azenha2011 modelling thermo chimo mech

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Engineering Structures 33 (2011) 3442–3453

Contents lists available at SciVerse ScienceDirect

Engineering Structures

journal homepage: www.elsevier.com/locate/engstruct

Thermo–hygro–mechanical modelling of self-induced stresses during the servicelife of RC structures

Miguel Azenha a,∗, Carlos Sousa b,1, Rui Faria b,2, Afonso Neves b,3

a ISISE — Institute for Sustainability and Innovation in Structural Engineering, University of Minho, School of Engineering, Civil Engineering Department, Azurém Campus, 4800-058

Guimarães, Portugalb LABEST — Laboratory for the Concrete Technology and Structural Behaviour, Faculdade de Engenharia da Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal

a r t i c l e i n f o

Article history:

Received 26 November 2010

Received in revised form

21 April 2011

Accepted 1 July 2011

Available online 2 August 2011

Keywords:

Cement hydrationService life conditions

Differential shrinkage

Cracking

Numerical simulation

a b s t r a c t

Current practices of structural design in reinforced concrete (RC) structures usually account for stressescaused by phenomena such as heat of hydration and drying shrinkage in a quite simplified manner. Thepresent paper aims to evaluate the consequences of explicitly considering self-induced stresses, whichactually vary significantly within structural cross-sections, combined with stresses caused by externalloads. Theused numericalframework involvesthe explicit calculationof the temperature fieldin concrete,with proper account forthe heat of hydration of cement. Simultaneously, themoisturefield in concrete iscomputed in order to ascertain the relative humidity changes in the pore structure caused by drying, andthe inherent shrinkage strains. Stress calculations are made with due consideration of the evolution of mechanical properties of concrete as a function of the equivalent age, as well as relevant phenomena likecreep, concrete cracking and influence of reinforcement. Two separate groups of numerical applicationsare presented, checking influence of the self-induced stresses: a unrestrained concrete prism usually usedfor shrinkage measurement, and concrete slabs subjected to external loads. Particularly for the second setof applications, the obtained results (with explicit consideration of the differential effects of self-inducedstresses)are compared, in terms of cracking loads andcrackpropagation, to those that would be obtained

by using the simplified design approach based on considering uniform shrinkage fields in concrete. It isfound that the behaviour of both formulations is quite similar after crack stabilization, but may be quitedistinct in the crack propagation phase.

© 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The existing methodologies for practical analysis and design of RC structures comprise several simplifying assumptions, namelyconcerning the stress state induced by heat of hydration anddrying shrinkage. Actually, due to the exothermic nature of cement hydration reactions, concrete members endure non-uniform internal temperature distributions, usually with hotter

regions in the core and cooler ones close to the external surfaces.Total or partial restrictions to the volumetric deformationsassociated to these thermal fields induce stresses in concrete,whose importance is usually disregarded by designers, mostof the time without a quantitative notion of their magnitude.

∗ Corresponding author. Tel.: +351 93 840 4554; fax: +351 253 510 217.

E-mail addresses: [email protected], [email protected]

(M. Azenha), [email protected] (C. Sousa), [email protected] (R. Faria),

[email protected] (A. Neves).1 Tel.: +351 22 508 1914.2 Tel.: +351 22 508 1950.3 Tel.: +351 22 508 1879.

Their quantification is, however, possible with recourse tothermo–mechanical methodologies that allow the computationof temperatures induced by heat liberation, and using them asthermal loads in the numerical simulation of stress fields.

Regarding drying shrinkage, and according to code provi-sions [1,2], its structural effects are taken into account by usingreference shrinkage strains (based on the geometry of the concretemember, environmental relative humidity and concrete grade

class), which thereafter are applied uniformly to the structurein order to ascertain the corresponding stresses. This is clearly asimplification for practical calculations, since drying occurs non-uniformly in concrete, which generates stresses in the presence of restraints (which are external, caused by reinforcement or due tocross-sectional effects). The quantification of the differential dry-ing shrinkage calls for the necessity of knowing the moisture fieldswithin concrete, as well as the relation between moisture lossesand the corresponding volumetric variations. Since most generallyRC structures suffer effects of both heat of hydration and dryingshrinkage, it is important to have numerical simulation tools thatallow the computation of thermal, moisture and mechanical fieldsfor the computation of stress states during the service life.

0141-0296/$ – see front matter © 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2011.07.008

http://dx.doi.org/10.1016/j.engstruct.2011.07.008http://www.elsevier.com/locate/engstructhttp://www.elsevier.com/locate/engstructmailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]://dx.doi.org/10.1016/j.engstruct.2011.07.008http://dx.doi.org/10.1016/j.engstruct.2011.07.008mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]://www.elsevier.com/locate/engstructhttp://www.elsevier.com/locate/engstructhttp://dx.doi.org/10.1016/j.engstruct.2011.07.008


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M. Azenha et al. / Engineering Structures 33 (2011) 3442–3453 3443

Several authors conducted research in the field of numericalsimulation of self-induced stresses during the service life of struc-tures. Kwak et al. [3], Kwak and Ha [4] studied non-structuralcracking in RC walls using an approach similar to the one adoptedin this paper, with explicit consideration of non-uniform tem-perature and moisture distributions, aiming the evaluation of post-cracking behaviour. Bernard and Brühwiler [5] and Martinolaet al. [6] have explicitly calculated moisture fields and the corre-sponding drying shrinkage in the evaluation of debonding poten-tial between concrete layers cast at different ages. Grasley et al. [7]monitored internal RH in concrete specimens and evaluated theself-induced stresses in both linearly restrained and free speci-mens. Oh and Cha [8], Oh and Choi [9] established a numericalframework for analysis of thermal, moisture and mechanical fields,applying it to the study of concrete decks of composite bridges.Chen and Mahadevan [10] used a thermo–hygro–mechanicalframework for the cracking analysis of a 1 × 1 × 0.5 m3 concreteblock using a smeared approach. Granger et al. [11] used a hy-gro–mechanical approach to evaluate skin micro-cracking of con-crete with recourse to contact analyses. Gawin et al. [12] deviseda numerical framework for thermo–hygro analysis in which con-crete stresses dueto creep andshrinkagewere calculated using theeffective stress concept.

