Prediction of Splitting Tensile Strength from Cylinder Compressive Strength of Concrete by Support Vector Machine

Compressive strength and splitting tensile strength are both important parameters that are utilized for characterization concrete mechanical properties. This paper aims to show a possible applicability of support vector machine (SVM) to predict the splitting tensile strength of concrete from compressive strength of concrete, a SVM model was built, trained, and tested using the available experimental data gathered from the literature. All of the results predicted by the SVM model are compared with results obtained from experimental data, and we found that the predicted splitting tensile strength of concrete is in good agreement with the experimental data. The splitting tensile strength results predicted by SVM are also compared to those obtained by using empirical results of the building codes and various models. These comparisons show that SVM has strong potential as a feasible tool for predicting splitting tensile strength from compressive strength.


Introduction
Compressive strength (  ) and splitting tensile strength ( spt ) are two significant indexes in the design of concrete structure.Tensile strength is important for plain concrete structures such as dam under earthquake excitations.Other structures for example pavement slabs and airfield runways, which are designed based on bending strength, are subjected to tensile stresses.Therefore, in the design of these structures, tensile strength is more important than compressive strength [1,2].Ideally, the splitting tensile strength is measured directly on concrete samples under uniform stresses.However, this is not always easy from an experimental point of view.To avoid the demanding and time-consuming direct measurements of the splitting tensile strength, engineers and researchers have tried to predict the splitting tensile using theoretical and empirical approaches based on compressive strength.Generally, tensile strength of concrete was often assumed proportional to the square root of its compressive strength [3].However, there have been very few published works dealing with experimental and analytical researches of the relation of  spt and   of concretes [4].
Larrad and Malier [5] found that the calculated  spt obtained from the French regulations were in good agreement with experimental data.Kim et al. [6,7] found that the ACI model overestimates the value for concrete with compressive strength less than 20 MPa and underestimates the value for concrete with compressive strength over 30 MPa.Zain et al. [2] determined relationships between tensile strength with compressive strength, concrete age and water/binder (W/B).Shaaban and Gesund [8] investigated the splitting tensile strength and compressive strength of steel fiber reinforced concrete but, unfortunately, did not analyze the relationship between the tensile strength and compressive strength.Choi and Yuan [4] investigated the relationship between the tensile strength and compression strength of glass fiber reinforced concrete and polypropylene fiber reinforced concrete.Xu and Shi [1] developed the formulation between the  spt and   of steel fiber reinforced concrete.Mathematical regression models are usually imperfect description of complex physical phenomena.The accuracy of the estimated result depends on the size of available data.Saridemir [9] proposed for  spt prediction from cylinder   of concrete or age of specimen (AS) and cylinder   of concrete by gene expression programming (GEP).The support vector machine (SVM) is a new, efficient, and novel approach to improve generalization performance, and it can attain a global minimum.SVM achieves good generalization ability by adopting a structural risk minimization induction principle that aims at minimizing a bound on generalization error of a model rather than minimizing the error on the training data only.It has the ability to avoid overtraining and has better generalization capability than artificial neural networks (ANN) model.Moreover, the SVM can always be updated to get better results by presenting new training examples as new data become available.
In this paper, support vector machine is applied to predict the splitting tensile strength of concrete using the datum collected by Saridemir [9].Compressive strength used to predict the splitting tensile strength is considered as input in SVM model that is compared with experimental data and other methods.The mean absolute parentage error (MAPE), root-mean-squared error (RMSE), and -square ( 2 ) are used as the criteria to compare the performance of SVM models and other models.

