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

Authors: Kezhen Yan; Hongbing Xu; Guanghui Shen; Pei Liu
Journal: Advances in Materials Science and Engineering
 (2013)

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## 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.


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## Body

## 1. Introduction

Compressive strength (fc) and splitting tensile strength (fspt) 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   fspt and  fc  of concretes [4].Generally, the splitting tensile strength can be established from compressive strength. National building codes propose various formulas for the splitting tensile strengthfsptand compressive strengthfc. Various relationships for concrete were given as follows:
(1)CEB-FIP(1991)fspt=0.3(fc)2/3,(2)ACI363R-92(1992)fspt=0.59(fc)1/2,(3)ACI318-99(1999)fspt=0.56(fc)1/2,
Where  fsptandfcare expressed in MPa.Larrad and Malier [5] found that the calculatedfspt 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 fspt and fc 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 forfsptprediction from cylinder fc of concrete or age of specimen (AS) and cylinder fc 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{xi,yi},i=1,2,…n,xi∈Rd,yi∈R, where xi is an input vector, yi  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:
(4)f(x)=ω·x+b,
where,ω∈Rnandb∈R,ω is an adjustable weight vector,bis the scalar threshold;Rn 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(5)12∥ω∥2+C∑i=1n(ξi+ξi*).Subject to(6)yi-[(ω·xi)+b]≤ε+ξi,(7)[(ω·xi)+b]-yi≤ε+ξi*,(8)ξi≥0,ξi*≥0,i=1,2,…n,
whereC>0is a penalty factor,ξandξ*are slack variables. ε is the insensitive loss function and can be described in the following way:
(9)Lε(y)={|f(x)-y|-ε|y-f(x)|≥ε0otherwise.By introducing Lagrangian multipliers and maximizing (3), the dual optimization problem can be expressed asMaximize(10)∑i=1nyi(αi*-αi)-ε∑i=1n(αi*+αi)-12∑i=1n∑j=1n(αi*-αi)(αj*-αj)(xi,xj).Subject to(11)∑i=1nαi=∑i=1nαi*,0≤αi*≤c,0≤αi≤c,
where theαi,αi* are called Lagrangian multipliers. When the Lagrangian multipliers (αi,αi*) 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 transformationϕ(x) replaces the input x in (7), and the regression function can be written as
(12)f(x)=∑i=1nsv(αi-αi*)k(xi,xj)+b,
wherensvis the number of support vectors and k(xi,xj)=ϕ(xi)·ϕ(xj),k(xi,xj)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(13)K(x,y)=[(x+y)+1]d,d=1,2,…,n,(2)
radial basis function (RBF)(14)K(x,y)=exp{-γ∥x-y∥2}γ≥0,(3)
sigmoid kernel function(15)K(x,y)=tan[φ(x·y)+θ],whereαi,αi*satisfyαiαi*=0,αi≥0,  and  αi*≥0.


## 3. SVM for Predictingfspt of Concrete

As mentioned previously, many methods have been proposed for the prediction offsptfrom compressive strength of concrete; This study attempts to utilize SVM for the prediction of concrete offspt. SVM-I model is developed for150×300mmcylinderfsptprediction from150×300mmcylinderfcof concrete, and SVM-II model is developed for 100×200mm and 150×200mm cylinder fspt prediction from100×200mmcylinderfcof 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:
(16)X=(Xi-Xmin)(Xmax-Xmin),
where Xmax and Xmin 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:
(17)MAPE=1n[∑i=1n|ti-oi|∑i=1nti×100%],RMSE=1n∑i=1n(ti-oi),R2=(n∑tioi-∑ti∑oi)2(n∑ti2-(∑ti2))(n∑oi2-(∑oi)2),
where tiis the experimental value,oiis the predicted value andnis total number of data.Table 1
Parameters of SVM-I and SVM-II.
Kernel
SVM-I
SVM-II
Radial basis function
Polynomial
Radial basis function
Polynomial
C
500000
160000
820000
870000
ε
0.10
0.10
0.04
0.05
Parameter
γ
=
0.01
d
=
2
γ
=
0.10
d
=
3


## 4. Results and Discussion

There is no doubt that the splitting tensile strengthfspt 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 fspt of concrete from compressive strength. In this paper,  fspt 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  fspt  from thefcof 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.