In regard to predicting the service life performance of ordinaryRC structures with advanced numerical techniques, none of the above mentioned works focused however on addressingcomparisons with results obtained with simplified approachescommonly adopted in design (based on codes such as Model Code1990 [2]), to evaluate how RC structures are designed nowadays.

This is precisely the main purpose of the present paper,commencing with a general description of the developed ther-mal–hygro finite element (FE) framework, which is thereafter usedwith the mechanical part of the FE code DIANA [13].

Two sets of applications are considered subsequently. Thefirst one concerns to the analysis of a concrete drying shrink-age prism, to check relevance of the self-induced stresses due

to thermal–hygro phenomena. The second example focuses onthe behaviour of RC slab-like members, where influence of non-uniformity of the self-induced stresses due to heat of hydrationand drying shrinkage are evaluated with the members perform-ing under different conditions: (i) under imposed longitudinal de-formations, or (ii) under imposed curvatures applied at distinctages. Conclusions are extracted in regard to the influences of self-induced stresses on the cracking loads, as well as on the perfor-mances of the RC members during the crack propagation phases.

2. Thermo–hygro–mechanical model

2.1. Thermal submodel

Computation of the transient temperature fields in concrete ismade through the implementation in a FE code of the heat balanceequation

k∇ · (∇ T )+ Q̇ = ρc Ṫ (1)

where k is the thermal conductivity, ρc is the volumetric specificheat and T is the temperature. Q̇ is the volumetric heat generationrate due to cement hydration, formulated as an Arrhenius typelaw [14]

Q̇ = Af (α) e−E aRT (2)

where A is a rate constant, E a is the apparent activation energy, α is

thedegree of heat development (ratiobetween the heat Q releasedup to time t and the total heat Q final released upon completion of

cement hydration), R = 8.314 J mol−1 K−1 is the Boltzmann’sconstant and f (α) is a normalized function for heat.

The formulation of boundary conditions

qT = h (T b − T e) (3)

where qT is the heat flux per unit area, T b is the boundary surfacetemperature and T e is the environmental temperature, comprisesa mixed convection–radiation boundary transfer coefficient h [15].

The calculated temperatures are used as inputs for the mechan-ical model, where thermal strains of concrete are computed ac-counting for the material coefficient of thermal expansion (αT ).Even though this coefficient is known to be variable during the firsthours after casting [16,17], the applications envisaged in this paperdo not involve significant temperature changes at early ages, andthus it was decided to use a constant value for αT .

Also, the evolution of temperatures at each point in the struc-ture allows computing the equivalent age t eq (useful for updatingthe age-dependent mechanical properties), based on the followingArrhenius-type equation

t eq =

∫ t 0

e−

E aR

1T (τ )

− 1T ref

dτ (4)

where T ref stands for the reference temperature (usually 20 °C).Specific details regarding the numerical implementation of thisformulation using the FE Method can be found elsewhere [18].

2.2. Moisture submodel

Modellingof themoisturestates insideporous materialssuch asconcrete requires the selection of an appropriate driving potential.Several approaches exist, using the pore pressure, the pore relativehumidity (H ), the global water content, or even separating thevapour water from the liquid water. For the applications discussedherein the averaged pore relative humidity H has been selectedas the driving potential, and accordingly the moisture state in thepore structure is expressed according to equation [19]

Ḣ = ∇ · (DH ∇ H )+ Ḣ S (5)

where 0 ≤ H ≤ 1, DH is the moisture diffusion coefficient and Ḣ Srelates to the H drop in the pore structure due to self-desiccation.As the applications addressed in this paper pertain to ordinaryconcrete, with relatively high water-to-cement ratios, the effect of Ḣ S has been neglected. CoefficientDH was considered to dependonH as defined in MC90 [2]

DH = D1

[αH +

1 − αH

1 + [(1 − H ) / (1 − H C )]n

] with αH =

D0

D1(6)

where D1 and D0 are the values of DH for H = 1 and H = 0,respectively, H C is the pore relative humidity at which DH = 0.5D1

and exponent n is a material property.Moisture fluxes qH across the boundaries are modelled throughthe equation

qH = hH (H b − H e) (7)

where hH is the moisture boundary coefficient, H b is the boundaryporerelative humidity and H e standsfor the environmental relativehumidity. The computed moisture field H is transformed into afield of unrestrained potential shrinkage εsh,pot according to [3]

εsh,pot = εsh,∞

0.97 − 1.895 (H − 0.2)3

(8)

where εsh,∞ is the ultimate drying shrinkage strain, that is, uponcomplete drying at infinite time (valid in the scope of this paper, asself-desiccation due to autogenous shrinkage is being neglected).

It should be highlighted that εsh,pot does not necessarily representthe real shrinkage strain at a given point; in fact, it is the shrinkage


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3444 M. Azenha et al. / Engineering Structures 33 (2011) 3442–3453

Fig. 1. Concrete stress–strain diagrams normal to the crack: (a) outside the

effective area; (b) inside the effective area.

strain that would potentially occur at this point if no restraint todeformation was present. The εsh,pot field is to be applied in themechanical model as an imposed strain, and the real shrinkagestrain can then be calculated with due account to the presentrestraints.

Details about the implementation of the moisture model can befound in [18].

2.3. Mechanical submodel

The mechanical analyses were performed using the DIANA FE

software, adopting as inputs from the thermo–hygro models theequivalent ages t eq and the thermal and unrestrained potentialshrinkage strains.

Basic creep of concrete was accounted for through the useof the Double Power Law (DPL), which has a reasonably goodperformance on both early age and long term time spans [20]

J (t , t ′) =1

E 0(t ′)+

φ1

E 0(t ′)(t ′)−m(t − t ′)n (9)

where J (t , t ′) is the compliance function at time t for a loadapplied at instant t ′, E 0(t

′) is the asymptotic elastic modulus, andφ1, m and n are material parameters. Even though the DPL wasdevised for basic creep estimation of bulk concrete members, it isdeemed feasible to the approach of this numerical framework, inaccordance to the strategy also adopted in Ref. [7].

