*3.4. LR*

LR of ML is a method of predicting values based on a linear function similar to the LR method [40]. However, because various optimization algorithms and cross-validation are employed, LR can achieve a higher predictive power than normal LR. In LR, the number of input data is indicated as n; each input datum is expressed as *x*(*i*) = *<sup>x</sup>*1(*i*), *<sup>x</sup>*2(*i*) using index *i*, and the label (actual value) is expressed as *<sup>y</sup>*(*i*). During model training, the error is typically determined using a loss function, such as the expression defined in Equation (9):

$$\text{loss } function = l^{(i)}(\omega, b) = \frac{1}{2} (\mathfrak{g}^{(i)} - \mathfrak{g}^{(i)})^2 \tag{9}$$

where 1/2 (a constant value) is added to enhance the ease of computation by yielding a value of 1 when the 2D function is differentiated. The smaller the error value, the higher the similarity between the

predicted value and the actual value; if the two values are equal, the error will be zero. LR is used for training a model by determining the model parameter <sup>ω</sup><sup>∗</sup>, *b*∗ = *argminL*(<sup>ω</sup>, *b*), which minimizes the average loss of input data.

### **4. Results and Discussion**

### *4.1. Experimental Variability of Static and Dynamic Tests*

In this study, the coefficient of variation (COV), which is equal to the standard deviation (i.e., σ divided by the average values of the specimen set μ), was used to evaluate the experimental variability of the static and dynamic measurements of the concrete specimens. Outliers were confirmed using the modified Z-score method and excluded from the analyses [41]. The statistical parameters—COV and μ—of the obtained static and dynamic experimental results are presented in Table 2.


**Table 2.** Summary of statistical analysis.

The COV of *Ec* ranged from 2.99% to 8.50%, and the COV of *fc* ranged from 2.41% to 5.34%. When determining the COV of a slightly higher *Ec*, the opposite side of the test specimen was slightly skewed such that the deformation during compression was nonuniform. The error due to the test appeared to have a more significant effect on the *Ec* results than the *fc* results. In Figure 7 and Table 3, the theoretical (ACI 209R) and experimental values for *Ec* and fc are compared. As presented in Figure 7 and Table 3, concrete with FA and GBFS has characteristics such that the development of the initial elastic modulus and strength develop later than those of normal concrete, but the long-term elastic modulus and strength are increased. In addition, it has the advantage of being suitable for mass concrete because it generates less heat of hydration.

**Figure 7.** Comparison of experimental and theoretical values for Ec and fc. (**a**) Static elastic modulus; (**b**) Compressive strength.


**Table 3.** Comparison of experimental (mean) and theoretical values for Ec and fc.

The COV values of *Ed.Vp* determined through the P-waves ranged from 2.17% to 5.99%, and the COV values of *Ed.LT* and *Ed.TR* obtained through the longitudinal and transverse resonance frequency tests were 2.13–4.37% and 2.51–4.69%. The COV values of *Ed.Vp*, *Ed.LT*, and *Ed.TR* were reasonably consistent; however, the COV values of *Ed.Vs* determined through the S-waves ranged from 1.87% to 13.59%, which were higher than those obtained using the other methods. This was because it was often difficult to accurately detect the initial arrival time of the S-waves in early-age concrete, owing to the interference between P-waves and S-waves. Furthermore, even when the S-wave transducer was used, low-amplitude P-waves always appeared with the S-waves in the time domain. This phenomenon was significant owing to the effects of moisture and voids as well as the relatively low compressive strength in the early-age concrete specimens. Therefore, these data sets were not included in the analyses. A thorough effort is required to determine the *Vs* of concrete with relatively low compressive strength.

### *4.2. Relationship between Static and Dynamic Elastic Moduli*

Various equations have been proposed for predicting *Ec* using *Ed*. Lydon and Balendran proposed the following empirical Equation (10) [11]:

$$E\_c = 0.83 E\_d \text{(MPa)} \tag{10}$$

Another empirical Equation (11) was specified in British Standards BS 8110 Part 2 [12]:

$$E\_c = 1.25E\_d - 19 \,\text{\AA}(\text{MPa}) \tag{11}$$

It should be noted that Equation (11) is not applicable to concrete containing a lightweight aggregate or more than 500 kg per m<sup>3</sup> of cement.

