HAL Id: hal-01421765https://hal.archives-ouvertes.fr/hal-01421765

Submitted on 3 Feb 2017

HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, estdestinée au dépôt et à la diffusion de documentsscientifiques de niveau recherche, publiés ou non,émanant des établissements d’enseignement et derecherche français ou étrangers, des laboratoirespublics ou privés.

Copyright

Modelling drying and loading effect in structuralconcrete repair

Laurent Molez, Yves Berthaud, Benoît Bissonnette, Denis Beaupré

To cite this version:Laurent Molez, Yves Berthaud, Benoît Bissonnette, Denis Beaupré. Modelling drying and loadingeffect in structural concrete repair. Computational Modelling of Concrete Structures, Mar 2003, St.Johann Im Pongau, Austria. �hal-01421765�

Modelling drying and loading effect in structural concrete repair

Laurent Molez & Yves BerthaudLaboratoire de Mécanique et Technologie, ENS Cachan / CNRS / Université Paris 6, France

Benoît Bissonnette & Denis BeaupréCentre de Recherche sur les Infrastructures en Béton, Université Laval, Canada

Drying shrinkage may be a significant cause of deterioration of thin concrete repairs. Shrinkage induced stressescan be partially relieved by tensile creep. Hydration effects and materials properties may have a significant in-fluence on concrete repair behaviour. In this paper, a numerical tool that takes into account these differentphenomena is presented. The aim of this numerical model is to obtain an analysis tool to complete experi-mental tests. Experimental results obtained on small lab specimens permit to calibrate numerical coefficients.Numerical results on larger specimens show a good agreement with experimental data.

1 INTRODUCTION

Second half of twentieth century is marked by an in-tense activity of construction of concrete infrastruc-tures. These structures have a limited lifespan, andseveral of them require repairs today. Thus the field ofconcrete repairs knows, since ten years, a significantrise in the sector of construction. For owners, achiev-ing durable repairs is of primary importance becausethe costs involved are generally very high. Data pre-sented at table 1 give an idea of the extent of work inthe field of repairs. Experience shows, however, thatdesign of durable concrete repairs can be as complexas the design of new structures because each damagedstructure imposes its own set of conditions (Emmonsand Vaysburd 1994)

In this study, concrete repair is defined as a thinconcrete layer (' 50 to 150 mm) cast on an exist-ing structure to replace the deteriorated part close tothe surface. These two concrete layers have differ-

Table 1: Value of construction work in some countriesFrance1 UK2 Canada3 USA4

billion of�

£ $ CAD $ USBuilding new 47,3 551

repairs 44 294Nonbuilding new 17 103

repairs 9,5 67All repairs 14,9 161 Ministère de l’Équipement (2001)2 Cusson (2001)3 Emberson and Mays (1996)4 US department of commerce (2000)

Dessication of the cement paste

Temperature change

External loads

Shrinkage strain

Thermal strain

Mechanical strain

Restriction to the free strain - due to end conditions

- due to a non-uniform strain over the section

STRESSRelaxation due to the

viscous behaviour of the concrete

Cracking in the overstressed

locations

Figure 1: Schematical illustration of the stress build-up in repairs

ent mechanical and physical characteristics. The oldone is quasi-stable with time. The new one changewith time (hydration, drying, shrinkage, creep...). Inthe figure 1, Saucier et al. (1997) resume the effectof hygral, thermal and mechanical change. Shrinkageis identified to be the most severe parameter (Saucieret al. 1991; Bissonnette 1997; Laurence 2001). In theother hand, tensile creep can partly relax the inducedstrains (Bissonnette 1997). The cracking of repair ma-terials can cause the most serious deterioration pro-cesses leading to repair failures, since the aggressiveagents can then penetrate through these cracks.

We first present the numerical model developed in

this research study. Hydration process is not mod-elled, but its effects are taken into account in theevolution of physical and mechanical characteristics.Endogenous shrinkage is considered. The drying ismodelled by a non-linear diffusion equation, and dry-ing shrinkage is assumed to be proportional to rela-tive humidity changes. Mechanical behaviour is intro-duced with respect of basic and drying creeps. A non-linear continuum damage model is applied to con-crete rupture. In second time, numerical results arecompared to experimental ones from previous study(Molez et al. 2001; Molez 2003).

