Modeling basic creep of concrete since setting time

Accepted Manuscript Modeling basic creep of concrete since setting time B. Delsaute, J.-M. Torrenti, S. Staquet PII:

S0958-9465(16)30260-8

DOI:

10.1016/j.cemconcomp.2017.07.023

Reference:

CECO 2873

To appear in:

Cement and Concrete Composites

Received Date: 14 June 2016 Revised Date:

19 July 2017

Accepted Date: 21 July 2017

Please cite this article as: B. Delsaute, J.-M. Torrenti, S. Staquet, Modeling basic creep of concrete since setting time, Cement and Concrete Composites (2017), doi: 10.1016/j.cemconcomp.2017.07.023. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

Modeling basic creep of concrete since setting time

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B. Delsaute1,2, J.-M. Torrenti2, S. Staquet1

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1 Université Libre de Bruxelles (ULB), BATir, LGC, Avenue F.D. Roosevelt 50 CP194/04, 1050

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Bruxelles, Belgium.

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2 Université Paris Est, IFSTTAR, Boulevard Newton, 77447 Marne la Vallée Cedex 2, France.

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Corresponding author: Brice Delsaute ([email protected])

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Abstract

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Modeling the early age evolution of concrete properties is necessary to predict the early age behaviour

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of structures. In case of restrained shrinkage or application of prestress load [1], creep plays an

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important role in the determination of the effective stress. The difficulty lies in the fact that the

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modeling of creep must be based on experimental data at early age and this data must be obtained

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automatically because the hardening process of the concrete takes place rapidly during the first hours

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and also the first days. This paper presents a new methodology to model basic creep in compression

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since setting. Two kinds of tests are used: classical loadings and repeated minute-scale-duration

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loadings. The classical test is used to characterize the creep function for one age at loading and the

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repeated minute-scale-duration loadings test is used to define two ageing factors for the creep

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function. A new model based on the physical mechanisms and the two ageing factors is presented. A

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comparison with the Model Code 2010 is done and an advanced way to consider ageing with the

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Model Code 2010 is presented.

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Keywords: Concrete, very early age, basic creep, modeling

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Introduction

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Nowadays, the construction phases of modern concrete structures (including high-rise buildings,

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bridge piers, and storage tanks) are challenging due to their high performance requirements.

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ACCEPTED MANUSCRIPT Consequently, for the design of concrete structures it is important to model accurately the early age

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behavior of concrete, which influences the whole service life. Even though the mechanical behavior

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of hardened concrete can usually be accurately estimated, it is not always the case for the early age

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behavior of concrete, when the mechanical properties change rapidly in function of the advancement

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of the hydration reaction. Among all the usual parameters (including measures of strengths and E-

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modulus) needed for the design of the concrete structures, creep and relaxation must also be taken into

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consideration.

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For general concrete structures built in several phases, the evolution of the restrained strains is similar

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and is composed of two periods: a heating period followed by a cooling one (Figure 1a). The heating

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period begins just after the initial setting [2] when the mechanical properties of concrete start to

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develop. During and after the setting, a large amount of heat is produced by the hydration reaction

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which leads to an increase of the temperature inside the concrete element and thus to an increase of

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the thermal strain. In the meantime, autogenous strain starts to develop. No systematic tendency can

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be given for the autogenous strain, because during the heating period, autogenous deformation results

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in swelling or shrinkage according to the mixtures proportions and content [3,4]. However thermal

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strains are generally higher than autogenous strain (especially for massive structures) and thus a

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general swelling of the concrete occurs during the heating period. The cooling period starts when heat

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of hydration decreases rapidly or, depending on the massivity of the structure, when the formwork is

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removed. During this period, both autogenous and thermal strains decrease. As a result of the

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restriction of the strains of concrete, stresses are induced (Figure 1b). During the heating period,

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concrete element is in compression and inversely during the cooling period the concrete is submitted

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to tension. However, temperatures are nonuniform in most cases. It is possible to have both tension

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and compression during cooling. Contraction of cooling surfaces can induce compression in the

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interior, for example.

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Creep and relaxation often play a positive role for the design of concrete structures at early age.

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However their impact at very early age as part of restrained deformation could be negative as

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highlighted on Figure 1b. In a general view, no consideration of creep/relaxation leads to a global

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ACCEPTED MANUSCRIPT overestimate of the stress. Thus creep/relaxation seems to play a general positive role for the design of

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concrete structures at early age. However, at very early age, the creep/relaxation amplitude is very

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significant and reduces strongly the compressive stresses caused by thermal expansion. Thereafter,

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during the cooling period, stresses may switch rapidly to tension. During this period, an

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underestimation of the creep/relaxation phenomenon leads to an underestimation of the tensile

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stresses which can cause cracking in the concrete structure. Hence it is important to model correctly

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the basic creep since final setting time.

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State of the art

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During this last decade, different creep models have been developed for early age concrete. These

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models are generally based on the decomposition of the total strain

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a sum of the elastic strain

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as expressed in Equation 1. The effects of drying are significant (see e.g. [5]), but are not considered

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in the following state of the art. For a constant uniaxial stress , the uniaxial mechanical strain can be

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defined as in Equation 2 where t is the age of the concrete, t’ the age of the concrete at loading, J(t,t’)

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is the compliance function, E(t’) is the elastic modulus and C(t,t’) is the specific creep. Generally

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authors use the creep deformation, the creep compliance, the specific creep or the creep coefficient

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of a loaded concrete sample in

, the autogenous strain

and the thermal strain

TE D

, the creep strain

to model creep. The link between these three last parameters is given in Equation 3.

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1 2

3

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As concrete creep at early age is still not well understood, mathematical expressions are used to fit

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experimental data. Several models developed for hardened concrete are given in [6]. Models are often

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composed of two terms: an amplitude term and a kinetic term. For consideration of ageing both

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parameters can be expressed in function of the age at loading [6–12], the advancement degree of

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reaction at loading [13,14] or the equivalent age at loading [15–17]. The advancement degree of

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reaction

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the heat release

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based on the Arrhenius equation in order to take into account the main temperature effect on the

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hydration process. Equation 5 defines the equivalent time which is function of the age of the material

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t, the evolution of the temperature T (°C), a reference temperature

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gas constant R (=8.314 J/mol/K) and the apparent activation energy (material parameter in J/mol).

is defined as the relative amount of hydrated cement and can be computed as a function of is

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(Equation 4). The equivalent time

(generally 20°C), the universal

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and the heat release at an infinite time

4

5

Ageing of the creep function can be linked to other parameters such as the mechanical strength

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[18,19] and the effect of the stress level could be included [19].. For these models no real physical

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mechanism is associated to the amplitude or kinetic terms. With only one amplitude term and one

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kinetic term, it is assumed in these models that only one mechanism is responsible of creep or that all

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mechanisms evolve with same kinetics but different amplitudes.

