## 4 Predictions for Compressive Strength Development of HVFA UhPC
### Empirical Equations for Compressive Strength Development  
As presented in Figure 7, the compressive strength development of UHPC with age can be characterized as a parabola, which agrees with the observation that the increasing rate of compressive strength development decreases with the increase of age. Therefore, the compressive strength development of UHPC can be identified by using a parabolic time function, which is a common approach for normal-strength concrete. To formulate straightforward empirical equations for compressive strength development of UHPC, the present study follows the exponential function specified in the _fib_ 2010 model [27]:  
\[f_{c}(t)/f_{c}^{\prime}=EXP\left[S_{l}\left(1-\left(\frac{28}{t}\right)^{0.5} \right)\right] \tag{1}\]  
where \(S_{l}\) is the parameter to identify the slope of the parabolic curve at different ages (\(t\); in days). The compressive strength of UHPC with fly ash at age of 28 days is significantly affected by R\({}_{f}\), _W/B_, and early heat curing, as presented in Figure 7. Considering these influencing parameters, non-linear regression analysis (NLRA) was conducted using the current 200 datasets. In establishing the fundamental models for \(f_{c}^{\prime}\), each influencing parameter investigated was combined and adjusted repeatedly by the trial-and-error approach until a relatively higher correlation coefficient (R\({}_{2}\)) was obtained. The influence of early heat curing on \(f_{c}^{\prime}\) was reflected by introducing a maturity concept. From the NLRA approach, \(f_{c}^{\prime}\) for UHPC with fly ash can be estimated as follows (Figure 8):  
\[f_{c}^{\prime}\ =\ 6.4\left[\xi(M/M_{0})^{1.3}/(W/B)\right]^{0.4}\!f_{c0} \tag{2}\]  
where \(\xi\) is the coefficeient depending on the fly ash content R\({}_{f}\) as follows  
\[\xi\ =\ \left(1\ +\ R_{f}\right)^{0.1}\ \text{for}\ R_{f}\ \leq\ 0.3 \tag{3}\]  
\[\xi\ =\ \left(1\ -\ R_{f}\right)\text{for}\ R_{f}\ >\ 0.3 \tag{4}\]  
Figure 8: Regression analysis for the 28-day compressive strength of UHPC. \(M\) is the maturity function (=\(\sum\{\Delta t\times(T+10)\}\)) of the time interval (\(\Delta t\))-temperature (\(T\): in \({}^{\circ}\)C) combination, \(M_{o}\) is the reference value (= 1036 \({}^{\circ}\)C-days) for maturity of concrete cured under standard condition of 27 \({}^{\circ}\)C, and \(f_{c0}\) is the reference value for compressive strength of concrete. In Equation (2), the coefficient \(\xi\) is to explain the effect of fly ash on \(f_{c}^{\prime}\), indicating that \(f_{c}^{\prime}\) increases with the increase of \(R_{I}\) up to 30%, beyond which \(f_{c}^{\prime}\) decreases gradually, as shown in Figure 8. Note that it is quite hard to achieve a high value of \(R_{2}\) while establishing the model for \(f_{c}^{\prime}\) because of a relatively great scatter of the current test data with regard to the variations of \(R_{I}\) and heat curing ages. The value of parameter \(S_{I}\) in Equation (1) for each specimen was determined through regression analysis using the present test data until the best fitting curve was obtained. UHPC specimens cured under heat curing exhibited very consistent shape for the normalized compressive strength development, irrespectively of \(W\)/\(B\) and \(R_{I}\), as shown in Figure 7b. The values of \(S_{I}\) determined for UHPC specimens under heat curing ranged between 0.01 and 0.03. Therefore, the present study fixed the \(S_{I}\) value to be 0.02 for UHPC cured under heat curing condition. Meanwhile, the value of \(S_{I}\) determined for UHPC specimens cured under standard condition was affected by \(R_{I}\) and \(W\)/\(B\), varying from 1.67 to 9.28. With the increase in \(R_{I}\) and \(W\)/\(B\), a higher \(S_{I}\) value was obtained.