2 NUMERICAL MODEL2.1 Hydration effectsHydration mechanisms are not modelled, but their ef-fects have to be taken into account. Degree of hydra-tion ξ can be evaluated from the experimental evolu-tion of compressive strength fc(t) (Ulm 1999):

ξ(t) = ξ0 + (1− ξ0)fc(t)

fc(∞)(1)

where ξ0 is the percolation threshold as defined byAcker (1987) and fc(∞) is the compressive strengthat an infinite time.

Thereafter, the evolution of strength will be ex-pressed according to the following equation:

R(t) = R(∞)tm1

m2 + tm1

(2)

where R(∞) is the strength at an infinite time, andm1, m2 parameters determined from experimental re-sults (figure 2).

0

10

20

30

40

50

0 5 10 15 20 25 30

BO

BAP

BO

Time (d)

Com

pres

sive

str

engt

h -

f c (

MP

a)

Figure 2: Experimental (dots) and calculated (lines) evolution ofcompressive strength for an ordinary concrete (BO) and a self-levelling concrete (BAP).

2.2 Thermal effectsWe assumed that ambient temperature is constant.The thickness of new concrete layer is small thentemperature growth due to chemical reactions is low.Consequently, in all this study, thermal effects are ne-glected.

2.3 Drying modelMass conservation equation, expressed according tothe mass water content w, is:

∂w

∂t= −div(Jv + Jl) +

∂wad

∂t(3)

where Jv is the flux of water in vapour form and Jl

the flux of water in liquid form. The derivative termof wad represent the variation of water mass contentdue to self-desiccation.

Using Fick law and ideal gas law, vapour flux canexpressed according to the relative humidity hr. Sim-ilarly, using Darcy law and Kevin law, the liquidflux can be written in terms of hr. In additions, ex-perimental results (Baroghel-Bouny 1994; Therrien2001) show that desorption isotherms are linear (i.e.the slope of the curve w = w(hr) is constant) for0.4 < hr < 0.95. Then we obtain the classical diffu-sion equation:

∂hr

∂t= div(D(hr)gradhr) +

∂had

∂t(4)

Thereafter, we will neglect the variation of relativehumidity ∂had/∂t due to hydration reactions. How-ever associated shrinkage (part of endogenous shrink-age) is introduced further.

The macroscopic diffusion coefficient depend onhr. When hr is important, humidity transfers takesplace in water of capillary spaces. In opposition, whenrelative humidity is low, transfer is of vapour form andsurface diffusion occurs in areas of hindered adsorp-tion. According to theses observations, we choose thefollowing expression of D(hr) (Bažant 1988) :

D(hr) = D0(t)

a+1− a

1 +(

1−hr1−hc

)n

(5)

where a, n and hc are adjustable parameters specificto material. hc would be the humidity threshold wheretransfer change from first phenomenon to the second.Those parameters can be identified with experimentaldata for weight loss (figure 3).

Structure of the porous network evolves during hy-dration. Some results (Therrien 2001) show that diffu-sion coefficient is dependent on maturity of concrete.However, it is difficult to obtain an experimental rela-tion D(ξ). So, we fit the evolution of D to the inverse

of the evolution of strength.

D0(t) = D0(t∞)m2 + tm1

tm1

(6)

Boundary conditions of convective type are im-posed at the surface Γ in contact with air at relativehumidity ha

r . Convective flux is then written:

JΓ = β(hΓr − har) (7)

0

20

40

60

80

100

120

0 50 100 150 200

modelexperimental 1experimental 2

C

Time (d)

Wei

ght l

oss

(g)

Figure 3: Comparison of experimental and calculated weightloss.

2.4 ShrinkageShrinkage of concrete is due to chemical reactionsand drying of porous media. Thus, an endogenousshrinkage and a drying shrinkage can be identified.

In numerical simulations, experimental results ofendogenous shrinkage are directly used. At each step,a homogeneous deformation is imposed in repair con-crete layer.

Drying shrinkage is obtained from variation of rel-ative humidity through a coefficient of hydrous dilata-tion:

εds = αh(t)hr1 (8)

This relation does not reflect micro-mechanisms ofshrinkage (capillary depression, Gibbs-Bangham the-ory and disjoining pressure theory), but a theoreticalstudy (Lassabatere 1994) has shown that it gives a notso bad approximation.

According to concrete structuration, αh(t) shoulddepend on the maturity of material. Some experi-mental results (Therrien 2001) show this dependence.However, it is difficult to obtain an experimental evo-lution law. Thus, αh is considered constant and deter-mined for each concrete (figure 4).

