Hindawi Publishing CorporationAdvances in Materials Science and EngineeringVolume 2013 Article ID 597257 13 pageshttpdxdoiorg1011552013597257

Research ArticlePrediction of Splitting Tensile Strength from CylinderCompressive Strength of Concrete by Support Vector Machine

Kezhen Yan Hongbing Xu Guanghui Shen and Pei Liu

College of Civil Engineering Hunan University Changsha 410082 China

Correspondence should be addressed to Kezhen Yan yankz2004163com

Received 13 May 2013 Revised 24 July 2013 Accepted 25 July 2013

Academic Editor Pavel Lejcek

Copyright copy 2013 Kezhen Yan et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

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

1 Introduction

Compressive strength (119891119888) and splitting tensile strength (119891spt)

are two significant indexes in the design of concrete structureTensile strength is important for plain concrete structuressuch as dam under earthquake excitations Other structuresfor example pavement slabs and airfield runways which aredesigned based on bending strength are subjected to tensilestresses Therefore in the design of these structures tensilestrength is more important than compressive strength [1 2]Ideally the splitting tensile strength is measured directly onconcrete samples under uniform stresses However this is notalways easy from an experimental point of view To avoidthe demanding and time-consuming direct measurementsof the splitting tensile strength engineers and researchershave tried to predict the splitting tensile using theoreticaland empirical approaches based on compressive strengthGenerally tensile strength of concrete was often assumedproportional to the square root of its compressive strength[3] However there have been very few published worksdealing with experimental and analytical researches of therelation of 119891spt and 119891

119888of concretes [4]

Generally the splitting tensile strength can be establishedfrom compressive strength National building codes propose

various formulas for the splitting tensile strength 119891spt andcompressive strength 119891

119888 Various relationships for concrete

were given as follows

CEB-FIP (1991)

119891spt = 03(119891119888)23

(1)

ACI363R-92 (1992)

119891spt = 059(119891119888)12

(2)

ACI318-99 (1999)

119891spt = 056(119891119888)12

(3)

where 119891spt and 119891119888are expressed in MPa

Larrad and Malier [5] found that the calculated 119891sptobtained from the French regulations were in good agree-ment with experimental data Kim et al [6 7] foundthat the ACI model overestimates the value for concretewith compressive strength less than 20MPa and underesti-mates the value for concrete with compressive strength over30MPa Zain et al [2] determined relationships betweentensile strength with compressive strength concrete age and

2 Advances in Materials Science and Engineering

waterbinder (WB) Shaaban and Gesund [8] investigatedthe splitting tensile strength and compressive strength of steelfiber reinforced concrete but unfortunately did not analyzethe relationship between the tensile strength and compressivestrength Choi and Yuan [4] investigated the relationshipbetween the tensile strength and compression strength ofglass fiber reinforced concrete and polypropylene fiber rein-forced concrete Xu and Shi [1] developed the formulationbetween the 119891spt and 119891

119888of steel fiber reinforced con-

crete Mathematical regression models are usually imperfectdescription of complex physical phenomena The accuracyof the estimated result depends on the size of available dataSaridemir [9] proposed for 119891spt prediction from cylinder119891119888of concrete or age of specimen (AS) and cylinder 119891

119888

of concrete by gene expression programming (GEP) Thesupport vector machine (SVM) is a new efficient and novelapproach to improve generalization performance and it canattain a global minimum SVM achieves good generalizationability by adopting a structural risk minimization inductionprinciple that aims at minimizing a bound on generalizationerror of a model rather than minimizing the error on thetraining data only It has the ability to avoid overtrainingand has better generalization capability than artificial neuralnetworks (ANN) model Moreover the SVM can alwaysbe updated to get better results by presenting new trainingexamples as new data become available

In this paper support vectormachine is applied to predictthe splitting tensile strength of concrete using the datumcollected by Saridemir [9] Compressive strength used topredict the splitting tensile strength is considered as inputin SVM model that is compared with experimental data andother methods The mean absolute parentage error (MAPE)root-mean-squared error (RMSE) and 119877-square (1198772) areused as the criteria to compare the performance of SVMmodels and other models

