## 4 Results and Discussion
### Relationship among Static and Initial Chord and Dynamic Elastic Modulus
This trend is considered to decrease the difference between the _Ed_ value and _Ei_ in the theoretical equation, as the moisture content inside the concrete is small (because the w/c ratio is small). The difference between the dynamic elastic modulus (_Ed_) and initial chord elastic modulus (_Ei_) values from the three theoretical equations was confirmed using the method shown in Figure 9, and an analysis was conducted to confirm how accurately the _Ei_ could be predicted by correcting for the _Eds_. The correction factors and MAPE according to the mixes and ages are summarized in Table 6. Although the MAPE with the _Ei_ could be lowered to 4.69%, 2.81%, 3.12%, and 3.43% (for days 4, 8, 14, 28, respectively) with corrections based on mixes and ages for the _Eds_ values, this process was quite complicated and difficult. Therefore, the entire data were applied for the _ASTM.LT_, _f1_f2_LT_, and _ASTM.TR_, average correction factors of 0.80, 0.79, and 0.82. Compared to the actual _Ei_ value, the MAPE values of _ASTM.LT_, _f1_f2_LT_, and _ASTM.TR_ according to the correction factors showed errors of 6.40%, 6.50%, and 6.43%, respectively. However, the _Ei_ is not suitable for predicting the entire section, as it is nonlinear with the theoretical equations. Figure 8: Comparison of Ed and Ei by ages and mixes according to ASTM.LT. (**a**) Mix 1 (20 MPa), (**b**) Mix 2 (40 MPa). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Type of Errors** & **ASTM.LT** & **f1,f2.LT** & **ASTM.TR** \\ \hline Mean square error (MSE) & \(2.53\times 10^{7}\) & \(2.95\times 10^{7}\) & \(1.88\times 10^{7}\) \\ Root MSE (RMSE) & 5031 & 5427 & 4338 \\ Mean absolute percentage error (MAPE) & 26.32\% & 28.39\% & 22.59\% \\ R & 0.9572 & 0.9556 & 0.9551 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison of errors and correlations between _Ei_ and dynamic elastic moduli (_Eds_). Next, as a method for predicting the \(Ei\), an analysis was conducted with the fundamentally collected frequencies in the resonance frequency test. Figure 9 shows the relationship between the first resonance frequency in the longitudinal mode and the E-modulus. The first to fourth frequencies are defined as _f1_ to _f4_. For f1, the ASTM equation has an exponential function form. The \(Ei\) and \(Ec\) have the same form. Based on these results, the \(Ei\) and \(Ec\) could be predicted using the ASTM method (correction coefficient, etc. ), and the \(Ei\) and \(Ec\) could be predicted using the resonance frequencies (such as _f1_ and _f2_) collected in the resonance frequency test. Therefore, an analysis was conducted with the _f1_ to _f4_ resonance frequencies collected through the resonance frequency test. Figure 10a shows the relationship between the four frequencies in the longitudinal (LT) mode and initial chord elastic modulus (\(Ei\)), and Figure 10b shows the relationship between the three frequencies in transverse (TR) mode and \(Ei\). The number of used data and frequency range for the frequency combination of LT and TR modes are summarized in Tables 7 and 8. The data with unclear peaks or inconsistent with others in the frequency domain were excluded from the analysis. As both the LT and TR modes increased from _f1_ to _f4_, the variance of the data increased, and the consistency of data decreased. _f1_ and _f2_ of the LT and TR modes have relatively small coefficient of variation (COV) values and are statistically stable, whereas the trend can be identified from _f3_, but the statistical stability is