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Among many previous studies like those reported in [12–15,17,19–22], several theories were

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developed to clarify mechanisms related to creep behavior. However each theory alone does not allow

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explaining all experimental observations. Globally each theory can be linked to two mechanisms:

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direct mechanisms linked to the cement paste and responsible of the highest part of the creep

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amplitude and indirect mechanisms linked to the heterogeneity of the concrete. Direct mechanisms are

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related to the water mobility and to the solidification of the material and can be separated in short and

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long terms phenomena [23–26]. The short term phenomenon is reversible with a small characteristic

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time of about 10 days, is linked to a stress-induced water movement towards the largest diameter

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pores and to the solidification [26,27] of the material, and occurs under increasing volume for uniaxial

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compression. The long term phenomenon is irreversible with a high characteristic time and related to

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viscous flow in the hydrates and occurs under almost constant volume. The creep rate of this long

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term phenomenon evolves as a power function

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according to [31], between -0.72 and -0.69 according to results of [32] on concrete and an exponent

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between -0.86 and -0.6 on cement paste according to results of [22]. Nanoindentation tests were

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carried out on C-S-H by Vandamme, et al [33]. It was shown that C-S-H exhibits a logarithmic creep

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which is in agreement with results obtained on concrete [18,34]. Vandamme [35] compared also this

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logarithmic behavior with other heterogeneous and porous materials with porosity including several

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orders of magnitude (soils and wood). For these non-ageing materials, a logarithmic long-term creep

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was also observed. It can then be assumed that this long term creep is not linked to a hydration

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process or any chemical specificity of the C-S-H. The indirect mechanisms are due to micro-cracks

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which occur progressively in the cement paste and at the interface between cement paste and

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inclusions. Their presence can cause a redistribution of the stresses in the material [36].

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On basis of the results coming from the literature presented above, it could be considered that creep is

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separated in three functions which correspond to the short term creep, the long term creep and a term

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related to the formation of micro-cracks. In the model B3 of Bazant [31,37], both short and long term

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creep are considered separately as expressed in Equation 6 and 7. Complete details about the different

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parameters of the model can be found in the references [31,37]. The short term creep is linked to the

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solidification theory [27,38] and corresponds to the terms with parameter

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The long term creep is linked to the microprestress theory [26,39] and corresponds to the term with

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parameter

between -1 and -0.9

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[28–30] with an exponent

and

in the equations.

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in the equation.

6

7

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Another way to model the viscoelastic behaviour of concrete is the use of rheological model where

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the creep behavior is associated to spring and dashpot in series or in parallel. There is an infinite

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number of associations of springs and dashpots, several examples are given in [40]. Bazant advises to

ACCEPTED MANUSCRIPT use Kelvin Voigt chains in series for creep [41] and Maxwell chains in parallel for relaxation [42]. To

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consider ageing creep, spring and dashpot parameters have to evolve according to the age of the

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concrete. De Schutter [13] linked the creep coefficient to a Kelvin-Voigt chain by considering that the

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spring and the dashpot evolve according to the amplitude term of a model depending on the

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advancement degree of hydration. This model has the advantage of being very simple because it uses

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only two parameters. However this model is limited for early age study and for a limited duration of

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loading. To extend the use of the model of De Schutter, Benboudjema and Torrenti [43] generalized

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this model by means of several Kelvin-Voigt chains in series with different springs and dashpots. The

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ageing of each element of each Kelvin-Voigt chain depends on the evolution of one parameter which

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is function of the advancement degree of hydration. In case of complex loadings with partial or total

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unloadings, Briffaut [21] added a dashpot in series to the three Kelvin-Voigt chains where the creep

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strains predicted by the additional dashpot are totally irreversible. Then the Kelvin-Voigt chains are

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linked to the short term creep and the dashpot is linked to the long term creep. A more general

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rheological scheme of the concrete behavior is given in Figure 2 [26,44] where elastic, creep (short

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term and long term), shrinkage and thermal strain are represented on a same rheological model.

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Another way to consider ageing by using rheological model is done by Hermerschmidt [45] who uses

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4 Maxwell units and 1 single spring in parallel to model tensile creep test with several histories of

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loading. The influence of the hardening process on the viscoelastic behavior is carried out by

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increasing the stiffness of the single spring and the viscosities of the dashpots according to the age of

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the concrete.

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Therefore several approaches were proposed in the literature and in the design codes to model creep

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behavior for early age or for hardened concrete.

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Research significance

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This paper is a continuation of the previous work by Delsaute, et al. [46] and reports on a new

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strategy for the modeling of the basic creep since setting time. This strategy uses as a basis the results

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from two kinds of test: classical creep test with permanent load applied during one week or more and

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ACCEPTED MANUSCRIPT repeated minute-scale-duration loadings. With the consideration of the existing models and the

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physical mechanisms presented above, new models are developed in order to accurately reproduce the

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basic creep since the final setting time. For that purpose, experimental results on an ordinary concrete

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studied in isothermal conditions at 20°C and in sealed conditions reported in [46] are used. A

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comparison between the experimental results and the Model Code 2010 is carried out in order to point

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out a certain deficiency in application to very early age and to suggest a modification which may

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eliminate this deficiency. This research is original since there is no model for the basic creep which is

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simultaneously based on experimental results at very early age and considers the different physical

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mechanisms occurring during the hydration process.

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Experimental details

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Materials and mixtures

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The results used in the present paper were performed on an ordinary concrete for which mix

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proportions are given in Table 1. All materials come from the same batch of production. An ordinary

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Portland cement of type CEMI 52.5 N was used. Its chemical composition is given in [46]. Siliceous

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sand and gravel coming from Sandrancourt (France) were used. Sand and gravel were dried. The

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effective water-to-cement ratio is 0.45.

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Components

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Sand (Sandrancourt 0/4) (kg /m3)

739

Gravel (Sandrancourt 6.3/20) (kg /m3)

1072

Added water (kg /m3)

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Wadded/C

0.54

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CEMI 52.5 N – SR 3 CE PM-CP2 NF (kg /m3)

ACCEPTED MANUSCRIPT Weff/C

0.45

Table 1 – Mixture proportions and materials properties of the concrete

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Different mechanical properties have already been characterized at IFSTTAR and ULB. The

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evolution of the tensile and compressive strengths, the heat release, the Young’s modulus and the time

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of setting are presented in [47,48] and creep properties are presented in [46].

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Assessment of the viscoelastic properties

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The devices used for the characterisation of the creep properties are a TSTM, the BTJASPE and 16

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compressive creep rigs [46–49]. All tests were performed in a climatic chamber with a temperature of

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20 ± 1 °C and a relative humidity of 50 ± 5 %. Special attention was taken in order to assure that no

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loss of water occurred during the test. Specimens tested on compressive creep rigs were weighted

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before and after the test. With two layers of self-adhesive aluminium sheet, no mass change was

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observed for typical duration of loading of two weeks. With the TSTM and the BTJASPE, it is

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assumed that no drying occurred due to the continuous presence of the mould during the whole test

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duration.