Popovics proposed a more general equation for lightweight concrete and normal-weight concrete considering the effect of density [13]:

$$E\_{\mathfrak{c}} = \frac{446.09 E\_d^{1.4}}{\omega\_{\mathfrak{c}}} (\text{MPa}) \tag{12}$$

where ω*c* is the density of hardened concrete (kg/m3).

Figure 8 presents the relationship between *Ed* and *Ec* determined using the four methods; based on the figure, it is evident that *Ed* and *Ec* have a nonlinear relationship. In addition, the modulus of elasticity of the concrete, including FA and GBFS, is low in the early-age ranges and at low strength, so it may exhibit nonlinearity. Subsequently, Equations (10)–(12) and the general regression equations were substituted to compare the experimental results with the values obtained using the proposed equations. The equation recommended in BS 8110 Part 2 was located at the top of the data and fits the *Ed.Vp* data; moreover, the Lydon–Balendran equation was located at the bottom of the data and fits the *Ed.Vs* data. The Popovics equation and the general regression equation were centrally located and fit the *Ed.LT* and *Ed.TR* data. As shown in Figure 8, it can be confirmed that the four equations were suitable

for predicting *Ed* values; however, as the existing equations were not limited to a specific method, the calculated and measured *Ec* values of the entire set of data were compared for each equation.

**Figure 8.** Relationship between *Ec* and *Ed*.

The MSE and MAPE were used to verify the errors between the predicted and measured values. The MSE and MAPE are defined by Equations (13) and (14), respectively:

$$MSE = \frac{1}{n} \sum\_{i=1}^{n} (A\_i - P\_i)^2 \tag{13}$$

$$MAPE = \frac{1}{n} \sum\_{i=1}^{n} \left| \frac{A\_i - P\_i}{A\_i} \right| \tag{14}$$

where n is the number of experimental data, *Ai* is the measured value, and *Pi* is the predicted value. The MSE, MAPE, and R values for the calculated and measured *Ec* values of the entire data set are listed in Table 4. Among the four equations, the Popovics equation yielded the lowest MAPE value, and the error in its *Ec* prediction was small. For the general regression equation, the MSE value was lower than that for the Popovics equation; however, its MAPE was higher because *Ed* and *Ec* have a nonlinear relationship. Therefore, the results of the general regression equation at a low elastic modulus were higher than those predicted using the Popovics equation. Because of the nonlinear relationship between *Ed* and *Ec*, the general regression equation was considered to generate a larger error than the Popovics equation within the low elastic modulus range. In addition, as the MAPE and R values for *Ed.LT* and *Ed.TR* in the Popovics and general regression equations exhibited a small error range, the prediction error of *Ec* could be minimized when the Popovics equation was applied only to the resonance frequency test.


**Table 4.** Correlation analysis for predicted and measured *Ec* values obtained using the proposed equations and the general regression equation.

As the existing equations were not applied to a specific *Ed*, the *Ec* value may have a large error value if the equations are incorrectly applied. General regression using individual *Ed* values was analyzed because the *Ed* measured using the four NDE methods possessed different characteristics. Figure 9 presents the overall regression results for each *Ed* value. The MSE, MAPE, and R values between the calculated *Ec* and measured *Ec* are listed in Table 5. The data for *Ed.Vp* and *Ed.Vs* exhibited a significant variance, resulting in MAPE values of 8.59% and 12.95%, respectively. However, the data for *Ed.LT* and *Ed.TR* exhibited a small variance; hence, their MAPE values were reduced to approximately 7%.

**Figure 9.** Comparison of individual *Ed* and *Ec* values.

**Table 5.** Analysis of correlation between predicted *Ec* and measured *Ec* using general regression for each variable.


### *4.3. Comparison between Static and Prediction Elastic Moduli*

Recently, studies have been conducted to improve prediction accuracy and data reliability, using ML methods. In this study, four ML methods were applied by combining the four predicted *Ed* value

sets to develop a prediction model that is superior to existing models and the general regression model. SVM, ANN, ensemble, and LR were used as the ML methods, and *Ec* prediction and analyses were performed for 15 different sets of combinations. Table 6 lists the MSE and MAPE values of the predicted and measured *Ec* values for the 15 combinations using ML, where A denotes *Ed.Vp*, B denotes *Ed.Vs*, C denotes *Ed.LT*, and D denotes *Ed.TR*.


