## 1. Introduction

To evaluate the compressive strength of concrete in cast-in-place or existing structures, the standard destructive test is the most reliable method. Typically, standard cylindrical samples are made on site and sent to the laboratory for compressive strength testing. However, the test results may not be representative, because many factors such as the type and size of the aggregate and cement, the fine aggregate modulus, the water to cement ratio, and the interfacial transition zone, etc., are not considered [

1]. Furthermore, the drilling of core samples from existing concrete members is not always possible on site and the concrete member might be damaged by the process. In this respect, one suitable alternative is to apply non-destructive evaluation (NDE) methods to estimate the concrete strength when core sampling is not preferable. The most widely used NDE method for the strength evaluation of concrete is the ultrasonic pulse velocity test, which measures the velocity between two transducers on both sides of a specimen. The velocity can be calculated based on the assumption that the wave path is known in advance. Many attempts to apply the pressure wave velocity Vp (km/s) as a measure of the concrete compressive strength have been made, using field convenience and simple equipment.

The estimation of concrete strength is typically based on various empirical relations between the concrete strength and nondestructive variables, such as the ultrasonic pulse velocity and rebound number. NDE tests typically have their own empirical relations to connect the test parameter and concrete strength, but these depend strongly on factors such as the water to cement ratio, type and size of the aggregate and cement, and the amount of aggregate. Thus, they cannot be applied universally and require supplemental methods. Furthermore, the typical approach using regression analysis, such as with an exponential function, has not been successful [

1].

The proposed equations have not been accurate for strength estimations because the strength may depend on factors other than the P-wave velocity (Vp). There are many factors that affect concrete strength, in addition to the ultrasonic pulse velocity. For example, the type and size of the aggregate affect the relationship between Vp and the concrete strength. Concrete with the largest aggregate content tends to have the highest pulse velocity [

2]. Furthermore, a higher water content results in a higher ultrasonic velocity of the concrete [

3,

4]. However, a higher water content typically corresponds to a lower strength of the concrete [

5]. This inconsistency can make the interpretation of the ultrasound results difficult. Trtnik et al. [

6] investigated the effects of many factors, including the type, size, and shape of the aggregate, the concrete cast temperature, and the water to cement ratio, on the ultrasonic velocity. They concluded that the aggregate properties were the most effective factors.

For normal-strength concrete, many regression methods, such as linear or nonlinear regressions, have been applied to estimate the concrete strength. However, such regression methods cannot directly predict the concrete strength. Recently, various studies have been performed to obtain more accurate predictions of the concrete strength using machine learning algorithms, such as support vector machine (SVM) and artificial neural network (ANN) methods [

7,

8,

9,

10]. Prasad et al. analyzed the performance of an ANN to estimate the 28-day concrete strength of normal and high-strength concretes [

9]. In their work, factors such as the water to cement ratio, aggregate to cement ratio, and the amount of cement were used as input parameters, and the concrete strength was the output of the network.

As an alternative to ANN, an SVM was developed that could effectively classify data and minimize the risk [

11]. Unlike ANN, which operates based on the minimization of the training error, the SVM was created based on the minimization of the upper bound of the generalization error, which summarized the training error and confidential term [

12]. The aim of the SVM method is to find the global optimum rather than local optima by solving the nonlinear problem in a high-dimensional region. Associated with an insensitive loss function, the SVM method was designed to process nonlinear regression problems, such as wind velocity estimation [

13], traffic flow prediction [

14], financial time-series prediction [

15], and electricity load prediction [

16]. In particular, the SVM has proven to be successful in the prediction of concrete strength. Yan and Shi [

17] used an SVM model to accurately predict the strength and elasticity modulus of concrete. Ahmadi-Nedushan [

18] also predicted these properties with the conventional method and a high-performance model using an adjustable fuzzy neural network, proving that this method is highly reliable. Yuvaraj et al. [

19] predicted the fracture properties of concrete beams using an SVM model and compared them against experimental results. Gencel et al. [

20] adopted an ANN and linear regression algorithm to study the abrasion resistance of concrete with different constituents. The results demonstrated that the ANN was more reliable than the conventional linear regression algorithm.

The capacity of the SVM depends mainly on a penalty factor, a kernel function parameter, and the width of an insensitive loss function. In this sense, the key to the application of the SVM to the prediction of concrete strength is to set the most appropriate parameters. Although previous researchers have tried to optimize the parameters, no approach has been able to provide optimal setting criteria.

This study began with an effort to determine major factors other than the P-wave velocity that are related to concrete strength. It is difficult to estimate the concrete strength indirectly from the mix design or existing buildings, and the predicted strength may not be accurate for newly constructed buildings. Typically, it is most reasonable to measure the current strength based on the physical information identified by reliable methods, such as the rebound hardness, the ultrasonic velocity, and electromagnetic waves. In this regard, the aim of this study was to increase the accuracy of strength evaluations by incorporating the shear wave (S-wave) and Rayleigh wave (R-wave) velocities, because P-waves with small energies and waveforms may not yield accurate predictions of the concrete strength. From the measurements of S- and R-waves, it is possible to predict other properties, such as the shear modulus and Poisson’s ratio, and to estimate the P-wave itself. Also, the S-wave is not affected by the water in concrete and R-wave has the advantage of being sensitive to concrete surfaces [

2,

3].