f
c (MPa)
f
spt (MPa)
GEP-I [9]
ACI 363R
ACI 318
CEB-FIP
Regression [9]
RBF
Polynomial
2
6.33
3.07
-
0.72
-
0.04
-
0.20
-
0.41
-
0.52
-
0.30
-
0.29
35.82
3.12
-
0.18
0.41
0.23
0.14
-
0.02
0.34
0.01
30.93
3.21
-
0.57
0.07
-
0.10
-
0.25
-
0.39
-
0.33
-
0.30
37.42
2.07
0.97
1.54
1.36
1.29
1.12
0.00
1.14
29.10
3.01
-
0.49
0.17
0.01
-
0.17
-
0.29
0.01
-
0.17
28.03
3.41
-
0.95
-
0.29
-
0.45
-
0.64
-
0.76
-
0.79
-
0.53
36.98
3.25
-
0.24
0.34
0.16
0.08
-
0.09
0.37
-
0.12
31.65
3.39
-
0.71
-
0.07
-
0.24
-
0.39
-
0.53
-
1.07
-
0.40
38.58
3.16
-
0.05
0.50
0.32
0.27
0.09
0.37
0.16
81.00
6.60
-
1.10
-
1.29
-
1.56
-
0.98
-
1.40
-
0.06
-
1.19
68.00
5.90
-
1.11
-
1.03
-
1.28
-
0.90
-
1.24
-
1.76
-
0.99
60.00
5.00
-
0.65
-
0.43
-
0.66
-
0.40
-
0.70
-
0.22
-
0.60
54.00
4.70
-
0.69
-
0.36
-
0.58
-
0.41
-
0.68
-
0.15
-
0.68
39.08
3.11
0.04
0.58
0.39
0.34
0.17
0.76
0.13
22.33
2.29
-
0.20
0.50
0.36
0.09
0.01
0.25
0.39
23.23
2.19
-
0.04
0.65
0.51
0.25
0.16
0.32
0.49
22.89
2.38
-
0.25
0.44
0.30
0.04
-
0.05
0.36
0.34
22.16
2.17
-
0.09
0.61
0.47
0.20
0.11
0.34
0.50
20.48
2.15
-
0.18
0.52
0.38
0.10
0.02
0.07
0.48
20.82
1.99
0.00
0.70
0.57
0.28
0.21
0.02
0.65
22.83
2.41
-
0.28
0.41
0.27
0.00
-
0.08
0.10
0.25
25.14
2.62
-
0.34
0.34
0.19
-
0.05
-
0.14
0.12
0.12
25.60
2.69
-
0.38
0.30
0.14
-
0.08
-
0.19
-
0.15
0.09
35.20
3.20
-
0.29
0.30
0.12
0.02
-
0.13
0.39
-
0.10
44.40
2.90
0.57
1.03
0.83
0.86
0.65
0.76
0.62
37.60
2.40
0.66
1.22
1.03
0.97
0.80
0.00
0.84
41.80
3.60
-
0.29
0.21
0.02
0.01
-
0.18
0.12
-
0.20
42.00
3.50
-
0.17
0.32
0.13
0.12
-
0.07
0.47
-
0.10
38.30
2.70
0.40
0.95
0.77
0.71
0.53
0.41
0.51
55.40
3.40
0.71
0.99
0.77
0.96
0.69
0.00
0.78
54.00
2.80
1.23
1.54
1.32
1.49
1.22
0.00
1.22
48.60
2.80
0.91
1.31
1.10
1.20
0.96
0.00
0.94
56.50
4.10
0.07
0.33
0.11
0.32
0.04
-
0.06
0.09
47.00
3.90
-
0.28
0.14
-
0.06
0.01
-
0.22
0.19
-
0.22
45.69
4.19
-
0.65
-
0.20
-
0.40
-
0.36
-
0.57
-
0.37
-
0.59
41.71
3.09
0.22
0.72
0.53
0.52
0.32
0.32
0.33
42.49
3.74
-
0.38
0.11
-
0.09
-
0.09
-
0.29
-
0.53
-
0.27
33.69
2.93
-
0.11
0.49
0.32
0.20
0.05
0.40
0.16
37.30
3.14
-
0.10
0.46
0.28
0.21
0.04
0.49
0.07
24.75
2.62
-
0.37
0.32
0.17
-
0.07
-
0.17
-
0.21
0.11
26.96
2.68
-
0.28
0.38
0.23
0.02
-
0.09
0.36
0.07
22.14
2.35
-
0.27
0.43
0.28
0.02
-
0.07
0.16
0.32
23.91
2.24
-
0.04
0.64
0.50
0.25
0.16
0.03
0.44
41.00
3.90
-
0.63
-
0.12
-
0.31
-
0.33
-
0.52
-
0.94
-
0.49
42.00
4.00
-
0.67
-
0.18
-
0.37
-
0.38
-
0.57
-
0.03
-
0.60
57.68
4.26
-
0.04
0.22
-
0.01
0.22
-
0.07
0.57
-
0.02
45.75
4.10
-
0.57
-
0.11
-
0.31
-
0.26
-
0.48
-
0.28
-
0.45
38.10
3.62
-
0.54
0.02
-
0.16
-
0.22
-
0.40
-
0.19
-
0.37
31.95
3.31
-
0.61
0.02
-
0.14
-
0.29
-
0.43
-
0.27
-
0.40
18.20
1.90
-
0.10
0.62
0.49
0.18
0.12
-
0.24
0.68
28.70
2.70
-
0.20