**Table 6.** *Ec* results for the 15 combinations predicted using four machine learning (ML) methods.

For the results of the *Ec* prediction using the single *Ed*, C had the lowest MAPE value, and the prediction error increased in the order of D, A, and B. Depending on the applied ML method, the MAPE values differed by approximately 1–2%; however, the order remained the same. The MAPE values of individual variables predicted using the general regression of Table 4 ranged from 7 to 13%; however, the MAPE values of individual variables predicted using ML ranged from 4.5 to 8%. When the ML techniques were used, the reduction in the prediction errors was 2.5–5% greater than that when using general regression; hence, using ML in *Ec* prediction could result in a more accurate prediction. Based on the analysis of the combinations of two variables, the overall MAPE values were lower than those of single variables, and the BC combination was optimal for predicting *Ec*. The prediction accuracy improved because the longitudinal wave (C) and the transverse wave (B) complemented each other. For the combinations of three *Ed* variables, the ABC combination and the BCD combination exhibited the lowest MSE and MAPE values, respectively, based on the ML methods. The optimal combination varied depending on the applied ML method; however, the BC combination significantly contributed toward *Ec* prediction because BC was included in both optimal combinations. For the 15 combination sets, the accuracy of the ANN and ensemble methods improved as the number of variables increased, and the SVM and linear regression methods had similar prediction accuracies in two or more combination sets. Overall, the ensemble method yielded the highest accuracy for all combination sets, among the four methods.

Although an analysis of Table 5 confirmed that the BC combination contributed significantly toward the predicted *Ec* values, it was difficult to determine in detail the individual variables that contributed toward this prediction. Therefore, the relative importance (RI) using the weight of the ANN was calculated to determine the detailed contribution of each variable involved in the combination [42]:

$$RI\_X = \sum\_{y=1}^{m} \omega\_{xy} \omega\_{yz} \tag{15}$$

In Equation (15), RI is the relative importance of the variable, <sup>ω</sup>*xy* is the weight between the input neuron and the hidden neuron, and <sup>ω</sup>*yz* is the weight between the hidden neuron and output neuron.

Table 7 lists the RI values for 11 combination sets using the ANN method. The order of RI in the combination was C, D, A, and B, which was similar to the order of the MAPE for individual variables using ML and the order of general regression. Based on the RI analysis, the RI value of B had a minimal ratio; however, when B is combined with C, it should be used as a combination rather than a single variable because it improves the accuracy by 0.6%.


**Table 7.** Comparison of relative importance (RI) values in the ANN.

Figure 10 displays the relationship between the measured *Ec* and the ratio of the predicted *Ec* to the measured *Ec* obtained using the four ML methods. The combinations were selected based on the ensemble method, which achieved the highest accuracy. Among all the ML methods, the ratios of the predicted values with two or more variables exhibited smaller variations than those of the single variables. The results generated via the ensemble method were closer to a ratio of 1, compared with those obtained using other methods, and the data distribution was concentrated. Additionally, in the existing linear prediction, it is difficult to consider the error according to age, but the error can be minimized with training by the ML method ANN and the ensemble method. Figure 11 compares the predicted values from the ensemble method with the experimental values for the concrete age. As confirmed from the figure, there is a good correlation, and some data at an early age are out of the line of equality because ultrasonic pulse velocity and resonance frequency methods are suitable for hardened concrete, so it is necessary to develop ML in consideration of concrete age in the future.

**Figure 10.** Relationship between measured *Ec* and the ratio of predicted *Ec* to measured *Ec* using ML methods: (**a**) SVM; (**b**) ANN; (**c**) Ensemble; (**d**) LR.

**Figure 11.** Relationship between measured *Ec* (experimental values) and predicted *Ec* (theoretical values) by the ensemble.

Table 8 lists the prediction results for each section of the seven ensemble combinations, where A denotes *Ed.Vp*, B denotes *Ed.Vs*, C denotes *Ed.LT*, and D denotes *Ed.TR*; the section was categorized into three ranges. The differences in error for each section were not large; however, the error was generally high at a low elastic modulus. In particular, the error for *Ec* predicted using B was high at a low modulus of elasticity, because the low elastic modulus data contained numerous early-age data. By contrast, based on the results presented in Table 6, the contribution of *Vs* was large because the accuracy improved when B was included in the combination. Additionally, it was useful for 0.4% when it was 15,000 MPa or more, except for the early-age elastic modulus. Therefore, it could improve the accuracy when performing resonance frequency tests and determining *Vs*, which are the most reliable and widely used methods.