In this research, an artificial-intelligence-based approach is proposed that incorporates SVM and ANN models as alternatives to regression approaches [

21,

22,

23]. In addition, more core samples were used to enhance the reliability of the analysis. Although fewer than 20 samples were used in most of the previous studies, 72 core samples were used in this study. The compressive strength tests were performed on 72 core samples, and three types of ultrasonic tests were performed before the core samples were produced. Based on the data, ANN and SVM models were developed using the MATLAB software, and concrete strengths were predicted more accurately than those predicted using the regression approach. Thus, the P-, S-, and R-wave ultrasonic pulse velocities were used as the SVM and ANN model inputs, and the actual compressive strengths from the destructive tests were used as the model output.

## 5. Analysis of Results

The optimum parameters were obtained for the SVM and ANN regression algorithms using an optimum search approach corresponding to the best generalized performance for the trained model based on measured performance criteria, such as the mean square error, as listed in

Table 10.

For the SVM, the model was built using all the data (three types of ultrasonic velocities), and the errors with the target values (core strength) were analyzed for accuracy evaluation. For the ANN, 70% of the total data was used to construct the model, 15% was used for data verification, and 15% was used for testing. The ANN model constructed in this manner was compared with the SVM for the entire data set for comparative evaluation.

The prediction accuracies of the SVM and ANN models were evaluated using the correlation coefficient (R), mean absolute error (MAE), mean relative error, and mean squared error (MSE), defined as follows:

where

${\mathrm{y}}_{\mathrm{i}}$ is the target value,

${\hat{\mathrm{y}}}_{\mathrm{i}}$ is the predicted value, and the n is the number of data.

For the comparison, the SVM and ANN models were constructed with different input variables: (a) P-wave velocity, (b) P- and S-wave velocities, and (c) P-, S-, and R-wave velocities.

The output variable was the actual core strength for all the models. After training with the 72 test sample data, all the samples in the testing data set were used to examine the model prediction accuracy. The model prediction results are summarized in

Table 11.

As summarized in

Table 11, the correlation did not increase significantly as the number of input variables increased from one to three. However, as the number of variables increased, the prediction accuracy improved more than the correlation. Overall, the SVM and ANN models with three input variables (P-, S-, and R-wave velocities) yielded the best prediction results in terms of the MAE, MRE, and MSE. For the MAE, which mainly measures the performance of the machine learning, the error was reduced by more than 50% when three variables were used instead of one variable in the SVM and ANN models. Compared with the linear regression, there was no significant difference between the SVM and ANN with one variable (P-wave velocity). Thus, more types of ultrasonic velocities, rather than the predictive method, can provide a more accurate prediction.

Relative comparisons between the SVM and ANN showed a 4.13% improvement in the MSE when three variables were used.

Figure 9 shows the correlation between the actual and predicted strengths. Both SVM and ANN can make good predictions, also the SVM was more highly correlated with the actual strength than the ANN, and it was more concentrated on the equality line.

Figure 10 shows the ratio of the predicted value to the actual value. The data from the SVM model was closer to a ratio of 1 than that of the ANN model, and the data was highly concentrated.

The accessibility of predicted strength by SVM and ANN into the target strength is presented in

Figure 11, and there was less data variance for the SVM model than the ANN model. Understanding the nonlinearity of the predicted model was difficult owing to the lack of data in the 50–60 MPa range. The predicted and measured strengths exhibited abrupt changes above 50 MPa, and thus, the ability of the model to simulate the nonlinearity at normal and high strengths is important.

In

Figure 12, the empirical cumulative distribution function provided by MATLAB was applied to the predicted data, and a ±25% deviation was plotted. First, it was confirmed that below 50 MPa, the data were below the distribution curve, and above 50 MPa, they were above this curve. This was because the velocities of the ultrasonic waves changed significantly from 50 MPa because of the change in the mechanical properties between the normal and high strengths.

The convergence efficiency and prediction accuracy of the ANN method for the data variables are shown in

Figure 13. It was confirmed that convergence was faster, and the MSE (performance) decreased as the number of ultrasonic velocity types increased. Although the prediction accuracy for the validation and test data sets is large due to the limited number of data and nonlinearity of concrete, the differences of prediction in validation and test data sets are greatly reduced and the accuracy is greatly increased with Vp, Vs, and Vr.

This study analyzed the performances and accuracies of the SVM and ANN models with three types of ultrasonic velocities. As in previous studies, the SVM was more accurate than the ANN because of the inherent drawbacks of the ANN, such as the slow convergence, less generalizable performance, tendency to find only local minima, and over-fitting problems. The generalization of the developed model was difficult owing to the nonlinearity in the normal- and high-strength concrete, and it may be more reasonable to use individual models for these concrete types. Moreover, as the concrete strength decreased (in the low strength concrete), various factors, such as the type of aggregate and cement, water to cement ratio, and aggregate interface specificity, must be involved in addition to the ultrasonic velocity. Nevertheless, the mix design information of an existing structure is generally not easily obtained. In this sense, the measurement of Vp, Vs, and Vr can predict the actual strength more accurately even though the economic cost including the device and the human effort for the proposed measurements is more expensive than the traditional methods. Furthermore, these velocities can capture the nonlinearities present for normal- and high-strength concretes.