THE NEW MODEL FOR CREEP OF CONCRETE IN FIP … NEW MODEL FOR CREEP OF CONCRETE IN FIP MODEL CODE...

THE NEW MODEL FOR CREEP OF CONCRETE IN FIP MODEL CODE 2010

F. Mola, Politecnico di Milano, Italy

L.M. Pellegrini, ECSD, Italy

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37th Conference on Our World in Concrete & Structures

Singapore, August 29-31, 2012

__________________________

†ECSD, Srl, Milano, Italy

THE NEW MODEL FOR CREEP OF CONCRETE IN FIP 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 improved formulations to define the model for the long-term behaviour of concrete. In this way some inconsistencies present in the older CEB/FIP Model Code 1990 (MC 90) have been removed. In the new formulation the total creep deformation of concrete has been introduced as sum of two contributions: basic and drying creep, for each of which a specific time evolution function is provided. In the present paper, after a brief recall of the basic principles of linear viscoelasticity, the differences between the new and the previous formulation will be firstly carried out. Then, with reference to a case study regarding a homogeneous viscoelastic structure interacting with elastic restraints, the results derived from structural analysis performed by using MC 10 and MC 90 will be compared.

1. INTRODUCTION The new fib Model Code 2010, /1/, introduced new and improved formulations to define the model for

the long-term behaviour of concrete in service life loading conditions. In comparison with the older CEB/FIP Model Code 90, /2/, new functions have been implemented, allowing some interesting aspects of the long term behaviour of concrete to be better clarified. In particular, long term deformations 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 is assumed to depend on the mean strength of concrete, whereas the latter depends on the thickness of the concrete element and the external humidity. In the new MC 10, a method to take into account the effects of temperature on the time of loading is also suggested, as well as modified values for the creep coefficient, in a non linear form, so that also situations where the stress values in concrete are more 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 prerequisites of the limiting models i.e. the non ageing model of Kelvin – Voigt, /3/, the ageing model of Dischinger, /4/, and the composite one of Arutyunyan, /5/, the models proposed in Europe in the three 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 of MC 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.

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 that

period the most important problem involving creep in structural analysis was the evaluation of the evolution 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 displacements and 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 time dependent form according to the mathematical approach stated by Volterra, /8/, the problem connected to structural analysis of concrete structures was not properly investigated. Only a certain number of special cases had been studied without deriving a consistent and reliable formulation governing the time dependent structural analysis of concrete structures.

The relationship introduced by Mc Henry was:

�� · ��, ��

�� · ��, �� · �� (1)

where ε(t) is the total deformation of concrete under a variable stress which can be written in the form: �� (2)

having indicated by εe(t) the elastic, instantaneous part of the deformation and by εc(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 develop independently of the applied stress, the preceding expression can be generalized in the subsequent way:

�� · ��, ��

�� · ��, �� · �� (3)

�� (4)

where εσ(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 the discontinuous 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 of concrete structures.

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

ε(t) and the stress σ(t) which can be solved with respect σ(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 Volterra integral equation:

� ��,�� · ��, �� · �� 1�

� (6)

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

following expression:

�� !�� "�� · #��, �� (7)


Eqs. (5), (6), (7) describe in a consistent way the long term behaviour of concrete according to

viscoelastic theory and can be adopted in structural analysis. From the mechanical point of view, we observe that the viscoelastic model is described by two functions, namely J(t,t’), R(t,t’) respectively representing by virtue of eqs. (1), (7) the strain and the stress generated by a unit stress or by a unit deformation applied at time t’.

