5.2 Time dependent deformations
5.2.2 Elevated temperatures
During the experiments of creep and shrinkage conducted at elevated temperatures, the relative humidity in the concrete samples was also measured (see Appendix B.4.2).
Based on the knowledge acquired in these investigations, the formulations presented before can be improved in order to include the influence of elevated temperatures. In the following segments additional factors that account for the influence of temperature are presented to complement the prediction of shrinkage and creep based on the time- development of the relative humidity in the concrete poresh. These formulations can be assumed to be valid for a temperature range between 20 °C and approx. 100 °C.
5.2.2.1 Basic shrinkage
Similar to the experiments conducted at 20 °C, the shrinkage measurements at elevated temperatures were conducted on advanced age concrete samples for which no further basic shrinkage was expected to occur. It can be assumed that elevated temperatures influence basic shrinkage by accelerating the process without affecting the final value of deformation. A logical approach to include these effects can be taken from [N15].
5.2.2.2 Drying shrinkage
As presented before, variations of the mean relative humidity of the concrete pores can be related to changes in the drying shrinkage of concrete by multiplying them by the factorKcdsdescribed in Eq. 5.15. This factor is dependent on the w/c-ratio of the con- crete mixture and can be considered as independent of temperature. In order to account for the influence of temperature, Eq. 5.14 is multiplied by the factorKcdsT and becomes:
dεcds=KcdsT·Kcds·dh (5.26)
The factorKcdsT was calibrated according to the measurements conducted at tempera- tures higher than the reference of 20 °C. The calibration led to the following equation.
KcdsT =1.2−0.01·T (5.27)
According to Eq. 5.27, with increasing temperature, the factor relating changes in the relative humidity of the concrete pores with drying shrinkage deformation decreases.
In Fig. 5.7 the shrinkage deformations measured on companion specimens during the loading period of the creep tests conducted on the concrete MRC and MHC at 70 °C and 65 % RH are plotted with the calculations carried out based on Eq. 5.26.
1 1 0
- 5 0
0
5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0
Shrinkage deformation, εcds [µm/m]
P r e c o n d i t i o n
R H / 2 0 ° C m e a s u r e d c a l c u l a t e d
6 5 %
8 5 %
9 5 %
6 0 C o n c r e t e M R C
1 1 0
- 5 0
0
5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 0
Shrinkage deformation, εcds [µm/m]
D u r a t i o n o f d r y i n g d u r i n g l o a d i n g , t - ts [ d ] P r e c o n d i t i o n
R H / 2 0 ° C m e a s u r e d c a l c u l a t e d
6 5 %
8 5 %
C o n c r e t e M H C
6 0
Figure 5.7: Drying shrinkage calculated with the mean relative humidity measured in the concrete pores in comparison with the measured values for the concrete MRC (top) and MHC (bottom) on companion specimens during the creep tests conducted at 70 °C and 65 % RH
5.2 Time dependent deformations
The developed model of shrinkage at elevated temperatures can also follow the measured data very accurately. The calculated curves in Fig. 5.7 start at later points in time (1 to 3 days) than the measurements of shrinkage because, for the calculations, the measure- ments of relative humidity are only useful if the temperature within the concrete samples has already stabilised. During the incrementation of the temperature, the relative humid- ity of the concrete pores changes but these changes shall not be related with shrinkage deformations. As it was discussed by the analysis of the measurements of shrinkage at elevated temperatures in Chapter 4.4.3, at the moment of application of the load, the temperature of the samples was around 2 to 3 °C below the desired test temperature and it took around 24 h for the samples to reach the temperature of the chamber and even longer (up to 3 d) for the relative humidity of the concrete pores to reach a maximum value and start decreasing.
In addition to the comparisons from Fig. 5.7, similar calculations, providing also satis- factory results, were carried out for the concrete MLC at a testing temperature of 70 °C and the concrete MRC tested at 40 °C (see Appendix C.2).
5.2.2.3 Creep
Following the formulations presented before, creep at 20 °C can be modelled based on the development of the mean relative humidity of the concrete pores during the loading period. The relative humidity in the concrete pores is influenced by temperature, and therefore basic creep and drying creep as described by Eqs. 5.19 and 5.22 shall be in- fluenced by temperature too. Hence, new factors KbcT and KcdsT need to be consider in these equations in order to account for the influence of temperature leading to the following equations:
dϕbc(t,t0) =KbcT· Kbc1
(fcm)0.7· 1 (t−t0) +Kbc2
·dt (5.28)
dϕdc(t,t0) =KcdsT·Kcds·Kdc·dh (5.29) Eq. 5.28 results from multiplying Eq. 5.19 by the factorKbcT to account for the influence of temperature on the development of basic creep. In case of drying creep, temperature affects the development of drying shrinkage and consequently also drying creep, as as- sumed in Eq. 5.21. Therefore, the influence of temperature on drying creep is already covered if the influence of temperature on drying shrinkage is considered by multiplying Eq. 5.22 by the factorKcdsT as presented in Eq. 5.29.