Cracking was reproduced via a smeared crack approach, withthe total strain tensor ε being decomposed into an elastic compo-nent εe anda crack strainεcr that is,ε = εe+εcr [21]. A stress–straintension model with softening, based on the Crack Band Theory(CBT) developed by Bazant and Oh [22], was employed to describethe crack formation and crack growth in plain concrete. The evo-lution of the stress normal to the crack, σ n, and the correspondingnormal strain εn is depicted in Fig. 1(a), and it involves the con-crete tensile strength f ct and the fracture energy GF . Consistentlywith the CBT the specific fracture energy GF /h is adopted, wherethe characteristic length h of the current FE is dependent on itssize, to ensure objective results with respect to the refinement of FE mesh adopted in the discretization.

The stiffness increase due to bond between concrete and

reinforcement (tension-stiffening effect) was accounted for bymodifying the σ n–εn relation for the effective area of concretesurrounding the reinforcing bars, Ac ,ef , according to MC90 [2]. InFig. 1(b) concrete softening inside Ac ,ef is reproduced by a multi-linear diagram defined by GF and the average crack spacing ls, asrecommended by Feenstra and de Borst [23]; accordingly, σ n andεn are assumed as averaged over ls. During the stabilized crackingphase the average tensile stress carried by the concrete betweencracks takes a constant value of βt f ct, as suggested in [23–25].Bearing in mind the applications intended for the present paper,and in accordance to the recommendations of MC90 [2], β t wastaken as 0.4. The horizontal branch of theσ n–εn diagram of Fig.1(b)ends when the average strain εn reaches ε yc , the latter beingquantified as [23]:

ε yc = εsy − βt f ctE sρs

. (10)

In this equation E s is the Young’s modulus of steel, ρs is thereinforcement ratio with respect to Ac ,ef , and εsy is the yieldingstrain of steel. The tension-stiffening effect vanishes when theaverage strain reaches εsy.

An elastic–perfectly plastic diagram was adopted for thereinforcing steel, with a yielding stress f sy = E sεsy.

Given the fact that for the present paper the structuralbehaviour is to be considered solely for service life conditions, alinear elastic behaviour was assumed for concrete in compression.

2.4. Coupling between the submodels

The physical phenomena involved in the three fields understudy (thermal, hygro and mechanical) interact with each other.However, in the macro-level approach of this paper severalsimplifications are implicit, and the purpose of this section is toprovide further information on the degree of coupling between thethree submodels just presented.

It is important to remark that the thermal model providessome further information than strictly the nodal FE temperatures,namely in what concerns the state of the chemical hydrationreactions, since equivalent ages and a degree of heat developmentare inherently calculated also.

Thermal and moisture phenomena occur at rather dissimilartime scales, because the thermal diffusivity of concrete is in theorder of k/ρc = 10−6 m2 s−1, whilst the counterpart for mois-ture flow is around 10−8–10−10 m2 s−1 [26]. Moisture fluxeswithin concrete transfer heat energy, as well as the evaporation/condensation of water involved in moisture transport. However,the mentioned differences in diffusivities imply that variations intemperature due to thermal fluxes occur almost instantly withinconcrete, by comparison with the thermal variations due to mois-ture fluxes. This provides grounds forthe simplification assumed inthis work, which assumes that the thermal field is not influencedby the moisture one.

Moisture equilibrium in the pore structure of concrete and dif-

fusion properties are, however, strongly dependent on temper-ature. But in this work the diffusion parameters were derivedfor an environmental temperature of 20 °C, and the applicationsto be presented later concern only to moisture movements un-der isothermal conditions at this temperature. Therefore, the usedmodel does not account for this dependency of the moisture fieldwith regards to the thermal one. The simplification of consider-ing constant environmental temperature conditions holds valid forthe first application envisaged in this paper, as drying shrinkageprisms are usually maintained inside climatic chambers undercon-stant T and H . However, for the second application this simplifica-tion hasimplications on theaccuracyof predictions of temperaturefields at early ages, and also on the effect of variable T on moisturediffusion. Nonetheless, it was decided to do so to allow focusing

on non-thermal related issues, without having to cancel effects of temperature.

Moisture state of concrete during early ages plays a fundamen-tal role in cement hydration: it is known that under H < 80% hy-dration reactions almost stop, and that for H between 80% and 97%they are strongly hindered [27,28]. Therefore, upon moisture de-pression at the concrete surfaces exposed to evaporation cementhydration is locally retarded, a relevant issue that several authorshave put forward in numerical models [20,29–31]. Yet, this situa-tion is disregarded in the applications of the present paper, sinceexposure to drying is prevented until the age of 28 days. If otherkinds of applications were envisaged, namely involving formworkremoval at earlier ages (7 days or less), or comprising high perfor-mance concrete with significant self-desiccation, the formulation

presented in this paper would have to be adapted, to account forthe degree of hydration dependency on the available water.


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Fig. 2. (a) Material parameters for heat generation; (b) E -modulus evolution.

The mechanical field is influencedby the thermal submodel notonly due to the volumetric changes induced by the temperaturevariations, but also because the evolution of the mechanicalproperties of concrete depend on the equivalent age field; seeEq. (4). Also, theshrinkagestrains computed in themoisturemodelare important data for the mechanical model; see Eq. (8).

Conversely, the mechanical field mayinfluence the thermal and

moisture fields in several ways:– Volumetric variations of concrete may cause it to separate from

boundaries, changing boundary coefficients for both thermaland moisture transports. However, this interaction is usuallyconsidered negligible and thus disregarded.

– Cracking may affect transport phenomena in concrete, butdue to the usually small crack widths it is feasible to assumethat they only marginally affect heat transport. In the caseof moisture cracking, it does in fact increase the moisturetransport coefficients [32–34], but this effect can be consideredsmall enough as to be disregarded without significantlyaffecting the quality of numerical predictions.

As a result of what has been stated, the approach of one-

directional coupling of the involved phenomena is justified: theheat and moisture fields are initially calculated, and the resultingdata is used as input to the mechanical field whose calculations areperformed separately.

3. Applications

3.1. Overview/material properties

The applications to be presented regard firstly to an unre-strained drying shrinkage prism, and then to a slab under the com-bined effects of self-induced stresses (heat of hydration and dryingshrinkage) and externally applied uniformly distributed loads.Material properties in all the examples concern solely to normal

strength concrete, which is still the most widely used in currentconstruction.