0.46
0.30
0.11
-
0.01
-
0.20
0.23
23.90
2.30
-
0.11
0.58
0.44
0.19
0.10
-
0.03
0.38
43.70
3.30
0.11
0.60
0.40
0.42
0.22
0.44
0.21
24.60
2.30
-
0.07
0.63
0.48
0.24
0.14
-
0.01
0.45
43.00
3.50
-
0.13
0.37
0.17
0.18
-
0.02
0.00
0.00
14.20
1.80
-
0.28
0.42
0.31
-
0.04
-
0.08
-
0.29
0.63
36.70
3.20
-
0.21
0.37
0.19
0.11
-
0.05
-
0.48
0.01
22.70
1.70
0.41
1.11
0.97
0.71
0.62
0.35
0.95
35.30
2.40
0.51
1.11
0.93
0.83
0.67
0.00
0.74
13.60
1.30
0.18
0.88
0.77
0.41
0.38
0.79
1.12
28.60
1.70
0.79
1.46
1.29
1.11
0.99
1.40
1.14
26.60
2.40
-
0.04
0.64
0.49
0.27
0.17
-
0.01
0.42
43.50
2.90
0.50
0.99
0.79
0.81
0.61
0.52
0.57
30.90
2.40
0.24
0.88
0.71
0.55
0.42
0.48
0.51
49.90
3.80
-
0.03
0.37
0.16
0.27
0.03
0.39
0.04
32.20
3.00
-
0.28
0.35
0.18
0.04
-
0.10
-
0.09
0.01
57.80
3.50
0.72
0.99
0.76
0.98
0.70
0.33
0.81
26.80
2.40
-
0.02
0.65
0.50
0.29
0.18
0.47
0.44
36.50
2.90
0.08
0.66
0.48
0.40
0.24
0.27
0.26
65.60
4.40
0.26
0.38
0.14
0.48
0.15
0.14
0.36
30.30
2.40
0.20
0.85
0.68
0.52
0.39
1.37
0.54
52.80
3.80
0.14
0.49
0.27
0.42
0.17
-
0.39
0.23
40.60
3.50
-
0.27
0.26
0.07
0.04
-
0.14
0.15
-
0.13
52.30
4.20
-
0.29
0.07
-
0.15
0.00
-
0.26
0.10
-
0.24
41.20
3.20
0.06
0.59
0.39
0.38
0.19
0.65
0.22
72.80
5.10
-
0.05
-
0.07
-
0.32
0.13
-
0.24
0.03
-
0.05
41.20
3.20
0.06
0.59
0.39
0.38
0.19
0.65
0.22
72.80
5.10
-
0.05
-
0.07
-
0.32
0.13
-
0.24
0.03
-
0.05
72.80
5.10
-
0.05
-
0.07
-
0.32
0.13
-
0.24
0.03
-
0.05
41.80
3.30
0.00
0.51
0.32
0.31
0.12
0.42
0.10
68.40
4.10
0.71
0.78
0.53
0.92
0.57
0.38
0.76
42.30
2.90
0.43
0.94
0.74
0.74
0.54
0.00
0.57
59.10
3.80
0.50
0.74
0.51
0.75
0.46
0.25
0.60
41.30
3.70
-
0.43
0.09
-
0.10
-
0.12
-
0.31
-
0.19
-
0.27
59.70
4.10
0.23
0.46
0.23
0.48
0.19
0.14
0.30
42.30
3.20
0.13
0.64
0.44
0.44
0.24
0.71
0.27
75.10
5.10
0.08
0.01
-
0.25
0.24
-
0.14
0.67
-
0.05
48.50
3.70
-
0.01
0.41
0.20
0.29
0.06
0.08
0.09
81.90
4.80
0.75
0.54
0.27
0.86
0.44
0.94
0.53
41.30
3.30
-
0.03
0.49
0.30
0.28
0.09
0.21
0.13
67.10
4.40
0.34
0.43
0.19
0.55
0.22
0.68
0.36
42.00
3.20
0.11
0.62
0.43
0.42
0.23
0.77
0.20
70.10
4.30
0.61
0.64
0.39
0.80
0.45
0.65
0.58
44.80
3.40
0.08
0.55
0.35
0.38
0.17
0.36
0.21
65.40
4.00
0.65
0.77
0.53
0.87
0.54
0.67
0.70
38.70
3.50
-
0.39
0.17
-
0.02
-
0.07
-
0.24
0.40
-
0.21
56.40
4.30
-
0.16
0.13
-
0.09
0.11
-
0.16
-
0.25
-
0.14
42.40
3.20
0.13
0.64
0.45
0.45
0.25
0.41
0.25
67.70
4.80
-
0.03
0.05
-
0.19
0.18
-
0.16
0.27
-
0.03
50.70
3.60
0.22
0.60
0.39
0.51
0.26
0.48
0.26
81.10
4.80
0.70
0.51
0.24
0.82
0.41
0.62
0.52
42.00
3.20
0.11
0.62
0.43
0.42
0.23
0.77
0.20
48.90
3.60
0.11
0.53
0.32
0.41
0.18
0.61
0.21
85.30
5.80
-
0.07
-
0.35
-
0.63
0.01
-
0.42
0.32
-
0.37
52.30
4.20
-
0.29
0.07
-
0.15
0.00
-
0.26
0.10
-
0.24
85.10
5.80
-
0.08
-
0.36
-
0.63
0.00
-
0.43
-