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Compressive creep rigs are used to apply permanent loading during at least 6 days with a

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stress/strength ratio of 40% at the age at loading. For each age, tests were carried out on 3 samples to

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demonstrate the good repeatability of the results. For the monitoring of the first term of the short term

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creep, a new test so-called repeated minute-scale-duration loadings test was developed. Just after the

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setting, every 30 minutes a load corresponding to 20% of the compressive strength is applied for 5

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minutes. During the 5-minutes load duration, the specific creep is computed from raw force and

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displacement measurements after having removed the free strain from the dummy specimen (thermal

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and autogenous strain). Complete details about the protocol of loading and the devices can be found

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on the reference [46].

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Previous results and observations

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In a previous research work [46], results on compressive creep are reported for ages at loading of 15 –

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20 – 24 – 40 and 72h00 (Figure 3a). It was observed that: -

The amplitude of the basic creep coefficient is strongly influenced by ageing during the first

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hours after loading. Earlier is the application of the loading and higher is the basic creep

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coefficient. On the contrary, no ageing effect is detected for the kinetic of the basic creep

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coefficient (Figure 3b).

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-

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After the first hours of loading, no significant effect of the ageing is observed on the kinetic and the amplitude of the basic creep coefficient. However a general trend shows that, after

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few hours of loading, ageing increases the amplitude of the basic creep coefficient as it is

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shown in Figure 3c. All basic creep coefficients are set to zero after two hours of loading to

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highlight this observation.

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Results from one repeated minute-scale-duration loadings test are presented in Figure 4. The specific

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creep curves coming from each minute-scale-duration loading are superimposed according to the age

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after loading in Figure 4a. Two parameters were studied: the evolution of the kinetic and the

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amplitude according to the age at loading. In Figure 4b, each creep curves are normalized by their

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value obtained after 5 minutes of loading. Thus each normalized creep curve has a value of 1 after

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duration of loading of 5 minutes. Each normalized creep curves are very close and no ageing effect is

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detected. In Figure 4c, the amplitude of the creep coefficient after 5 minutes of loading (coming from

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the repeated minute-scale-duration loadings test) is compared to the amplitude of the creep coefficient

201

after 2 hours of loading (coming from the permanent loading test). Both amplitudes follow the same

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evolution according to the age at loading. Thus it was concluded that the first term of the short term

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creep can be defined with repeated minute-scale-duration loadings test and is divided in two terms: a

204

dimensionless kinetic term which is constant and an amplitude term which is function of the age of

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concrete at loading.

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Through a comparison of the experimental results and outcomes from microstructural simulation

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carried out with VCCTL (version 9.5) [46], it is noted that short term creep is divided in two parts:

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-

A first term is linked to the state of the cement paste when the load is applied and more specifically to the volume fraction of capillary pores of the cement paste (diameter between 5

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and 9 µm) and also to the volume fraction of the CSH when the load is applied. A second

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term is linked to the development of the CSH and the theory of solidification of Bazant

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[26,27]. The consolidation creep depends on the evolution of the properties of cement paste.

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Modeling

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Part1: Modeling of the repeated minute-scale-duration loadings test

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Two models were developed to best fit the creep coefficient for each repeated minute-scale-duration

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loadings. A first model was introduced in [50]. The model is given in Equation 8. The model is a

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power law and is inspired from the work of Gutsch [17] and De Schutter [13]. The model has the

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advantage to consider only two parameters: a parameter

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coefficient after 5 minutes of loading and a parameter

220

coefficient.

linked to the amplitude of the creep

linked to the kinetic evolution of the creep

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where

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the Model Code 2010 (Equation 9). The model is also based on an amplitude parameter

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to the amplitude of the creep coefficient after 5 minutes of loading and a parameter

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kinetic evolution of the creep coefficient.

8

. The second model developed is a logarithmic law and is inspired from and linked linked to the

9

ACCEPTED MANUSCRIPT where

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Firstly, each creep curve was fitted with three Kelvin-Voigt chains in order to remove noise from the

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measurement. The value of the fitted curve after 5 minutes of loading corresponds to the amplitude

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parameter for each age of loading. Secondly, the value of the kinetic coefficient is computed with

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method of least squares (by using the fminsearch function in Matlab©). To compare the power and

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logarithmic expression in term of performance, the error is computed by considering the experimental

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and modeling value of the creep coefficient for each cycle at the same time according to Equation 10.

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. The calculation of both parameters was carried out in two steps.

and

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are respectively the number of cycles during the test and the number of measuring points

for each cycle. The error is 0.0020 for the power law and 0.0014 for the logarithmic law. A reduction

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of error by a factor 1.5 is observed. This is coherent with results of long duration which show a

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logarithmic trend for each age of loading since setting time during the first hours of loading. Then,

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only results from the logarithmic law will be considered on the next parts of this paper.

10

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Coefficient of the logarithmic model is given in Figure 5. The amplitude parameter decreases strongly

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with the age at loading and has an evolution inversely proportional to the age at loading as indicated

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in Equation 11. The kinetic parameter is relatively constant as expected with experimental results

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obtained. Then, for this concrete, a constant value of 0.35 is considered for the kinetic parameter

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The evolution of both parameters is given in function of the age at loading and not in function of the

242

advancement degree of reaction as it has been done in [13] due to technical reasons. Calorimetry

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testing in isothermal, semi-adiabatic or adiabatic condition is generally used to define the evolution of

244

the advancement degree of reaction. These technics are very accurate for the early age but not for later

245

ages. Indeed, after several weeks the heat flow of the hydration of the cement paste is very low and is

246

very difficult to assess experimentally. Thus after few weeks, the advancement degree of hydration

247

does not evolve significantly whereas the mechanical properties such as the compressive strength and

248

the elastic modulus continue to evolve significantly [47]. That is the reason why it has been chosen

.

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ACCEPTED MANUSCRIPT 249

that the expression of all parameters is given according to the equivalent time. A value of 35 kJ/mol is

250

used for the activation energy [51]. In Figure 6, modeled and experimental data are compared for

251

several ages at loading. A very good agreement is obtained between both.

where a = 11.26, b = -19.62 and

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Part 2 : Modeling of the permanent loading

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Based on experimental observations explained in section “Previous results and observations”, new

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approaches are used for the modeling of the basic creep function at early age.

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= 1h.

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a. Rheological model with ageing factor

Several Kelvin-Voigt units in series were used. The stiffness E and the viscosity

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retardation

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TE D

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lower than 2 hours depend on the evolution of the amplitude parameter

of the chain with time

which is determined by

means of a minute-scale-duration loadings test. The Figure 7 illustrates the different models proposed

261

before by De Schutter [13], Benboudjema & Torrenti [43] and the one developed in this study.