2 Support Vector Machine

SVM is a machine-learning algorithm based on statisticallearning theory The main idea of the SVM is to transformthe input space into a high-dimensional space by a nonlineartransformation defined by an inner product function SVMcalculation takes the form of a problem in convex quadraticoptimization ensuring that the solution is optimal It is betterthan the traditional artificial neural networks which is basedon the traditional minimization principle of experience risk[10] The SVM has a good ability to generalize and resolvesome practical problems such as small samples nonlinearityand high-dimensional input space

In this section a brief description of the process ofconstructing a SVM for a regression problem is presentedThere are three distinct characteristics to consider when anSVM is used to solve a regression problem [11] First theSVM estimates the regression by a set of linear functions thatare defined in a high-dimensional space Second the SVMcarries out the regression estimation by risk minimizationwhere the risk is measured using Vapnikrsquos 120576-insensitive lossfunction Third the SVM uses a risk function consisting

Table 1 Parameters of SVM-I and SVM-II

Kernel

SVM-I SVM-IIRadialbasis

functionPolynomial

Radialbasis

functionPolynomial

119862 500000 160000 820000 870000120576 010 010 004 005Parameter 120574 = 001 119889 = 2 120574 = 010 119889 = 3

of the empirical error and a regularization term whichis derived from the structural risk minimization (SRM)principle

Consider a set of training samples 119909119894 119910119894 119894 = 1 2

119899 119909119894isin 119877119889 119910

119894isin 119877 where 119909

119894is an input vector 119910

119894is the

corresponding output value and 119899 is the number of trainingsamples The regression problem is to select a function thatpredicts the actual value of 119910 as closely as possible with aprecision of 120576 Therefore the purpose of the SVM is to seekthe optimum regression function

119891 (119909) = 120596 sdot 119909 + 119887 (4)

where 120596 isin 119877119899 and 119887 isin 119877 120596 is an adjustable weightvector 119887 is the scalar threshold 119877119899 is 119899-dimensional vectorspace and 119909 is one-dimensional vector space

Following statistical theory SVM determines the regres-sion function by minimizing an objective function Theparameters 120596 and 119887 of the regression function are estimatedby minimizing the regularized risk function as follows

Minimize

1

21205962 + 119862

119899

sum119894=1

(120585119894+ 120585lowast119894) (5)

Subject to

119910119894minus [(120596 sdot 119909

119894) + 119887] le 120576 + 120585

119894 (6)

[(120596 sdot 119909119894) + 119887] minus 119910

119894le 120576 + 120585lowast

119894 (7)

120585119894ge 0 120585lowast

119894ge 0 119894 = 1 2 119899 (8)

where 119862 gt 0 is a penalty factor 120585 and 120585lowast are slack variables120576 is the insensitive loss function and can be described in thefollowing way

119871120576(119910) =

1003816100381610038161003816119891 (119909) minus 1199101003816100381610038161003816 minus 120576

1003816100381610038161003816119910 minus 119891 (119909)1003816100381610038161003816 ge 120576

0 otherwise(9)

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

Maximize119899

sum119894=1

119910119894(120572lowast119894minus 120572119894) minus 120576119899

sum119894=1

(120572lowast119894+ 120572119894)

minus1

2

119899

sum119894=1

119899

sum119895=1

(120572lowast119894minus 120572119894) (120572lowast119895minus 120572119895) (119909119894 119909119895)

(10)

Advances in Materials Science and Engineering 3

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

119891119888(MPa) 119891spt (MPa) GEP-I [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

