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THE NEW MODEL FOR CREEP OF CONCRETE INFIP MODEL CODE 2010

F. Mola, Politecnico di Milano, ItalyL.M. Pellegrin i, ECSD, Italy

37thConference on OUR WORLD IN CONCRETE & STRUCTURES: 29 - 31 August 2012,Singapore

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37thConference on Our World in Concrete & Structures

Singapore, August 29-31, 2012

__________________________

ECSD, Srl, Milano, Italy

THE NEW MODEL FOR CREEP OF CONCRETE INFIP MODEL CODE 2010

F. Mola* and L.M. Pellegrini

*Politecnico di Milano, Italy

Keywords: Concrete, creep, relaxation, long term behaviour, Model Code

Abstract. The new fib Model Code 2010 (MC 10) has introduced new and improvedformulations to define the model for the long-term behaviour of concrete. In this waysome inconsistencies present in the older CEB/FIP Model Code 1990 (MC 90) havebeen removed. In the new formulation the total creep deformation of concrete hasbeen introduced as sum of two contributions: basic and drying creep, for each ofwhich a specific time evolution function is provided. In the present paper, after abrief recall of the basic principles of linear viscoelasticity, the differences betweenthe new and the previous formulation will be firstly carried out. Then, with referenceto a case study regarding a homogeneous viscoelastic structure interacting withelastic restraints, the results derived from structural analysis performed by using MC10 and MC 90 will be compared.

1. INTRODUCTION

The new fib Model Code 2010, /1/, introduced new and improved formulations to define the model forthe long-term behaviour of concrete in service life loading conditions. In comparison with the olderCEB/FIP Model Code 90, /2/, new functions have been implemented, allowing some interestingaspects of the long term behaviour of concrete to be better clarified. In particular, long termdeformations in concrete are now expressed as the sum of basic and drying creep deformations.

The former develops on condition of no moisture exchange with the ambient environment and isassumed to depend on the mean strength of concrete, whereas the latter depends on the thicknessof the concrete element and the external humidity. In the new MC 10, a method to take into accountthe effects of temperature on the time of loading is also suggested, as well as modified values for thecreep coefficient, in a non linear form, so that also situations where the stress values in concrete aremore than 40% of the mean strength can be duly investigated.

After a detailed discussion of the basic principles of linear viscoelasticity and of the main prerequisitesof the limiting models i.e. the non ageing model of Kelvin Voigt, /3/, the ageing model ofDischinger, /4/, and the composite one of Arutyunyan, /5/, the models proposed in Europe in thethree Model Codes, namely CEB/FIP Model Code 1978, /6/, CEB/FIP Model Code 1990 and fib

Model Code 2010 will be discussed in detail and a first in-depth insight of the new formulations ofMC 10 and of the differences between the new and the previous ones will be carried out.

A simple case study will be finally investigated in order to point out the differences between the results

deriving from the use in structural analysis of MC 90 and MC 10.


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F. Mola and L.M. Pellegrini

2. BASIC PRINCIPLEs OF LINEAR VISCOELASTICITY

In Europe concrete creep become relevant in describing structural behaviour in the Thirties. In thatperiod the most important problem involving creep in structural analysis was the evaluation of theevolution in time of the postensioning force in prestressed elements.

Even though this problem was prominent, other problems were of great importance, in particular the

behaviour of reinforced concrete structures subjected to imposed deformations and displacementsand the mutual interaction between the delayed deformations of concrete and creep.

Up to the mid of the Fourties, when Mc Henry, /7/, expressed the stress-strain relationship in a timedependent form according to the mathematical approach stated by Volterra, /8/, the problemconnected to structural analysis of concrete structures was not properly investigated. Only a certainnumber of special cases had been studied without deriving a consistent and reliable formulationgoverning the time dependent structural analysis of concrete structures.