0.34
-
0.42
49.70
4.20
-
0.44
-
0.04
-
0.25
-
0.14
-
0.38
-
0.49
-
0.36
71.20
5.30
-
0.33
-
0.32
-
0.57
-
0.15
-
0.50
-
0.52
-
0.33
51.90
3.80
0.09
0.45
0.23
0.37
0.12
0.36
0.17
87.70
6.10
-
0.24
-
0.57
-
0.86
-
0.18
-
0.63
-
0.05
-
0.66
51.90
3.80
0.09
0.45
0.23
0.37
0.12
0.36
0.17
87.70
6.10
-
0.24
-
0.57
-
0.86
-
0.18
-
0.63
-
0.05
-
0.66
51.90
3.80
0.09
0.45
0.23
0.37
0.12
0.36
0.17
87.70
6.10
-
0.24
-
0.57
-
0.86
-
0.18
-
0.63
-
0.05
-
0.66
53.80
4.10
-
0.10
0.23
0.01
0.18
-
0.09
-
0.10
-
0.06
85.30
5.10
0.63
0.35
0.07
0.71
0.28
1.02
0.33
58.70
4.40
-
0.13
0.12
-
0.11
0.13
-
0.16
-
0.26
-
0.09
87.10
5.60
0.22
-
0.09
-
0.37
0.29
-
0.15
-
0.09
-
0.20
49.30
4.50
-
0.76
-
0.36
-
0.57
-
0.47
-
0.70
-
1.35
-
0.69
66.20
5.10
-
0.41
-
0.30
-
0.54
-
0.19
-
0.52
-
0.23
-
0.40
53.10
3.80
0.16
0.50
0.28
0.44
0.18
0.50
0.33
84.30
5.60
0.07
-
0.18
-
0.46
0.17
-
0.26
0.10
-
0.15
86.70
6.03
0.03
-
0.54
-
0.80
-
0.16
-
0.60
0.02
-
0.54
56.10
4.30
-
0.17
0.12
-
0.11
0.10
-
0.18
-
0.58
-
0.17
52.10
3.80
0.10
0.46
0.24
0.38
0.13
0.11
0.16
78.50
5.00
0.36
0.23
-
0.04
0.50
0.10
0.87
0.29
49.90
4.10
-
0.33
0.07
-
0.14
-
0.03
-
0.27
0.09
-
0.26
87.50
5.10
0.75
0.42
0.14
0.81
0.37
0.48
0.29
55.10
4.40
-
0.33
-
0.02
-
0.24
-
0.06
-
0.33
-
0.11
-
0.23
75.40
4.00
1.22
1.12
0.86
1.35
0.97
0.00
1.05
24.67
2.74
-
0.50
0.19
0.04
-
0.20
-
0.29
-
0.33
0.02
23.01
2.79
-
0.66
0.04
-
0.10
-
0.36
-
0.45
0.00
-
0.09
32.03
3.03
-
0.32
0.31
0.14
0.00
-
0.14
-
0.64
-
0.03
28.78
2.93
-
0.43
0.24
0.07
-
0.11
-
0.23
-
0.43
0.00
33.60
2.03
0.77
1.39
1.22
1.09
0.95
1.30
1.06
23.30
3.04
-
0.89
-
0.19
-
0.34
-
0.59
-
0.68
-
1.25
-
0.31
GEP-I model is developed by Saridemir [9] based on gene expression programming. Regression is regression-based formulation results by Saridemir [9].Table 3
Comparison of experimental results to testing results of RBF, polynomial of SVM-I, and other models.
f
c (MPa)
f
spt (MPa)
GEP-I [9]
ACI 363R
ACI 318
CEB-FIP
Regression [9]
RBF
Polynomial
4
4.80
3.80
-
0.31
0.15
-
0.05
-
0.02
-
0.23
-
0.01
-
0.33
56.20
4.40
-
0.25
0.02
-
0.20
0.00
-
0.27
-
0.41
-
0.20
53.40
3.60
0.39
0.71
0.49
0.65
0.39
0.33
0.39
56.70
4.60
-
0.42
-
0.16
-
0.38
-
0.17
-
0.45
-
0.09
-
0.36
64.50
4.80
-
0.18
-
0.06
-
0.30
0.02
-
0.30
-
0.18
-
0.12
66.50
4.70
0.03
0.11
-
0.13
0.22
-
0.11
0.10
0.01
65.40
4.30
0.37
0.47
0.23
0.57
0.24
0.21
0.43
73.80
4.90
0.23
0.17
-
0.09
0.38
0.01
0.22
0.14
74.00
5.30
-
0.16
-
0.22
-
0.48
-
0.01
-
0.39
-
0.26
-
0.15
72.50
5.00
0.06
0.02
-
0.23
0.22
-
0.15
0.24
-
0.06
74.60
4.90
0.28
0.20
-
0.06
0.42
0.04
0.06
0.15
77.90
5.50
-
0.14
-
0.29
-
0.56
-
0.03
-
0.42
-
0.36
-
0.19
79.10
5.20
0.22
0.05
-
0.22
0.33
-
0.07
0.01
0.15
84.20
5.70
0.00
-
0.29
-
0.56
0.06
-
0.37
-
0.34
-
0.28
83.30