J(t,t’) and R(t,t’) are respectively called Creep Function and Relaxation Function and are connected by the integral relationship (6) defining the so called convolution integral. J(t,t’) and R(t,t’) can be expressed by means of the subsequent relationship:

��, �� $%�� · !1 � &��, ��" (8)

#��, �� '�� · !1 � (��, ��" (9) where E(t’) is the elastic modulus and &��, ��, (��, �� respectively represent the ratio between the

delayed deformation or the delayed stress and the related elastic parts. It is noteworthy to observe that the creep deformation increases monotonically in time in a more pronounced way when the load is applied at early age, so that for the creep function the following inequalities hold:

�)��,��

�� * 0

�)��,�� , 0 (10)

&��, �� 0, &��, �� - 0, ��, �� 1/'��

From eqs. (8), (9), remembering eq. (6), (10), we immediately obtain $

)��,�� #��, �� '��, and for the Relaxation Function we can write the subsequent inequalities:

��,�� , 0

��,��

�� * 0 (11)

(��, �� 0, (��, �� / 0, #��, �� '�� The now briefly discussed properties of linear viscoelasticity point out that the knowledge of the Creep

Function J(t,t’) is enough to define the long term behaviour of concrete, so that in order to correctly proceed the definition of the Creep Function becomes mandatory.

The research towards a reliable function J(t,t’), able to refinedly describe the creep behaviour of concrete represented, and also nowadays still represents, one of the most investigated problems in concrete experimental and theoretical analysis, so that new models are frequently suggested, as is the case of the model illustrated in the final draft of fib MC 10.

Before entering in discussing the most important Creep Models adopted in Europe in the last decades it is interesting to point out a very outstanding property of the linear viscoelastic law (5). For this let us assume � �� 0 and derive with respect t the two members of eq. (5). We obtain:

0��

� � �� · ��, �� · �)��,��

��

� (12)

Eq. (12) points out that the integral form (5) cannot be in general reduced to a differential form as the

derivative of J(t,t’) with respect t depends on the two variables t, t’. Consequently, by differentiating eq. (5) with respect t we obtain a new integral form in t, t’.

In order to transform eq. (5) in a differential form, a special formulation for the J(t,t’) function is needed. At this point, let us suppose that function J(t,t’) satisfies the mass-action law expressed in mathematical form by the following relationship:

�)��,��

�� 1 · !�∞, �� , ��" (13)


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 α > 0 to the quantity of creep has to be developed from t to t = ∞. The validity of eq. (12) requires that:

lim�67 8�)��,�� 8 � 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: 0��

� � �� · $

%�� 1 · � �� · �∞, �� 1 · �� (15)

so differentiating eq.(15) we finally obtain 90��

�9 � 1 · 0��

� · : $%�� · ��

� ; � 1 · �� · �∞, �� (16)

Eq. (16) represents a linear differential equation with variable coefficients for which the principle of

superposition 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 the McHenry form (5) can be exactly applied. For other models, not satisfying eq. (13), eq. (5) represents only an approximate form and its application in some cases can drive to inconsistent results. For this application 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 the

Kelvin-Voigt model exhibiting the following Creep Function:

<=��, �� $% � >?

% · �1 � @A��A�� B⁄ � (17)

where E, constant in time, is the elastic modulus, ϕ∞ is the final value of the creep coefficient and τ* is the retardation time.

Introducing eq. (17) in eq. (13) we obtain: >?%·�B · @A��A�� B⁄ � >?

% · @A��A�� B⁄ · 1 (18)

so, assuming 1 � $�B , 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:


lim�67 &<=��, �� &7 (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 express eq. (5) in the following form:

��∞� � ��∞� · �$D>?�% � ��∞� (20)

so that assuming that ��∞� / ∞, and introducing the effective modulus '� � %�$D>?� , eq. (20) can be

expressed by means of the following pseudo-elastic form:

��∞� � ��7�%� � ��∞� (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 time development 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 E�)��8�F�� >?

%·�B .

Neverless the experimental results show that E�)��8�F�� G ∞, so to satisfy this property �B G 0 is needed.

This assumption, according to eq. (17), would give <=��, �� $D>?% so the time development of

creep cannot be described. This is essentially the basic reason which does not allow to assume the creep function defined by an exponential form.

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

H��, �� $%�� >?

%9I · !�@AJ��A�� @AJ��A��" (22)

where: E(t’) is the elastic modulus at loading time t’ E28 is the elastic modulus at t = 28 days

1/β is the retardation time �� is the first possible loading time. Introducing eq. (22) in eq. (13) we derive: >?%9I · K · @AJ��A�� >?

% · @AJ��A�� (23)

so that assuming α = β the (D) model satisfies the mass-action law and allows the application of McHenry principle of superposition in an exact way. Furthermore for � 6 ∞ it results:

lim�67 H��, �� $%�� >?

