2 MATERIAL PROPERTIES
2.1.5 Time-dependent behaviour
Definitions
When concrete is subjected to a sustained stress, the resulting strain can be divided into the following three components:
• Instantaneous elastic strain: When the stress is applied to the concrete it causes an instantaneous elastic strain, which can be expressed as follows (see Section 2.1.3):
εc = fc
Ec (2-10)
where fc = applied stress
εc = instantaneous elastic strain Ec = Young’s modulus of the concrete
• Shrinkage strain: In the absence of temperature variations, shrinkage is defined as that part of the time-dependent strain which is independent of stress. Shrinkage therefore corresponds to the time-dependent strain which occurs in the absence of stress.
• Creep strain: Creep is defined as the component of the time-dependent strain which is dependent on the applied stress. Although this definition has been used for many years, it is important to point out that, strictly speaking, it is not correct because it implies that creep and shrinkage are independent phenomena which are additive when they occur simultaneously (Ref. 2-31). It is well known that creep and shrinkage are not independent, the effect of shrinkage on creep being to increase its magnitude. In order to use the mass of experimental data obtained on the basis of the assumption that creep and shrinkage are independent, Neville (Ref. 2-31) suggests that creep should be defined as the time-dependent strain which takes place in excess of shrinkage.
The consequence of this definition is that the total creep must be considered as consisting of two components:
- Basic creep, which is the component of creep which occurs under conditions where there is no moisture exchange with the ambient medium.
Shrinkage from t0
Shrinkage of an unloaded specimen
Shrinkage Nominal elastic strain
Nominal elastic strain
Nominal elastic strain
True elastic strain
Creep on the basis of additive definition
Drying creep Creep
Basic creep t0
t0
t0
t0
1
1
1 2
3
3
Total creep 2 Age t
Time (t-t0)
Time (t-t0)
Time (t-t0)
Strain StrainStrainStrain
(a) Shrinkage of an unloaded companion specimen
(b) Change in strain of a loaded and drying specimen
(c) Creep of a loaded specimen in hygral equilibrium with the ambient medium
(d) Change in strain of a loaded and drying specimen
Figure 2-14: Definition of time-dependent deformations of concrete (Ref. 2-31).
- Drying creep, which is the component of creep influenced by the drying process.
These definitions are illustrated in Fig. 2-14, in which the various components of strain are shown for a concrete specimen subjected to a sustained low-level compressive stress (i.e. less than 40% of its short-term compressive strength). It should be noted that the elastic strain of the concrete reduces with time because the elastic modulus increases with age. Strictly speaking, creep should be determined on the basis of the elastic strain at the time under consideration and not the time at which the load is applied. Although both methods can be used, the change in elastic strain is not accounted for under normal circumstances because the difference is usually small and because this approach is more convenient for structural analysis.
Factors which Influence Creep and Shrinkage
Creep and shrinkage of concrete can be ascribed to the movement of water within the crystalline structure of the cement paste and loss of water to the surrounding environment by evaporation. The factors which influence creep and shrinkage can be grouped into two broad categories: Intrinsic factors, which deal with the actual composition of the concrete as well as the influence of stress, and extrinsic factors, which account for the state of the environment to which the concrete is exposed.
A partial list of these factors includes (Ref. 2-32):
• Water-cement ratio: Both creep and shrinkage are increased by an increase in the water-cement ratio, partially because the evaporable water is increased, and because of more and larger capillary pores.
• Aggregate: Since the seat of creep and shrinkage is to be found in the cement paste, the aggregate tends to restrain the deformation of the paste induced by creep and shrinkage. Hence, an increase in the aggregate-cement ratio will lead to lower values of creep and shrinkage. Aggregates which have higher values for the modulus of elasticity can offer greater restraint to potential creep and shrinkage of the paste and therefore tend to yield concrete which creeps and shrinks less. The use of more porous aggregates leads to increased creep and shrinkage, possibly because an increase of porosity can facilitate moisture transfer within the concrete. However, it should be noted that aggregates with higher porosity tend to have a lower modulus of elasticity.
• Cement type: The influence of the type of cement on creep appears to be related in part to its effect on the rate of strength development which, in turn, depends on the composition and fineness of grinding (Ref. 2-1). The magnitude of the creep of concrete made with the following cements occurs in an increasing order: high-aluminium, rapid-hardening, ordinary Portland, Portland blast-furnace, low-heat and Portland-pozzolana.
