Models of Linear Creep

1.2 Creep

1.2.3 Models of Linear Creep

For the study of the phenomenon of creep, models of theoretical mechanics have been initially applied. Starting from the basic ones consisting of Hooke’sspringand Newton’sdamper, the former being transposed into the linear relationship between force and displacement, the latter into the linear relationship between force and velocity, the fundamental combinations have been used (the one in series by Maxwell and the one in parallel by Kelvin) in order to formulate composite schemes able to simulate the principal characteristics of the phenomenon.

Hereditary models are for example quoted, derived from the scheme by Voigt or the one by Zerner (see Fig.1.19a, b), which have the deformation law such as

Fig. 1.19 Mechanical models for creep

eðt;toÞ ¼ coþc1 1e^bðttoÞ

h i

n o

;

where the constantsc_o,c₁andbdepend on the characteristics of spring and damper.

The Extreme Theoretical Models

Mechanical models were not able to simulate certain important aspects of the phenomenon of creep, such as the ones related to ageing. Other theoretical models have been therefore formulated mathematically. Some, as the one by Dishinger presented hereafter, take into account the maturation of creep characteristics with the concrete ageing. Among these the two theoretical models that represent the extreme interpretations of the creep behaviour are further discussed.

A first model is deduced from the observation of how, for concrete at early stages, the trend of creep curves at a given instant, that is the speed of development, is significantly independent from the past duration of loading (see Fig.1.20a). The slope of diagrams would therefore depend only on the timetof measurement and not on the timet_oof load application:

@uðt;t_oÞ

@t ¼CðtÞ:

Fig. 1.20 Extreme theoretical models for creep

From this equation, integrating betweentoand tone obtains uðt;t_oÞ ¼cðtÞ cðt_oÞ:

In practice forc(t), setting the origin of time at the minimum age at which the first loading of concrete is possible, an exponential law typical of exhaustion phenomena is assumed. Adding the elastic part, the creep function becomes

vðt;toÞ ¼ 1 Eo

1þu_oe^btoe^bt

; wherebis related to the fading speed of the phenomenon.

This model corresponds to theextreme theory of ageingby Dishinger–Whitney, according to which the final value u_∞ of the creep coefficient decreases expo- nentially with the age of load application with respect to the one uo of the fist possible event:

u₁¼u_oe^bto:

It can also be noted how, applying the superposition principle for a loading and unloading event (see Fig.1.20b), only the irreversible part of the strain remains after unloading. The Dishinger–Whitney model is therefore not able to represent the delayed elasticity.

A second model is deduced from the observation of how, for very aged con- cretes, the creep curves remain substantially the same for every successive event (see Fig.1.20c). The deformation e at time t of measurement would therefore depend only on the loading durationt−t_oand the curves relative to events started at successive times would simply be translated along thex-axis, instead of along the y-axis as for the previous model. Assuming the usual exponential function and adding the elastic part one has

vðt;toÞ ¼ 1

E_o 1þu₁ 1e^bðttoÞ

h i

n o

;

where the creep coefficient at infinite time remains the same for every successive loading event. This model corresponds to theextreme hereditary theoryby Kelvin– Voigt. Applying the superposition principle for a loading and unloading event (see Fig.1.19d), after unloading, one obtains the slow complete release of every strain.

Kelvin–Voigt model is therefore capable of representing only the delayed elasticity and not the irreversible part of the residual strain.

Between the two extreme models, the one of themodified hereditary theories can be proposed associating a coefficientu₁ ¼u₁ðtoÞas a function of the loading time to the law expressed in terms of the durationt−t_o. It can be set, for example,

vðt;toÞ ¼ 1 Eo

1þu_oe^ato 1e^bðttoÞ

h i

n o

:

According to such model, successive loading events have similar but reduced creep curves.

The interest of extreme or modified models, for the viscoelastic behaviour of concrete, lies in the simplicity of their analytical expression, which allows in several cases the formal integration of the solving equations of the studied problems. The approximations of related results are more or less technically acceptable, given also the incertitude related to the assumption on the correct values of the parametersE_c andu_∞.