%9I · @AJ��A�� (24)

showing that the final value of the creep deformation decreases with the time of loading up to become

zero when �� 6 ∞. This property defines the creep model as an only ageing material. It is important to observe that for this model also we see that it is not possible to comply with the experimental evidence of having a creep function rapidly increasing after the loading time. We can so conclude that (KV) and (D) models even though they satisfy the mass-action law and strictly comply the principle of superposition, they can describe only two limit behaviour of creep concrete i.e. the non ageing and the ageing material. Furthermore they cannot comply with the experimental results showing a rapid increase of the creep function after the loading time. A model able to strictly satisfy the principle of superposition and to describe both non ageing and ageing behaviours was introduced by Arutyunyan.


The creep function assumed by Arutyunyan is:

L��, �� $%�� M&7�%� � >?�N�

�� O · P1 � @AQ��A��R (25)

where &7�%�, &7�L�

are respectively the non ageing and the ageing part of the creep deformation and 1/γ is the retardation time. The Arutyunyan model introduced in eq. (13) gives:

M&7�%� � >?�N�� O · S · @AQ��A�� 1 · M&7�%� � >?�N�

�� O · @AQ��A�� (26)

so that assuming α = γ the model satisfies the mass-action law and strictly complies the superposition principle. Despite the capability of taking into account the non ageing and the ageing properties of the creep deformation the exponential form of the eq. (25) prevents also in this case to properly describe 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 of concrete. On the other hand rearranging eq. (13) in the form:

�)��,��

�� 1 · ��, �� 1 · �∞, �� (27)

the solution of the differential eq. (27) can be expressed in the subsequent way: ��, �� T@AU� � �∞, �� (28) with the initial condition:

��, �� $%�� (29)

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

��, �� : $%�� ∞, ��; · @AU��A�� ∞, �� (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 the principle of superposition are not suitable for describing the rapid time development of the creep law in times near the loading time. To do this different models have to be chosen, exhibiting a creep law expressed in the integral form (5) which cannot be reduced to a differential form. This kind of models do not strictly satisfy the principle of superposition, so that some shortcomings can take place and their 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-action

law 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 in

time up to zero value for t = ∞. The deformation of (D) model remains constant and the one of (A)

1 1

1 1 1

1


model reduces but does not reach zero for t = ∞. Indicating by “delayed elasticity” the deformation developing after the unloading we can define (KV) model a total reversible model, (D) model an irreversible 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 were

suggested in Europe by Nielsen, /9/, Rüsch Hilsdorf and Jungwirth, /10/, and CEB/FIP Model Code 1978. The creep law adopted in MC 78 was a generalization of the (KV) and (D) models, expressed in a sum form according to the subsequent expression:

��, �� $%�� $

%9I · &��, �� (31)

&��, �� KV�� & · K �� &W · PKW�� KW��R (32)

The three terms at right member of eq. (32) respectively represent: KV�� the sudden creep deformation at time of loading; K �� the developing in time of the non ageing part of creep; KW�� KW�� the developing in time of the ageing part of creep.

The corresponding expressions were as follows:

KV�� 0.8 · :1 � WZ[��WZ[�7�; & � 0.4

K �� : �A��A��D]^_;$/`.^

&W � &W$ · &W^ &W$ � 4.45 � 0.035 · #c (33)

&W^ � @d`.`·$�ef·g�A�.hfij� Akl mj��.nooi

9.o pq

KW�� : ��D<n�g��;

<9�g��

r$�s�� @:f.�9j� Dkl �t.uv·g�n.9f;

r^�s�� @:�.��$``·g�An.nj�Akl �$.��v·g��.9wfx;

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


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 Jy��

%9I . This way of

operating is debetable as creep is not an istantaneous deformation, anyhow the assumption made allowed to directly use the two ageing and non ageing models, for which different time functions with respect 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 the notional thickness h0. Some results taken from an extended analytical investigation performed by Chiorino et al., /11/, are reported in Fig.3-4.