5.90
-
0.25
-
0.52
-
0.79
-
0.18
-
0.60
-
0.60
-
0.39
80.10
5.30
0.15
-
0.02
-
0.29
0.27
-
0.13
-
0.22
0.10
86.50
5.50
0.32
-
0.01
-
0.29
0.37
-
0.07
-
0.27
0.12
102.00
5.50
1.15
0.46
0.16
1.05
0.53
0.37
0.34
101.00
6.50
0.09
-
0.57
-
0.87
0.01
-
0.51
-
0.45
-
0.60
111.00
6.20
0.92
0.02
-
0.30
0.73
0.16
0.40
-
0.17
94.50
5.80
0.45
-
0.06
-
0.36
0.42
-
0.06
0.08
-
0.01
118.00
6.20
1.28
0.21
-
0.12
1.02
0.41
0.64
-
0.14
19.00
2.30
-
0.43
0.27
0.14
-
0.16
-
0.23
-
0.04
-
1.00
20.00
2.40
-
0.46
0.24
0.10
-
0.19
-
0.26
-
0.13
-
0.96
28.60
2.90
-
0.40
0.26
0.09
-
0.09
-
0.21
0.11
-
0.67
28.90
3.00
-
0.48
0.17
0.01
-
0.17
-
0.30
0.08
-
0.82
36.10
3.30
-
0.33
0.24
0.06
-
0.02
-
0.19
-
0.11
-
0.48
44.50
3.70
-
0.23
0.24
0.04
0.07
-
0.14
0.07
-
0.26
23.00
2.50
-
0.36
0.33
0.19
-
0.07
-
0.16
-
0.37
-
0.82
33.00
3.50
-
0.72
-
0.11
-
0.28
-
0.41
-
0.56
-
0.43
-
1.02
58.00
4.20
0.05
0.29
0.06
0.30
0.01
-
0.25
0.20
69.00
4.50
0.37
0.40
0.15
0.55
0.20
0.12
0.44
36.00
3.30
-
0.34
0.24
0.06
-
0.03
-
0.19
-
0.04
-
0.45
42.80
3.90
-
0.53
-
0.04
-
0.24
-
0.23
-
0.43
-
0.19
-
0.63
59.30
4.00
0.33
0.54
0.31
0.56
0.27
0.29
0.40
63.30
4.50
0.05
0.19
-
0.04
0.26
-
0.05
-
0.18
0.20
67.70
4.70
0.10
0.15
-
0.09
0.28
-
0.06
0.26
0.07
77.00
5.10
0.21
0.08
-
0.19
0.33
-
0.06
-
0.05
0.01
49.90
4.30
-
0.51
-
0.13
-
0.34
-
0.23
-
0.47
-
0.32
-
0.45
96.70
5.70
0.63
0.10
-
0.19
0.62
0.12
0.05
0.01
61.70
4.60
-
0.14
0.03
-
0.20
0.08
-
0.22
-
0.36
-
0.08
73.20
5.10
0.00
-
0.05
-
0.31
0.15
-
0.22
0.17
-
0.03
86.50
5.50
0.32
-
0.01
-
0.29
0.37
-
0.07
-
0.27
0.12
92.30
5.90
0.23
-
0.23
-
0.52
0.23
-
0.25
0.10
-
0.19Table 4
Comparison of experimental results to training results of RBF, polynomial of SVM-II, and other models.
f
c (MPa)
f
spt (MPa)
GEP-III [9]
ACI 363R
ACI 318
CEB-FIP
Regression [9]
RBF
Polynomial
7
7.90
5.45
-
0.08
-
0.24
-
0.51
0.02
-
0.33
0.48
0.32
57.10
3.13
1.03
1.33
1.10
1.32
1.15
1.74
1.45
71.60
4.90
0.11
0.09
-
0.16
0.27
-
0.02
0.81
0.51
44.10
3.91
-
0.52
0.01
-
0.19
-
0.17
-
0.23
0.28
-
0.17
64.80
4.57
0.04
0.18
-
0.06
0.27
0.03
0.73
0.42
42.60
3.66
-
0.36
0.19
0.00
0.00
-
0.05
0.47
-
0.04
30.40
2.79
-
0.26
0.46
0.30
0.13
0.17
0.46
0.00
46.80
3.89
-
0.34
0.15
-
0.06
0.01
-
0.08
0.49
0.04
89.50
5.91
0.11
-
0.33
-
0.61
0.09
-
0.36
0.73
0.41
66.90
4.55
0.19
0.28
0.03
0.39
0.14
0.96
0.59
76.10
5.47
-
0.21
-
0.32
-
0.58
-
0.08
-
0.42
0.41
0.22
53.60
4.37
-
0.41
-
0.05
-
0.27
-
0.11
-
0.25
0.47
-
0.02
64.80
5.07
-
0.46
-
0.32
-
0.56
-
0.23
-
0.47
0.23
-
0.08
45.80
4.02
-
0.53
-
0.03
-
0.23
-
0.18
-
0.26
0.40
-
0.16
37.60
3.16
-
0.17
0.46
0.27
0.21
0.19
0.63
0.14
53.10
4.08
-
0.15
0.22
0.00
0.16
0.02
0.78
0.24
34.90
3.50
-
0.68
-
0.01
-
0.19
-
0.30
-
0.29
0.12
-
0.40
53.90
5.40
-
1.42
-
1.07
-