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In order to optimize the number of Kelvin-Voigt units, the modeling of the creep curves were carried

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out in several steps. First the retardation time

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modeling was carried out with 10 Kelvin-Voigt units in series for which the retardation time

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defined according to Equation 12 which is adapted from the work of Bazant [41] in which

266

retardation time of the first Kelvin-Voigt chain and has a value of 1 minute. The value of the

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coefficient

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retardation time is the value of the maximum duration of loading considered in the modelling. As only

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data of the first 145 hours after loading are considered, the value of the last Kelvin-Voigt unit

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of each chain must be defined. For that, a first is

is the

depends on the value of the characteristic time of the first and last chain. The maximum

ACCEPTED MANUSCRIPT 270 271

corresponds to a value of 145 hours. A same weight is given for each creep curve. Then, the value of is computed by considering

and

and is equal to 2.24.

12

The stiffness of the spring of each unit (expressed in MPa) is imposed to be positive by using

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Equation 13 in which the parameter d is computed by using method of least squares (with the

274

fminsearch function of Matlab©) by considering each creep curve at the same time. Then units having

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a spring with high stiffness are removed because limited creep is produced by those units. Only 4

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Kelvin-Voigt chains are necessary to model the ageing of each creep curve for a period of 145 hours.

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13

Finally a new computation of the four Kelvin-Voigt chains is done by considering the four

280

characteristic times defined just before and by using again the method of least squares as done before.

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Values of the parameters of the 4 Kelvin-Voigt chains are given in Table 2 and the results of the

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modeling are compared to experimental results in Figure 8.

0.34

24.58

0.94

27.33

(h)

(h) 19.32

1.66

145

2.80

Table 2 Parameter values of the basic creep coefficient for 4KV model.

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(h)

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A good agreement is found between the experimental results and the modeling. This way to model

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creep curve has the advantage to be easily implemented in finite element software. For the study of

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loading of short duration, as the case of restrained shrinkage, this way to model creep is convenient.

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However it has also several disadvantages:

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-

The modeling is not able to predict creep for long duration of loading.

ACCEPTED MANUSCRIPT 289

-

290 291

The modeling needs several creep tests of long duration to be calibrated. If only two tests are used to fit the parameters, the model cannot accurately predict results for other ages.

-

The model does not highlight correctly the mechanisms which occur during the loading and does not distinguish between reversible and irreversible creep.

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b. Model with separation between short term and long term creep

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To improve the model, physical mechanisms must be taken into account. Long term creep was

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identified by several authors [25,31,34] as logarithmic. It is particularly highlighted by comparing the

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evolution of the time derivative of the creep function of several creep curves according to the age of

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the concrete (Figure 9). For a given composition, the time derivative of the creep function follows a

298

same power function trend after several days for early age loading or after several weeks for loading

299

at later ages. Similar observations are done with the experimental results of the concrete studied in

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this article for very early age loadings and loading applied at an age of 28 days [52] as shown in

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Figure 10. With very early age loading, long term creep is identified easily, accurately and with a

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short duration of loading. The long term creep corresponds to a power expression and is defined by

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equations 14 and 15.

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=-1.048. The parameter

15

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where

corresponds to the long term creep rate for an

305

age after loading of 1 hour. The amplitude of the long term creep is mainly governed by the exponent

306

parameter . The value of

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Figure 11, the long term creep is removed from the experimental results by using Equation 15 in order

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to isolate the short term creep. For very early age loading, the short term creep evolves strongly

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during the first hours after loading with a logarithmic trend and after approximately ten hours no short

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term creep occurs anymore. For later ages at loading, a logarithmic evolution is observed during the

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first week of loading. Short term creep can then be separated into two components:

AC C

= 9E-6 MPa-1 and

14

is in agreement with observation of Bazant [31] (between -1 and -0.9). In

ACCEPTED MANUSCRIPT 312

-

Initial short term creep,

, which is represented by a logarithmic function associated with

the actual state of the material (capillarity porosity, CSH) when the load is applied. The

314

function is linked to the amplitude term of the minute-scale-duration loadings test which is

315

strongly correlated to the evolution of the largest diameter pores. The function has two

316

materials parameters.

317

-

RI PT

313

Solidification creep,

, which is linked to the solidification of the material and associated to

the quantity of formed CSH during loading and ipso facto to the decrease of the capillarity

319

pores.

SC

318

16

Firstly the initial short term creep is modelled by a logarithmic expression as given in Equation 16

321

where

322

kinetic parameter. Both parameters are function of the concrete composition. The equation used to

323

model the initial short term creep is inspired from the second term of the model B3 [31] as for the

324

modeling of the creep coefficient with the minute-scale-duration loadings test. The modeling of the

325

initial short term creep is compared to experimental results for which the long term creep is removed

326

in Figure 12. For early age loading, a very good agreement is observed during the first hours of

327

loading. For later ages (here loadings at 2 or 3 days), the agreement is good till one week of loading.

328

In Figure 12, it is also shown that the model developed for minute-scale-duration loadings test

329

(Equation 11) is not able to predict alone the creep behaviour for long duration. This can easily be

330

explained by the fact that only the initial state of the material is considered and not the mechanisms

331

which evolve during the hardening process as the solidification of the material and the viscous flow in

332

the hydrates. The results presented here are in good agreement with the observation carried out by

333

Irfan, et al. [53] who show that creep results from repeated minute-scale-duration loadings on cement

334

paste are enough to predict the evolution of the creep function for several days.

335

Secondly, the initial short term creep is removed also from the experimental results in order to

336

identify the solidification part of the creep for each age of loading. In Figure 12, results of this

337

subtraction are given. For early age loading, the solidification term creep is very important and

= 0.08. The parameter

is an amplitude parameter and the parameter

is a

AC C

EP

TE D

= 20 and

M AN U

320

ACCEPTED MANUSCRIPT evolves strongly during the first days of loading. For later ages of loading, the amplitude of this

339

parameter seems very low. In order to compare only the kinetics aspect of the solidification term

340

creep, each solidification creep curve is normalized at a time corresponding to an age after loading of

341

145 hours. In Figure 13, it is observed that the kinetics evolution of the solidification term creep does

342

not depend on the age at loading of the concrete. Results from the loading applied at an age of 72

343

hours are not included in Figure 13 because for later ages corresponding to an high advancement

344

degree of reaction (the advancement degree of reaction for an age of 72 hours has a value of 0.69), the

345

solidification phenomenon is very low and therefore is very difficult to analyse. The kinetic of the

346

solidification term creep is modelled by the use of Kelvin-Voigt chains in series which best fits the

347

experimental data (Figure 13). Three Kelvin-Voigt chains in series are used for the modeling of the

348

solidification term. Values of the parameters of the Kelvin-Voigt chains are given in Table 3. (h) 1

M AN U

SC

RI PT

338

(MPa)

(h)

(MPa)

(h)

(MPa)

8344450

11.94

2788404

142.5

1153883

Table 3 - Solidification creep parameters for 3 KV model

350

Then, the evolution of the amplitude of the solidification term creep is given in Figure 14. The

351

amplitude is defined as the value of the solidification term creep at an age after loading of 145 hours.