The relationship introduced by Mc Henry was:

, , (1)where (t) is the total deformation of concrete under a variable stress which can be written in the form:

(2)having indicated by e(t) the elastic, instantaneous part of the deformation and by c(t) the delayed part

due to creep.According to eq. (2) we observe that eq. (1) involves only the deformation due to an applied stress. If

other deformations are present, i.e. shrinkage, temperature, imposed displacements, which developindependently of the applied stress, the preceding expression can be generalized in the subsequentway:

, , (3)

(4)where (t), coinciding with the left member of eq. (1) is the deformation due to an applied stress and is the stress indipendent strain. In eqs. (1), (3) the integral at right member postulates a stress

varying in time in a continuous way. In this case the integral assumes the form of a Riemann integral.When the stress varies in a non continuous way, in particular in a form resulting by the superposition of

a continuous function in time and a finite number of stress instantaneous steps, the integrals of eqs.(1), (3) can be transformed in Stieltjes integrals, including in the integral form the continuous and thediscontinuous part of the stress. In this way we finally obtain:

, (5)Eq. (5) expresses the Mc Henry principle of superposition in its most general form and describes the

time behaviour of the linear viscoelastic solid usually assumed in the long term structural analysis ofconcrete structures.

Mathematically speaking eq. (5) represents a Volterra integral equation relating the total deformation

(t) and the stress (t) which can be solved with respect (t) when the strain dependent on the stress history is known.

At this regard we introduce the resolving kernel R(t,t) of eq. (5), solution of the subsequent Volterraintegral equation:

, , 1 (6)Applying the principle of superposition we can then express the solution of eq. (5) according to the

following expression:

, (7)


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According to Fig. 1, eq. (13) embodies the physical property that the speed of creep phenomenon at a

generic time t is proportional by means of a constant > 0 to the quantity of creep has to be

developed from t to t = . The validity of eq. (12) requires that:

lim , 0 (14)or, in other words, the final value of the creep deformation is limited, J(, t) < .

Fig. 1 The mass action law

Remembering eq. (10) and introducing eq. (13) in eq.(12) we derive:

, (15)

so differentiating eq.(15) we finally obtain

, (16)

Eq. (16) represents a linear differential equation with variable coefficients for which the principle ofsuperposition is strictly satisfied. We can so conclude that only for models associated to a creep

function satisfying eq. (13), i.e. the mass-action law, the principle of superposition expressed in theMcHenry form (5) can be exactly applied. For other models, not satisfying eq. (13), eq. (5) representsonly an approximate form and its application in some cases can drive to inconsistent results. For thisapplication of eq. (5) requires some restrictions in particular:

- the stress level in concrete has to be less than 0.4 fcm;- strain of decreasing magnitude does not take place although the stress may decrease;- no significant change in moisture content distribution develops during creep;- no large sudden stress increasing long after the initial loading takes place.

3. LIMITING CREEP MODELS

The first creep model able to describe, albeit approximately, the creep behaviour of concrete, was theKelvin-Voigt model exhibiting the following Creep Function:

, 1 (17)where E, constant in time, is the elastic modulus, is the final value of the creep coefficient and * is

the retardation time.Introducing eq. (17) in eq. (13) we obtain:

(18)so, assuming , the creep function JKV satisfies the mass-action law. The Kelvin-Voigt model

allows to use the principle of superposition both in integral and differential form in an exact way.Another very important property of the model derives from the subsequent relationship:


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lim , (19)which points out that the final creep deformation is independent from the concrete age. For this reason

the (KV) model defines concrete as a non ageing viscoelastic material. Eq. (19) allows to expresseq. (5) in the following form:

(20)so that assuming that , and introducing the effective modulus , eq. (20) can be

expressed by means of the following pseudo-elastic form:

(21)allowing to perform the structural analysis at final time in a very simple and straightforward way.

Besides these interesting properties the (KV) model is unable to properly describe the timedevelopment of the delayed behaviour of concrete as it is strongly affected by ageing and presents a

rapid increase in times near to the loading time. This requirement can be indeed satisfied by the(KV) model assuming a convenient retardation time as .