-700

-600

-500

-400

-300

-200

-100

0

0 50 100 150 200 250 300 350

model - 3d curedmodel - 28d curedexperimental 1 - 3d curedexperimental 2 - 3d cured

C

Time (d)

Tot

al s

hrin

kage

(µm

/m)

Figure 4: Evolution of total shrinkage. Comparison of experi-ments and numerical simulation.

2.5 CreepCreep of concrete can be separated in two parts: basiccreep and drying creep. The origins of compressive ortensile creep are discussed in Bažant (1988), Bisson-nette (1997), or Molez (2003).

Basic creep:Basic creep can be written according to viscoelas-

ticity theory. A compliance function J is defined by:

ε(t) = J(t0, t)σ0 (9)

where t0 is the age at loading σ0, and t− t0 the loadduration.

For a stresses history, creep strain can be written:

ε(t) = ε(t0) +∫ t

t0

BJ(τ, t0)σ(τ)dτ (10)

where B is the fourth order tensor of Poisson’s terms.Adopting a Dirichlet series expression for compli-

ance function (Bažant 1988):

J(t0, t) =1

C0(t0)+

n∑

µ=1

1

Cµ(t0)

(

1− exp(−t− t0τµ

)

) (11)

makes it possible to free from storage of stresses his-tory, and strain can be calculated from the precedingstep. Previous expression can be explained in termsKelvin chain (figure 5) with ageing modulus Eµ =

Cµ− τµCµ and constant dashpot viscosity ηµ = τµCµ.Strain evolution is calculated step by step. For step

[tn, tn+1], one can write:

∆ε = ε(tn+1)− ε(tn)

E�η�

E�

η�

E�

η

E

η�

E�µ

µσ σ

Figure 5: Kelvin chain equivalent to the proposed model

If Cµ and σ = ∆σ/δt are constant during this step,integral can be easily calculated (see Molez (2003) fordetailed calculus):

∆ε =

[

1

C0(tn + ∆t/2 )+

n∑

µ=1

1−(

1− exp(−∆tτµ))

τµ/∆t

Cµ(tn + ∆t/2 )

B∆σ+

n∑

µ=1

(

1− exp(−∆t

τµ)

)

∆ε∗µ(tn)

(12)with the recursive expression:

∆ε∗µ(tn) =1− exp(−∆t

τµ)τµ/∆t

Cµ(tn − ∆t/2 )B∆σ+

∆ε∗µ(tn−1) exp(−∆t

τµ)

(13)

Thus the increment of deformation is calculable ac-cording to the increment of load and the history ofloading defined in previous step. The increment of ba-sic creep is obtained by subtracting the increment ofinstantaneous strain from previous equation. We ob-tain:

∆εbc=

n∑

µ=1

1−(

1− exp(−∆tτµ))

τµ/∆t

Cµ(tn + ∆t/2 )

B∆σ+

∆εhist

(14)Characteristic relaxation times τµ are fixed a pri-

ori according to a logarithmic series (Bažant 1988;Granger 1997):

τµ = τ1 10µ−1 (15)

and Cµ parameters are obtained by the method of leastsquares. An example of results is given in figure 6.

Drying creep:Creep experimental tests show that an extra creep

can be measured when test specimen is allowed todry. This phenomenon is called drying creep. Differ-ent interpretations are proposed. Bažant and Chern(1985) suggest a mechanism of stress-induced shrink-age. On the contrary, drying creep can be interpretedas a drying-induced creep (Bažant 1988). These dif-ferent mechanisms can be written mathematically as

�

� �

� � �

� � �

��� �� �� �� � � �� ��� �

� ��� � � � �� ��� ��� � �� ��� ��� � �� � ��� ��� � �

� ���� � ��!���" #$%$% &' ()

*�+ ,.-0/ 1 2

� � 354 6 7 6

Figure 6: Experimental (dots) and calculated (lines) basic creepof an ordinary concrete

a single equation:

εdc = κfsBσ |hr| (16)

Coefficient κfs is determined from basic and dryingcreep test. We can notice that the sign of (εdc)ij de-pends only on the sign of (σ)ij . So in uniaxial trac-tion, we notice an increase of drying creep whether itis in moistening or drying conditions.

Then, total creep can be obtained by adding basicand drying creep (figure 7).