352

The solidification mechanism is not directly dependent of the state of the material when the load is

353

applied. This is dependent of how the material will harden during its hydration. That is why a

354

macroscopic parameter such as the strength and the E-modulus or a microscopic parameter such as the

355

porosity and the CSH cannot directly be used to characterize the amplitude of the solidification creep.

356

As the amplitude of the solidification is linked to a mechanism which evolves during the hydration,

357

this parameter should be linked to the evolution rate of one parameter such as the CSH at

358

microstructural scale [46] or the elastic modulus at macroscopic scale. Parameters such as the

359

compressive strength could not be considered because there are not relative to the general behaviour

360

of the concrete but relative to the behaviour of the material at failure which is not what is studied here.

361

In Figure 14, the amplitude of the solidification term creep is plotted according to the time derivative

362

of the elastic modulus. A linear relation is observed between both parameters. Therefore it could be

AC C

EP

TE D

349

ACCEPTED MANUSCRIPT 363

concluded that both parameters evolve with a same kinetics and that the solidification term creep and

364

the time derivative of the elastic modulus are directly proportional.

365

Finally the solidification term creep can be defined by the multiplication of two functions: -

One function which is linked to the kinetics evolution of the solidification and is independent

367 368

RI PT

366

of the age at loading. Three Kelvin-Voigt chains are used to model it. -

One function which is linked to the amplitude of the solidification, dependent of the age at

369

loading and directly proportional to the evolution of the time derivative of the elastic

370

modulus. This amplitude term is noticed

SC

17

M AN U

371 372

Where

= -0.054. In Figure 15, the contribution of the very short term, the

373

solidification term and the long term are summed and compared to the experimental results. A very

374

good agreement for each age at loading is observed. It is then concluded that the creep function is

375

divided in three terms and can be defined since the setting with one early age loading test of long

376

duration and one minute-scale-duration loadings test which begins just after setting. The methodology

377

used to identify each term has three steps:

EP

TE D

= 0.6615 and

and is given in Equation 17.

1. The long term creep is defined with one creep test for which the load is applied at early age

379

during one week or more. After one day or more, the creep compliance rate follows a power

380

AC C

378

law which corresponds to the long term creep.

381

2. The initial short term creep is obtained after having removed the long term creep and

382

corresponds to a logarithmic law. The kinetic is constant and the amplitude depends on the

383

amplitude parameter defined with the minute-scale-duration loadings test. This term fits well

384

the results obtained during the first hours of loading or more. This term is function of the

385

actual state of the material (capillarity porosity, CSH) when the load is applied. The function

ACCEPTED MANUSCRIPT 386

is linked to the amplitude term of the minute-scale-duration loadings test which is strongly

387

correlated to the evolution of the largest diameter pores. 3. The solidification creep is obtained after having removed the long term and the initial short

389

term creep. The kinetic is constant and the amplitude depends on the time derivative of the

390

elastic modulus at the age of loading.

RI PT

388

391 c. Model Code 2010

SC

392 393

In the recent Model Code 2010 (MC2010) [18], basic creep is expressed as the multiplication of an

394

amplitude term

395

kinetic term

396

cement (Equation 18 and 19). For one composition defined, all parameters are constant except the age

397

at loading which is the only parameter which considers the ageing. When the time approaches infinity,

398

the creep compliance rate approaches a value corresponding to the inverse of the time. Then MC2010

399

considers the long term creep,

M AN U

which is linked to the mean compressive strength at an age of 28 days and a

TE D

which is function of the age at loading, the age after loading and the type of

EP

18

19

where

depends on the type of cement and is equal to 1 for CEM 52.5 N,

401

compressive strength at an age of 28 days (here

402

are compared to the experimental results. For age at loading of 40 and 72 hours, the predicted values

403

of MC2010 are quite close to the experimental results. However for earlier loadings, a significant

404

difference is observed. To understand this difference, results of the elastic modulus and its time

405

derivative are compared to the predictive value of MC2010 (Equation 20 and 21) in Figure 17.

406

Several significant differences are highlighted. The elastic modulus is highly overestimated till an age

407

of 20 hours. The time derivative of the elastic modulus is always underestimated and particularly

AC C

400

is the mean

= 48 MPa). In Figure 16, results from MC2010

ACCEPTED MANUSCRIPT during the two first days. This underestimation is also noted on the amplitude of the creep compliance

409

(Figure 16). As mechanical properties are not well predicted for the very early age, it is normal that

410

MC2010 is not able to predict correctly a parameter such as creep compliance at very early age. In

411

order to improve the prediction of the creep compliance it is necessary to adapt the model by changing

412

the effect of the age at loading.

21

Where

in days,

=36300 MPa is the elastic

414

modulus at an age of 28 days and depends on the concrete grade, s=0.2 depends on the strength class

415

of cement.

M AN U

413

416

is the elastic modulus in MPa at an age

20

SC

RI PT

408

d. Adapted Model Code 2010

Le Roy et al. [34] put forward that MC2010 can be expressed as in Equation 22. The parameter

418

constant for one concrete composition and is the only parameter which changes according to the age

419

at loading. By adjusting both parameters for different set of results coming from tests carried out on

420

ordinary and high performance concrete, Le Roy et al. show that MC2010 could be able to predict the

421

basic creep coefficient. The same methodology is used on the concrete studied in this paper and

422

results of the fitting of the creep curve are given in Figure 18. The model is able to fit very well the

423

experimental results. A value of =217.8 GPa is found. The value of the parameter

424

age of loading is given in Table 4.

EP

AC C

for the different

22

425

426

is

TE D

417

15

20

24

40

72

2.49E-07

8.20E-07

3.90E-04

1.62E-02

4.76E-02

Table 4 - Value of parameter

from Equation 22.

ACCEPTED MANUSCRIPT 427

Results of the parameter

are compared to the inverse of the time derivative of the elastic modulus in

428

Figure 19. A power law linking the parameter

429

highlighted. Therefore the time derivative of the elastic modulus seems to be a very good indicator of

430

the ageing of the creep function. As for the model separating short and long term creep , this

431

observation can be explained by the link between the rate of the elastic modulus and the velocity of

432

the hardening of the cement paste and thus to the solidification of the material. In comparison, the

433

predictive values of are plotted according to the inverse of the time derivative of the elastic modulus

434

predicted by MC2010 in Figure 20. An excellent linear correlation is done between both parameters.