**Table 8.** MAPE for each section of *Ec* predicted using the ensemble method.

### *4.4. Relationship between Static Elastic Modulus and Compressive Strength*

The *Ec* of concrete is determined through static uniaxial compressive tests performed according to ASTM C469/C469M-14 requirements. However, *Ec* is typically determined using compression strength values based on the design codes. Therefore, various equations have been proposed to express the

relationship between *Ec* and *fc*. Equation (16), which is proposed in the CEB-FIP and Eurocode 2 codes, is typically applied to normal and high-strength concretes [14]:

$$E\_c = 22\,000 \left( f\_c / 10 \right)^{1/3} \left( \text{MPa} \right) \tag{16}$$

Equation (17), which is recommended by the ACI 318 Committee for *fc* values less than 38 MPa, is expressed as follows [15]:

$$E\_c = 0.43 f\_c^{0.5} w\_c^{1.5} \text{(MPa)}\tag{17}$$

where ω*c* is the density (1440 ≤ ω*c* ≤ 2560 kg/m3).

 Equation (18), suggested by the ACI 363 Committee, is used when the value of *fc* lies between 21 MPa and 83 MPa [16]:

$$E\_{\mathbb{C}} = \left(3320f\_{\mathbb{C}}^{0.5} + 6900\right) (\omega\_{\mathbb{C}}/2300)^{1.5} (\text{MPa})\tag{18}$$

Equation (19) was proposed by Noguchi et al.; correction factors are applied for ordinary and high-strength concretes based on extensive experimental data [43]:

$$E\_c = k\_1 k\_2 33,500 (f\_c/60)^{1/3} (\omega\_c/2400)^2 (\text{MPa}) \tag{19}$$

where *k*1 and *k*2 are the correction factors for the aggregate and SCM, respectively.

Figure 12 depicts the relationship between *Ec* and *fc* in accordance with the ASTM C469 and ASTM C39/C39M-14a standards. Equations (16)–(19) were substituted for comparing the predicted values with the measured values. For Equation (19), *k*1 = *k*2 = 0.95 was employed because materials similar to crushed cobblestone and ground-granulated blast-furnace slag were used.

**Figure 12.** Comparison of the relationships between *Ec* and *fc*.

The MAPEs between the *Ec* calculated using Equations (16)–(18) and the measured *Ec* were 70%, 27%, and 35%, respectively. However, Equation (19) yielded an MAPE of 10%; hence, it was the most suitable equation for predicting *Ec* values. The equation proposed by Noguchi et al. was more suitable than the equations used for ordinary Portland cement, such as CEB-FIP, ACI 318, and ACI 369; this is because the aggregate and SCMs were similar and appropriate correction factors were applied. Therefore, *Ec* and *fc* exhibited a strong correlation, and the values were within a reasonable range.

### *4.5. Relationship between Dynamic Elastic Modulus and Compressive Strength*

Figure 13 presents the relationship between *fc* and *Ed*. The relationship between *Ed* and *fc* was linear when the compressive strength exceeded 20 MPa; however, this relationship was nonlinear at lower strength values, especially due to the concrete mix with FA and GBFS. The *fc* value was calculated using the general regression equations for individual *Ed* values, and the results of the comparison with the measured *fc* values are listed in Table 9. As the *fc* values calculated using all the *Ed* values resulted in large MSE and MAPE values, it was unsuitable to use *Ed* values to compute *fc*. Based on the results of the analysis of individual *Ed* values, the data for A and B exhibited a large variance and MAPE values of 23.54% and 23.64%, respectively. However, the data for C and D exhibited a small variance and MAPE values of approximately 15% each. The MSE difference between A and B was more than twice the A value; however, their MAPE values were almost equal (Table 9). The reason for this difference was that A exhibited a significant variance for all ranges, whereas B exhibited a small variance, except for the early-age ranges.

**Figure 13.** Analysis of general regression for individual *Ed* and *fc*.

**Table 9.** Comparison of MSE, MAPE, and R values for predicted *fc* and measured *fc*.


#### *4.6. Comparison between Predicted and Measured Values of Compressive Strength*

The MSE and MAPE values for the predicted and measured *fc* results for 15 combinations were obtained using four ML methods (Table 10 and Figure 14). For the *fc* prediction results using individual *Ed* values, the MAPE value of C was the lowest, and the prediction error increased in the order of D, A, and B. The MAPE values of the individual *Ed* predicted using the general regression equation presented in Table 9 ranged from 15.3% to 23.5%. However, the MAPE values of the individual *Ed* predicted using ML varied from 6.2% to 22.5%. The SVM and LR methods reduced the MAPE by 2%, as compared to the general regression. However, when the ANN and ensemble methods were used, the MAPE decreased by 8%. Therefore, the ANN and ensemble methods can be used to predict *fc* more accurately. The SVM and LR methods exhibited a significant variation in MAPE for two or more combinations, and the ANN and ensemble methods exhibited decreases in the MAPE as the number of variables increased. For the ANN and ensemble methods, the optimal combinations using two and three variables were BC, ABC, and BCD. Hence, the BC combination contributed significantly to the

accurate prediction of *fc*, because all the optimal combinations included BC. This observation was similar to that for the predicted results of *Ec*.