Neverless the experimental results show that , so to satisfy this property 0 is needed.

This assumption, according to eq. (17), would give , so the time development ofcreep cannot be described. This is essentially the basic reason which does not allow to assume thecreep function defined by an exponential form.

Another model, very popular in Europe, is the Dischinger model exhibiting the subsequent creepfunction:

, (22)

where:E(t) is the elastic modulus at loading time tE28 is the elastic modulus at t = 28 days

1/ is the retardation time is the first possible loading time.Introducing eq. (22) in eq. (13) we derive:

(23)so that assuming = the (D) model satisfies the mass-action law and allows the application of

McHenry principle of superposition in an exact way. Furthermore for

it results:

lim , (24)

showing that the final value of the creep deformation decreases with the time of loading up to becomezero when . This property defines the creep model as an only ageing material. It is importantto observe that for this model also we see that it is not possible to comply with the experimentalevidence of having a creep function rapidly increasing after the loading time. We can so concludethat (KV) and (D) models even though they satisfy the mass-action law and strictly comply theprinciple of superposition, they can describe only two limit behaviour of creep concrete i.e. the nonageing and the ageing material. Furthermore they cannot comply with the experimental resultsshowing a rapid increase of the creep function after the loading time. A model able to strictly satisfythe principle of superposition and to describe both non ageing and ageing behaviours was

introduced byArutyunyan.


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The creep function assumed byArutyunyanis:

, 1 (25)

where

,

are respectively the non ageing and the ageing part of the creep deformation and 1/is

the retardation time. TheArutyunyanmodel introduced in eq. (13) gives:

(26)

so that assuming = the model satisfies the mass-action law and strictly complies the superpositionprinciple. Despite the capability of taking into account the non ageing and the ageing properties ofthe creep deformation the exponential form of the eq. (25) prevents also in this case to properlydescribe the rapid increasing of the creep deformation just after the loading time t, so that

Arutyunyan model does not allow to reach a satisfactory description of the creep deformation ofconcrete. On the other hand rearranging eq. (13) in the form:

,

, , (27)the solution of the differential eq. (27) can be expressed in the subsequent way:

, , (28)with the initial condition:

, (29)Combining eqs. (28), (29) we finally derive:

, , , (30)Eq. (30) points out that the models satisfying the mass-action law are always described by a creep

function of exponential form, so we can conclude that the materials satisfying in an exact way theprinciple of superposition are not suitable for describing the rapid time development of the creep lawin times near the loading time. To do this different models have to be chosen, exhibiting a creep lawexpressed in the integral form (5) which cannot be reduced to a differential form. This kind of modelsdo not strictly satisfy the principle of superposition, so that some shortcomings can take place andtheir consequences have to be carefully detected, investigated and controlled.

Fig. 2 Creep behaviour of concrete under a loading-unloading process: (KV), (D), (A) models

The basic character of the three models satisfying the principle of superposition and the mass-actionlaw is pointed out in Fig.2, where the deformation associated to a loading-unloading process is

reported. We observe that in condition of zero stress the deformation of the (KV) model reduces intime up to zero value for t = . The deformation of (D) model remains constant and the one of (A)


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model reduces but does not reach zero for t = . Indicating by delayed elasticity the deformationdeveloping after the unloading we can define (KV) model a total reversible model, (D) model anirreversible model and (A) a partially reversible model.

4. THE MODELS IN THE EUROPEAN MODEL CODES

4.1 The Model Code 1978

Creep models oriented to take into account the rapid initial increase of the creep deformation weresuggested in Europe by Nielsen, /9/, Rsch Hilsdorf and Jungwirth, /10/, and CEB/FIP Model Code1978. The creep law adopted in MC 78 was a generalization of the (KV) and (D) models, expressedin a sum form according to the subsequent expression:

, , (31), (32)The three terms at right member of eq. (32) respectively represent: the sudden creep deformation at time of loading; the developing in time of the non ageing part of creep; the developing in time of the ageing part of creep.The corresponding expressions were as follows:

0.8 1 0.4

/.

