Creep in Concrete

Abstract

Creep in concrete is a long-term deformation under sustained loading. It’s influenced by many factors including constituent materials, environmental conditions among others. Whenever there is an alteration in the convectional concrete preparation process, the creep characteristics need to be realistically assessed. Creep prediction models have been developed for determining creep in convectional plain concrete. It has been shown that creep changes when constituents of concrete production are changed hence realistic assessment needs to be done. Creep of some modified concrete have been presented in this chapter.

1. Introduction

1.1 Concrete definition

Concrete is one of the most used building materials. It is a composite material made from readily available constituents that is aggregates, sand, cement, and water. Concrete is a versatile material and can be mixed to meet a variety of special needs and formed to virtually any shape. There has been a growing demand of concrete worldwide with 27 billion tonnes of concrete produced annually and the amount expected to be four times in 2050 than it was in 1990 [1]. This has put pressure on the existing natural resources for its production and threatening their depletion. It has also raised environmental concerns on sustainability of concrete production by convectional materials.

With the above concerns, research has been undertaken on alternative ingredients for production of concrete. Many industrial wastes have been studied and found that some possess similar properties to the constituents of concrete. These industrial wastes have been recycled as substitutes of either cement or aggregates. They include fly ash slag, ground granulated blast furnace slag (GGBS), waste glass and even ground vehicle tires among others. Thus, concrete production can significantly reduce environmental impacts of industrial waste whilst improving the properties of concrete.

1.2 Creep definition

Creep is defined as increase in strain under a sustained constant stress after considering other time-dependent deformations not associated with stress e.g., shrinkage and swelling. When hardened concrete is loaded, it deforms; partly because of elastic strain and partly because of plastic strain or permanent deformation as shown in the Figure 1.

Figure 1.

Time dependent creep deformation.

Creep testing is conducted on unsealed and sealed specimens with unsealed being the most used method of creep testing. These specimens, without an applied stress have volumetric changes due to drying and autogenous shrinkage. Hence, the total deformation of unsealed specimens is because of an applied stress producing elastic defamation, shrinkage and creep. Creep includes drying and basic creep; whereby, drying creep is the total deformation minus elastic deformation whereas basic creep is total deformation of loaded, sealed specimens minus elastic deformation and autogenous shrinkage. Autogenous shrinkage and basic creep require testing of both sealed and unsealed specimens [2].

2. Mechanism of creep

Creep of concrete is influenced by all its constituents as well as the loading time, duration of loading and environmental conditions. In general, the constituents used, age of loading, type and duration of loading mainly affect magnitude of creep. Environmental conditions affect not only amplitude but also the development of creep. Concrete behaves as a viscoelastic material with early age at loading and longer loading duration increasing creep.

Aggregates being stiffer than hydrated cement paste do not creep. Hence, a higher content of stiffer aggregates of notional size restrains concrete creep. Creep is also related to cement paste content though the relationship is not linear. Other relevant factors of concrete creep are; water-cement ratio, type of cement and its fineness, compressive strength, stress-strength ratio, environmental conditions, size of structural a member and admixtures and additives [3].

The long-term creep deformation mechanism in cement gel is that involving narrowing of intercrystallite spaces. On application of load, there is instantaneous elastic response from both the solid and liquid systems of the concrete matrix. Under sustained load the compressed liquid begins to migrate from higher to lower stressed areas. This is accompanied by the transfer of load from liquid to the surrounding solid. It is believed that after several days of sustained load pressure on capillary water disappears. As the process of hydration is progressing; growth of solid phase and the expense of liquid phase gradually changes the parameter governing extent and rate of creep [4].

3. Mathematical modelling of creep

The necessity of adapting any creep theory for the practical calculations of concrete constructions requires the presentation of the relationships between concrete deformations and time, in the form mathematical functions. These mathematical functions should be in good conformity with experimental results.