**Table 10.** Predicted *fc* results for 15 combination sets using the four ML methods.

**Figure 14.** Comparison of MAPE for combinations using four ML methods.

Table 11 and Figure 15 list the RI values for two or more combinations using Equation (15) derived from the ANN method. The order of RI in the combination sets was C, D, A, and B, which was similar to the trend of the MAPE order for individual variables using ML, the order for the general regression, and the RI order of the *Ec* predicted values.


**Table 11.** Comparison of RI values using the ANN method.

**Figure 15.** Comparison of RI values using the ANN method.

Figure 16 depicts the relationship between the measured *fc* and the ratio of the predicted *fc* to the measured *fc* using the four ML methods. The combination sets were selected based on the ensemble method, which achieved the highest accuracy. As B was affected by moisture and voids during the early ages, the MAPE of B was high for the early-age data range; however, at higher strength values (35–50 MPa), B yielded a lower MAPE than A. The results for the ANN and ensemble methods were close to a ratio of 1 for all the ranges, and the variance in the data was small and dense; this resulted in an excellent prediction accuracy. However, the results obtained using the SVM and LR methods had a large dispersion at strengths of 20 MPa and lower. This was because the nonlinearity for the range below 20 MPa was high, as shown in Figures 12 and 13. Furthermore, as the basic functions of SVM and LR are linear, the function might be generalized for the high-strength range, which exhibits linearity and contains numerous data. Therefore, errors in the prediction for low-strength ranges may be high. As a result, the existing linear predictions are difficult for considering the slow development of strength at a low age, but this can be overcome through the use of ML such as ANNs and ensembles and the combination of various *Eds*. On the other hand, the values predicted by ensemble are presented with the experimental results in Figure 17 for the comparison analysis. Similarly, to in Figure 12, it can be seen that the error in early-aged concrete is relatively large; thus, it can be recognized that this approach is more effective for hardened concrete after 14 days of age.

The MAPE values for various *fc* ranges of the seven combination sets predicted using the ensemble method are presented in Table 12 and Figure 18. The MAPE values for the ranges of 0–15 and 35–50 MPa were similar; however, those for the range of 15–35 MPa increased. For the 15–35 MPa strength range, because the later-age data of Mix 1 and the early-age data of Mix 2 were mixed, the prediction error may be higher, owing to the differences in the mix proportions and curing ages of the data set.

**Figure 16.** Relationship between the measured *Ec* and the ratio of predicted *fc* to measured *fc* using the four ML methods. (**a**) SVM; (**b**) ANN; (**c**) Ensemble; (**d**) LR.

**Figure 17.** Relationship between measured *fc* (experimental values) and predicted *fc* (theoretical values) by the ensemble method.


**Table 12.** MAPE values for each section of *fc* predicted using the ensemble method.

**Figure 18.** MAPE values for each section of *fc* predicted using the ensemble method.

In this study, the correlations between *Ed* and *Ec*, and *Ed* and *fc*, were analyzed using four types of *Ed*. Additionally, four ML methods—SVM, ANN, ensemble, and LR—were applied to analyze the accuracy and RI of each variable combination. The relationship between *Ed*-*Ec* and *Ed*-*fc* was nonlinear in the low-elastic-modulus and low-strength ranges. The *Ec* and *fc* values predicted using the ML methods were more accurate than those obtained using general regression. The ML methods were advantageous over general regression because they allow correction for nonlinearities in the ranges of the early-age strength and elastic modulus. The ensemble method achieved the best performance in predicting *Ec* and *fc*, among the four ML methods. C produced the highest accuracy when used for a single variable; however, it was difficult to predict *Ed* and *fc* as a single variable for B, because the variance in the data was significant, owing to the presence of moisture and voids at the early age. However, B could yield the best accuracy in predicting *Ec* and *fc* when combined with C.