In terms of heat of hydration generation, the thermal output of a dosage of 250 kg m−3 of cement type I 52.5R is represented inFig. 2(a), which also indicates the applicable material parameters

A, E a and Q final. Other assumed relevant thermal properties of concrete are k = 2.6 W m−1 K−1 and ρc = 2400 kJ m−3 K−1.The boundary transfer coefficient corresponding to a direct contactbetween concrete and the environment is expressed by thestandard value h = 10 W m−2 K−1, and the initial temperature of concrete is 20 °C. The values used for thermal properties k, ρc andh were adopted in accordance to the conditions of a previous casestudy of a wind tower foundation [35], where ordinary concretewas used in outdoor conditions, with an average wind speed under

5 km/h. The effects of solar radiation and night cooling have beenneglected in this study.

For the moisture model a moist curing during 28 days isconsidered in all the calculations in this paper, and the diffusionparameters reported by Kim and Lee [19] for a similar situationare adopted: D1 = 6.17 × 10

−10 m2 s−1, αH = 0.047 andH C = 0.8. Also based on the previous reference, the moisturetransfer boundary coefficient is null during the first 28 days (moistcuring conditions), and switches to hH = 5.81 × 10

−9 m s−1

thenceforward.εsh∞ inEq. (8) is considered as 500×10−6

, (feasiblevalue of final drying shrinkage upon total drying, according toseveral codes [1,2]), and for the initial moisture conditions H =100% was considered. After the initial moist curing conditionsat 20 °C, environmental conditions are constant throughout theanalyses, with T = 20 °C and H = 50% after the age of 28 days.

For the mechanical analyses the following concrete propertiesare adopted: f ct = 2.6 MPa, GF = 0.085 N/mm, αT = 10

−5 anda Poisson’s ratio ν = 0.2. These values correspond to a normalstrength concrete of the class C25, with a maximum aggregatesize of 20 mm, according to the Model Code 1990 [2]. Evolutionof concrete E -modulus as a function of the equivalent age atthe reference temperature of 20 °C is represented in Fig. 2(b)(E -modulus considered constant for ages above 28 days). Creep

parameters for the DPL are the ones obtained experimentally byAtrushi [36]: φ1 = 1.88, m = 0.2 and n = 0.125. Reinforcementis characterized by a Young’s modulus E s = 200 GPa, a yield stress

f sy = 500 MPa and a coefficient of thermal expansion equal to theone of concrete.

3.2. Drying shrinkage prism

3.2.1. Geometry and numerical model

The ‘structural’ performance of a drying shrinkage prism withdimensions 200×200×600 mm3, usual in material research, is tobe analysed in this section. Disregarding influence of self-weightFig. 3(a) reproduces the prism and their symmetry planes, whichallows just one-fourth of the prism to be reproduced numerically

according to Fig. 3(b) (surfaces labelled as ‘‘sym’’ regard tosymmetry planes, and the ones marked as ‘‘bou’’ correspond toboundary surfaces in contact with the environment). The FE meshfor the drying shrinkage prism is represented in Fig. 3(c), whereit can be realized that a finer refinement is considered in theboundaries’ neighbourhoods, to cope with the higher thermal andmoisture gradients expected at these locations. The symmetryplanes perform as isolated in the normal directions, both for thethermal and moisture submodels; for the mechanical analyses,displacements normal to the symmetry planes are prevented. Allcalculations (thermal, moisture and mechanical) are conductedwith the same 8-nodded FE brick (with 2 × 2 × 2 Gauss points),although for the thermal and moisture analyses 4-nodded surfaceFE (with 2 × 2 Gauss points) are also considered to account for

the fluxes normal to the boundaries. Time steps are selected in thesequence that follows: 40 steps of 1 h, 20 steps of 4 h, 20 steps


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Fig. 3. Prism: (a) symmetry planes; (b) schematic representation; (c) FE mesh.

Fig. 4. Results for a selected group of points on the prism.

Fig. 5. Evolution of H on the prism.

of 16 h, 30 steps of 24 h and 230 steps of 96 h. This results in atotal time of analysis that extends for∼968 days (23 240 h), whichwas found to be enough for theprisms to attainalmost steady stateconditions.

3.2.2. Results of the thermal and moisture submodels

Results pertaining to the temperature evolution during thecement hydration phase, and for points P1–P5 of the prism aredepicted in Fig. 4(a). The highest temperature rise occurs at P1,the centre of the prism, and has a small value (less than 6 °C).Conversely, the lowest temperature rise occurs at the corner of the

specimen, at P5, amounting to less than 3°

C, with the temperaturegradients in the specimen remaining always below 4 °C. All

temperature gradientsvanisharound the age of 3 days, after whichthe temperature within the prism becomes constant (T = 20 °C).Evolution of the relative humidity H within the specimen occurs ina much slower fashion, as plotted in Fig. 4(b) for points P1–P5 andreproduced via the contours of Fig. 5. It is clear that the surfacepoints P3 and P5 dry much faster than the inner points right afterthe exposure at the age of 28 days. P1, the geometrical centre, isthe slowest to converge to the environmental relative humidityof 50%, reaching H = 51.8% at the end of the analysis. Thesegradients of H are responsible for unrestrained potential shrinkagestrains calculated according to Eq. (8), which are bound to induce

self-equilibrated stresses as a consequence of the necessary straincompatibility between the inner and outer parts of the prism.


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Fig. 6. Left column: vectors of principal stresses (red colour corresponding to tensile stresses); right column: stresses σ y (Pa). (For interpretation of the references to colour

in this figure legend, the reader is referred to the web version of this article.)