Presentation of creep functions should satisfy some determined conditions. The creep function denoted by C (t, τ) expresses creep deformation at any moment caused by unit stress at a time τ. These established conditions are given in Eqs. (1)–(7) as follows;

1. Creep at the time of loading is equal to zero for concrete of any age.

   $$C\left(t,t\right)=C\left(t,\tau \right)\cong 0$$E1

2. For all time ‘t’ whereby t > τ; creep is more than 0.

   $$C\left(t,\tau \right)>0\mathit{\text{for}}t>\tau$$E2

3. The rate of increase of creep deformation at constant time loading diminishes with increase of ‘t’ and equals zero at ‘t’ tends to infinity.

   $$\underset{t\to \infty }{lim}\frac{\partial C\left(t,\tau \right)}{\partial t}=0$$E3

4. The rate of decrease of creep after unloading diminishes with increasing ‘t’ and for $t=\infty$ the value is zero. However, the creep function diminishes at the same time.

5. The creep function diminishes uniformly with increasing concrete age.

$$\frac{\partial C\left(t,\tau \right)}{\partial \tau }<0$$E4

The most general creep theory of concrete developed by several authors assumes partial deformation reversibility. Its course it represented in the Figure 2; whereby the curve the unloading tends to a determined final value rather than zero.

Figure 2.

General form of creep theory.

The expression $\phi \left(\tau \right)$ is introduced to the creep function, in the form of

$$C\left(t,\tau \right)=\phi \left(\tau \right).f\left(t-\tau \right)$$E5

Ageing of concrete is also considered. The expression $f\left(t-\tau \right)$ is a loading duration function and allows partial reversibility deformations to be considered. Function $\phi \left(\tau \right)$ accepts the form;

$$\phi \left(\tau \right)=\left(C_0+\frac{A_1}{\tau }\right)$$E6

Whereby C₀ is final value for τ → ∞

Finally, the creep theory function is of the form;

$$C\left(t,\tau \right)=\left(C_0+\frac{A_1}{\tau }\right)\left[1-e^{-y\left(t-\tau \right)}\right]$$E7

Whereby C₀, A₁ and y are determined in such a way that the theoretical and experimental curve match [5]. Creep functions lead to integral equations which after certain transformations can be reduced to easily solvable differential equations.

4. Creep prediction models

Various creep prediction models have been developed by researchers based on a large number of tests. The most commonly used models are; ACI-209 model developed by American Concrete institute. CEB-FIB model developed by Euro-international concrete committee. B3 model developed by Bazant Z. P and S. Baweja and the GL2000 model developed by Gardener and Lockman. All these models are fitting or empirical formulas from test data. These models look at free shrinkage, creep strain and elastic deformations. They relate creep strain to the loading conditions by using either creep compliance, specific creep or creep coefficient. Each model has its own reasonable explanation method based on theoretical evaluation and test data and therefore the explanation of their results has different formulas [6].

4.1 ACI model

Its procedure is applicable to normal and lightweight concrete (using types I and III cement, moist and steam-cured conditions) under standard conditions. This model is sensitive to water content.

Its mathematical formulations for elastic modulus, creep compliance and specific creep are shown in Eqs. (8)–(10) [7];

$$E_c\left(t\right)=0.04326\sqrt{{\rho }^3{f}_{\mathit{cm}}\left(\frac{t}{a+\mathit{bt}}\right)}$$E8

$$J\left(t,t_0\right)=\left[\frac{1+\mathrm{Ф}\left(t,t_0\right)}{E_c\left(t_0\right)}\right]$$E9

$$\mathrm{Ф}\left(t,t_0\right)=\frac{v_u{\left(t-t_o\right)}^{\phi }}{d+{\left(t-t_0\right)}^{\phi }}$$E10

4.2 CEB-FIB

Its prediction is restricted to ordinary structural concrete with 28 days mean cylinder strength ranging between 12 and 80 MPa, mean temperature of 5–30°C and mean relative humidity of 40–100%. This model is sensitive to relative humidity. The elastic modulus and creep compliance functions are given in Eqs. (11)–(15) as follows [7];

$$E_c\left(t_0\right)=E_c\sqrt{\mathit{exp}\left\{s\left(1-\frac{18}{t_o}\right)\right\}}$$E11

$$J\left(t,t_0\right)=\frac{1}{E_c\left(t_0\right)}+\frac{\mathrm{Ф}\left(t,t_0\right)}{E_c}$$E12

$$=\frac{1}{E_c\left(t_0\right)}+\frac{\mathrm{Ф}\left(t,t_0\right)}{E_c}$$E13

$$\mathrm{Ф}\left(t,t_0\right)=\left[1+\frac{1-h}{0.46{\left(\frac{2A_c}{100\mu }\right)}^{\frac{1}{3}}}\right]\left[\frac{5.3}{\sqrt{0.1f_{\mathit{cm}}}}\right]\left[\frac{1}{0.1+{\left(t_0\right)}^{0.2}}\right]\ast \left[\frac{{\left(t-t_0\right)}^3}{\left[{\beta }_H+{\left(t-t_0\right)}^3\right]}\right]$$E14

$${\beta }_H=150\left[1+{\left(1.2h\right)}^{18}\right]\left(\frac{2A_c}{100\mu }\right)+250\le 1.50000$$E15

