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

Abstract

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.

1. Introduction

Compressive strength (𝑓_𝑐) and splitting tensile strength (𝑓_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 𝑓_spt and 𝑓_𝑐  of concretes [4].

Generally, the splitting tensile strength can be established from compressive strength. National building codes propose various formulas for the splitting tensile strength𝑓_sptand compressive strength𝑓_𝑐. Various relationships for concrete were given as follows: ()𝑓CEB-FIP1991_spt=0.3(𝑓_𝑐)^2/3,()𝑓ACI363R-921992_spt=0.59(𝑓_𝑐)^1/2,()𝑓ACI318-991999_spt=0.56(𝑓_𝑐)^1/2, (1)(2)(3) Where  𝑓_sptand𝑓_𝑐are expressed in MPa.

Larrad and Malier [5] found that the calculated𝑓_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 𝑓_spt and 𝑓_𝑐 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 for𝑓_sptprediction from cylinder 𝑓_𝑐 of concrete or age of specimen (AS) and cylinder 𝑓_𝑐 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 𝑅-square (𝑅²) are used as the criteria to compare the performance of SVM models and other models.

2. 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’s𝜀-insensitive loss function. Third, the SVM uses a risk function consisting of the empirical error and a regularization term, which is derived from the structural risk minimization (SRM) principle.

Consider a set of training samples {𝑥_𝑖,𝑦_𝑖},𝑖=1,2,…𝑛,𝑥_𝑖∈𝑅^𝑑,𝑦_𝑖∈𝑅, where 𝑥_𝑖 is an input vector, 𝑦_𝑖 is the corresponding output value, and𝑛is the number of training samples. The regression problem is to select a function that predicts the actual value of 𝑦 as closely as possible, with a precision of 𝜀. Therefore, the purpose of the SVM is to seek the optimum regression function: 𝑓(𝑥)=𝜔⋅𝑥+𝑏, (4) where,𝜔∈𝑅^𝑛and𝑏∈𝑅,𝜔 is an adjustable weight vector,𝑏is the scalar threshold;𝑅^𝑛 is 𝑛-dimensional vector space, and 𝑥 is one-dimensional vector space.

Following statistical theory, SVM determines the regression function by minimizing an objective function. The parameters𝜔and𝑏of the regression function are estimated by minimizing the regularized risk function as follows:

Minimize 12‖𝜔‖²+𝐶^𝑛_𝑖=1𝜉_𝑖+𝜉^∗_𝑖. (5)

Subject to 𝑦_𝑖−𝜔⋅𝑥_𝑖+𝑏≤𝜀+𝜉_𝑖,𝜔⋅𝑥_𝑖+𝑏−𝑦_𝑖≤𝜀+𝜉^∗_𝑖,𝜉_𝑖≥0,𝜉^∗_𝑖≥0,𝑖=1,2,…𝑛, (6)(7)(8) where𝐶>0is a penalty factor,𝜉and𝜉^∗are slack variables. 𝜀 is the insensitive loss function and can be described in the following way: 𝐿_𝜀𝑦=𝑓(𝑥)−𝑦−𝜀𝑦−𝑓(𝑥)≥𝜀0otherwise. (9)

By introducing Lagrangian multipliers and maximizing (3), the dual optimization problem can be expressed as

Maximize ^𝑛_𝑖=1𝑦_𝑖𝛼^∗_𝑖−𝛼_𝑖−𝜀^𝑛_𝑖=1𝛼^∗_𝑖+𝛼_𝑖−12^𝑛𝑛_𝑖=1𝑗=1𝛼^∗_𝑖−𝛼_𝑖𝛼^∗_𝑗−𝛼_𝑗𝑥_𝑖,𝑥_𝑗. (10)

Subject to ^𝑛_𝑖=1𝛼_𝑖=^𝑛_𝑖=1𝛼^∗_𝑖,0≤𝛼^∗_𝑖≤𝑐,0≤𝛼_𝑖≤𝑐, (11) where the𝛼_𝑖,𝛼^∗_𝑖 are called Lagrangian multipliers. When the Lagrangian multipliers (𝛼_𝑖,𝛼^∗_𝑖) 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 𝜙(𝑥) replaces the input 𝑥 in (7), and the regression function can be written as 𝑓(𝑥)=^nsv_𝑖=1𝛼_𝑖−𝛼^∗_𝑖𝑘𝑥_𝑖,𝑥_𝑗+𝑏, (12) wherensvis the number of support vectors and 𝑘(𝑥_𝑖,𝑥_𝑗)=𝜙(𝑥_𝑖)⋅𝜙(𝑥_𝑗),𝑘(𝑥_𝑖,𝑥_𝑗)is a kernel function. Any function satisfying Merce’s condition can be used as the kernel function [12, 13]. Some popular kernel functions are(1)polynomial kernel function 𝐾𝑥,𝑦=𝑥+𝑦+1^𝑑,𝑑=1,2,…,𝑛, (13)(2)radial basis function (RBF) 𝐾𝑥,𝑦=exp−𝛾𝑥−𝑦²𝛾≥0, (14)(3)sigmoid kernel function 𝐾𝑥,𝑦=tan𝜑𝑥⋅𝑦+𝜃, (15)where𝛼_𝑖,𝛼^∗_𝑖satisfy𝛼_𝑖𝛼^∗_𝑖=0,𝛼_𝑖≥0,  and  𝛼^∗_𝑖≥0.

