## 3 Results
### Developing Master Curves for UPV and Dynamic Modulus
The resulting models are presented in Tables 6 and 7 and the following equations: \[\delta_{UPV}=357.91{(CS)}^{2}+229.99 CS-14.4A^{2}+176.97A-54.71T^{2}+154.53T-59.81\] (6) \[\delta_{DM}=4.6{(CS)}^{2}+0.28 CS+0.51{(UW)}^{2}-4.51 UW-1.45 WC^{2}+0.48 WC-0.55\] (7) where \(\delta_{DM}\) and \(\delta_{UPV}\) are the shift factors for the dynamic modulus, \(\text{E}_{\text{d}}\) (RTG) and UPV, respectively; \(CS\) is the compressive strength in MPa; \(UW\) is the unit weight (\(\text{Kg}/\text{m}^{3}\)); \(A\) is the air content (%); \(T\) is the average concrete temperature (\({}^{\circ}\)C) during the curing period; and \(WC\) is the water-cement ratio. As can be observed from the results, the prediction accuracies for the selected models were 91% and 94% for the UPV and the dynamic modulus shift factor cases, respectively. Figure 2: _Cont._  
Figure 3: Master curves for (**a**) UPV, (**b**) Eq.. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Mix No.**} & \multicolumn{2}{c}{**UPV vs. Temp-Time**} & \multicolumn{2}{c}{**Eq (RTC) vs. Temp-Time**} \\  & \multicolumn{2}{c}{**(y = \(a^{*}\) _Ln_ (x) + _b_)**} & \multicolumn{2}{c}{**(y = \(a^{*}\) _Ln_ (x) + _b’_)**} \\ \cline{2-5}  & \(b\) & **R\({}^{2}\)** & \(b\) & **R\({}^{2}\)** \\ \hline
1 & 2915.7 & 0.82 & 16.86 & 0.95 \\
2 & 3243.0 & 0.95 & 22.17 & 0.97 \\
3 & 3131.8 & 0.91 & 18.42 & 0.96 \\
4 & 3299.8 & 0.96 & 21.44 & 0.95 \\
5 & 3089.6 & 0.92 & 17.38 & 0.97 \\
6 & 3107.0 & 0.96 & 16.82 & 0.96 \\
7 & 3197.9 & 0.97 & 18.42 & 0.95 \\
8 & 3128.1 & 0.92 & 17.09 & 0.97 \\
9 & 3009.0 & 0.94 & 13.45 & 0.95 \\
10 & 3233.5 & 0.99 & 17.73 & 0.99 \\
11 & 3288.1 & 0.95 & 17.44 & 0.96 \\ \hline \hline \end{tabular}  
* \(a=224.26\); ** \(a^{\prime}=2.5\); \(p<0.05\) for all cases.  
\end{table} 
Table 3: Modified equations of best fit for each mixture. Figure 2: Relationship between (**a**) UPV and (**b**) Eq with temp–time factor with constant “\(a\)”. In Figure 4, the predicted shift factors of UPV and dynamic modulus are compared with the actual values. Since data from eleven mixtures were used in relation to the number of variables in the shift functions, some deviations from the line of equality (\(y=x\)) were expected. Nevertheless, overall, there is good agreement between predicted and actual values (values close to the 45\({}^{\circ}\) line of equality). In fact, this is reflected in the developed master curves presented in Figure 3, in which all the shifted data are on the trend lines (R\({}^{2}=1\)). \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{3}{*}{**Model**} & \multirow{3}{*}{**Intercept**} & \multicolumn{4}{c}{**Compressive Strength**} & \multirow{3}{*}{**Unit Weight**} & \multirow{3}{*}{**Air Content**} & \multicolumn{4}{c}{**Average Temperature**} & \multirow{3}{*}{**R\({}^{2}\)**} & \multirow{3}{*}{**R\({}^{2}\)**} & \multirow{3}{*}{**RSE**} & \multirow{3}{*}{**\(p\)-Value**} \\ \cline{3-3} \cline{5-12}  & & \(a\)\({}^{*}\) & \(b\)\({}^{*}\) & \(a\) & \(b\) & \(a\) & \(b\) & \(a\) & \(b\) & & & \\ \hline \(1\) & \(-59.81\) & 426.79 & 464.11 & 134.62 & \(-392.67\) & \(-36.75\) & \(-42.21\) & \(-35.93\) & 395.65 & 0.95 & 60.8 & 0.19 \\ \(2\) & \(-59.81\) & 357.91 & 229.99 & 0 & 0 & \(-14.40\) & 176.97 & 54.71 & 154.53 & 0.91 & 57.06 & 0.05 \\ \(3\) & \(-59.81\) & 137.77 & \(-169.0\) & \(-349.81\) & \(-714.24\) & 479.38 & \(-738.61\) & 0 & 0 & 0.81 & 80.88 & 0.16 \\ \hline \hline \end{tabular} \({}^{*}\) The equations are in the form of \(\text{ax}^{2}+\text{bx}\).  
\end{table} 
Table 6: Models for predicting UPV shift factor.