4.3 B3 model

The prediction of material parameters for this model is limited to Portland cement concretes having w/c ratio of 0.30–0.85, a/c ratio of 2.5–13.5, cement content of 160–720 kg/m³ and mean cylinder compressive strength varying between 17 and 70 MPa. Its mathematical formulations are given are sin Eqs. (16)–(26) as follows [7];

$$E_c\left(t\right)=4.734\sqrt{\frac{f_{\mathit{cm}}t}{4+0.85t}}$$E16

$$J\left(t,t_0\right)=\frac{127}{\sqrt{f_{\mathit{cm}}}}+C_o\left(t,t_0\right)+C_d\left(t,t_0,t_c\right)$$E17

$$C_0\left(t,t_0\right)=185.4c^{0.5}{\left(f_{\mathit{cm}}\right)}^{-0.9}Q\left(t,t_0\right)+53.766{\left(w/c\right)}^4{c}^{0.5}{\left(f_{\mathit{cm}}\right)}^{-0.9}$$E18

$$\ast ln\left\{1+{\left(t-t_0\right)}^n\right\}+20.3{\left(a/c\right)}^{-0.7}ln\left(t,t_0\right)$$E19

$$Q\left(t,t_0\right)={\left[0.086{\left(t_0\right)}^{2/9}+1.21{\left(t_0\right)}^{4/9}\right]}^{-1}$$E20

$$\ast {\left[+1(\frac{{\left[0.086{\left(t_0\right)}^{2/9}+1.21{\left(t_0\right)}^{4/9}\right]}^{-1}}{{\left(t_0\right)}^{-m}ln\left\{1+{\left(t-t_0\right)}^n\right\}}r\left(t_0\right)\right]}^{-1/r\left(t_0\right)}$$E21

$$r\left(t_0\right)=1.7{\left(t_0\right)}^{0.12}+8$$E22

$$m=0.5n=0.1$$E23

$$C_d\left(t,t_0,t_c\right)=7.57\ast {10}^5{\left(f_{\mathit{cm}}\right)}^{-1{{\in }_{\mathit{shu}}}^{-0.6}}\sqrt{e^{-8H\left(t\right)}-e^{-8H\left(t_0\right)}}$$E24

$$H\left(t\right)=\left[1-\left(1-h\right)s\left(t\right)\right]0\le H\left(t\right)\le 1$$E25

$$\mathrm{Ф}\left(t,t_0\right)=E_c\left(t_0\right)J\left(t,t_0\right)-1$$E26

4.4 GL 2000 model

This model is applicable to concretes of w/c ratio 0.40–0.60 and characteristic compressive strength below 70 MPa. Its mathematical formulations are given in Eqs. (27)–(30) as follows [7];

$$E_c\left(t\right)=3500+4300\sqrt{\left[\frac{f_{\mathit{cm}}{t}^{0.75}}{a_1+a_2{t}^{0.75}}\right]}$$E27

$$J\left(t,t_0\right)=\frac{1}{E_c\left(t_0\right)}+\mathit{\text{Specific creep}}$$E28

$$\mathit{\text{Specific creep}}=\frac{\mathrm{Ф}\left(t,t_0\right)}{E_c}$$E29

$$\mathrm{Ф}\left(t,t_0\right)=\begin{array}{l}\mathrm{Ф}\left(t_c\right)[\left\{\frac{2.0{\left(t-t_0\right)}^{0.3}}{{\left(t-t_0\right)}^{0.3}+1.4}\right\}+\sqrt{\left\{\frac{7(t-t_0}{t_0(t-t_0+7}\right\}}\\ +2.5\left(1-0.086h^2\right)\sqrt{\left\{\frac{t-t_0}{t-t_0}+0.15{\left(V/S\right)}^2\right\}}]\end{array}$$E30

The notations for all equations are given in Appendix E.