Reference 2-1 suggests that the type of cement affects shrinkage mainly through variations in C3A content, and that fineness of grinding has a negligible effect on shrinkage, except when the cement is extremely fine or extremely coarse. It appears that concretes containing Portland blast furnace cement (PBFC) and rapid-hardening Portland cement generally tend to shrink more than concrete containing ordinary Portland cement.
• Admixtures: The effect of admixtures on creep and shrinkage appears to be highly variable, depending on the specific admixture and cement used, as well as a number of other factors which include exposure conditions, age at loading and time under load (Ref. 2-1). It is important to note that the use of certain admixtures can significantly increase the creep and shrinkage of concrete.
• Member size and shape: The volume to exposed surface ratio of a member can be used as a general parameter for describing the influence of the size and shape of the member on creep and shrinkage. A larger value of this ratio represents a thicker (larger) member which has a longer diffusion path for moisture loss. Consequently, creep and shrinkage reduce with an increase in the volume to surface ratio, i.e. as the member becomes larger, with creep approaching the value of basic creep for very large members. As far as creep is concerned, it is most probably only drying creep which is affected by a variation of the size and shape of the member because basic creep remains unaffected by loss of moisture from the concrete and, as such, is independent of
the size and shape of the member. Evidently shrinkage is affected to a greater extent than creep by the size and shape of the member.
• Magnitude of the applied stress: Creep strains are approximately proportional to the magnitude of the applied sustained stress for values less than 50% of the cube strength. For most practical structures, creep may therefore be considered to be linearly related to stress within the service load range.
• Age of loading: The age of the concrete when it is loaded has an important influence on the magnitude of creep, the effect being to increase creep with earlier ages at loading. The manner in which the age at loading influences creep seems to be related to the manner in which it affects the development of strength and the degree of hydration. For these reasons, creep has been found to correlate well with maturity.
• Temperature: Creep is apparently not a monotonic function of temperature and passes a maximum in the vicinity of 50°C. Beyond this point creep reduces with temperature up to about 120°C after which it, once again, increases with temperature (Ref. 2-1). It also appears that the creep of specimens heated just prior to loading is more significantly influenced by temperature than that of specimens cured at the test temperature, because of improved hydration in the latter case.
Tests by England and Ross (Ref. 2-33) indicated that the effect of temperature on creep is greater in the range of 20-60°C than in the range 100-140°C. Shrinkage is also increased at higher temperatures during drying.
• Relative humidity: Both creep and shrinkage are increased with a decrease of the ambient relative humidity. It appears that it is not the relative humidity which is the influencing factor with regard to creep, but rather the process of drying while under load. This is confirmed by the fact that the effect of relative humidity is much smaller if the concrete has already reached hygral equilibrium before loading and, furthermore, that creep is strongly dependent on relative humidity when the concrete is allowed to dry while under load. At 100% relative humidity the concrete absorbs water and swells slightly (as opposed to shrinking).
Creep: behaviour and prediction
The development of creep with time is shown in Fig. 2-15, which shows that most of the creep develops within a fairly short time period after the application of the load. SABS 0100 (Ref. 2-14) suggests that, under conditions of constant relative humidity, 40, 60 and 80% of the final creep develops during the first month, the first 6 months and the first 30 months under load, respectively.
It should be noted that the final creep is defined by SABS 0100 as the creep strain after 30 years.
Evidently creep continues for a very long time, and even at ages of the order of 30 years small, but measurable, creep rates have been reported (Ref. 2-34).
Instantaneous recovery Creep recovery Residual deformation Strain on application
of load Creep
Time since application of load (days) Specimen under constant load Load removed
Strain(10)-6
500
0
0 50 100 150 200
1000 1500
Figure 2-15: Creep and creep recovery of concrete (Ref. 2-31).
Removal of the sustained stress is accompanied by an instantaneous strain recovery in the concrete, which is normally smaller than the instantaneous elastic strain associated with the application of the stress. As shown in Fig. 2-15, the instantaneous recovery is followed by a time-dependent recovery of strain, termed creep recovery, which tends to a finite value. The magnitude of the creep recovery is usually smaller than that of the creep at the time of removal of the stress. An exception occurs if the concrete is old when the stress is applied, in which case the creep recovery can have the same magnitude as the creep.