4.45 0.035 (33) .

. ..

. . . .. ..

Fig. 3 Creep Function, MC 1978 Fig. 4 Relaxation Function, MC 1978


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The basic character of CEB 1978 Creep Model was the computation of the rapid increase of the creep

deformation after the loading time by introducing the instantaneous deformation

. This way ofoperating is debetable as creep is not an istantaneous deformation, anyhow the assumption madeallowed to directly use the two ageing and non ageing models, for which different time functions withrespect the exponential forms were assumed. Another interesting aspect regards the computation of

the external conditions which are taken into account by the level of relative humidity RH and by thenotional thickness h0. Some results taken from an extended analytical investigation performed byChiorino et al., /11/, are reported in Fig.3-4.


The shortcomings deriving from the introduction of an initial creep deformation were eliminated in theCEB/FIP Model Code 90, where the creep model was calibrated on a product function having theform:

, ,

, 1 . / . . (34) . .h = 2Ac/u

.

150 1 1.2 250 1500

In the set of eqs. (34) fcm, RH, h respectively represent the mean value of the compressive strength at28 days, the relative humidity expressed in percent, the notional thickness, ratio between the area ofthe cross section and the half of the perimeter in contact with the atmosphere. RH 0, h0, t1, areprescribed reference values i.e. RH0=100%; h0= 100 mm; t1= 1 day.


The MC 90 introduces some important prerequisites. First of all we can easily observe that it results:

lim, (35)


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and according to eq. (6)

lim, (36)so that the rapid increase of creep at early time, as illustrated in Fig. 5-6, is accounted for by the vertical

direction of the tangent of the creep curve for . As a second point we observe that the twofunctions , , the first expressing the creep coefficient for and the seconddescribing the developing in time of the creep function, are both depending on the relative humidityand by the notional thickness. This means that the creep deformation is thoroughly accounted asdrying creep, neglecting to express the basic part of the total creep deformation.


The creep MC 90 model even though it represented an improvement with respect MC 78 model, wasstill lacking of important prerequisites.

The inconsistencies of MC 90 have been accommodated in the new fib MC 10 creep Model. The twomost important additions introduced regard the separation of the total creep deformation in the twocontributions related to basic creep and drying creep and the absence of an asymptotical limit for the

function expressing the time developing of the basic creep. The analytical form of the creep functionof fib MC 10 model is:

, , , (37)with , , , basic creep and drying creep part of the creep coefficient , having

expression:

, , , (38)where

, ,

..

, 0.035 1

, , (39) .

.

. .

,

. .

1.5 250 1500

.


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In the set of eqs. (39) fcm, RH, h respectively represent the mean compressive strength at 28 days inMPa, the relative humidity in % and the notional thickness in mm.

The linear form of creep function of eq. (37) can be used for || 0.4 .In the interval 0 . 4 | | 0.6 the following non linear form holds:

, , .. 0.4 0.6 | | (40)


Fig.7 shows that the basic creep function is not limited in time as it results lim , , whilefor the drying creep we have lim , 1 . In fact it is not known in advance if creepapproaches a finite value so the introduction of a creep coefficient expressed by a logarithmic formhas been suggested in order to predict with good approximation the behaviour of concrete up to fiftyyears of loading. From the mathematical point of view the introduction of an unlimited creep functiondoes not guarantee the existence and the uniqueness of the solution of eq. (6). In any case anunlimited creep function drives to the following result:

lim , 0 (41)expressing that the stress induced by a unit deformation applied at t is totally dissipated at final time.

On the other hand, taking into account that the relaxation function is monotonically decreasing intime, eq. (41) allows to observe that for a creep model exhibiting an unlimited creep function therelated relaxation function satisfies the inequality:

, 0, (42)Furthermore, taking into account that the logarithmic function develops very slowly in time, we see that

the increase of creep from 50 years to 150 years of duration of loading will not exceed 10% of thecreep after 50 years. We can so conclude that the introduction of an unlimited creep function allowsto better approximate the concrete behaviour, does not introduce significant numericalinconsistencies in the numerical solution of eq. (6) and avoids the physically absurd result ofobtaining negative values for R(t,t).