3.2.3. Results of the mechanical submodel

The computed stress field led to the principal stress vectors

depicted on the left column of Fig. 6 (red colour representingmaximum tensile stresses), andthe contours of thenormal stressesσ y (longitudinal to the specimen) shown on the right. During

the first 28 days only thermally-induced stresses occur, as thespecimen is under moist curing, and so, along this phase the

thermal field evolves as follows: (i) the prism is cast at 20 °C; (ii)then the temperature rises non-uniformly (higher temperatures

in the core and lower in the surface); (iii) after the maximum

temperature is reached the prism starts cooling (faster on the

surface and slower in the core). As a consequence of cooling fromhigher temperatures, the core tends to contract more than the

surface of the prism, so the latter becomes compressed and theinterior endures tensile stresses. The stress distribution at theend of this hydration-induced thermal process can be seen inFig. 6 for the age of t = 27.9 days: since the thermal gradientswere less than 4 °C, the resulting tensile stresses at the core arenegligible, amounting to 0.05 MPa, which allows to assume thatat the age of 28 days the prism starts to dry from an almoststress-free state.

The results computed for the prism right after exposure todrying are shown in Fig. 6 for t = 28.1 days. External surfacesexperience a quite sudden desiccation (in accordance to Fig. 4(b)),which induces strong potentialshrinkage strains at these locations,

whilst the core remains with high relative humidity, thus havingmuch lower tendency to shrink. As a result of this, the external


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Fig. 7. Stresses σ y in points P1–P5.

Fig. 8. Evolution of cracking strains (vectors normal to the cracks, with length

proportional to the crack opening).

surfaces endure tensile stresses, self-equilibrated by compressivestresses in the interior of the prism. Given the fact that thedesiccated depth at the beginning of drying is very small, therelative stiffness of this desiccated ‘skin’ is much lower thanthe internal non-shrinking concrete core. Therefore, high tensilestresses develop at the ‘skin’ – particularly in the vicinity of theedges, where drying is stronger –, reaching values as high as1.5 MPa at t = 28.1 days. This tendency is intensified untilthe tensile strength of concrete f ct = 2.6 MPa is reached, andcracking starts at the edges of the prism at t = 30 days, as itcan be confirmed in Fig. 7 (where evolutions of the σ y stresses arereproduced) and in Fig. 8 (where vectors proportional to the crackstrains are shown). The very low maximum crack strain predictedat this age, of circa 1× 10−4, indicates that the cracks are virtually

non-visible, which is consistent with the findings reported inRefs. [7,11,34,37]. This process keeps intensifying until the age of approximately 42 days, when crack openings are maximum (Fig. 8)and tensile stresses are generalized on the ‘skin’ ( Fig. 6).

For ages greater than t = 42 days, and due to the progressionof drying towards the core of the prism, a reversal of theconcrete stress signs occurs, and the maximum σ y tensile stressprogressively migrate to the interior of the specimen (see Fig. 6from t = 42 days until t = 964 days). This tendency can beconfirmedalso in Fig. 7, where the surface points P2 and P3, havinghigh tensile stresses at t = 42 days, experience a reductionof thosestresses along time, until they actually reverse at t ≈ 400 days,stabilizing around −0.5 MPa (compression) at t = 964 days. Theopposite occursin thecore pointsP1 andP4, which reach theend of

the analysis with tensile normal stresses of about 0.5 MPa. Locatedin the corner, and thus with a very low restraint to deformation,

Fig. 9. Computed strains at the centre of the prism vs. predictions using EC2.

point P5 endures almost negligible stresses during the wholeperiod of analysis. The mentioned process of stress reversal in thespecimen is also visible in the crack openings reproduced in Fig. 8,which decrease since approximately t = 42 days, and even closeafter t = 324 days. This tendency of surface shrinkage cracksto close along drying had been already documented for similar

conditions by Granger et al. [11].Feasibility of the results obtained for the analysed prism will

now be checked by comparing thenumericallypredicted evolutionof the shrinkage strains with the one that would be expectableupon application of Eurocode 2 (EC2) formulae. For such a goal,the final shrinkage in the EC2 formulae was adjusted to coincidewith the εsh,pot that arises from considering εsh,∞ = 500 × 10

−6

and H = 50% in Eq. (8), that is εsh,pot = 459.4 × 10−6. This

comparison does not aim to provide an exact match, since EC2 justprovides, in a simplified way, reference values for the evolution of strains. The predictable shrinkage strain evolution obtained fromapplication of EC2, together with the one computed at the core of the specimen by using the numerical thermo–hygro–mechanicalmodel here proposed, are depicted in Fig. 9. The resemblance of

both evolution curves is notorious.The analyses in this section allowed checking the feasibility of

the proposed model for calculating shrinkage-induced concretestrains, in view of existing regulation approaches (EC2).

The EC2 shrinkage model assumes uniform strain distributionover the cross section. If this assumption was considered, nostresses would occur in the prism. Therefore, shrinkage prismsare frequently used without a clear perception of the actual stressand strain distribution over the specimen. The developed modelallowed us to quantify the stress field in this kind of specimens,which, as demonstrated, experiences relevant variations alongtime that cannot be considered negligible as often assumed. Eventhough the shrinkage induced self-equilibrated stresses cannot beestimated by simple hand calculations (as would be desirable from

the practitioner point of view), the exposed results are expected toprovide a deeper perception of the actual behaviour of shrinkageprisms.

3.3. RC slab

3.3.1. Overview

The second example to be analysed is a 0.25 m thick RC slab,with the purpose of checking how the self-induced stresses (due toheat of hydration and drying shrinkage) will affect its mechanicalresponse to externally applied axial forces or bending moments.Particularly, it will be checked how those self-induced stressesinfluence the cracking loads and the stiffness of the slab during thecrack propagation phase.

The evolutions of both the temperature and the moisture fieldsare presented first, as they are the sources of the self-induced


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Fig. 10. FE mesh for the mechanical simulation.

stresses. For the thermo–hygro analyses, interactions with theenvironment are supposed to occur only through the bottom andtopsurfaces, with 1D heat andmoisturefluxes perpendicular to theslab’s middle plane.

Then, taking into account the results of the thermo–hygroanalyses, the mechanical response of the slab is evaluated anddiscussed. Three different situations are considered:

– At first the slab is subjected to the self-induced stresses only(neither external loads nor external restraints exist).

– In the second situation the combined effects of self-inducedstresses and axial forces are discussed. The slab is subjectedto an imposed axial deformation, after a time interval duringwhich it was left to shrink freely.

– Finally, the combined effects of self-induced stresses andbending moments are evaluated by prescribing rotations to theslab ends, after a time interval during which it was left to shrinkfreely.