8

9 8

: 8 8

: 9 8

8�: 8; 8�< 8= 8 9 8�> 8�? 8

? @�A B C D E? @�A F�G @ E: @�A F�G @ E; H @�A F�G @ EI JKLMN MLLOKKJP QRSRS TU VW

X�Y Z\[0] ^ _

9 8 `5a b c b

: 8 8 `5a b c b

Figure 7: Experimental (dots) and calculated (lines) total creep(basic + drying creep) of an ordinary concrete

2.6 Mechanical damageTotal strains εtotal are supposed to be the sum of in-stantaneous strains εi , shrinkage strains εs and creepstrains εc:

εtotal = εi + εs + εc (17)

The stress-strain relation of damaged material iswritten:

σ = Cd (εtotal

− εs − εc) = Cd εi

(18)

In the present analysis, we use an isotropic scalardamage model (Mazars 1984). In this model, the me-chanical effect of progressive micro-cracking due toexternal loads is described by a single internal vari-able which degrades the Young’s modulus of the ma-terial. The constitutive relations are:

σij = (1− d)Λijkl(εi)kl (19)

where σij and (εi)kl are the components of the stressand strain tensors respectively (i, j, k, l ∈ [1, 3]), Λijkl

are the initial stiffness moduli, and d is the damagevariable. The material is initially isotropic, with Eand ν the initial Young’s modulus and Poisson’s ra-tio respectively. Damage is a function of the positivestrains which means that it is mainly due to microcracks opening in tension mode. In order to avoidill-posedness due to strain softening, the mechani-cal model has to be enriched with an internal length(Pijaudier-Cabot and Bažant 1987).

3 NUMERICAL EXAMPLESIn this section, we present some results of numericalsimulations of concrete repairs. Test configuration isthe same one as that used in previous experimentalstudy (Molez et al. 2001; Molez 2003).

1000

70 60

2 n˚15

n˚1016 @ 120 mm

Repair

2050

Figure 8: Test specimens

Figure 8 shows the geometry of the investigatedconcrete repair. A 2D analysis of one half of thebeam, using four-noded plane stress elements, wasperformed (figure 9).

Examples of calculated humidity distribution, cor-responding deformation and induced damage areshown in figures 12, 13 and 14.

Numerical results can be compared to experimen-tal ones. During experimental tests, crack width anddeflection has been measured for different configura-tions: drying only, drying and loading at 33% of ulti-mate load, and flexural testing. Those results are com-pare in figures 10 and 11.

These numerical results show a good agreementwith experimental ones.

Repair - damage model

- creep model

- drying model

- aging

Old concrete - damage model

- drying model

- non-aging

Old concrete - elastic

- drying model

- non-aging

Steel reinforcement - ideal elastic-plastic law

Figure 9: FE mesh

�

� �

� � �

� � �

� � � � � � � � � � � �

� � � � � � � �� � � � � � �

� � ����� � �

� ���� ��� !" µ

#$

Figure 10: Comparison of experimental (dots) and calculated(lines) average crack width for a repaired beam submitted to dry-ing and loading.

4 CONCLUSIONSTo be able to make a complex analysis (to com-plete experimental study) of the behaviour of struc-tural concrete repairs, the main phenomena acting inthose structures have been integrated in a finite ele-ments code.

So, we can take into account the evolution of me-chanical properties during hydration process and in-duced volume change (endogenous shrinkage). Thedrying of the material is modelled by a non linearlaw. Furthermore, the diffusion law is coupled withthe evolution of hydration process by a diffusion coef-ficient which depends on the maturity of the material.Drying shrinkage is connected with the variation ofhumidity by a proportionality law that is experimen-tally verified for relative humidity included between0.5 and 0.95. Viscoelastic behaviour is modelled bya Kelvin chain. The effects of hydration are takeninto account with variable coefficients of the Kelvinchain. The influence of drying on creep is introducedby means of an extra term proportional to variationof relative humidity and imposed stress. A damagemodel is used to reproduce the mechanical effect ofprogressive micro-cracking of concrete.

The identification phase shows that the numericaltool well reproduces the behaviour of small lab spec-imens, and also bigger structures.

� ��

���

�� �

���

�� �

���

�� ������ �

� �� ��

��� �

�� ���� ������� ���� ��

�� � �� �� �

���� �"!#$%'&)(*)*,+

Figure11:

Flexuralbehaviour:

experimental

(dots)and

calcu-lated

(lines)results.

