tc = 5.8 %.

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Concerning the beam P621 constitutive concrete, Young’s modulus is 30.5 GPa, thetensile strength is 4 MPa and the compressive strength is 42.9 MPa. For the beam P421concrete, the Young’s modulus is 38.6 GPa, the tensile strength is 6.2 MPa and thecompressive strength is 68.1 MPa. Regarding the steel, Young’s modulus is 195 GPaand the yielding stress is 309 MPa. And, Poisson’s ratio has been chosen equal to 0.2for the concrete and 0.3 for the steel. Moreover, an isotropic hardening modulus equalto 0.05 has been assumed (expressed in terms of percentage of the steel in Young’smodulus).

4.2. Modeling

First, only half of the beam has been modelled for symmetrical reasons. This choicecan be discussed because the corrosion repartition along the steel rebars is not symmet-ric. The finite element mesh for the half beam is composed of 12 beam elements in theaxial direction. The beam cross-sections are meshed thanks to four-node quadrilateralelements including four Gauss points per element. One four-node quadrilateral elementhas been considered for the steel rebar and four for the concrete (Figure 7 for beamsP621 and P412).

The load is displacement controlled in order to ensure the description of the postpeak region and to provide a numerical robustness. The concrete has been modelled bythe proposed uniaxial version of the constitutive law seen previously and developed by(Richard et al., 2010b; Richard, 2010). The steel is characterised by a classic elastoplas-tic model including isotropic hardening (Ouglova, 2004) and the steel/concrete interface

Figure 7. Mesh of beams P621 (right) and P412 (left).

Figure 6. Cross section schematic view of the beams.

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by the constitutive model briefly presented in the previous section (Richard et al.,2010a; Richard, 2010).

The identification of material parameters associated to the steel/concrete interface,concrete and steel constitutive laws are not detailed in this article. Ouglova (2004),Richard et al. (2010b) and Richard et al. (2010a) performed an accurate sensitivity anal-ysis of the steel, concrete and steel/concrete interface models respectively and their con-clusions and recommendations have been used in the present work. Material parametersfor the concrete and steel are chosen according to the concrete and steel experimentalcharacteristics respectively. The material parameters related to the steel/concrete inter-face model are more delicate to identify than the concrete one (Tables 1 and 2). That isrealised thanks to an inverse analysis according to the experimental results.

4.3. Results and comparisons

As seen previously, two kinds of loading are simulated, a monotonic loading on thebeam P621 and a cyclic one on the beam P412. Several calculations were performed inorder to evaluate the corrosion effects on the mechanical behaviour of the beams. Thecorrosion effects analysed are the cross-section loss and the bond strength variation. Forthat, different corrosion degrees Tc in terms of cross-section loss are taken into account.The corrosion degrees are chosen and in caption one gets, Tc equal to 0% represents «ricint_0 », Tc= 10% corresponds to « ricint_10 », Tc= 20% is equivalent to « ricint_20» and Tc= 50% is connected to « ricint_50 ». In addition to this first corrosion effect,the bond strength variation which is the second corrosion effect considered is studied atthe same time. The numerical results compared to the experimental one are given inFigures 8–11.

Table 1. Material parameters related to the steel/concrete interface model used.

Material parameter

Value

P621 P412

Shear coefficient (l) 13267 106 Pa 13267 106 PaBond strength (Ad) … J-1.m3 … J-1.m3

Initial threshold (Y0) 98 J.m-3 98 J.m-3

Kinematic hardening (c) 1.95 109 Pa 7. 109 PaNon linear hardening (a) 7 10-7 Pa-1 7 10-7 Pa-1

Anchorage length (lIa) 0.2 0.2

Table 2. Material parameters related to the concrete model deriving from (Richard et al., 2010b).

Material parameter

Value

P621 P412

Shear coefficient (l) 12708 106 Pa 12708 106 PaTensile brittleness (ADir) 5.2 10-3 J-1.m3 3.5 10-3 J-1.m3

Compressive brittleness (AInd) 9.7 10-6 J-1.m3 4.3 10-6 J-1.m3

Initial threshold (Y0) 262 J.m-3 124 J.m-3

Kinematic hardening (c) 8.7 109 Pa 8. 109 PaNon linear hardening (a) 4.2 10-7 Pa-1 6. 10-7 Pa-1

Anchorage length (lca) 0.2 0.2

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Figures 8–11 represent the load mid-span/deflection curves obtained for various cor-rosion degrees Tc and bond strengths Ad for the beams P621 and P412 respectively.The graphs a and b of Figures 8 and 10 show the curves with a bond strength coeffi-cient Ad of 0.1 10-3 and of 1.0 10-3 respectively. And, the graphs d and c of Figures 9and 11 exhibit the curves with a cross-section loss Tc of 20% and of 50% respectively.