4.2 The Model Code 1990 The shortcomings deriving from the introduction of an initial creep deformation were eliminated in the

CEB/FIP Model Code 90, where the creep model was calibrated on a product function having the form:

��, �� $%�� $

%9I · &��, �� &��, �� &�� · K�� &�� &�z · K�{�|� · K��

&�z � 1 � $A�z �z�⁄�.`t·�g g�⁄ �n/h

K�{�|� � v.]�WZ[ WZ[�⁄ ��.f (34)

K�� $�.$D�� n⁄ ��.9

h = 2Ac/u

K�� : ��A�� n⁄J}D��A�� n⁄ ;�.]

Kz � 150 · M1 � ~1.2 · �z�z��$_O · g

g� � 250 , 1500 In the set of eqs. (34) fcm, RH, h respectively represent the mean value of the compressive strength at

28 days, the relative humidity expressed in percent, the notional thickness, ratio between the area of the cross section and the half of the perimeter in contact with the atmosphere. RH0, h0, t1, are prescribed reference values i.e. RH0 =100%; h0 = 100 mm; t1 = 1 day.

Fig. 5 – Creep Function, MC 1990 Fig. 6 – Relaxation Function, MC 1990 The MC 90 introduces some important prerequisites. First of all we can easily observe that it results:

lim�6�� )��,�� ∞ (35)


and according to eq. (6)

lim�6�� ,�� ∞ (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 � 6 ��. As a second point we observe that the two functions K��, K�� , the first expressing the creep coefficient for � 6 ∞ and the second describing the developing in time of the creep function, are both depending on the relative humidity and by the notional thickness. This means that the creep deformation is thoroughly accounted as drying creep, neglecting to express the basic part of the total creep deformation.

4.3 The Model Code 2010 The creep MC 90 model even though it represented an improvement with respect MC 78 model, was

still lacking of important prerequisites. The inconsistencies of MC 90 have been accommodated in the new fib MC 10 creep Model. The two

most important additions introduced regard the separation of the total creep deformation in the two contributions 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 function of fib MC 10 model is:

��, �� $%�� $

%9I · !&��, �� & ��, ��" (37)

with &��, ��, & ��, �� basic creep and drying creep part of the creep coefficient &��, �� having

expression: &��, �� &��, �� & ��, �� (38) where &��, �� K��{�|� · K��, ��

K��{�|� � $._�WZ[��.i

K��, �� d� ]��y�� 0.035�^ · �� 1q

& ��, �� K ��{�|� · K�#c� · K �� · K ��, �� (39)

K ��{�|� � `$^�WZ[�n.x

K�#c� � $A�z $��⁄��.$·g $��⁄h

K �� $�.$D~�y�� .9

K ��, �� : ��A��J�D��A��;

Q��

S�� $^.]D h.f

��y��

K� � 1.5 · s � 250 · 1WZ[ , 1500 · 1WZ[

1WZ[ � ~ ]vWZ[��.v


In the set of eqs. (39) fcm, RH, h respectively represent the mean compressive strength at 28 days in

MPa, 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:

&��, �� &��, ��@!$.v·��A�.`�" 0.4 / �� , 0.6 �� |�|WZ[�� (40)

Fig. 7 – Creep Function, MC 2010 Fig. 8 – Relaxation Function, MC 2010 Fig.7 shows that the basic creep function is not limited in time as it results lim�67 K��, �� ∞, while

for the drying creep we have lim�67 K ��, �� 1. In fact it is not known in advance if creep approaches a finite value so the introduction of a creep coefficient expressed by a logarithmic form has been suggested in order to predict with good approximation the behaviour of concrete up to fifty years of loading. From the mathematical point of view the introduction of an unlimited creep function does not guarantee the existence and the uniqueness of the solution of eq. (6). In any case an unlimited creep function drives to the following result:

lim�67 #��, �� 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 in time, eq. (41) allows to observe that for a creep model exhibiting an unlimited creep function the related 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 the creep after 50 years. We can so conclude that the introduction of an unlimited creep function allows to better approximate the concrete behaviour, does not introduce significant numerical inconsistencies in the numerical solution of eq. (6) and avoids the physically absurd result of obtaining 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 t0 = 14 days and t0 = 28 days. Fig.9 deals with a concrete having fck = 32 MPa; h0 = 200 mm; RH = 70%. We observe that MC 10 curves develop in a more straightforward form and give slightly higher values of the creep coefficient for t = 10

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

faster way up to about 104 days, 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 are smaller for MC 10. This result shows for this model an higher negative correlation between creep and compressive strength.