1.29
-
1.12
-
1.27
-
0.62
-
1.00
46.50
3.80
-
0.27
0.22
0.02
0.08
-
0.01
0.60
0.07
13.80
1.70
-
0.29
0.49
0.38
0.03
0.17
-
0.42
-
0.17
67.10
4.80
-
0.05
0.03
-
0.21
0.15
-
0.10
0.64
0.35
20.30
3.40
-
1.53
-
0.74
-
0.88
-
1.17
-
1.06
-
1.26
-
1.36
22.40
2.10
-
0.09
0.69
0.55
0.28
0.38
0.38
0.10
61.40
4.30
0.12
0.32
0.09
0.37
0.16
0.79
0.51
33.10
3.00
-
0.29
0.39
0.22
0.09
0.11
0.53
-
0.03
79.00
5.60
-
0.17
-
0.36
-
0.62
-
0.08
-
0.44
0.46
0.21
43.60
3.60
-
0.24
0.30
0.10
0.12
0.06
0.59
0.10
25.60
2.42
-
0.20
0.57
0.41
0.19
0.26
0.31
0.02
26.20
2.62
-
0.36
0.40
0.25
0.03
0.10
0.23
-
0.15
16.90
1.88
-
0.25
0.55
0.42
0.10
0.23
-
0.13
-
0.10
14.90
1.71
-
0.22
0.57
0.45
0.11
0.25
-
0.38
-
0.08
27.30
2.81
-
0.48
0.27
0.12
-
0.09
-
0.03
0.19
-
0.26
28.40
2.87
-
0.47
0.27
0.11
-
0.08
-
0.02
0.22
-
0.23
16.90
1.91
-
0.28
0.52
0.39
0.07
0.20
-
0.16
-
0.13
16.90
2.11
-
0.48
0.32
0.19
-
0.13
0.00
-
0.36
-
0.33
30.90
2.92
-
0.35
0.36
0.19
0.03
0.07
0.31
-
0.10
32.30
3.08
-
0.43
0.27
0.10
-
0.04
-
0.01
0.36
-
0.17
19.60
2.22
-
0.40
0.39
0.26
-
0.04
0.08
-
0.10
-
0.22
19.10
2.08
-
0.30
0.50
0.37
0.06
0.18
-
0.11
-
0.10
35.30
3.24
-
0.40
0.27
0.09
-
0.01
-
0.01
0.38
-
0.11
38.40
3.36
-
0.32
0.30
0.11
0.05
0.04
0.57
-
0.03
22.50
2.47
-
0.45
0.33
0.19
-
0.08
0.02
-
0.05
-
0.27
23.40
2.57
-
0.49
0.28
0.14
-
0.12
-
0.02
-
0.08
-
0.29
31.00
2.90
-
0.33
0.38
0.22
0.06
0.10
0.35
-
0.06
17.00
2.30
-
0.66
0.13
0.01
-
0.32
-
0.18
-
0.60
-
0.51
47.00
4.00
-
0.44
0.04
-
0.16
-
0.09
-
0.18
0.44
-
0.09
37.00
3.60
-
0.65
-
0.01
-
0.19
-
0.27
-
0.28
0.21
-
0.35
42.00
4.00
-
0.74
-
0.18
-
0.37
-
0.38
-
0.42
0.16
-
0.39
54.00
4.60
-
0.62
-
0.26
-
0.48
-
0.31
-
0.46
0.21
-
0.24
42.00
4.40
-
1.14
-
0.58
-
0.77
-
0.78
-
0.82
-
0.24
-
0.79
62.00
5.20
-
0.75
-
0.55
-
0.79
-
0.50
-
0.72
0.03
-
0.35
67.00
4.25
0.49
0.58
0.33
0.70
0.44
1.15
0.91
72.50
5.10
-
0.04
-
0.08
-
0.33
0.12
-
0.19
0.56
0.38
74.50
5.20
-
0.03
-
0.11
-
0.37
0.11
-
0.21
0.59
0.38
79.50
5.75
-
0.29
-
0.49
-
0.76
-
0.20
-
0.57
0.30
0.11
78.00
5.15
0.22
0.06
-
0.20
0.33
-
0.03
0.83
0.60
85.00
5.75
0.02
-
0.31
-
0.59
0.05
-
0.36
0.69
0.44
86.00
4.80
1.03
0.67
0.39
1.05
0.62
1.67
1.40
90.50
4.40
1.68
1.21
0.93
1.65
1.19
2.32
2.04
54.80
3.34
0.69
1.03
0.81
0.99
0.83
1.61
1.06
32.90
2.85
-
0.16
0.53
0.36
0.23
0.25
0.68
0.11
34.50
2.92
-
0.13
0.55
0.37
0.26
0.27
0.66
0.13
53.60
3.48
0.48
0.84
0.62
0.78
0.64
1.36
0.87
74.10
5.26
-
0.11
-
0.18
-
0.44
0.03
-
0.29
0.58
0.30
67.70
4.81
-
0.03
0.04
-
0.20
0.17
-
0.09
0.68
0.37
84.20
6.57
-
0.85
-
1.16
-
1.43
-
0.81
-
1.21
-
0.25
-
0.46
75.30
5.35
-
0.13
-
0.23
-
0.49
0.00
-
0.33
0.48
0.28
102.30
7.29
-
0.55
-
1.32
-
1.63
-
0.73
-
1.29
0.52
-
0.31
40.10
3.30
-
0.16
0.44
0.25
0.21
0.18
0.70
0.15
40.50
3.40
-