The cracking loads of the previous situations are compared tothe ones obtained when shrinkage is accounted for by assuminga uniform distribution along the slab thickness. Relevance of thiscomparison lies in the fact that a uniform shrinkage distribution istraditionally assumed in design, thus the potential inaccuracies of this simplification are worth being evaluated.

3.3.2. Geometry and numerical model

Fig. 10 reproduces the geometry of the element to be analysed,with a thickness of 0.25 m, reinforced on the top and bottom faceswith anamount ofsteel equal to10.83cm2/m (12 mmrebars, witha 29 mm cover), with E s = 200 GPa and f sy = 500 MPa. A 1 mwide slab strip is considered, as it is assumed that the imposeddeformations are applied in the longitudinal direction only, i.e. norestraint is considered in the direction perpendicular to the planeshown in Fig. 10. The structureunder analysis is 1.79 m long, whichcorresponds to 10timesthe average crack spacing forthe stabilizedcracking phase, sr ,m = 0.179 m, quantified according to the MC90definition sr ,m = 0.67 × φs/(3.6 × ρs,ef ) (φs = 12 mm is therebar diameter; ρs,ef is the reinforcement ratio reported to the

‘effective area of concrete’, that is, the area of concrete effectivelybonded to steel, where tension-stiffening phenomenon develops).The assumed slab lengthwas chosen just forconvenience, as in factit plays no role in theproblems under study.The leftmost extremityof the slab is considered fixed as depicted in Fig. 10.

Concrete is discretized by using 4-noded plane stress FE (in14 layers across the thickness, as reproduced in Fig. 10), whereasfor the reinforcement, 2-noded truss FE are adopted. The effectivearea of concrete that surrounds each reinforcement layer isdecomposed into five 1.75 cm tall FE, whereas the remaining coreof the slab (7.5 cm thick) is discretized with 4 FE (see Fig. 10).The length of the concrete FEs is 17.9 cm, which coincides withthe average crack spacing sr ,m. Furthermore, linear constraintsare imposed to all vertically aligned nodes, so as to assure that

plane sections before deformation remain plane after deformation(Bernoulli’s hypothesis). This prevents occurrence of sectional

distortions in the element, as the imposed strains associated to thetemperature and shrinkage fields and the concrete maturity varynonlinearly across the thickness of the slab. The adopted FE lengthand the imposition of the Bernoulli’s condition are consistent withthe adopted models for simulation of the tension-stiffening effect,as the σ –ε relationships considered for the concrete are assumedto correspond to an average behaviour between cracks. In thisway the numerical model replicates the intended integral of thelongitudinal concrete strains over the average distance betweensuccessive cracks.

If a constant tensile strength was assumed for the concrete inall the vertical alignments of FE (see Fig. 10), then cracking wouldoccur simultaneously over all the slab length when subjected to animposed deformation. This result would not be realistic becauseexperimental results of tensile tests of RC ties with imposeddeformation reveal that cracks do notarise simultaneously andthelast crack occurs for a load approximately 30% higher than thatof the first crack [2,25,38]. Consistency with experimental resultswas kept by assuming that f ct grows linearly from 2.6 MPa at theleftmost vertical FE alignment up to 1.3 × 2.6 = 3.38 MPa atthe rightmost one, but keeping the same value of f ct in all the 14concrete FE of each vertical alignment.

The thermo–hygro calculations were carried out by using thesame 2D mesh (4-nodded elements) used for the mechanical

analysis, which is enough to simulate the actual heat/moistureflows which behave as 1D.

All analyses were conducted since the instant of casting untilthe age of 48 years.

3.3.3. Results of the thermal and moisture submodels

The small thickness of the slab, as well as its relatively lowcement content, contribute to a small temperature rise in concretedue to heat of hydration. In fact, upon observation of Fig. 11(a),where the temperature evolutions at the surface and at the coreareplotted, one can conclude that both the maximum temperature(32 °C) and the maximum temperature gradient (2.3 °C) occurat the same time (∼10 h). Bearing these results in mind, andassuming the absence of external restraints to deformation, it is

highly unlikely to expect significant stresses to be associated withthe heat of hydration release.

With the moisture field computed according to Eq. (5) the plotsdepicted in Fig. 11(b) were obtained for two points: at the surfaceand in the core. After the 28 days period, until which all pointsof the slab remained at H = 100%, exposure to an environmentwith H = 50% occurred, leading to a sudden drop in the relativehumidity at the horizontal surfaces of the slab. The decrease in H progresses along time towards the core, causing it to be almostin equilibrium with the environmental H at the age of 3000 days(8 years). The gradient of H along the slab thickness increases from0% up to a maximum of 20% at the age of 88 days, and remainsrelatively constant until about theage of 200days, time after whichit decreases to 1% at 3000 days.

The potential free shrinkage (εsh,pot) evolutions along the slabthickness that result from the computed H , which will be inputs


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a b

Fig. 11. Results at the surface and the interior (core): (a) temperatures; (b) relative humidity.

Fig. 12. Potential shrinkage along the slab’s thickness.

to the mechanical model, are shown in Fig. 12: the well-knowntendency of concrete to progressively shrink from the surface tothe core is fully captured by the numerical model.

3.3.4. Results of the mechanical submodelSelf-induced stresses acting alone.The developed stresses on the slab strip due to the hydration-

induced temperatures and the potential shrinkage are reproducedin Fig. 13 until the age of 4000 days. As the rising and decreas-ing of temperature occur mainly within the first 4 days, theircontribution to the stress development may be commentedseparately from the one due to the drying shrinkage, which isonly activated at the age of 28 days. As during the initial phaseof cement hydration the temperature in the core is higher than inthe surface, the former tends to expand more and thus it becomescompressed,whilst the surface endures tensile stresses (Fig. 13(a)).This tendency occurs until the age of 10 h, but is inverted fromthen on since the core will cool more than the surface; this causesa sign reversal of stresses, with compressive stresses developingin the surface and tensile stresses in the core. Due to the verylow gradients of temperature within the cross-section, and to theinexistence of external restraints, the developed thermal stressesare quite low (less than 0.1 MPa).