2633 307 minus072 minus004 minus020 minus041 minus052 minus030 minus029

3582 312 minus018 041 023 014 minus002 034 001

3093 321 minus057 007 minus010 minus025 minus039 minus033 minus030

3742 207 097 154 136 129 112 000 114

2910 301 minus049 017 001 minus017 minus029 001 minus017

2803 341 minus095 minus029 minus045 minus064 minus076 minus079 minus053

3698 325 minus024 034 016 008 minus009 037 minus012

3165 339 minus071 minus007 minus024 minus039 minus053 minus107 minus040

3858 316 minus005 050 032 027 009 037 016

8100 660 minus110 minus129 minus156 minus098 minus140 minus006 minus119

6800 590 minus111 minus103 minus128 minus090 minus124 minus176 minus099

6000 500 minus065 minus043 minus066 minus040 minus070 minus022 minus060

5400 470 minus069 minus036 minus058 minus041 minus068 minus015 minus068

3908 311 004 058 039 034 017 076 013

2233 229 minus020 050 036 009 001 025 039

2323 219 minus004 065 051 025 016 032 049

2289 238 minus025 044 030 004 minus005 036 034

2216 217 minus009 061 047 020 011 034 050

2048 215 minus018 052 038 010 002 007 048

2082 199 000 070 057 028 021 002 065

2283 241 minus028 041 027 000 minus008 010 025

2514 262 minus034 034 019 minus005 minus014 012 012

2560 269 minus038 030 014 minus008 minus019 minus015 009

3520 320 minus029 030 012 002 minus013 039 minus010

4440 290 057 103 083 086 065 076 062

3760 240 066 122 103 097 080 000 084

4180 360 minus029 021 002 001 minus018 012 minus020

4200 350 minus017 032 013 012 minus007 047 minus010

3830 270 040 095 077 071 053 041 051

5540 340 071 099 077 096 069 000 078

5400 280 123 154 132 149 122 000 122

4860 280 091 131 110 120 096 000 094

5650 410 007 033 011 032 004 minus006 009

4700 390 minus028 014 minus006 001 minus022 019 minus022

4569 419 minus065 minus020 minus040 minus036 minus057 minus037 minus059

4171 309 022 072 053 052 032 032 033

4249 374 minus038 011 minus009 minus009 minus029 minus053 minus027

3369 293 minus011 049 032 020 005 040 016

3730 314 minus010 046 028 021 004 049 007

2475 262 minus037 032 017 minus007 minus017 minus021 011

2696 268 minus028 038 023 002 minus009 036 007

2214 235 minus027 043 028 002 minus007 016 032

2391 224 minus004 064 050 025 016 003 044

4100 390 minus063 minus012 minus031 minus033 minus052 minus094 minus049

4200 400 minus067 minus018 minus037 minus038 minus057 minus003 minus060

5768 426 minus004 022 minus001 022 minus007 057 minus002

4575 410 minus057 minus011 minus031 minus026 minus048 minus028 minus045

3810 362 minus054 002 minus016 minus022 minus040 minus019 minus037

3195 331 minus061 002 minus014 minus029 minus043 minus027 minus040

4 Advances in Materials Science and Engineering

Table 2 Continued

119891119888(MPa) 119891spt (MPa) GEP-I [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