0.50

0.52

0.55

0.57

0.60

0.62

0.65

0.68

0.70

0.73

0.75

0.77

0.80

0.82

0.85

0.88

0.90

0.93

0.95

0.98

1.0

Humidity

Figure12:

Distribution

ofhum

idity:initial

stateand

after497

daysof

drying

Damage

0.0

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.0

Figure13:Induced

damage

distribution:finalstate

EPXX

-1.00E-03

-7.50E-04

-5.00E-04

-2.50E-04

-5.42E-20

2.50E-04

5.00E-04

7.50E-04

1.00E-03

1.25E-03

1.50E-03

1.75E-03

2.00E-03

2.25E-03

2.50E-03

2.75E-03

3.00E-03

3.25E-03

3.50E-03

3.75E-03

4.00E-03

Figure14:Induced

deformation

distribution:finalstate.

REFERENCESAcker, P. (1987). The mechanical behavior of concrete : con-

tributions of the physico-chemical approach. Thèse dedoctorat, École nationale des ponts et chaussées.

Baroghel-Bouny, V. (1994). Caractérisation des pâtes de ci-ment et des bétons - Méthodes, analyse, interprétations.Thèse de doctorat, École nationale des ponts et chausées.

Bažant, Z. P. (Ed.) (1988). Mathematical modeling of creepand shrinkage of concrete.

Bažant, Z. P. and J. C. Chern (1985). Concrete creep at vari-able humidity: constitutive law and mechanism. Materi-als and structures 18(103).

Bissonnette, B. (1997). Le fluage en traction: un aspect im-portant de la problématique des réparations minces enbéton. Ph. D. thesis, Université Laval, Québec, Canada.

Cusson, D. (2001). La réparation durable des structures debéton passe par l’éducation et le transfert de la technolo-gie. Innovation en construction 6(3). IRC, CNRC-NRC.

Emberson, N. K. and G. C. Mays (1996). Significance ofproperty mismatch in the patch repair of structural con-crete. part 1,2,3. Magazine of concrete research 48(174).

Emmons, P. H. and A. M. Vaysburd (1994). Factor affectingthe durability of concrete repair: the contractor’s view-point. Construction and building materials 8(1).

Granger, L. (1997). Calcul des ouvrages de construction,Chapter 14 : Ouvrages en béton soumis aux effets du flu-age, pp. 614–654. Hermès. sous la direction de M. Prat.

Lassabatere, T. (1994). Couplage hydromécanique en milieuporeux non saturé avec changement de phase : applica-tion au retrait de dessiccation. Thèse de doctorat, Ecolenationale des ponts et chaussée, Paris, France.

Laurence, O. (2001). La fissuration due au retrait restreintdans les réparations minces en béton: apports combinésde l’expérimentation et de la modélisation. Thèse dedoctorat, Université Laval, Canada, École nationale desponts et chaussées, France.

Mazars, J. (1984). Application de la mécanique del’endommagement au comportement non linéaire et à larupture de béton de structure. Thèse de doctorat, Univer-sité Paris VI, France.

Ministère de l’Équipement (2001). Grands agrégatséconomiques de la construction. France.

Molez, L. (2003). Thèse de doctorat, École normalesupérieure de Cachan, France et Université Laval(Québec), Canada.

Molez, L., Y. Berthaud, B. Bissonnette, and D. Beaupré(2001, 28 may - 1 june). Structural concrete repair be-havior: coupled effect of drying and loading. In de Borst,Mazars, Pijaudier-Cabot, and van Mier (Eds.), Fracturemechanics of concrete structures, pp. 263–270. FraM-CoS: A. A. Balkema Publishers.

Pijaudier-Cabot, G. and Z. P. Bažant (1987). Nonlocal dam-age theory. J. of Engrg. Mech. 113.

Saucier, F., F. Claireaux, D. Cusson, and M. Pigeon (1997).The challenge of numerical modeling of strains andstresses in concrete repairs. Cement and concrete re-search 27(8).

Saucier, F., M. Pigeon, and M. Cameron (1991). Air-voidstability, part v: temperature, general analysisand perfor-mance index. ACI materials journal 88(1).

Therrien, J. (2001). Étude de l’évolution des propriétés detransfert hygrométrique d’un mortier au jeune âge.

Ulm, F.-J. (1999). Couplages thermochémomécanique dansles bétons :un premier bilan. Mémoire d’habilitation àdiriger des recherches, Université de Marne la Vallée.Rapport provisoire.

US department of commerce (2000). 1997 economic cen-sus, construction. Economics and statistics administra-tion, US Census Bureau.