Linear creep theory can be applied to most practical structures within the service load range. This theory leads to the conclusion that creep strain is linearly related to the instantaneous elastic strain under constant sustained stress and under constant environmental conditions. Using this approach, the creep strain is given by
(2-11) where ecr(t) = creep strain, as a function of time t
ec = instantaneous elastic strain, given by Equation (2-10) f(t) = creep coefficient, as a function of time t
t = time, measured from the time at which the sustained stress is applied t0
For most practical cases, the long-time value of f(t) can vary between 1.5 and 3.5. The 30 year creep coefficient f30 can be obtained from Fig. 2-16, which is taken from SABS 0100 (Ref. 2-14).
This figure gives f30 as a function of the ambient relative humidity, the age at loading and the effective thickness of the section which, for the purposes of Fig. 2-16, is defined as twice the cross-sectional area of the member divided by the exposed perimeter. More comprehensive procedures for determining f(t), which explicitly include a greater number of factors that influence creep, are given in Ref. 2-7 and Refs. 2-35 through 2-37.
ecr( )t =f( )t ec
20 30
150 300 Airconditioned area(offices) Coastalarea
Inland
* Relevant values for outdoor exposure may be determined through the Weather Bureau, Department of Environmental Affairs
30 Year creep
Figure 2-16: Effects of relative humidity, age of concrete at loading and section thickness on the creep coefficient (Ref. 2-14).
Equation 2-11 expresses the creep strain as a linear function of the instantaneous elastic strain which, in turn, is dependent on the magnitude of the modulus of elasticity Ecof the concrete (see Eq. 2-10).
It is therefore clear that f(t) is implicitly defined in terms of Ec. Because some of the procedures for estimating f(t) base the calculation of the instantaneous elastic strain on the magnitude of Ecat the time at which the concrete is loaded (Refs. 2-14 and 2-37) while others base it on the magnitude at 28 days (Refs. 2-35 and 2-36), great care should be exercised to determine exactly which value of Ec should be used. This observation also emphasizes the fact that different procedures should never be combined to estimate creep strains.
The creep strain is often expressed in terms of specific creep (defined as the creep strain per unit stress) as follows:
(2-12) where C(t) = specific creep, as a function of time t
fc = sustained concrete stress
The specific creep can be expressed in terms of the creep coefficient by equating Equations (2-11) and (2-12), and using Eq. (2-10). Thus
so that
(2-13)
Shrinkage: behaviour and prediction
The development of shrinkage with time is shown in Fig. 2-17 where it may be seen that, as in the case of creep, the rate of shrinkage reduces with time, and that a measurable rate can still be obtained after 20 years. The rate at which shrinkage develops depends on the conditions of drying: Most of the shrinkage can take place within a period of 3 months under adverse drying conditions, while the concrete may not shrink at all if it always remains wet. It is reported in Ref. 2-34 that for concrete stored in air at 50% relative humidity and at 21°C (70°F) there are indications that creep and shrinkage develop at similar rates. For the purpose of estimating prestressing losses, SABS 0100 (Ref. 2-14) suggests that 50% and 75% of the total shrinkage takes place within the first month and within the first six months after the transfer of prestress, respectively. Note that the total shrinkage, referred to by SABS 0100 above, excludes the shrinkage which takes place before transfer. Although the time period associated with the total shrinkage is usually ill-defined, it appears reasonable to take it as the design life of the structure.
For the types of concrete generally used for prestressed concrete, the magnitude of the shrinkage strain will normally vary between 0.0002 and 0.0006. Figure 2-18 gives the shrinkage strain after 6 months and after 30 years as function of the ambient relative humidity and the effective section thickness (defined as for creep, see Fig. 2-16), as recommended by SABS 0100. These values apply to concrete with an original water content of 190 l/m3. If the concrete has a water content which differs from this value, but which lies within the range 150 to 230l/m3, then the shrinkage obtained from Fig. 2-18 must be adjusted in proportion to the water content.
More comprehensive procedures for determining the shrinkage strain are presented in Ref. 2-7 and Refs. 2-35 through 2-37. These procedures explicitly include a greater number of factors which influence shrinkage.
ecr( )t =C t f( ) c
C t f t t f
c c E
c c
( ) ( ) ( )
=f e =f
C t t
Ec ( ) ( )
= f
800
Time reckoned since end of wet curing at the age of 28 days
´ 6 6 Month shrinkage 10
for an effective
Figure 2-18: Drying shrinkage for normal density concrete (Ref. 2-14).