3. SVM for Predicting 𝑓_spt of Concrete

As mentioned previously, many methods have been proposed for the prediction of𝑓_sptfrom compressive strength of concrete; This study attempts to utilize SVM for the prediction of concrete of𝑓_spt. SVM-I model is developed for150×300mmcylinder𝑓_sptprediction from150×300mmcylinder𝑓_𝑐of concrete, and SVM-II model is developed for 100×200mm and 150×200mm cylinder 𝑓_spt prediction from100×200mmcylinder𝑓_𝑐of concrete. The experimental data which are taken from studies [9, 14, 15] are used in this study. Among 184 experimental data, 138 data were randomly selected as the training set for SVM-I model and among 168 cases, 126 cases were randomly selected as the training set for SVM-II model, the rest are considered as testing data set. The data are normalized before being used in the model as follows: 𝑋𝑋=^𝑖−𝑋_min𝑋_max−𝑋_min, (16) where 𝑋_max and 𝑋_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,𝐶and𝜀and other kernel-specific parameters have been chosen by a trial-and-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 (𝑅²) were used as the criteria between the experimental and predicted values, which are, according to the equations,as follow: 1MAPE=𝑛∑^𝑛_𝑖=1𝑡_𝑖−𝑜_𝑖∑^𝑛_𝑖=1𝑡_𝑖×100%,RMSE=1𝑛^𝑛_𝑖=1𝑡_𝑖−𝑜_𝑖,𝑅²=𝑛∑𝑡_𝑖𝑜_𝑖−∑𝑡_𝑖∑𝑜_𝑖²𝑛∑𝑡²_𝑖−∑𝑡²_𝑖𝑛∑𝑜²_𝑖−∑𝑜_𝑖², (17) where 𝑡_𝑖is the experimental value,𝑜_𝑖is the predicted value and𝑛is total number of data.

Table 1 

Parameters of SVM-I and SVM-II.

4. Results and Discussion

There is no doubt that the splitting tensile strength 𝑓_spt with an increase in the compression strength of concrete, but there is no agreement on the precise form of the relationship. Codes propose different formulas for prediction cylinder 𝑓_spt of concrete from compressive strength. In this paper, 𝑓_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 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 𝑓_spt from the𝑓_𝑐of 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.

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 5 

Comparison of experimental results to testing results of RBF, polynomial of SVM-II, and other models.

Figure 1 

Comparison of experimental results to training results of SVM-I.

Figure 2 

Comparison of experimental results to testing results of SVM-I.

Figure 3 

Comparison of experimental results to training results of SVM-II.

Figure 4 

Comparison of experimental results to testing results of SVM-II.

The 𝑅-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. 𝑅-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 𝑓_spt of concrete from the corresponding 150 × 300 mm cylinder compressive strength 𝑓_𝑐 and 100 × 200 mm cylinder 𝑓_spt from the corresponding 100 × 200 mm cylinder compressive strength 𝑓_𝑐 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 𝑅² of experimental 𝑓_spt between the SVM model and other methods. The MAPE, RMSE,  and 𝑅² 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 𝑅²can be regarded to be the best model with the assumption of analysis. Generally, 𝑅² 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.

Table 6 

Error measurement for 𝑓_spt from 150 × 300 mm cylinder 𝑓_𝑐.

Table 7 

Error measurement for 𝑓_spt from 100 × 200 mm cylinder 𝑓_𝑐.

5. 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.

In this paper, radical basis function and polynomial are adopted for predicting splitting tensile strength from150×300mmcylinder and100×200mmcylinder𝑓_𝑐 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, and𝑅²show 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.

Acknowledgments

This project was granted financial support from the National Natural Science Foundation of China (no. 50808077 and no. 51278188) and the Fundamental Research Funds for the Central Universities. The authors are grateful for support.