Regarding the relevance of the longitudinal stresses inducedby drying shrinkage, the situation is substantially different, asobservable in Fig. 13(b) and in Fig. 14 between the age of exposure (28 days) and the time at which H gradients reachtheir maximum within the cross-section (88 days): the surface,which was slightly compressed due to the initial temperaturedevelopment, goes into tension due to the rapid increase of the

surface potential shrinkage after exposure, and the core becomescompressed. At the age of 88 days the surface tensile stresses arelarge enough to cause cracking, approaching the tensile strengthof concrete f ct = 2.6 MPa (Fig. 14). From then on the relativehumidity gradient within the cross-section decreases (due to theprogressive reduction of H in the interior), and thus the surfacetensile stress alleviates, whilst the core endures a contraction thatis partially restrained by the reinforcement and the outer concretelayers, which changes its stress state from compression to tension.At the age of 2000 days (∼5.5 years) the concrete normal stressin the surface is almost 0 MPa and the tensile stress in the corereaches 1 MPa (Fig. 13(b)).

It is instructive to compare the detailed numerical analysis justdescribed with the one disregarding the effect of hydration heatrelease, and adopting the usual design approach of prescribing

shrinkage as a uniform strain field. The prescribed uniformshrinkage is equal to the average strain obtained with thedifferential shrinkage model. The results of the two analyses areplotted in Fig. 14, which put into evidence the rather differenttrends and values of the obtained stresses, particularly evidentat the age of maximum H gradient (88 days), with the standardprocedure being clearly non-conservative as far as the concretecracking risk at the surface is concerned. Even for a late age as48 years noticeable differences are observable between the twoapproaches for the self-induced stresses, although in such scenariothe standard procedure stands on the safe side with regards tosurface cracking of concrete.

Self-induced stresses combined with an imposed axial deformation.The present section analyses the consequences of imposing an

increasing axial deformation εm after the slab had been allowedto develop self-induced stresses, the latter acting alone up to someextent. Influence of these thermo–hygro stresses is checked by im-posing εm (as an imposed deformation at the rightmost extrem-ity of the slab) at three different ages (scenarios): (i) t = 28 days

a b

Fig. 13. Concrete longitudinal normal stresses: (a) until 4 days; (b) until 4000 days.


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Fig. 14. Concrete normal stresses, considering both differential (continuous lines)

and uniform (dashed lines) drying shrinkage.

Fig. 15. N –εm curves for imposed axial deformations at three different ages.

(but just before the beginning of drying), (ii) t = 88 days (instantof maximum H gradient) and (iii) t = 48 years (‘infinite’ time). Thecorresponding relationships between εm and the axial forces N in-stalled on the slab may be observed in Fig. 15. For guidance dashedlines are also included in this figure, defining the uncracked stiff-ness of the slab (state 1), or totally disregarding the contributionof concrete (state 2). Configurations of the obtained graphics arecoherent with the typical ones reported for experimental worksrunning under similar conditions [38,39].

Fig. 15 shows that the cracking load N cr, thatis, the axial force atthe beginning of the first descending branch of the N –εm diagram,is markedly influenced by the previously installed self-inducedstresses. For scenario (i) the cracking load computed with thenumerical model is practically coincident with the one obtainedthrough N cr = f ct Ac (1 + αρ) = 688 kN, α being the modularratio and ρ the reinforcement ratio. However, for scenario (ii),

t = 88 days, the effect of self-induced stresses causes N cr to reduceto560kN,andthe first branchof N –εm becomes non-linear(see theenlarged graph in Fig. 16), due to the process of crack formationthat follows the internal normal stress distribution within thecross-section. Thus, besides causing a decrease of the crackingload, self-induced stresses lead to a decrease of the stiffness of theascendingbranch of the N –εm relationship. If the axialdeformationis prescribed at ‘infinite’ time (scenario (iii)) N cr becomes evenlower than when imposed at the age of 88 days. Naturally, theultimate tensile strength is the same for the three scenarios, as itonly depends on the yielding strength of reinforcement, andnot onthe previous service live history.

Still regarding to Fig. 15, it is worth to remark that in the phaseof stabilized cracking the N –εm diagram gradually approaches

the dashed straight line corresponding to state 2 as the axialdeformation εm is imposed at later ages, which indicates an

Fig.16. N –εm curves for imposed axial deformations at different ages, considering

both differential (continuous lines) and uniform (dashed lines) drying shrinkage.

Fig. 17. M –χm curves for imposed curvatures at three different ages.

apparent progressive decay of tension-stiffening contributionof concrete between cracks. This tendency, also reported byBischoff [40], is due to the shrinkage that occurs prior to the loadapplication.

If uniform shrinkage strains are assumed over the cross-sectionof the slab, instead of the more realistic differential shrinkagedistribution predicted by the numerical model proposed here, thenhigher cracking loads are obtained (see Fig. 16) regardless the ageof loading (for scenario (i) results practically coincide, since dryingshrinkage is notyet active). The lower N cr loads for the cases whereself-induced differential stresses are considered have to be relatedto the fact that cracking occurs sooner at the surface FE, due to thehigher tensile stresses induced by non-uniform drying shrinkage.

Self-induced stresses combined with an imposed curvature.The present section discusses the effects of superimposing

increasing curvatures χm to the self-induced stresses installedon the slab. Fig. 17 reproduces the evolution of the bendingmoment M and the curvature χm, the latter being prescribed viasymmetrical rotations imposed to both extremities of the slab(constant curvature along the span). Similarly to what was done inthe previous section, curvatures will be imposed to the structurethat had been left solely under the effect of self-induced stressesuntil three different ages: 28 days (scenario (iv)), 88 days (scenario(v)) and 48 years (scenario (vi)).

Disregarding influence of shrinkage, it is easy to compute theelastic cracking moment, which for the present case is equal to

M cr = 29.5 kN m. Nevertheless, from Fig. 17 it is possible tonotice that when the increasing curvature is imposed before onset


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Fig. 18. M –χm curves for imposed curvatures at different ages, considering both

differential (continuous lines) and uniform (dashed lines) drying shrinkage.

of concrete drying, at the age of 28 days (scenario(iv)), the bendingmoment that corresponds to the peak of the first ascending branch

of the M –χm curve exceeds considerably that value, reaching43.7 kN m.It shouldbe remarked,however,thatat this stage linear-elasticity is no longer valid, as evident on the detail of Fig. 18. Thefact that for slender slabs this first peak on the M –χm diagramexceeds the elastic M cr value is well known in the concrete field.Guinea [41], who tested concrete beams with different heights,stored under adequate conditions to prevent drying shrinkage,showed that the flexural tensile strength is significantly higherthan the axial or splitting tensile strengths. He demonstratedalso that non-linear calculations taking into account the tension-stiffening effect – as done in the present applications – are able toexplain the differences between the axial and the flexural tensilestrengths.