Different concrete models are used, the BAEL model (corresponding to « bael » incaption), the proposed concrete model (equivalent to « riccon » in caption). The steel/concrete interface model is used with different bond strength coefficients Ad , for Ad=0.1 10-3 represents « ricint_1 », Ad= 0.5 10-3 represents « ricint_2 », Ad= 1.0 10-3 repre-sents « ricint_3 », Ad= 4.0 10-3 represents « ricint_4 », Ad= 7.0 10-3 represents «

0 5 10 150

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Displacement (mm)

Loa

d (k

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riccon/ricint_0

riccon/ricint_10

riccon/ricint_20

riccon/ricint_50

bael/par-uni

riccon/par-uni

experience

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50

Displacement (mm)

Loa

d (k

N)

(a)

(b)

Figure 8. Load mid-span/deflection curves obtained for various corrosion degrees (expressed interms of cross section losses) at given bond strengths for the beam P621.

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ricint_5 » and Ad= 1.0 10-2 represents « ricint_6 ». Moreover, the low bond strengthcoefficients, Ad correspond to high bond strengths. As seen previously, this interfacemodel is the combination of an interface model (Richard, 2010) and an Ouglova steelmodel (representing « par-uni » in caption). That allows taking into account to thecross-section loss by means of the proposed interface model. The different cross-sectionlosses considered are seen previously.

The Figure 8 presents the numerical results in the case of the monotonic loading onthe beam P621 with various corrosion degrees Tc at a given bond strength coefficientAd . Figure 8a shows the evolution of the carrying capacity of the beam P621 accordingto different corrosion degrees Tc at a given bond strength coefficient Ad= 0.1 10-3. And

0 5 10 150

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Displacement (mm)

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riccon/ricint_1

riccon/ricint_2

riccon/ricint_3

riccon/ricint_4

riccon/ricint_5

riccon/ricint_6

bael/par-uni

riccon/par-uni

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Displacement (mm)

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d (k

N)

(d)

(c)

Figure 9. Load mid-span/deflection curves obtained for various bond strengths at givencorrosion degrees (expressed in terms of cross section losses) for the beam P621.

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Figure 8b, the evolution of the carrying capacity of the beam P621 according to differ-ent corrosion degrees Tc at a given bond strength coefficient Ad= 1.0 10-3. In Figures8a and 8b, it can be noticed that the increasing of the corrosion level Tc leads to adecreasing of the carrying capacity of the beam P621 whatever the bond strength.Moreover, the consideration of the corrosion at the interface influences also the shapeof the curves. Usually, three main stages can be observed with a perfect steel/concreteinterface like it can see thanks to the curve « riccon/par-uni ». The first is related to thelinear elastic part of the behaviour. The second stage is related to the concrete cracking.And the third and last stage represents the yielding of the steel rebars. But, taking intoconsideration the corrosion, these three stages are not really respected. The first stage isseen in the case of the perfect steel/concrete interface. Then, the concrete cracking starts

0 5 10 15 20-10

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riccon/par-uni

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Figure 10. Load mid-span/deflection curves obtained for various corrosion degrees (expressed interms of cross section losses) at given bond strengths for the beam P412).

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but during this cracking it is possible to observe a stress redistribution and especiallythe beginning of the steel/concrete interface damage. Therefore, the following stage isdifferent than the stage with a perfect interface. It is the steel/concrete interface deterio-ration stage. The bond strength is lower when the corrosion degree is high. Besides, thistrend is made worse when decreasing the bond strength (increasing the bond strengthcoefficient)

During the structural failure mode, the yielding of the steel rebars experimentallyoccurs after the concrete cracking and the steel/concrete interface damage. It is the reasonwhy the choice to not consider the influence of the steel yielding on the steel/concreteinterface has been made (Engström, 1992). It will be interesting to consider that from anumerical point of view. In Figures 9c and 9d, the numerical results come from the samekind of loading as the one of the Figure 8 on the same beam, P621 only with variousbond strengths coefficients Ad at a given corrosion degree Tc. Figure 9c shows the evolu-tion of the carrying capacities of the beam P621 according to different bond strengthcoefficients Ad at a given corrosion degree Tc= 20% And Figure 9d represents the evolu-tion of the carrying capacities of the beam P621 according to different bond strengthcoefficients Ad at a given corrosion degree Tc= 50%. It is possible too see the same trendas seen in the case of the Figure 8. Whatever the corrosion degree, a loss of carryingcapacity is observed with the reduction of the bond strength. And the trend makes worse.

In Figures 8 and 9, the interest of taking into account the corrosion effects areshowed. Do not consider the corrosion effects leads to an overestimation of the carryingcapacity of the structure. Besides, the importance of using a relevancy concrete modelin order to consider the concrete cracking correctly can be observed. Otherwise, the ini-tial rigidity cannot be taken as well as the stress redistribution during the concretecracking.