435

In Figure 21, data from experimental results and MC2010 are superimposed. Two kinds of trend are

436

observed. For very early age (between setting up to 40 hours), the relation between the parameter

437

and the inverse of the time derivative of the elastic modulus follows a power trend. Values by

438

MC2010 are not able to predict this trend. For higher ages (after 40 hours), experimental results and

439

predicted values of MC2010 are very close and follow a same linear trend. Through results presented

440

in the model separating the short and long term creep, this change of trend at an age at loading of 40

441

hours can be interpreted as an effect of the solidification of the material. Indeed, before an age of 40

442

hours, the amplitude of the solidification of the material is significant on the evolution of the creep

443

coefficient. For later ages at loading, the solidification of the material has a low impact on the value of

444

the evolution of the creep coefficient. The mathematical expression linking the parameter

445

time derivative of the E-modulus is given in Equation 23 and 24 where n=0.0884 GPa, q=22542 d,

446

p=-7.41 and

RI PT

SC

M AN U

TE D

EP

and the

=1 d.GPa-1.

AC C 447

and the time derivative of the elastic modulus is

23 24

ACCEPTED MANUSCRIPT 448

Conclusions This paper presents a new methodology to model basic creep in compression since setting by means

450

of minute-scale-duration loadings test with low duration of loading (5 minutes) and classical creep

451

tests with duration of loading of 6 days or more. Results from the minute-scale-duration loadings test

452

allow defining the elastic modulus and an ageing creep factor corresponding to the value of the

453

creep coefficient after 5 minutes of loadings. Based on experimental observations, a new approach is

454

developed for the modeling of the creep function.

455

Several approaches are used to model creep on long duration (6 days or more):

SC

RI PT

449

1. Four Kelvin-Voigt units in series are used. Only units with a characteristic time lower than 2

457

hours have an ageing factor which corresponds to the value of the creep coefficient after 5

458

minutes of loading. This way to model creep curve has the advantage to be easily

459

implemented in finite element software. For the study of loading of short duration, as the case

460

of restrained shrinkage, this way to model creep is convenient. For long duration of loading,

461

however, this way of modelling creep is not accurate. Several creep tests of long duration are

462

needed for calibration and the model does not distinguish between reversible and irreversible

463

creep.

TE D

M AN U

456

2. In order to consider the physical mechanisms occurring on concrete under loading during the

465

hydration process, creep is divided in three terms. Two terms are related to the short term

467 468

AC C

466

EP

464

creep and one term is related to the long term creep. Each term is identified since the setting in three steps:

a. The long term creep is defined with one creep test for which the load is applied at

469

early age during one week or more. After one day or more, the creep compliance rate

470

follows a power law which corresponds to the long term creep.

471

b. The initial short term creep is obtained after having removed the long term creep and

472

corresponds to a logarithmic law. The kinetic is constant and the amplitude depends

473

on the amplitude parameter defined with the minute-scale-duration loadings test. This

ACCEPTED MANUSCRIPT 474

term fits well the results obtained during the first hours of loading or more. This term

475

is function of the actual state of the material (capillarity porosity, CSH) when the load

476

is applied. c. The solidification creep is obtained after having removed the long term and the initial

478

short term creep. The kinetic is constant and the amplitude depends on the time

479

derivative of the elastic modulus at the age of loading.

RI PT

477

The Model Code 2010 is used and compared to experimental results. For ages at loading of 40

481

and 72 hours, the predicted values of MC2010 are quite close to the experimental results.

482

However for earlier loadings, a significant difference is observed. An adapted version of the

483

Model Code 2010 is proposed. In this version, the effect of the age at loading is considered by

484

using the inverse of the time derivative of the elastic modulus.

M AN U

SC

480

This work is however limited to medium stress levels (up to 40 % of compressive strength). A next

486

step of this research is the study of creep in tension, higher stress levels and the coupling between

487

damage and creep at early age. More sophisticated tests with more complex histories of loadings or

488

temperatures are still needed for computational purposes. The measurement of creep effects on

489

transverse strain is also of interest.

490

Acknowledgements

491

Special thanks are addressed to Claude Boulay and Florent Baby, from IFSTTAR, Bernard Espion

492

and Thierry Massart, from Université Libre de Bruxelles, BATir Department, and Farid Benboudjema

493

from ENS Cachan for their fruitful discussions.

494

References

495

[1]

AC C

EP

TE D

485

S. Asamoto, K. Kato, T. Maki, Effect of creep induction at an early age on subsequent

496

prestress loss and structural response of prestressed concrete beam, Constr. Build. Mater. 70

497

(2014) 158–164. doi:10.1016/j.conbuildmat.2014.07.028.

ACCEPTED MANUSCRIPT 498

[2]

499 500

P. Kumar Mehta, P.J.M. Monteiro, Concrete: Microstructure, Properties, and Materials, Fourth Edition, Third, The McGraw-Hill Companies, 2006.

[3]

A. Bentur, Terminology and definitions, in: A. Bentur (Ed.), Early Age Crack. Cem. Syst. Rep. RILEM Tech. Comm. 181-EAS - Early Age Shrinkage Induc. Stress. Crack. Cem. Syst.,

502

RILEM Publications SARL, 2003: pp. 13–15. doi:10.1617/2912143632.002.

503

[4]

RI PT

501

B. Delsaute, S. Staquet, Decoupling Thermal and Autogenous Strain of Concretes with different water/cement ratios during the hardening process, Adv. Civ. Eng. Mater. 6 (2017) 22.

505

doi:10.1520/ACEM20160063. [5]

507 508

K. Kovler, Drying creep of concrete in terms of the age-adjusted effective modulus method, Mag. Concr. Res. 49 (1997) 345–351. doi:10.1680/macr.1997.49.181.345.

[6]

509

M AN U

506

SC

504

Z.P. Bazant, E. Osman, Double power law for basic creep of concrete, Matériaux Constr. 9 (1976) 3–11. doi:10.1007/BF02478522.

[7]

A.D. Ross, Concrete creep data, Struct. Eng. 15 (1937).

511

[8]

W.R. Lorman, The theory of concrete creep, Proc. ASTM. 40 (1940) 1082–1102.

512

[9]

Thomas Thelford, Comité Eurointernational du Béton CEB/FIB Model Code 90, Design Code, 1993.

EP

513

TE D

510

[10]

H. Straub, Plastic flow in concrete arches, Proc. Am. Soc. Civ. Eng. 56 (1930) 49–114.

515

[11]

J.R. Shank, Plastic flow of Portland Cement Concrete, Ind. Eng. Chem. 27 (1935) 1011–1014.

516

[12]

517

D.S. Atrushi, Tensile and Compressive Creep of Early Age Concrete : Testing and Modelling,

PhD thesis, The Norwegian University of Sciences and Technology, Trondheim,Norway,

518 519

AC C

514

2003. [13]

G. De Schutter, Applicability of degree of hydration concept and maturity method for thermo-

520

visco-elastic behaviour of early age concrete, Cem. Concr. Compos. 26 (2004) 437–443.