1820 190 minus010 062 049 018 012 minus024 068

2870 270 minus020 046 030 011 minus001 minus020 023

2390 230 minus011 058 044 019 010 minus003 038

4370 330 011 060 040 042 022 044 021

2460 230 minus007 063 048 024 014 minus001 045

4300 350 minus013 037 017 018 minus002 000 000

1420 180 minus028 042 031 minus004 minus008 minus029 063

3670 320 minus021 037 019 011 minus005 minus048 001

2270 170 041 111 097 071 062 035 095

3530 240 051 111 093 083 067 000 074

1360 130 018 088 077 041 038 079 112

2860 170 079 146 129 111 099 140 114

2660 240 minus004 064 049 027 017 minus001 042

4350 290 050 099 079 081 061 052 057

3090 240 024 088 071 055 042 048 051

4990 380 minus003 037 016 027 003 039 004

3220 300 minus028 035 018 004 minus010 minus009 001

5780 350 072 099 076 098 070 033 081

2680 240 minus002 065 050 029 018 047 044

3650 290 008 066 048 040 024 027 026

6560 440 026 038 014 048 015 014 036

3030 240 020 085 068 052 039 137 054

5280 380 014 049 027 042 017 minus039 023

4060 350 minus027 026 007 004 minus014 015 minus013

5230 420 minus029 007 minus015 000 minus026 010 minus024

4120 320 006 059 039 038 019 065 022

7280 510 minus005 minus007 minus032 013 minus024 003 minus005

4120 320 006 059 039 038 019 065 022

7280 510 minus005 minus007 minus032 013 minus024 003 minus005

7280 510 minus005 minus007 minus032 013 minus024 003 minus005

4180 330 000 051 032 031 012 042 010

6840 410 071 078 053 092 057 038 076

4230 290 043 094 074 074 054 000 057

5910 380 050 074 051 075 046 025 060

4130 370 minus043 009 minus010 minus012 minus031 minus019 minus027

5970 410 023 046 023 048 019 014 030

4230 320 013 064 044 044 024 071 027

7510 510 008 001 minus025 024 minus014 067 minus005

4850 370 minus001 041 020 029 006 008 009

8190 480 075 054 027 086 044 094 053

4130 330 minus003 049 030 028 009 021 013

6710 440 034 043 019 055 022 068 036

4200 320 011 062 043 042 023 077 020

7010 430 061 064 039 080 045 065 058

4480 340 008 055 035 038 017 036 021

6540 400 065 077 053 087 054 067 070

3870 350 minus039 017 minus002 minus007 minus024 040 minus021

5640 430 minus016 013 minus009 011 minus016 minus025 minus014

4240 320 013 064 045 045 025 041 025

Advances in Materials Science and Engineering 5

Table 2 Continued

119891119888(MPa) 119891spt (MPa) GEP-I [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

6770 480 minus003 005 minus019 018 minus016 027 minus003

5070 360 022 060 039 051 026 048 026

8110 480 070 051 024 082 041 062 052

4200 320 011 062 043 042 023 077 020

4890 360 011 053 032 041 018 061 021

8530 580 minus007 minus035 minus063 001 minus042 032 minus037

5230 420 minus029 007 minus015 000 minus026 010 minus024

8510 580 minus008 minus036 minus063 000 minus043 minus034 minus042

4970 420 minus044 minus004 minus025 minus014 minus038 minus049 minus036

7120 530 minus033 minus032 minus057 minus015 minus050 minus052 minus033

5190 380 009 045 023 037 012 036 017

8770 610 minus024 minus057 minus086 minus018 minus063 minus005 minus066

5190 380 009 045 023 037 012 036 017

8770 610 minus024 minus057 minus086 minus018 minus063 minus005 minus066

5190 380 009 045 023 037 012 036 017

8770 610 minus024 minus057 minus086 minus018 minus063 minus005 minus066

5380 410 minus010 023 001 018 minus009 minus010 minus006

8530 510 063 035 007 071 028 102 033

5870 440 minus013 012 minus011 013 minus016 minus026 minus009

8710 560 022 minus009 minus037 029 minus015 minus009 minus020

4930 450 minus076 minus036 minus057 minus047 minus070 minus135 minus069

6620 510 minus041 minus030 minus054 minus019 minus052 minus023 minus040

5310 380 016 050 028 044 018 050 033

8430 560 007 minus018 minus046 017 minus026 010 minus015

8670 603 003 minus054 minus080 minus016 minus060 002 minus054

5610 430 minus017 012 minus011 010 minus018 minus058 minus017

5210 380 010 046 024 038 013 011 016

7850 500 036 023 minus004 050 010 087 029

4990 410 minus033 007 minus014 minus003 minus027 009 minus026

8750 510 075 042 014 081 037 048 029

5510 440 minus033 minus002 minus024 minus006 minus033 minus011 minus023

7540 400 122 112 086 135 097 000 105

2467 274 minus050 019 004 minus020 minus029 minus033 002

2301 279 minus066 004 minus010 minus036 minus045 000 minus009

3203 303 minus032 031 014 000 minus014 minus064 minus003

2878 293 minus043 024 007 minus011 minus023 minus043 000

3360 203 077 139 122 109 095 130 106

2330 304 minus089 minus019 minus034 minus059 minus068 minus125 minus031

GEP-I model is developed by Saridemir [9] based on gene expression programming Regression is regression-based formulation results by Saridemir [9]