Support Vector Machine
SVM is a machine-learning algorithm based on statistical learning theory.The main idea of the SVM is to transform the input space into a high-dimensional space by a nonlinear transformation defined by an inner product function.SVM calculation takes the form of a problem in convex quadratic optimization, ensuring that the solution is optimal.It is better than the traditional artificial neural networks, which is based on the traditional minimization principle of experience risk [10].The SVM has a good ability to generalize and resolve some practical problems such as small samples, nonlinearity, and high-dimensional input space.
In this section, a brief description of the process of constructing a SVM for a regression problem is presented.There are three distinct characteristics to consider when an SVM is used to solve a regression problem [11].First, the SVM estimates the regression by a set of linear functions that are defined in a high-dimensional space.Second, the SVM carries out the regression estimation by risk minimization, where the risk is measured using Vapnik's -insensitive loss function.Third, the SVM uses a risk function consisting where,  ∈   and  ∈ ,  is an adjustable weight vector,  is the scalar threshold;   is -dimensional vector space, and  is one-dimensional vector space.Following statistical theory, SVM determines the regression function by minimizing an objective function.The parameters  and  of the regression function are estimated by minimizing the regularized risk function as follows: Minimize Subject to where  > 0 is a penalty factor,  and  * are slack variables. is the insensitive loss function and can be described in the following way: By introducing Lagrangian multipliers and maximizing (3), the dual optimization problem can be expressed as Advances in Materials Science and Engineering 3

Subject to
where the   ,  *  are called Lagrangian multipliers.When the Lagrangian multipliers (  ,  *  ) are equal to zero, shows that the training object is irrelevant to the final solution; otherwise, the training samples with nonzero Lagrangian multipliers are called support vectors.
When linear regression is not appropriate, then the input data have to be mapped into a high-dimensional feature space through some nonlinear mapping.A nonlinear transformation () replaces the input  in (7), and the regression function can be written as where nsv is the number of support vectors and (  ,   ) = (  ) ⋅ (  ), (  ,   ) is a kernel function.Any function satisfying Merce's condition can be used as the kernel function [12,13].Some popular kernel functions are (2) radial basis function (RBF) (3) sigmoid kernel function where   ,  *  satisfy    *  = 0,   ≥ 0, and  *  ≥ 0.
where  max and  min are the maximum and minimum input values data, respectively.In case of SVM training, two types of kernel functions were used, namely, radical basis function (RBF) and polynomial function, in training process,  and  and other kernel-specific parameters have been chosen by a trialand-error approach.The best simulation performances of SVM are summarized in Table 1; in order to evaluate the abilities of SVM models and other models, mean absolute percentage error (MAPE), roof-mean-squared error (RMES), and R-square ( 2 ) were used as the criteria between the experimental and predicted values, which are, according to the equations,as follow: where   is the experimental value,   is the predicted value and  is total number of data.

Results and Discussion
There is no doubt that the splitting tensile strength  spt with an increase in the compression strength of concrete, but there is no agreement on the precise form of the relationship.2 and 3 for 150 × 300 mm cylinder concrete and in Tables 4 and 5 for  100 × 200 mm cylinder concrete.As seen from these results, SVM model is sufficiently close to the experimental data and is able to predict the  spt from the   of concrete.For convenient comparison purpose, the experimental and predicted results are plotted in Figures 1, 2, 3 and 4. It can be seen that the results from SVM method are in good agreement with the experimental results.
The -square errors in the trained SVM are 0.8115 and 0.8227 for RBF and polynomial function of SVM-I, 0.8370 and 0.8253 for those of SVM-II, respectively.-square errors in the tested SVM are 0.9422 and 0.9327 for RBF and polynomial function of SVM-I, 0.8815 and 0.8823 for those of SVM-II, respectively.The whole results obtained from the SVM models show a successful performance of the SVM models of predicting 150 × 300 mm cylinder  spt of concrete from the corresponding 150 × 300 mm cylinder compressive strength   and 100 × 200 mm cylinder  spt from the corresponding 100 × 200 mm cylinder compressive strength   of concrete for each of the training and testing sets.It is also clear that there are no major difference on performance between the radical basis function and the polynomial function kernels that used in this paper.However, in generally, the radical basis function kernel exhibits slightly better performance than the polynomial kernel.
The performance of the trained and tested sets is analyzed by computing MAPE, RMSE and  2 of experimental  spt between the SVM model and other methods.The MAPE, RMSE, and  2 are calculated for these methods.Statistical parameters of the training and testing sets of all methods are presented in Tables 6 and 7.The model having the smallest MAPE or RMSE and the highest  2 can be regarded to be the best model with the assumption of analysis.Generally,  2 of the SVM models are higher than other methods, and the MAPE and RMSE of SVM are smaller than other models.The analysis shows that the SVM model is better than other models and can predict the splitting tensile strength of concrete well.Furthermore, SVM model has good generalization capacity to avoid over training, and can always be updated to get better results by presenting new training examples as new data become available.Thus, SVM model can be regarded as a very effective method to predict splitting tensile strength of concrete from their compressive strength.