5. Review of creep of various concretes

5.1 Creep and shrinkage of concrete containing Iranian pozzolans

Ghodousi carried out a study on early -age creep and shrinkage of concrete containing Iranian Pozzolans. The pozzolans studies were silica fume (SFL), Trass (TL) and GGBS. These were compared with plain concrete (PL). ACI209, BS 8110-1986, CEB1970 prediction models and an estimation based on 28-day results were compared with experimental data. Silica fume concrete exhibited highest creep whereas plain concrete exhibited lowest creep for the 200 days of the test as shown the Figure 3.

Figure 3.

200-days creep strain of concrete in laboratory conditions.

In order to verify the accuracy of prediction models, predicted creep and shrinkage values as in Table 1 were compared with measured values up to 200 days. It was found that most of the test results for creep and shrinkage strains were higher than predicted values. However, the 28-day estimation results (i.e., short-term data) showed a relatively better prediction of creep and shrinkage strains as shown in the table below. It was clearly seen that the accuracy of prediction of creep can be considerably improved by undertaking short-term test results of 28-days and extrapolating results to get long term values.

Concrete Code	ACI	BS	CEB	Estimates Based on 28-Day Results	Lab. Results
PL	564	930	846	972	760
TL	850	1399	1273	778	900
SFL	553	912	829	1421	1160
PC1	489	829	743	1777	1425
SFC1	492	829	743	1276	1695
SC1	759	1197	1086	1085	1000

Table 1.

200-day creep values based on tests and prediction.

It was also observed that the type of pozzolanic material had major effects on creep and shrinkage. The large errors observed in predicting creep of concrete with pozzolanic materials point out that a more accurate model taking into account the effect of pozzolanic material content on the time function is required. Although the existing models include effects of many significant factors affecting creep, they have not taken into account the effect of pozzolanic materials in concrete. They concluded that for accurate predictions of creep, tests must be carried on prototype concrete and extrapolation for long term behaviour made [8].

5.2 Creep analysis of concrete containing rice husk ash

Zhi-hai carried out a study on compressive creep analysis of concrete containing rice husk ash experimentally. In the study the researcher varied the RHA content in concrete from 0 to 20%. The creep loading was 25% of the compressive strength of specimens at 28 days. Their results indicated that the specimen with higher RHA-to-binder ratio had a smaller creep strain as shown in Figure 4. However, the influence of RHA on creep was not linear as 10, 15 and 20% RHA in binder reduced the 60 days creep by 17, 30 and 33%. This indicated that 15% RHA was the best terms of creep reduction.

Figure 4.

Effect of RHA on creep strain of specimens.

The phenomenon of decrease in creep with increasing RHA content was attributed to the pore microstructure of RHA. Large capillary pores exist in concrete without RHA whereas smaller capillary pores exist in concrete with RHA. Hence RHA decreases porosity in concrete improving durability properties of the material. In addition, RHA being highly reactive, reacts with CaOH originated from cement hydration producing additional C-S-H which fills larger pores or voids. Also the porous RHA particles release absorbed water retained in its small pores which helps improve cement hydration reducing the size of interfacial transition zones and porosity of gel pores [9]. Therefore, in cooperation of RHA in concrete makes the concrete matrix to be dense and impervious hence reducing its susceptibility to creep deformation.

5.3 Creep and drying shrinkage of concrete containing GGBS

A study on compressive creep and shrinkage of concrete containing varying amounts of GGBS (0, 20, 40 and 60%) was carried out. Creep recovery was also observed. The results as shown in the Figure 5 indicated that although the shape of creep-time curves for all concrete was similar, there was an increase in creep with increase in the amount of GGBS. This was attributed to the free water available in the body of concrete. The chemical requirement of water is for cement hydration and a relatively smaller quantity for CaOH to react with GGBS. The second reaction being subsequent to the 1st leaves free water availability in GGBS being higher because a constant w/c is used for all mixes. This water gets squeezed out during loading. Therefore, a higher creep effect with increase in GGBS.

Figure 5.

Creep of GGBS concrete.