0.23
0.35
0.16
0.14
0.10
0.57
0.07
95.50
5.70
0.66
0.07
-
0.23
0.57
0.06
1.53
0.96
35.20
3.17
-
0.33
0.33
0.15
0.05
0.06
0.48
-
0.05
57.50
3.69
0.51
0.78
0.56
0.78
0.60
1.29
0.86
38.60
3.38
-
0.33
0.29
0.10
0.05
0.03
0.52
-
0.04
61.50
4.48
-
0.05
0.15
-
0.09
0.19
-
0.02
0.70
0.32
47.40
3.76
-
0.17
0.30
0.10
0.17
0.08
0.67
0.18
47.10
3.68
-
0.11
0.37
0.16
0.23
0.14
0.86
0.23
44.70
3.26
0.17
0.68
0.48
0.52
0.45
1.09
0.46
52.40
3.99
-
0.10
0.28
0.06
0.21
0.08
0.79
0.27
57.90
4.49
-
0.27
0.00
-
0.23
0.00
-
0.18
0.61
0.12
39.50
4.04
-
0.93
-
0.33
-
0.52
-
0.56
-
0.59
0.00
-
0.65
36.70
4.03
-
1.09
-
0.46
-
0.64
-
0.72
-
0.72
-
0.29
-
0.82
38.00
3.65
-
0.63
-
0.01
-
0.20
-
0.26
-
0.28
0.20
-
0.35
24.20
2.20
-
0.07
0.70
0.55
0.31
0.40
0.38
0.14
17.40
1.80
-
0.13
0.66
0.54
0.21
0.34
-
0.06
0.02
29.30
2.80
-
0.33
0.39
0.23
0.05
0.10
0.27
-
0.09
27.00
2.55
-
0.23
0.52
0.36
0.15
0.22
0.33
-
0.02
32.60
3.21
-
0.53
0.16
-
0.01
-
0.15
-
0.12
0.26
-
0.26
46.30
5.20
-
1.67
-
1.19
-
1.39
-
1.33
-
1.41
-
0.76
-
1.37
35.80
4.30
-
1.42
-
0.77
-
0.95
-
1.04
-
1.04
-
0.57
-
1.11
43.10
4.18
-
0.85
-
0.31
-
0.50
-
0.49
-
0.55
-
0.04
-
0.51
35.20
3.17
-
0.33
0.33
0.15
0.05
0.06
0.48
-
0.05
57.50
3.69
0.51
0.78
0.56
0.78
0.60
1.29
0.86
42.80
4.90
-
1.59
-
1.04
-
1.24
-
1.23
-
1.28
-
0.76
-
1.26
61.50
4.48
-
0.05
0.15
-
0.09
0.19
-
0.02
0.70
0.32
47.40
3.76
-
0.17
0.30
0.10
0.17
0.08
0.67
0.18
47.13
3.68
-
0.10
0.37
0.16
0.23
0.14
0.86
0.23
44.73
3.26
0.17
0.69
0.49
0.52
0.45
1.04
0.50
34.60
4.10
-
1.30
-
0.63
-
0.81
-
0.91
-
0.90
-
0.51
-
1.02
59.70
5.40
-
1.08
-
0.84
-
1.07
-
0.82
-
1.01
-
0.30
-
0.69
51.90
4.00
-
0.14
0.25
0.03
0.17
0.04
0.76
0.21
50.00
3.90
-
0.15
0.27
0.06
0.17
0.06
0.77
0.22
48.50
4.10
-
0.44
0.01
-
0.20
-
0.11
-
0.21
0.49
-
0.08
48.80
3.70
-
0.02
0.42
0.21
0.31
0.20
0.80
0.32
46.50
3.80
-
0.26
0.22
0.02
0.08
-
0.01
0.60
0.07
45.20
4.00
-
0.54
-
0.03
-
0.24
-
0.19
-
0.27
0.30
-
0.21
45.20
3.30
0.16
0.67
0.46
0.51
0.43
1.00
0.49
42.60
2.98
0.32
0.87
0.68
0.68
0.63
1.15
0.64
81.60
4.40
1.19
0.93
0.66
1.24
0.86
1.80
1.58
42.70
3.00
0.31
0.86
0.66
0.66
0.61
1.21
0.64
41.90
3.50
-
0.24
0.32
0.12
0.12
0.07
0.60
0.09
41.20
3.80
-
0.59
-
0.01
-
0.21
-
0.22
-
0.26
0.31
-
0.26
39.70
2.70
0.42
1.02
0.83
0.79
0.76
1.24
0.73
65.50
5.50
-
0.84
-
0.73
-
0.97
-
0.63
-
0.87
-
0.21
-
0.47
39.50
3.70
-
0.59
0.01
-
0.18
-
0.22
-
0.25
0.34
-
0.31
17.10
1.90
-
0.25
0.54
0.42
0.09
0.22
-
0.11
-
0.10
12.10
1.10
0.18
0.95
0.85
0.48
0.64
0.00
0.31
12.90
1.60
-
0.26
0.52
0.41
0.05
0.20
-
0.48
-
0.13
16.90
1.50
0.13
0.93
0.80
0.48
0.61
0.25
0.28
24.50
2.50
-
0.35
0.42
0.27
0.03
0.12
0.16
-
0.16Table 5
Comparison of experimental results to testing results of RBF, polynomial of SVM-II, and other models.