Conclusion
The splitting tensile strength of concrete estimations from compression strength has been obtained so far in the literature either through regression or other methods.This present study reports a new and influential approach for predicting the splitting tensile strength using SVM for the first time in the literature.The study conducted in this paper shows the feasibility of using a simple SVM to estimate the splitting tensile strength of concrete.After learning from a set of selected training data, involving compressive strengths of concrete collected from the previous literature, the SVM can be utilized to predict the splitting tensile strengths of the test data.

Advances in Materials Science and Engineering
In this paper, radical basis function and polynomial are adopted for predicting splitting tensile strength from 150 × 300 mm cylinder and 100 × 200 mm cylinder   of concrete.It is found that the splitting tensile strength obtained from the SVM is more accurate than those obtained from design codes and several researches' empirical equation when a comparison is made on the basis of the experimental data.Since the SVM is largely characterized by the type of its kernel function, it is necessary to choose the appropriate kernel for each particular application problem in order to guarantee satisfactory results.The results of radial basis function and polynomial indicate that RBF and polynomial kernel have the ability to predict the splitting tensile strength of concrete from compression strength with an acceptable degree of accuracy.
The statistical parameters of MAPE, RMSE, and  2 show that the proposed SVM model results have the best accuracy and can predict splitting tensile strength very close to experiment results.The use of SVM is very advantageous for the prediction of the splitting tensile strength of concrete from compression strength because it can perform nonlinear regression efficiently for high-dimensional data sets.Furthermore, its solution is global.The satisfactory predictions of the splitting tensile strength of concrete by the model indicate that SVM is a useful modeling tool for engineers and research scientists at concrete construction fields.

Figure 1 :
Figure 1: Comparison of experimental results to training results of SVM-I.

Figure 2 :
Figure 2: Comparison of experimental results to testing results of SVM-I.

Figure 3 :Figure 4 :
Figure 3: Comparison of experimental results to training results of SVM-II.

Table 1 :
Parameters of SVM-I and SVM-II.  ∈   ,   ∈ , where   is an input vector,   is the corresponding output value, and  is the number of training samples.The regression problem is to select a function that predicts the actual value of  as closely as possible, with a precision of .Therefore, the purpose of the SVM is to seek the optimum regression function:

Table 2 :
Comparison of experimental results to training results of RBF Polynomial of SVM-I, and other models.

Table 3 :
Comparison of experimental results to testing results of RBF, polynomial of SVM-I, and other models.

Table 4 :
Comparison of experimental results to training results of RBF, polynomial of SVM-II, and other models.

Table 4 :
Continued.As mentioned previously, many methods have been proposed for the prediction of  spt from compressive strength of concrete; This study attempts to utilize SVM for the prediction of concrete of  spt .SVM-I model is developed for 150 × 300 mm cylinder  spt prediction from 150 × 300 mm cylinder   of concrete, and SVM-II model is developed for 100 × 200 mm and 150 × 200 mm cylinder  spt prediction from 100 × 200 mm cylinder   of concrete.
3. SVM for Predicting  spt of Concrete Codes propose different formulas for prediction cylinder  spt of concrete from compressive strength.In this paper,  spt results are investigated for 150 × 300 mm cylinder concrete and 100 × 200 mm cylinder concrete separately.The errors (predicted values subtract measured values) computed by SVM model, other models, and measured values of splitting tensile strengths are shown in Tables