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 that

for concrete C32/40, the final values of the relaxation functions are smaller for MC 90 model as for this model the final values of creep are higher. On the contrary for concrete C60/75 this trend is inverted and for t0 = 14 days the final values of R (t,t0)/E28 are 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 the

improvements introduced by MC 10 model. In Fig. 13 the non dimensional deformation during a loading-unloading process is reported for the two models. We see that for the MC 90 model the variation in time of the deformation after the removal of loading is initially decreasing then it increases up to an asymptotical value. So the model shows a negative delayed elasticity which is not possible from a thermodynamical point of view. On the contrary the MC 10 shows a continuous decrease of the deformation connected to a positive delayed elasticity. In Fig. 14 we see that in MC 90 model for a C30/37 concrete loaded at very early age the relaxation function can become negative so violating the principles of thermodynamics. In a different way MC 10 shows a monotonically decreasing diagram of the relaxation function which remains positive, as required by the 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 physically

consistent 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 relaxation curves make both of them suitable for structural analysis. This point is illustrated in Fig.15-16, where for 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 are reported.


Fig. 15 – Varied Creep Functions Fig. 16 – Reduced Relaxation Functions Functions R*, /12/, allow to solve problems related to homogeneous viscoelastic structures interacting

with 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 Xe the reaction of the elastic restraint at initial time, the solution at time t

can be expressed as follows /13/:

�� :1 � �� 1� · �B��,��

%9I ; (43)

where ω is the coupling coefficient, given by the expression:

� � �nZ�nZD�n� (44)

with �$�, �$� respectively the elastic influence coefficients of the structure and of the elastic restraint.

Assuming for the case study �$� � �$� we obtain ω = 0.5, so from eq. (43) we reach:

�� 2 · �� · :1 � 0.5 · �B��,��%9I ; (45)

from Fig. 16, for ω = 0.5 we obtain: �� w�B �$�f,^_�

%9I � 0.451 �� n�B �$�f,^_�%9I � 0.394

so the final values of the reaction of the elastic restraint become: � �� u��10v� � 2 · �� · !1 � 0.5 · 0.451" � 1.55 · �� $��10v� � 2 · �� · !1 � 0.5 · 0.394" � 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: � �� u��10v�/� �� $��10v� � 0.963 �� u�|V¡ �10v� �� $�|V¡ �10v�⁄ � 1.154 showing how the two models can be retained quite similar as regards structural analysis.

4M/Ql 1

0.45

0.39


6. CONCLUSIONS The new fib MC 10 creep model introduces important improvements with respect the previous fib MC 90

model. Even though the two models do not differ in a marked extent in predicting the final values of the 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 10 introduces total creep deformation as the sum of two contributions, basic and drying creep. The first one, developing when moisture movement to or from the ambient environment is prevented and the second, associated to drying. Basic creep is described by a logarithmic function in time, so it is not limited, while drying creep, predicted by a hyperbolic exponential form is limited. In this way two major shortcomings present in MC 90 model i.e. the possibility of deriving negative values from the convolution integral and the developing of negative delayed elasticity during loading-unloading process are avoided.

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

7. REFERENCES /1/ fib Model Code 2010,Vol.1, Final Draft, fib bulletin 66, 2012 /2/ Comité Eurointernational du Béton 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 (in German)

/5/ Arutyunyan N. Kh., « Applications de la théorie du Fluage », Eyrolles, Paris, 1957 (in French) /6/ Comité Eurointernational du Béton CEB/FIP, Code Modèle pour les structures en béton, CEB

Bulletin 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., « Leçons 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/ Rüsch 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 (in German)

/11/ Chiorino et al., « Structural Effects of Time-dependent Behaviour of Concrete », CEB Bulletin 142/142 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 the International ECSN (European Concrete Standard in Practice), Symp., Copenhagen, 1997

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