Imposing the curvatures after onset of drying shrinkage,

markedly different M –χm diagrams are observed in Figs. 17 and18. For instance, for scenario (v) (age of loading equal to 88 days),the ascending branch of the M –χm curve exhibits a progressivelylower stiffness than that corresponding to scenario (iv) (absenceof shrinkage-induced stresses). Moreover, the M –χm diagram doesnot show descending branches during the crackpropagation phase,which is a consequence of the large self-induced tensile stresses(and eventually cracking) installed prior to the external loading.When these stresses are aggravated by the external bendingmoments, cracking develops along the entire length of the tensilesurface of the slab (in this case, all the surface FE on the outermosttensile layer crack when M reaches 8 kN m). As M increasesfurther cracking propagates inwards along the thickness of theslab, and the width of the existing cracks progressively increases,

but cracking cannotspread alongthe spandirection, whichjustifiesthe inexistence of descending branches in the M –χm diagram.

For the case in which an increasing curvature is imposed atthe age of 48 years (scenario (vi)), Fig. 18 shows that the peakof the first ascending branch of the M –χm diagram occurs for alower bending moment than the one for scenario (iv), thatis, in theabsence of self-induced stresses. Moreover, the teeth-shaped partof the M –χm diagram is more irregular for scenario (vi), associatedto the successive formation of new cracks, as the loading on theslab commences with a self-induced stress profile (Fig. 14) withcompression near the surfaces and a moderate tensile stress statein the core. Therefore, the crack formation along the span andacrossthe thickness occursin a more abrupt waythan forscenarios(iv) and (v).

Fig. 18 allows also to analyse the outcomes of the simplifiedapproach in which uniform shrinkage distributions are assumed

along the slab’s thickness, instead of the differential ones repro-duced by the continuous lines in Fig. 14: except for the case of loading at the age of 28 days, different M –χm diagrams areobtained during the crack propagation phase, which is relevant forthe service life performance of slabs.For thecase of loading appliedat 48 years the apparent cracking bending moment is smallerwhen computed with thesimplified shrinkagemodel than with thedifferentialone, in opposition to what wasobservedat theage of 88days. This is due to the fact that in the former case the differentialshrinkage model predicts occurrence of compressive stresses in thesurface areas, which retards crack formation.

4. Conclusions

A numerical framework for predicting the service life perfor-mance of RC structures was presented, which accounts for theself-induced effects due to the cement hydration heat release,as well as to the development of non-uniform drying shrinkagestrains. The field of concrete temperatures induced by cementhydration is solved with a thermal submodel, and the dryingshrinkage strains are computed on the basis of the relative humid-ity predicted in the concrete volume by a hygro submodel. Bothsubmodels are solved as uncoupled, feeding thereafter a mechan-ical submodel where concrete stresses are computed with due re-gard to the thermo–hygro strains, the equilibrium with externalloads, the evolving material properties of concrete (based on theequivalent age concept) and creep.

Two applications were presented to illustrate the plausibility of the numerical framework predictions, together with its potentialfor application in real structures: an unrestrained drying shrinkageprism and a RC slab subject to the combined effect of self-inducedstresses and externally applied loads.

Results obtained for the shrinkage prism showed non-uniformresidual stresses associated to heat of hydration and dryingshrinkage, with significant variations in their intensities, includingsign reversals. Evolution of the computed strains revealed quite

coherent with the shape of predictive curves of Eurocode 2, thusconfirming the plausibility of results.

The analysis of the RC slab, with due consideration of softeningand cracking of plain concrete, as well as tension-stiffening of RC,allowed interesting conclusions to be drawn in regard to the effectof self-induced stresses on the load–deformation curves when loadwas applied at differentages. It was observed that forloads appliedat instants when important self-induced surface tensile stresseswere already installed in concrete, the structural response of theslab was significantly different from that corresponding to a ‘zerostress’ initial stage, or even from that assuming the simplificationof uniform distribution of shrinkage strains. It was shown thatdifferential shrinkage stresses lead to a decrease of the calculatedaxial cracking loads, regardless the age of loading. Therefore,

the standard procedure suggested by design codes such as theEurocode 2 (based on uniform shrinkage strains along the crosssection) underestimates the concrete cracking risk. Codes could beimproved by suggesting reduced cracking loads for design checksin which lower cracking loads are unfavourable. Moreover, non-uniform shrinkage stresses also cause a decrease of the stiffnessof the ascending branch of the N –εm relationship, which is morepronounced forloads applied during the initial drying phase, whenimportant superficial tensile stresses occur due to differentialshrinkage.

The numerical results also showed that self-induced stressesare responsible for important stiffness reductions in slabs sub-

jected to increasing bending moments. Furthermore, the flexu-ral tensile strength of slabs is significantly affected by differential

shrinkage stresses. For thatreason, the calculationof cracking loadsbased on the flexural tensile strength (an hypothesis admitted by


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design codes such as the Eurocode 2) may frequently lead to over-estimations.

Overall plausibility of the obtained results provides good ex-pectancies in regards to the applicability of numerical frameworkslike the one proposed here. Upon further validation, this kind of numerical tools has a great potential for analysing the service-lifeperformance of real RC structures. An important field of applica-tion concerns to the evaluation of reinforcement needs in order tocontrol shrinkage-induced cracking.

Acknowledgements

Funding provided by thePortuguese Foundation forScience andTechnology to theResearchUnitsISISEand LABEST, as well as tothefirst and second authors through the Ph.D. grants SFRH/BD/13137/2003 and SFRH/BD/29125/2006, and to the research projectsPTDC/ECM/68430/2006 and PTDC/ECM/099250/2008, are grate-fully acknowledged.

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