# Analysis and Modelling of Shrinkage and Creep of Reactive Powder Concrete
## 4 Results and Discussion
### Shrinkage and Creep Model of RPC  
The existing shrinkage and creep models cannot accurately predict the shrinkage and creep of RPC, so it is necessary to develop new simple numerical models to predict the shrinkage and creep of RPC. Based on the ACI 209-82 shrinkage model [46], the general equations for predicting shrinkage of concrete are as follows:  
\[\varepsilon_{sh,c}(t,t_{c})=F_{sh}(t)\varepsilon_{shu,c} \tag{1}\]  
\[F_{sh}(t)=\frac{t-t_{c}}{b+t-t_{c}} \tag{2}\]  
In these equations, \(\varepsilon_{sh,c}(t,t_{c})\) is the shrinkage strain of concrete at time, \(t\) (days), with curing time \(t_{c}\) (days); \(F_{sh}(t)\) is the time function of shrinkage; \(\varepsilon_{shu,c}\) is the ultimate shrinkage strain; and \(b\) is constant which influences the rate of shrinkage with time. By applying regression fitting to the \begin{table}
\begin{tabular}{c c c c c} \hline
**Parameters** & **ACI 209-82** & **CEB-FIP 90** & **B3** & **GL 2000** \\ \hline \(f_{c}\)(MPa) & - & 20–90 & 17.2–69 & 16–82 \\ Cement content (kg/m\({}^{3}\)) & - & - & 160–719 & - \\ \(W/B\) & - & - & 0.35–0.85 & 0.40–0.60 \\ Relative humidity (\%) & 40–100 & 40–100 & 40–100 & 20–100 \\ \(t_{c}\) (Moist cured) & \(\geq\)7 days & \(\leq\)14 days & - & \(\geq\)1 day \\ \(t_{c}\) (Steam cured) & \(\geq\)1 day & \(\leq\)14 days & - & \(\geq\)1 day \\ \(\eta\) & \(\leq\)0.4 & \(\leq\)0.4 & \(\leq\)0.45 & \(\leq\)0.4 \\ \hline \end{tabular} Note: \(f_{c}\)(dB) is the compressive strength at 28 days, \(t_{c}\) is the curing age.  
\end{table} 
Table 4: Application scope of the existing models. Figure 8: Comparison of the experimental shrinkage strain results for RPC with shrinkage model predictions. Figure 9: Comparison of the experimental creep strain results for RPC with creep model predictions. shrinkage experimental data, a simple shrinkage model (Equations (3) and (4)) for RPC have been developed. In Equation (3), the ultimate shrinkage strain is 246 \(\mu\varepsilon\) corresponding to 780 \(\mu\varepsilon\) in normal strength concrete. The effects of steel fibre content on the ultimate value of shrinkage is considered by introducing coefficient \(k_{ss}\). A comparison of the proposed shrinkage model predictions and the experimental shrinkage results is shown in Figure 11. The correlation coefficients (\(R^{2}\)) for all the shrinkage samples are greater than 0.991. Therefore, the new shrinkage model can be used to predict the shrinkage of RPC after steam-heat curing with steel fibre content less than 2%. \[\varepsilon_{sh}(t,t_{c})=246k_{ss}\frac{(t-t_{c})}{(t-t_{c})+47} \tag{3}\]  
\[k_{ss}=0.025\rho_{s}^{2}-0.115\rho_{s}+1 \tag{4}\]  
where \(\varepsilon_{sh}(t,t_{c})\) is the shrinkage strain at time, \(t\), with curing time, \(t_{c}\); \(\rho_{s}\) is the volume fraction of steel fibre. Based on the ACI 209-82 creep model [47], the general equations for predicting creep of concrete can be written as:  
\[\varphi(t,t_{0})=F_{c}(t)\varphi_{cu,c} \tag{5}\]  
\[F_{c}(t)=\frac{(t-t_{0})^{A}}{B+(t-t_{0})^{A}} \tag{6}\]  
In these equations, \(\varphi_{c}(t,t_{c})\) is the creep coefficient of concrete at time, \(t\) (days), loading at time \(t_{0}\) (days); \(F_{c}(t)\) is the time function of creep; \(\varphi_{cu,c}\) is the ultimate creep strain; A and B are constant. By applying regression fitting to the creep experimental data, a simple creep model (Equations (7)-(9)) for RPC have been developed. In Equation (7), the ultimate creep coefficient is 0.82 corresponding to 2.35 in normal strength concrete. The steel fibre can influence both the ultimate value and the rate of creep. Therefore, the effects of fiber content on the ultimate value and development rate of creep are considered by coefficients \(k_{sc}\) and \(\alpha\), respectively. A comparison between the predictions of the proposed creep model and the experimental creep results is shown in Figure 10. The correlation coefficients (\(R^{2}\)) for all the creep samples are greater than 0.986.