Some comparison between MC 90 and MC 10 will illustrate in detail the concepts here discussed. In

Fig. 9-10 the diagrams of the creep functions are reported for two loading times, namely t 0= 14 daysand t0= 28 days. Fig.9 deals with a concrete having fck= 32 MPa; h0= 200 mm; RH = 70%. Weobserve that MC 10 curves develop in a more straightforward form and give slightly higher values ofthe creep coefficient for t = 10

5days, while the time development of creep in MC 90 proceeds in a

faster way up to about 104days, then it nearly flatters. On the contrary the creep curves of MC 10

show a non zero slope for t = 105 days. The basic character of the curves is also present for

concrete with fck= 60 MPa reported in Fig. 10. In this case nevertheless, the final values of creep aresmaller for MC 10. This result shows for this model an higher negative correlation between creepand compressive strength.


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Fig. 9 Creep Functions, fck= 32 MPa Fig. 10 Creep Functions, fck= 60 MPa

In Fig. 11-12 the relaxation curves for the same two materials are reported. In this case we observe thatfor concrete C32/40, the final values of the relaxation functions are smaller for MC 90 model as forthis model the final values of creep are higher. On the contrary for concrete C60/75 this trend isinverted and for t0= 14 days the final values of R (t,t0)/E28are higher in MC 10 model.

Fig. 11 Relaxation Functions, fck= 32 MPa Fig. 12 Relaxation Functions, fck= 60 MPa

Finally Fig. 13-14 point out two significant inconsistencies inherent to MC 90 model and show theimprovements introduced by MC 10 model. In Fig. 13 the non dimensional deformation during aloading-unloading process is reported for the two models. We see that for the MC 90 model thevariation in time of the deformation after the removal of loading is initially decreasing then itincreases up to an asymptotical value. So the model shows a negative delayed elasticity which is notpossible from a thermodynamical point of view. On the contrary the MC 10 shows a continuousdecrease of the deformation connected to a positive delayed elasticity. In Fig. 14 we see that in MC90 model for a C30/37 concrete loaded at very early age the relaxation function can becomenegative so violating the principles of thermodynamics. In a different way MC 10 shows amonotonically decreasing diagram of the relaxation function which remains positive, as required bythe basic principles of linear viscoelasticity.

Fig. 13 Loading-Unloading process Fig. 14 Relaxation Functions

5. CASE STUDY

The comparison now performed between the two models allows to state that MC 10 is physicallyconsistent and avoids some significant shortcomings which can take place when MC 90 is used.

Anyhow, the small numerical differences that the two models exhibit in term of creep and relaxationcurves make both of them suitable for structural analysis. This point is illustrated in Fig.15-16, wherefor a C32/40 concrete, h = 200 mm, RH = 70%, time of loading 28 days the Varied Creep Functions

J*(t,t0)E28 and the related solving kernels named Reduced Relaxation Functions R*(t,t0)/E28 arereported.


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Fig. 15 Varied Creep Functions Fig. 16 Reduced Relaxation Functions

Functions R*, /12/, allow to solve problems related to homogeneous viscoelastic structures interactingwith elastic restraints, like the one represented in Fig.17.

Fig. 17 Non-homogeneous viscoelastic structure Fig. 18 Bending moments at t=t0, t=

For this structure, indicating by Xethe reaction of the elastic restraint at initial time, the solution at time tcan be expressed as follows /13/:

1 1 , (43)

where is the coupling coefficient, given by the expression:

(44)with , respectively the elastic influence coefficients of the structure and of the elastic restraint.