Figures 10a and 10b present the numerical results in the case of the cyclic loadingon the beam P412 with various corrosion degrees Tc at a given bond strength coeffi-cient Ad which equal 0.1 10-3 and 1.0 10-3 respectively. In Figures 11c and 11d, areshowed the numerical results in the case of the cyclic loading on the beam P412 withvarious bond strength coefficients Ad at a given corrosion degree Tc of 20% and of50% respectively. In Figures 10a and 10b, it can be noticed that the loss of the bondstrength (the increasing of the bond strength coefficient Ad) leads to a decreasing of thecarrying capacity of the beam P412 whatever the corrosion level. In Figures 11c and11d, a loss of carrying capacity is observed when the bond strength decreases, whateverthe corrosion degree also. The same trend is noticed for Figures 10 and 11 as for theFigures 8 and 9 respectively. As a matter of fact, the trend is made worse when the cor-rosion degree increases or when the bond strength decreases (the bond strength coeffi-cient increases).

One can notice that the concrete model is capable considering the friction betweenthe crack lips and the steel/concrete interface model can take into account the frictionalsliding between the steel and the concrete although the hysteretic loops are bigger thanthe experimental ones. This hysteretic phenomenon is well associated to a loss of rigid-ity (the more the number of loops is high, the more the slope of rigidity is weak andthe more the area of the loops is enlarged).

As expected, the fact to consider a relevant concrete model capable of takingaccount of the nonlinearities due to the features of the concrete material like the perma-nent strains, hysteretic effects due to the frictional sliding between steel and concrete aswell as the quasi unilateral effect is observed. It can be seen that the initial rigidity iswell described.

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Besides, taking into account the existence of a steel/concrete interface influences themechanical behaviour of the beams P621 and P412. This fact is especially noticeableduring the concrete cracking stage in which the steel/concrete interface degradationstage occurs.

For high bond strengths and a low corrosion degree, the carrying capacity of thebeams P621 and P412 is influenced weakly. Furthermore, whatever the bond strength,the more the corrosion degree is high, the more the carrying capacity decreases. It caneven be seen that the carrying capacity of the beams tends to an asymptote when thebond strength decreases. All these observations made in the both cases monotonic andcyclic, is more intensified in the cyclic case than in the monotonic one due certainly to

Figure 11. Load mid-span/deflection curves obtained for various bond strengths at givencorrosion degrees (expressed in terms of cross section losses) for the beam P412.

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the steel/concrete interface which is much more degraded. In addition to this, we noticethat the load/displacement curve shape of the beam P412 behaviour is clearly betterdescribed with respect to the experimental, especially for a corrosion degree of 20%.

By only having the corrosion degree varied, we can notice a carrying capacity loss.These obtained numerical results were expected because this trend had already pointedout by Vu (2008). Moreover, in the case where only the bond strength which varies, acarrying capacity loss is noticed which is the same in the work of Ouglova (2004). Inshort, a bond strength loss connected with a corrosion degree growth in terms of cross-section loss is much more detrimental than if only one corrosion effect is considered.Obviously, when both the corrosion effects, cross-section loss and bond strength varia-tion are considered at the same time, this represents better the reality.

5. Conclusion

A simplified steel/concrete model capable to consider corrosion effects has been used inthe framework of the French research project « Benchmark des poutres de la Rance ».This model derives from a steel/concrete interface 3D model developed by Richardet al. (2010a) based on a multifiber approach. In fact, the model used is clearly able toreproduce a homogenised behaviour based on two elementary behaviours (steel/concreteinterface and steel rebar).

Besides, a simplified version of the concrete model developed by Richard, Rogation,Cremona, and Adelaide (2010b) has been used.

The efficiency of the simplified steel/concrete interface and concrete models hasbeen illustrated in our study on four-point bending tests of two beams P621 and P412tested respectively under monotonic and non-reverse cyclic loadings. A comparisonbetween the experimental data and the numerical results has been performed. Severalpoints from this comparison resulted. Firstly, the corrosion effects that is the loss ofbond strength and the reduction of steel cross-section leads to a decrease of the carryingcapacity of the beams. And the combination of the two effects makes the loss of struc-tural carrying capacity worse. Secondly, it is important to use a relevant concrete modelin order to take into account in a correct way, the initial rigidity, the concrete crackingand also the stress redistribution for instance. Thirdly, only the consideration of the cor-rosion at the steel/concrete interface allows being in agreement with the experience.

Although the strategy used in order to consider the steel/concrete interface includingcorrosion in the modelling of corroded RC structures do not allow taking into accountthe stirrups, the numerical results are in agreement with the experiments.

This approach is interesting because the computational cost saving involved is dras-tically reduced compared to full 2D- or 3D-finite elements ones. The main drawback ofthis approach is related to the obtaining of local information such as crack openings,crack spaces and crack tortuosity.

The use of the concrete model shows the relevancy of such a model because it takesinto account refined mechanisms related to concrete strain processes.

The aims of this study – to use a simplified approach simple, robust, efficient andallowing a computation cost saving and not to get local information such as crackingpattern, cracking openings, etc., the efficiency as well as the ability of the proposedapproach to handle practical structural cases – have been well illustrated. Besides, thisapproach seems to be helpful for civil engineers since reduced computational costs anda satisfying accuracy can be expected.

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AcknowledgementsThe investigations and results reported herein are supported by the National Research Agency(France) under the APPLET research program (grant ANR-06-RGCU-001-01).

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