521

doi:10.1016/S0958-9465(03)00067-2.

ACCEPTED MANUSCRIPT 522

[14]

W. Jiang, G. De Schutter, Y. Yuan, Degree of hydration based prediction of early age basic

523

creep and creep recovery of blended concrete, Cem. Concr. Compos. (2013).

524

doi:10.1016/j.cemconcomp.2013.10.012.

525

[15]

P. Laplante, Propriétés mécaniques des bétons durcissants: analyse comparée des bétons classiques et à très hautes performances, PhD thesis, Ecole Nationale des Ponts et Chaussées,

527

1993. [16]

529 530

I. Guenot, J.-M. Torrenti, P. Laplante, Stresses in Early-Age Concrete: Comparison of Different Creep Models, ACI Mater. J. 93 (1996) 254–259.

[17]

SC

528

RI PT

526

A.-W. Gutsch, Properties of early age concrete-Experiments and modelling, Mater. Struct. 35 (2002) 76–79. http://link.springer.com/article/10.1007/BF02482104 (accessed December 17,

532

2013). [18]

535

2010, Struct. Concr. 14 (2013) 320–334. doi:10.1002/suco.201200048. [19]

536 537

doi:10.1016/j.cemconcomp.2014.07.010. [20]

thesis, Ecole Nationale des Ponts et Chaussées, Paris, France, 1995. [21]

M. Briffaut, F. Benboudjema, J.-M. Torrenti, G. Nahas, Concrete early age basic creep:

[22]

540

Experiments and test of rheological modelling approaches, Constr. Build. Mater. 36 (2012)

541 542

R. Le Roy, Déformations instantanées et différées des bétons à hautes performances, PhD

AC C

538 539

J.P. Forth, Predicting the tensile creep of concrete, Cem. Concr. Compos. 55 (2014) 70–80.

TE D

534

H. Muller, I. Anders, B. R., Concrete: treatment of types and properties in fib Model Code

EP

533

M AN U

531

373–380. doi:10.1016/j.conbuildmat.2012.04.101. B.T. Tamtsia, J.J. Beaudoin, J. Marchand, The early age short-term creep of hardening cement

543

paste: load-induced hydration effects, Cem. Concr. Compos. 26 (2004) 481–489.

544

doi:10.1016/S0958-9465(03)00079-9.

545 546

[23]

F.-J. Ulm, F. Le Maou, C. Boulay, Creep and shrinkage coupling : New review of some evidence, in: ACI-RILEM Work. Creep Shrinkage Concr. Struct., 1999: pp. 21–37.

ACCEPTED MANUSCRIPT 547

[24]

O. Bernard, F.-J. Ulm, E. Lemarchand, A multiscale micromechanics-hydration model for the

548

early-age elastic properties of cement-based materials, Cem. Concr. Res. 33 (2003) 1293–

549

1309. doi:10.1016/S0008-8846(03)00039-5. [25]

[26]

[27]

556

115 (1989) 1691–1703. doi:10.1016/0008-8846(88)90028-2. [28]

557 558

[29]

[30]

[31]

[32]

[33]

571

M. Vandamme, F.-J. Ulm, Nanogranular origin of concrete creep., Proc. Natl. Acad. Sci. U. S.

A. 106 (2009) 10552–7. doi:10.1073/pnas.0901033106.

[34]

569 570

F.H. Wittmann, Useful Fundamentals of Shrinkage and Creep of Concrete, in: Concreep 10, 2015: pp. 84–93.

567 568

Z.P. Bažant, Creep and shrinkage prediction model for analysis and design of concrete structures - Model B-3, Mater. Struct. 28 (1995) 357–365.

565 566

O. Bernard, F.-J. Ulm, J.T. Germaine, Volume and deviator creep of calcium-leached cementbased materials, Cem. Concr. Res. 33 (2003) 1127–1136.

563 564

F.H. Wittmann, Creep and Shrinkage Mechanisms, in: Z.P. Bažant, F.H. Wittmann (Eds.), Creep Shrinkage Concr., 1982: pp. 129–161.

561 562

Z.P. Bažant, Double-power logarithmic law for concrete creep, Cem. Concr. Res. 14 (1984) 793–806.

559 560

M AN U

555

Z.P. Bazant, S. Prasannan, Solidification theory for aging creep. I: Formulation, J. Eng. Mech.

TE D

554

Concrete Creep. I Aging and Drying Effects, J. Eng. Mech. 123 (1997) 1188–1194.

SC

553

Z.P. Bažant, A.B. Hauggaard, S. Baweja, F.-J. Ulm, Microprestress-Solidification Theory for

EP

552

measurements, Nucl. Eng. Des. 203 (2001) 143–158. doi:10.1016/S0029-5493(00)00304-6.

RI PT

551

P. Acker, F.J. Ulm, Creep and shrinkage of concrete: Physical origins and practical

AC C

550

R. Le Roy, F. Le Maou, J.M. Torrenti, Long term basic creep behavior of high performance concrete: data and modelling, Mater. Struct. 50 (2017) 85. doi:10.1617/s11527-016-0948-8.

[35]

M. Vandamme, A Few Analogies Between the Creep of Cement and of Other Materials, in: Concreep 10, 2015: pp. 78–83.

ACCEPTED MANUSCRIPT [36]

573 574

temperatures, Eur. J. Environ. Civ. Eng. (2017) 1–10. doi:10.1080/19648189.2017.1280417. [37]

575 576

J.M. Torrenti, Basic creep of concrete-coupling between high stresses and elevated

Z.P. Bazant, S. Baweja, Creep and Shrinkage Prediction Model for Analysis and Design of Concrete Structures : Model B3, ACI Concr. Int. 83 (2001) 38–39.

[38]

Z.P. Bažant, S. Prasannan, Solidification Theory for Concrete Creep. II: Verification and

577

Application,

578

9399(1989)115:8(1691).

580 581

Mech.

115

(1989)

1704–1725.

doi:10.1061/(ASCE)0733-

Z. Bazant, A.B. Hauggaard, S. Baweja, Microprestress-Solidification Theory for Concrete

SC

[39]

Eng.

Creep. II: Algorithm and Verification, J. Eng. Mech. 123 (1997) 1195–1201. [40]

M AN U

579

J.

RI PT

572

Y.-S. Yoon, S.-K. Lee, M.-S. Lee, J.-K. Kim, K.-Y. Kwon, Rheological concrete creep

582

prediction model for prestressed concrete bridges constructed by the free cantilever method,

583

Mag. Concr. Res. 63 (2011) 645–653. doi:10.1680/macr.2011.63.9.645. [41]

[42]

587 588

Res. 4 (1974) 567–579. [43]

589 590

F. Benboudjema, J.M. Torrenti, Early-age behaviour of concrete nuclear containments, Nucl. Eng. Des. 238 (2008) 2495–2506. doi:10.1016/j.nucengdes.2008.04.009.