f
c (MPa)
f
spt (MPa)
GEP-III [9]
ACI 363R
ACI 318
CEB-FIP
Regression
RBF [9]
Polynomial
3
6.00
3.49
-
0.60
0.05
-
0.13
-
0.22
-
0.22
-
0.17
-
0.04
30.40
3.08
-
0.54
0.17
0.01
-
0.16
-
0.12
-
0.10
-
0.01
21.60
2.00
-
0.04
0.74
0.60
0.33
0.43
0.17
0.34
21.20
2.70
-
0.77
0.02
-
0.12
-
0.40
-
0.30
-
0.44
-
0.39
25.10
2.60
-
0.41
0.36
0.21
-
0.03
0.05
-
0.09
0.06
31.90
3.30
-
0.67
0.03
-
0.14
-
0.28
-
0.25
-
0.26
-
0.12
37.80
3.10
-
0.10
0.53
0.34
0.28
0.26
0.48
0.45
29.20
2.50
-
0.04
0.69
0.53
0.34
0.40
0.41
0.49
35.40
3.10
-
0.25
0.41
0.23
0.13
0.14
0.19
0.32
28.60
3.40
-
0.98
-
0.24
-
0.41
-
0.59
-
0.54
-
0.51
-
0.46
29.50
3.10
-
0.62
0.10
-
0.06
-
0.24
-
0.19
-
0.23
-
0.08
34.90
3.50
-
0.68
-
0.01
-
0.19
-
0.30
-
0.29
-
0.23
-
0.11
38.60
3.38
-
0.33
0.29
0.10
0.05
0.03
0.15
0.23
52.35
3.99
-
0.10
0.28
0.06
0.21
0.08
0.42
0.36
38.40
3.80
-
0.76
-
0.14
-
0.33
-
0.39
-
0.40
-
0.23
-
0.20
51.50
4.20
-
0.36
0.03
-
0.18
-
0.05
-
0.17
0.16
0.11
48.60
4.00
-
0.34
0.11
-
0.10
0.00
-
0.11
0.11
0.16
39.50
3.90
-
0.79
-
0.19
-
0.38
-
0.42
-
0.45
-
0.22
-
0.22
30.40
2.85
-
0.31
0.40
0.24
0.07
0.11
0.13
0.22
29.50
2.56
-
0.08
0.64
0.48
0.30
0.35
0.31
0.46
42.80
4.30
-
0.99
-
0.44
-
0.64
-
0.63
-
0.68
-
0.52
-
0.46
51.90
4.30
-
0.44
-
0.05
-
0.27
-
0.13
-
0.26
0.09
0.03
49.30
4.20
-
0.49
-
0.06
-
0.27
-
0.17
-
0.27
0.03
0.00
43.30
3.60
-
0.26
0.28
0.08
0.10
0.04
0.22
0.28
43.60
3.38
-
0.02
0.52
0.32
0.34
0.28
0.46
0.52
49.90
4.57
-
0.83
-
0.40
-
0.61
-
0.50
-
0.62
-
0.34
-
0.33
57.40
4.20
-
0.01
0.27
0.04
0.26
0.09
0.54
0.39
58.50
4.90
-
0.65
-
0.39
-
0.62
-
0.38
-
0.56
-
0.15
-
0.24
31.40
2.79
-
0.19
0.52
0.35
0.20
0.23
0.26
0.36
34.90
3.04
-
0.22
0.45
0.27
0.16
0.17
0.23
0.35
41.60
3.84
-
0.60
-
0.03
-
0.23
-
0.24
-
0.28
-
0.11
-
0.05
46.80
4.00
-
0.44
0.04
-
0.17
-
0.10
-
0.19
0.01
0.09
51.70
4.20
-
0.35
0.04
-
0.17
-
0.04
-
0.16
0.18
0.12
43.30
3.90
-
0.56
-
0.02
-
0.22
-
0.20
-
0.26
-
0.08
-
0.02
60.80
4.10
0.29
0.50
0.27
0.54
0.33
0.76
0.68
60.00
4.30
0.04
0.27
0.04
0.30
0.10
0.60
0.44
71.90
5.40
-
0.37
-
0.40
-
0.65
-
0.21
-
0.51
0.16
0.00
75.10
5.40
-
0.18
-
0.29
-
0.55
-
0.06
-
0.39
0.30
0.20
57.87
4.49
-
0.27
0.00
-
0.23
0.00
-
0.18
0.27
0.15
39.50
3.40
-
0.29
0.31
0.12
0.08
0.05
0.28
0.28Figure 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.TheR-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 fspt of concrete from the corresponding 150 × 300 mm cylinder compressive strength fc and 100 × 200 mm cylinder fspt from the corresponding 100 × 200 mm cylinder compressive strength fc 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 andR2 of experimental fspt 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 forfspt from 150 × 300 mm cylinder fc.
GEP-I [9]
ACI 363R
ACI 318
CEB-FIP
Regression [9]
RBF
Polynomial
RMSE of training set
0.4603
0.5995
0.5228
0.5027
0.4512
0.4978
0.4819
RMSE of testing set
0.4271
0.2735
0.3152
0.3754
0.2874
0.2748
0.4176
MAPE of training set
9.7554
13.647
11.4913
10.6213
9.5596
10.2434
10.5454
MAPE of testing set
7.3022
4.6139
5.5434
6.2059
5.286
4.987
6.9517
R
2 of training set
0.8282
0.8225
0.8224
0.8267
0.826
0.8115
0.8227
R
2 of testing set
0.9416
0.9486
0.9486
0.9471
0.9476
0.9422
0.9327Table 7
Error measurement forfspt from 100 × 200 mm cylinder fc.
GEP-III [9]
ACI 363R
ACI 318
CEB-FIP
Regression [9]
RBF
Polynomial
RMSE of training set
0.5886
0.539
0.5244
0.4987
0.4911
0.7287
0.562
RMSE of testing set
0.483
0.3348
0.3393
0.2803
0.3155
0.3082
0.2917
MAPE of training set
11.8729
11.6879
10.9223
9.2976
9.472
15.5902
10.6658
MAPE of testing set
10.8701
7.1383
7.604
6.2261
7.234
6.9713
6.2835
R
2 of training set
0.8198
0.8354
0.8353
0.83
0.8337
0.837
0.8253
R
2 of testing set
0.8778
0.8802
0.8793
0.88
0.8801
0.8815
0.8823


## 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 and100×200mmcylinderfc 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.



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*Source: 597257-2013-09-03.xml*