# Effect of Fibre Reinforcement on Creep in Early Age Concrete
## 3 Results and Discussion
### Creep Test
#### 3.2.1 Proposed Creep Formulation  
Comparing the evolution of the creep deformation of the FRC with the theoretical curve reflected in the EHE-08 standard [22], an important divergence is observed. The evolution of creep deformation undergone by FRC specimens loaded 24 h after their manufacture is the one that deviates the most from the theoretical curve defined in the EHE-08 standard. For the rest of the test specimens, loaded at the ages of three, seven and twenty-eight days, there is still a divergence between their experimental behaviour and that predicted by the EHE-08 standard, but this divergence is much smaller. The maximum divergence between experimental and theoretical creep deformation is recorded for the first stage of the creep process to which the FRC specimens have been subjected during the test. Equation (1) shows the formulation proposed by the EHE-08 standard for estimating creep deformation in concrete.  
Figure 6: Evolution of creep deformation undergone by FRC: (**a**) specimens loaded at 15 MPa; (**b**) specimens loaded at 20 MPa.  
\[\epsilon_{cor}(t,t_{0})=\frac{\sigma(to)}{E_{c,lo}}+\frac{\sigma(to)\cdot\varphi_{ HR}\cdot\beta(f_{cm})\cdot\beta(to)\cdot\beta_{c}(t-to)}{E_{c,28}} \tag{1}\]  
\[\beta(t_{0})^{\gamma}=\frac{1}{0,1+t_{0}{}^{\alpha(t_{0})}} \tag{2}\]  
\[\beta_{c}(t-t_{0})^{\gamma}=\left(\frac{(t-t_{0})}{\beta_{H}+(t-t_{0})}\right) ^{\alpha(t-t_{0})} \tag{3}\]  
where:  
- \(\sigma(to)\) is stress on specimens during creep test.  
- \(E_{c,lo}\) is the initial modulus of elasticity in specimens.  
- \(E_{c,28}\) is the modulus of elasticity of the specimens at 28 days.  
- \(\varphi_{HR}\) is the coefficient of influence of relative humidity [22].  
- \(\beta(f_{cm})\) is the coefficient of influence of concrete strength [22].  
- \(\alpha(t_{0})\) is the coefficient associated with the loading age proposed by the authors;  
- \(\alpha(t-t_{0})\) is the coefficient associated with the evolution of creep with time proposed by the authors.  
Figure 7 shows the evolution of the error of the model proposed by the authors with the variation of the value of the coefficients \"\(\alpha(t_{0})\)\" and \"\(\alpha(t-t_{0})\)\", while Figure 8 shows the range of optimal coefficients of the model proposed by the authors to minimize the deviation of the predictive model compared with the experimental results obtained in the laboratory.  
Figure 7: Analysis of the error induced by the variation of the exponent of the creep coefficients \(\alpha(t_{0})\) and \(\alpha(t-t_{0})\) in the model proposed by the authors to determine the creep strain undergone by FRC: (**a**) FRC specimens loaded at 24 h (**a**); at 3 d (**b**); at 7 d (**c**); and at 28 d (**d**).  
To determine the optimal creep coefficients, the authors have scanned all possible combinations of values of these coefficients and have obtained the error of the final proposed formulation for each load age analysed in the study. Figure 8 represents the traces of the interval corresponding to a maximum deviation of 5% for the different ages of FRC loading: (a) green trace for FRC loaded at 24 h; (b) cyan trace for FRC loaded at 3 d; (c) blue trace for FRC loaded at 7 d; (d) pink trace for FRC loaded at 28 days. The optimal interval is taken as the area in which the divergence intervals of less than 5% converge for all the load ages analysed in the creep test. The authors propose the following values for the coefficients associated with the loading age and the evolution of creep with time: \(\alpha(t_{0})\) = 0.13; \(\alpha(t-t_{0})\) = 0.39.