Assuming for the case study we obtain = 0.5, so from eq. (43) we reach: 2 1 0.5 , (45)from Fig. 16, for = 0.5 we obtain:

, 0.451

, 0.394

so the final values of the reaction of the elastic restraint become:

10 2 10.50.451 1.55 10 2 10.50.394 1.61 At initial time Xe= Q/2 and the diagrams of the bending moment at initial and final time are represented

in Fig. 18.For the restraint reaction and for the maximum bending moment at the mid-structure we reach the two

ratios:

10/ 10 0.963 10 10 1.154 showing how the two models can be retained quite similar as regards structural analysis.

4M/Ql1

0.45

0.39


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6. CONCLUSIONS

The new fib MC 10 creep model introduces important improvements with respect the previous fib MC 90model. Even though the two models do not differ in a marked extent in predicting the final values ofthe creep deformation and of the related relaxed stresses, fib MC 10 introduces some basic

concepts which make it more consistent from the physical point of view. At this regard MC 10introduces total creep deformation as the sum of two contributions, basic and drying creep. The firstone, developing when moisture movement to or from the ambient environment is prevented and thesecond, associated to drying. Basic creep is described by a logarithmic function in time, so it is notlimited, while drying creep, predicted by a hyperbolic exponential form is limited. In this way twomajor shortcomings present in MC 90 model i.e. the possibility of deriving negative values from theconvolution integral and the developing of negative delayed elasticity during loading-unloadingprocess are avoided.

Regarding the results deriving from structural analysis performed adopting the two models we observethat they give final values of creep coefficients very close so, when homogeneous viscoelasticstructures interacting with elastic restraints are dealt with, the limitation of the creep deformationoperated by the elastic restraint reduces in a major extent the differences existing between the twocreep models, making them quite similar for this purpose.

7. REFERENCES

/1/ fib Model Code 2010,Vol.1, Final Draft, fib bulletin 66, 2012/2/ Comit Eurointernational du Bton CEB/FIP Model Code 90, Design Code, Thomas Thelford,

London 1993/3/ Findley W. N., Lai J. S., Onaran K., Creep and Relaxation of Non Linear Viscoelastic Materials,

with an Introduction to Linear Viscoelasticity , North Holland, Amsterdam, 1976/4/ Dischinger, F., Untersuchungen uber die Knicksichereit, die Elastische Verformung und das

Kriechen des Betons bei Bogenbrucken, Der Bauingenieur, H. 33/34, H. 35/36, H. 39/40, 1937 (inGerman)

/5/ Arutyunyan N. Kh., Applications de la thorie du Fluage , Eyrolles, Paris, 1957 (in French)

/6/ Comit Eurointernational du Bton CEB/FIP, Code Modle pour les structures en bton, CEBBulletin 124/125F, Paris, 1978 (in French)

/7/ Mc Henry D., A new Aspect of Creep in Concrete and its Application to Design , Proc. ASTM,1943

/8/ Volterra V., Leons sur les Fonctions des Lignes , Gauthiers-Villars, Paris, 1913 (in French)/9/ Nielsen L. F., On the Applicability of Modified Dischinger Equations , Cement and Concrete

Research, Vol.7-1977, Vol.8-1978/10/ Rsch H., Hilsdorf H., Jungwirth D., Kritische Sichtung der Verfahren zur Berucksichtigung der

Einflusse von Kriechen und Schwinden des Betons , Beton und Stahlbetonbau, Vol. 68, 1973 (inGerman)

/11/ Chiorino et al., Structural Effects of Time-dependent Behaviour of Concrete , CEB Bulletin142/142 bis, Georgi, St. Saphorin, CH, 1984

/12/ Mola F., The Reduced Relaxation Function Method, an Innovative Approach to Creep Analysis in

Non Homogeneous Structures, International Conference on Concrete & Structures, Hong Kong,1993

/13/ Mola,F., Long Term Analysis of R.C. and P.C. Structures according to Eurocode2 , Proc. of theInternational ECSN (European Concrete Standard in Practice), Symp., Copenhagen, 1997


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