[44]

591 592

Z.P. Bažant, A. Asghari, Computation of Age-Dependent Relaxation spectra, Cem. Concr.

EP

586

Cem. Concr. Res. 4 (1974) 797–806.

TE D

585

Z.P. Bažant, A. Asghari, Computation of Kelvin chain retardation spectra of aging concrete,

AC C

584

M. Jirásek, P. Havlásek, Microprestress-solidification theory of concrete creep: Reformulation

and improvement, Cem. Concr. Res. 60 (2014) 51–62. doi:10.1016/j.cemconres.2014.03.008.

[45]

W. Hermerschmidt, H. Budelmann, Creep of Early Age Concrete under Variable Stress, in:

593

CONCREEP 10, American Society of Civil Engineers, Reston, VA, 2015: pp. 929–937.

594

doi:10.1061/9780784479346.111.

595 596

[46]

B. Delsaute, C. Boulay, S. Staquet, Creep testing of concrete since setting time by means of permanent and repeated minute-long loadings, Cem. Concr. Compos. 73 (2016) 75–88.

ACCEPTED MANUSCRIPT 597 598

doi:10.1016/j.cemconcomp.2016.07.005. [47]

C. Boulay, S. Staquet, B. Delsaute, J. Carette, M. Crespini, O. Yazoghli-Marzouk, et al., How

599

to monitor the modulus of elasticity of concrete, automatically since the earliest age?, Mater.

600

Struct. 47 (2014) 141–155. doi:10.1617/s11527-013-0051-3. [48]

B. Delsaute, C. Boulay, J. Granja, J. Carette, M. Azenha, C. Dumoulin, et al., Testing Concrete

RI PT

601 602

E-modulus at Very Early Ages Through Several Techniques: An Inter-laboratory Comparison,

603

Strain. 52 (2016) 91–109. doi:10.1111/str.12172. [49]

S. Staquet, B. Delsaute, A. Darquennes, B. Espion, Design of a Revisited Tstm System for

SC

604

Testing Concrete Since Setting Time Under Free and Restraint Conditions ., in: Concrack 3 -

606

RILEM-JCI Int. Work. Crack Control Mass Concr. Relat. Issues Concern. Early-Age Concr.

607

Struct., 2012: p. 12.

608

[50]

M AN U

605

B. Delsaute, J. Carette, S. Staquet, Monitoring of the creep and the relaxation at very early age: complementary results on the CEOS concrete, in: VIII Int. Conf. Fract. Mech. Concr. Concr.

610

Struct., 2013: pp. 453–458.

611

[51]

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S. Staquet, M. Azenha, C. Boulay, B. Delsaute, J. Carette, J. Granja, et al., Maturity testing through continuous measurement of e-modulus: an inter-laboratory and inter-technique study,

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in: Proc. ECO-CRETE Int. Symp. Sustain., 2014: p. 8. [52]

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under various levels of compressive stress, Cem. Concr. Res. 51 (2013) 32–37.

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P. Rossi, J.L. Tailhan, F. Le Maou, Creep strain versus residual strain of a concrete loaded

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doi:10.1016/j.cemconres.2013.04.005.

[53]

M. Irfan-ul-Hassan, B. Pichler, R. Reihsner, C. Hellmich, Elastic and creep properties of

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young cement paste, as determined from hourly repeated minute-long quasi-static tests, Cem.

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Concr. Res. 82 (2016) 36–49. doi:10.1016/j.cemconres.2015.11.007.

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List of figures

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Figure 1 - (a) Free strain inside concrete element. (b) Stress induced by restriction of the concrete

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displacement.

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Figure 2 - Rheological scheme of the concrete behavior [44] 628

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(b)

(c)

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Figure 3 – (a) Creep coefficient for several ages at loading – (b) First two hours after loading of the

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creep coefficient – (c) Creep coefficient set to zero at an age after loading of 2 hours [46].

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Figure 4 – Repeated minute-scale-duration loadings. a: for each minute-scale-duration loading, creep

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curves are superimposed and set to zero when the load is kept constant. b: for each minute-scaleduration loading, creep curves are normalized by their value obtained after 5 minutes of loading. c: Evolution of the amplitude of the creep coefficient after 5 minutes of loading (coming from the repeated minute-scale-duration loadings tests (RMSL)) and 2 hours of loading (coming from the permanent loading tests) according to the equivalent time [46].

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Figure 5 – Evolution of the amplitude and kinetic Figure 6 – Experimental data of the creep parameter of Equation 9 according to the coefficient and modeled curves for several ages at loading from Equation 9.

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equivalent age at loading.

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Figure 7 - Comparison between the approach of De Schutter [13], Benboudjema & Torrenti [43] and the new strategy developed

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Figure 8 – Comparison between 4 KV chains model developed with minute-scale-duration loading

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testing for duration of one week and experimental results of permanent loading tests.

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Figure 9 – Time derivative of the creep Figure 10 – Time derivative of the creep function function according to the age of the concrete according to the age of the concrete for results for a normal strength concrete (W/C=0.5) and obtained by Delsaute [46] and results obtained by a high strength concrete (W/C=0.33) from Rossi for an age at loading of 28 days [52]. [33]. 635

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Figure 11 – Evolution of the creep coefficient Figure 12 – Evolution of the creep coefficient with subtraction of the long term part and with subtraction of the long term and very short

lines correspond to the modelling)

consolidation creep (dashed lines).

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modeling of the very short term creep (dashed term part (continuous lines) and modeling of the

Figure 13 – Normalization at 145 hours of the Figure 14 – Comparison between the amplitude creep coefficient after subtraction of the long of the solidification term creep after 145 hours term creep and the very short term creep. of loading and the time derivative of the elastic Modeling of the kinetics of the solidification modulus according to the age at loading. term creep. 637

identification of the long term creep, the short creep

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solidification

creep

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term

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Figure 15 – Modeling of the creep coefficient by

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(continuous lines correspond to experiments and dashed lines to the modeling).

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Figure 16 – Comparison between predicted value Figure 17 – Comparison between predicted value of the creep compliance obtained with MC2010 of the elastic modulus and the time derivative of (dashed

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results the elastic modulus obtained with MC2010 and experimental results.

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Figure 18 – Comparison between experimental Figure 19 – Comparison between the time

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creep curves, creep curves coming from a model derivative of the elastic modulus and the inspired by the model code 2010 (continuous parameter of the adapted model code 2010 lines correspond to experiments and dashed lines to the modeling).

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Figure 20 – Evolution of the parameter τ from Figure 21 - Comparison of the relation between Model code 2010 according to the time derivative the inverse of the time derivative of the elastic of the elastic modulus.

modulus and τ from experimental results and Model Code 2010.

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