The range of parameters for creep models ACI, CEB, B3 and GL200 were not applicable for GGBS concrete. Therefore, a modification to incorporate the influence of GGBS was made. A new model was developed using multiple variable regression analysis if the test data. The creep coefficient was expressed as function of time, compressive strength and % of GGBS.

$$\mathrm{Ф}\left(t\right)=\frac{\alpha }{\sqrt{{f_c}^{\prime }}}(\frac{230t^{0.4})}{140+2.4t^{0.4})}$$E31

$$\alpha =exp\left(0.004p_s\right)$$E32

whereby

α is influence of GGBS on creep coefficient

t is time in days

f_c^’ is cylinder compressive strength of concrete

p_s is percentage of GGBS [10]

5.4 Experimental investigation of creep and shrinkage of R.C with influence of reinforcement ratio

Concrete creep and shrinkage with different ratios of reinforcements (0, 0.5, 1, 2 and 3.9%) was studied. Results of prediction by CEB, ACI, GL2000 and JTG D 62-2004 were compared with test data of plain concrete. The creep expressed by creep coefficient was found to increase faster at early age than latter age. The re-enforcement ratio was found to inhibit creep as shown in the Figure 6 below. However low ratio of reinforcement was found to have little influence on creep.

Figure 6.

Creep coefficient with varying reinforcement ratios.

Creep coefficients from the various models and the experiment showed a consistent trend with time. However, the prediction of ACI model was more accurate as shown in the Figure 7 below. Therefore, the ACI model was modified to fit the influence coefficient of reinforcement.

Figure 7.

Comparison of creep data with models.

ACI model was reevaluated with test data of plain concrete giving the following equation for creep coefficient.

$$\mathrm{Ф}(t,t_0=1.16x\frac{{\left(t-t_o\right)}^{0.95}}{25.87+{\left(t-t_o\right)}^{0.95}}$$E33

Whereby t₀ is loading time.

t is time 1.16 is a factor related to composition of material and environment.

Then the influence coefficient of reinforcement was obtained by combining with test results for the different ratios of reinforcement to give the following equation.

$$\mathrm{Ф}\left(t,t_0\right)=1.16x\frac{{\left(t-t_0\right)}^{0.95}}{25.87+{\left(t-t_0\right)}^{0.95}}{K}_p$$E34

$$K_p=0.83x{e}^{-0.072\rho }$$

Whereby K_p is reinforcement on creep influence

ῥ is reinforcement ratio.

6. Conclusions

Creep in concrete has been defined and the mechanism which governs creep. Creep prediction models with the mathematical formulations has also been given. Lastly creep of some concrete has been explained and how the variations occur when constituents of concrete have been changed. It’s recommended that more creep experiments be carried out with different types of concretes so as the increase the database of creep in concrete.

Conflict of interest

The author declares no conflict of interest.

Appendices and nomenclature

Ac	cross-sectional area
a, b	constants which depend on type on cement and curing
a1, a2	constants which depend on type of cement
a/c	aggregates cement ration by weight
Cd (t, to, tc)	compliance function for simultaneous drying
Co (t, to)	compliance function for basic creep (creep at constant moisture content)
c	cement content in concrete (kg/m3)
d	constant normally between 6 and 30 days
d10	10 days in absence of specific data for local aggregates and conditions
Ec	modulus of elasticity of concrete at 28 days (MPa)
Ec(t)	modulus of elasticity of concrete at age t (MPa)
Ec(to)	modulus of elasticity at age to (MPa)
fcm	measured mean compressive strength at 28 days (MPa)
H(t)	spatial average of pore relative humidity within the cross section
h	relative humidity of environment at ambient temperature, decimal
J (t, to)	creep compliance representing the total stress dependent strain per unit stress
S	constant which depends on type of cement
S(t)	time dependence factor
t	age of concrete at time of observation(days)
tc	age of concrete at which drying commenced
to	age of concrete at the time of loading (day)
V/S	volume to surface ratio(mm)
w	water content in concrete (kg/m3)
w/c	water-cement ratio by weight
βRH	constants depend on relative humidity
ϵshu	notional ultimate shrinkage strain (10−6)
μ	perimeter of member in contact with atmosphere (mm)
vu	coefficient of age at application of load
ῥ	unit weight of concrete (kg/m3)
Ф(t, to)	creep coefficient at time t
Ф(tc)	correction factor for drying before loading commenced
Ѱ	constant normally between 0.40 and 0.80