The factorKbcT was calibrated based on the conducted experiments. The calculations performed with the values presented in Table 5.3 showed good agreement with the mea- sured data.
Table 5.3: Factor accounting for the influence of temperature on basic creep
w/c-ratio
[-] 20 °C 40 °C 70 °C
MLC 0.40 1.0 - 3.1
MRC 0.50 1.0 2.5 3.0
MHC 0.60 1.0 - 2.5
KbcT Concrete Mixture
The calibration of the factorKbcT suggests that it is dependent on the w/c-ratio and the temperature. The dependency ofKbcT on the w/c-ratio can be expressed by a s-shaped function and the dependency of temperature is covered by including, within the function, two temperature-dependent parameters as proposed in Eq. 5.30.
KbcT=aKbcT·
"
bKbcT+ 1−bKbcT
1+ (2.2·w/c)−16
#
(5.30) The temperature-dependent parametersaKbcTandbKbcTare equal to 1.0 at 20 °C and vary with temperature according to Eqs. 5.31 and 5.32:
aKbcT=1+2.7
T−20 100
0.35
(5.31)
bKbcT=1.088−0.0044·T (5.32)
whereT is given in [°C]. Analysing the factor KbcT andKcdsT help to understand the behaviour of creep at elevated temperatures in comparison with 20 °C. The factor in- fluencing basic creepKbcT increases with increasing temperature, leading to basic creep deformations three times higher at 70 °C than at 20 °C (see Table 5.3). On the other hand, the factor influencing drying creepKcdsT decreases 50 % at 70 °C (see Eq. 5.27).
This means that for the same change on the relative humiditydhthe drying creep de- formations of concrete at elevated temperatures are lower than at 20 °C. This observation can be explained by analysing the storage capacity of the concrete. The higher the tem- perature, the lower the amount of water, that the concrete can store (see Chapter 2.1.2.2) which implies that changes on the relative humidity of the concrete pores induce less moisture loss at elevated temperatures than at 20 °C and therefore lower drying creep deformations are seen. Fig. 5.8 shows results from the creep tests conducted at 70 °C in comparison with the calculations based on Eqs. 5.28 and 5.29.
5.2 Time dependent deformations
0 . 0 1 0 . 1 1 1 0
05
1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0
Specific creep, εcc/σc [µm/(mMPa)]
P r e c o n d i t i o n
R H / 2 0 ° C m e a s u r e d c a l c u l a t e d
6 5 %
8 5 %
C o n c r e t e M L C
6 0
0 . 0 1 0 . 1 1 1 0
05
1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0
Specific creep, εcc/σc [µm/(mMpa)]
P r e c o n d i t i o n
R H / 2 0 ° C m e a s u r e d c a l c u l a t e d
6 5 %
8 5 %
9 5 %
C o n c r e t e M R C
6 0
0 . 0 1 0 . 1 1 1 0
05
1 0 1 5 2 0 2 5 3 0 3 5 4 0 4 5 5 0
Specific creep, εcc/σc [µm/(mMPa)]
D u r a t i o n o f l o a d i n g , t - t0 [ d ] P r e c o n d i t i o n
R H / 2 0 ° C m e a s u r e d c a l c u l a t e d
6 5 %
8 5 %
C o n c r e t e M H C
6 0
Figure 5.8: Specific creep calculated with the mean relative humidity measured in the concrete pores in comparison with the measured values for the concretes MLC, MRC and MHC during the creep tests conducted at 70 °C and 65 % RH
As new dependencies are added to the formulations, the accuracy of the results is af- fected. Nevertheless, as it can be seen in Fig. 5.8, the calculations achieved a good correlation with the measured data. Further comparisons with the results of the tests conducted at 40 °C for the concrete MRC, also showing acceptable correlations, are presented in Appendix C.3.
In conclusion, basic creep increases with increasing temperature while drying creep, although develops faster because of the acceleration of the drying process, reaches lower final values at elevated temperatures in comparison to 20 °C. These observations are in accordance with the results from Schwesinger et al. and Seki and Kawasumi [137, 139].
They found that the quotient between creep deformations at elevated temperatures and creep deformations at 20 °C is higher by sealed samples than by unsealed samples. This can be understood as follows: in case of sealed samples, only basic creep develops while by unsealed samples basic and drying creep take place. The quotient is therefore lower by unsealed samples because the drying creep component decreases with increasing temperature.