Empirical Models

Experimental results, as they became available, allowed to formulate empirical models capable of representing the various complex aspects of creep more accu- rately. It is to be noted that the relative experimentation is quite onerous. First of all it requires long durations, sometimes up to 30 years of loading. For a correct interpretation of results, it is necessary to adopt adequate measures in order to remove shrinkage deformation from the measurements and to distinguish different contributions. The study of the influence of ambient parameters for maturity and dimensional parameters for concrete shape presents relevant difficulties because of the number and the interdependence of the parameters themselves.

The one proposed by CEB–FIP MC 90 (see bull. CEB 213) can therefore be defined as amodified hereditary modelsince, withu_∞= u_∞(t_o), afinal amplitude decreasing with the concrete age at loading is associated to a hereditary function of growth in durationg(t− t_o), as the ageing theory requires.

The creep law is therefore expressed with the coefficient uðt;toÞ ¼u₁ðtoÞgðt;toÞ;

having in its two factors the main parameters that influence the phenomenon. The final value is further composed of three factors:

u₁ðt_oÞ ¼b_cb_hsu_o;

whereuo=uo(t_o) is thereference coefficientwhich gives, as a function of the age t_o at loading, the values relative to a standard situation (strength f_c= 28 MPa, relative humidity RH = 80%, equivalent thickness 2A_c/u = 150 mm).

Defining also

c¼fc=10 index of concrete classðkN=cm²Þ h¼RH=100 relative humidity ratio

s¼ ð2Ac=uÞ=100 index of equivalent thicknessðdmÞ

the following formulas are given:

b_c¼1:673 ffiffiffic

p ð¼1 forc¼2:8Þ

b_hs¼0:725 1þ 1h 0:46 ffiffi

s p3

ð¼1 forh¼0:8 ands¼1:5Þ u_o¼ 4:37

0:1þt⁰_o^:2 ð¼u₁forb_c¼b_hs¼1Þ:

In Tables1.12,1.13 and1.14, the numerical values of the above defined coef- ficientsbc,bhsanduoare reported. In particular for the calculation of the reference coefficientuoanominal age toof load application has to be assumed, correcting the effective ageto based on the average temperatureh of concrete in the time frame.

One therefore obtains

to¼b_Tto ðto ¼to forb_T ¼1Þ for

b_T¼e ^13:65

4000 273þh

ðb_T¼1 forh¼20 CÞ:

This last formula (or the related Table1.15) allows to take into account the effect of the accelerated maturation with a simple translation towards higher times of the loading age with whichu_o= u_o(t_o) is to be read.

The formulas reported above are given with fairly good reliability based on numerous experimental verifications that have been carried out (variance0.20).

Relevant incertitude remains in their application, related to the assumption at the design stage of the values of the parameters.

The calculations of creep effects are normally carried out in two extreme situations corresponding, respectively, to the initial stage withu= 0 and thefinal stage with u=u_∞. The first situation is analysed with elastic algorithms. The viscoelastic analysis in thefinal stage, or in the intermediate stages if required, requires the time functiong(t− t_o) on which the relative integrations are to be made. For such time law the available model is much less reliable than the others, especially at short terms.

The problem can be overcome if the approximations of a simplified analysis method are accepted, such as the one of the effective moduli presented at the end of this paragraph. In this case it is not necessary to know the creep time law; the value of itsfinal coefficient is sufficient.

The model proposed by CEB is anyway reported:

gðtt_oÞ ¼ ðttoÞ tþ ðttoÞ

0:3

;

where the parameter

t¼150 1h þ ð1:2hÞ¹⁸i

sþ250ðdaysÞ

can be assumed equal to 500 for the most common environmental and structural situations.

Design Nominal Values

For application purposes, nominal estimations of the creepfinal coefficient can be assumed at the design stage, conventionally referred to some standard situations.

Values ofu_∞(t_o) are reported hereafter for a nominal age of loadingt_o= 14 days and for a relative humidity UR = 60%, considering three representative classes, respectively, of low, medium and high strength, combined with equivalent thick- ness values between small and medium.

s= 1.0 s= 2.0 s= 3.0

C20/25 3.30 2.98 2.83

C35/43 2.66 2.41 2.28

C50/60 2.29 2.07 1.97