## 4 Results and Discussion
### Relationship among Static and Initial Chord and Dynamic Elastic Modulus  
The static elastic modulus (\(Ec\)) of concrete can be determined using a dynamic test method. Various empirical equations have been proposed for the prediction of \(Ec\) in several studies, using measured \(Ed\) values. [PERSON] considered the density of concrete in the relationship between Ed and \(Ec\) and proposed Equation (10) for lightweight concrete and general concrete. \(\omega_{c}\) represents the density of hardened concrete (kg/m\({}^{3}\)) [12]. \[E_{c}=\frac{446.09E_{d}{}^{1.4}}{\omega_{c}}(\text{MPa}) \tag{10}\]  
As another example, the British standard BS8110 Part 2 proposed Equation (11). This equation does not apply to concrete or lightweight aggregate concrete containing more than 500 kg of cement per 1 m\({}^{3}\) of concrete [13]. \[E_{c}=1.25E_{d}-19000(\text{MPa}) \tag{11}\]  
[PERSON] and [PERSON] proposed an empirical relationship (Equation (12)) between the Ed and Ec [14]. \[E_{c}=0.83E_{d}(\text{MPa}) \tag{12}\]  
The values of dynamic elastic modulus (_Ed_) as measured by the three methods in Figure 7 were almost similar, and the values followed _Eds (f1, f2.LT, ASTM.LT,_ and ASTM.TR_) calculated as the first and second resonance frequencies. Compared to the added empirical equation, the predicted static elastic modulus (_Ec_) and measured value of do not match, and in fact, the Ed value may vary greatly depending on the test methods and size and type of cylinder. Therefore, it is difficult to select the correct empirical equation for producing the minimum error for the various dynamic tests and cylinders. As expected, the initial chord elastic (_Ei_) had a large margin of error with the _Ed_, and was in the middle of the Ed and _Ec_. In addition, the _Ei_ is less variable and more consistent than the _Ed_ in relation to the _Ec_. Table 4 summarizes the correlations between Ed, _Ei_, and _Ec_. The correlation between the three _Ed_ values and _Ec_ was 0.93, and _Ei_ and _Ec_ were analyzed to have a greater correlation, at 0.96. Therefore, it is necessary to accurately predict the _Ei_ as having a greater correlation with _Ec_. Next, the E-moduli are compared, based on the most commonly used ASTM.LT value among the resonance frequency test for predicting the _Ed_. By comparing ASTM.LT values, other _Eds (ASTM.LT, f1,f2.LT, and ASTM.TR)_ show linear trend, but the _Eds_, initial chord elastic modulus (_Ei_), and static elastic modulus (_Ec_) have nonlinear relationship. _Ei_ and _Ec_ show similar trends and data distributions; thus, it is more appropriate to use the _Ei_ rather than _Eds_, according to the theoretical equation for _Ec_ prediction. In Table 5, the errors and correlations of _Ed_ and _Ei_ are analyzed. MSE, RMSE, and MAPE were used as methods for confirming errors. The MSE and MAPE are defined in Equations (13) and (14), respectively, where n is the number of data, \(A_{i}\) is the _Ei_ value, and \(P_{i}\) is the _Ed_ value. \[MSE=\frac{1}{n}\sum_{i=1}^{n}(A_{i}-P_{i})^{2} \tag{13}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Correlation** & **ASTM.LT-Ec** & **f1,f2.LT-Ec** & **ASTM.TR-Ec** & **Initial Chord Elastic Modulus (Ei)-Ec** \\ \hline Value & 0.9376 & 0.9362 & 0.9364 & 0.9645 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Correlation between static elastic modulus (_Ec_) and E-modulus. Figure 7: Comparison of static elastic modulus (_Ec_), _Ei_, and dynamic elastic modulus (_Ed_). \[MAPE=\frac{1}{n}\sum_{i=1}^{n}\left|\frac{A_{i}-P_{i}}{A_{i}}\right| \tag{14}\]  
As the three _Eds_ have almost similar values and trends for the same cylinder, the correlation with \(Ei\) was close to 0.95. ASTM.TR had the smallest MAPE with the _Ei_. Figure 8 shows the relationships between the _ASTM.LT_ and initial chord elastic modulus (_Ei_) according to mixes and ages. The MAPEs of the dynamic elastic modulus (_Ed_) and _Ei_ were large at low age and decreased as age increased. In addition, the difference between the Ed and _Ei_ in Mix 2, a higher strength mixture, was smaller.