Subject to119899

sum119894=1

120572119894=119899

sum119894=1

120572lowast119894 0 le 120572lowast

119894le 119888 0 le 120572

119894le 119888 (11)

where the 120572119894 120572lowast119894are called Lagrangian multipliers When

the Lagrangian multipliers (120572119894 120572lowast119894) are equal to zero shows

that the training object is irrelevant to the final solutionotherwise the training samples with nonzero Lagrangianmultipliers are called support vectors

When linear regression is not appropriate then the inputdata have to bemapped into a high-dimensional feature space

through some nonlinear mapping A nonlinear transforma-tion 120601(119909) replaces the input 119909 in (7) and the regressionfunction can be written as

119891 (119909) =nsvsum119894=1

(120572119894minus 120572lowast119894) 119896 (119909119894 119909119895) + 119887 (12)

where nsv is the number of support vectors and 119896(119909119894 119909119895) =

120601(119909119894) sdot 120601(119909

119895) 119896(119909

119894 119909119895) is a kernel function Any function sat-

isfying Mercersquos condition can be used as the kernel function[12 13] Some popular kernel functions are

6 Advances in Materials Science and Engineering

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

119891119888(MPa) 119891spt (MPa) GEP-I [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

4480 380 minus031 015 minus005 minus002 minus023 minus001 minus033

5620 440 minus025 002 minus020 000 minus027 minus041 minus020

5340 360 039 071 049 065 039 033 039

5670 460 minus042 minus016 minus038 minus017 minus045 minus009 minus036

6450 480 minus018 minus006 minus030 002 minus030 minus018 minus012

6650 470 003 011 minus013 022 minus011 010 001

6540 430 037 047 023 057 024 021 043

7380 490 023 017 minus009 038 001 022 014

7400 530 minus016 minus022 minus048 minus001 minus039 minus026 minus015

7250 500 006 002 minus023 022 minus015 024 minus006

7460 490 028 020 minus006 042 004 006 015

7790 550 minus014 minus029 minus056 minus003 minus042 minus036 minus019

7910 520 022 005 minus022 033 minus007 001 015

8420 570 000 minus029 minus056 006 minus037 minus034 minus028

8330 590 minus025 minus052 minus079 minus018 minus060 minus060 minus039

8010 530 015 minus002 minus029 027 minus013 minus022 010

8650 550 032 minus001 minus029 037 minus007 minus027 012

10200 550 115 046 016 105 053 037 034

10100 650 009 minus057 minus087 001 minus051 minus045 minus060

11100 620 092 002 minus030 073 016 040 minus017

9450 580 045 minus006 minus036 042 minus006 008 minus001

11800 620 128 021 minus012 102 041 064 minus014

1900 230 minus043 027 014 minus016 minus023 minus004 minus100

2000 240 minus046 024 010 minus019 minus026 minus013 minus096

2860 290 minus040 026 009 minus009 minus021 011 minus067

2890 300 minus048 017 001 minus017 minus030 008 minus082

3610 330 minus033 024 006 minus002 minus019 minus011 minus048

4450 370 minus023 024 004 007 minus014 007 minus026

2300 250 minus036 033 019 minus007 minus016 minus037 minus082

3300 350 minus072 minus011 minus028 minus041 minus056 minus043 minus102

5800 420 005 029 006 030 001 minus025 020

6900 450 037 040 015 055 020 012 044

3600 330 minus034 024 006 minus003 minus019 minus004 minus045

4280 390 minus053 minus004 minus024 minus023 minus043 minus019 minus063

5930 400 033 054 031 056 027 029 040

6330 450 005 019 minus004 026 minus005 minus018 020

6770 470 010 015 minus009 028 minus006 026 007

7700 510 021 008 minus019 033 minus006 minus005 001

4990 430 minus051 minus013 minus034 minus023 minus047 minus032 minus045

9670 570 063 010 minus019 062 012 005 001

6170 460 minus014 003 minus020 008 minus022 minus036 minus008

7320 510 000 minus005 minus031 015 minus022 017 minus003

8650 550 032 minus001 minus029 037 minus007 minus027 012

9230 590 023 minus023 minus052 023 minus025 010 minus019

(1) polynomial kernel function

119870(119909 119910) = [(119909 + 119910) + 1]119889

119889 = 1 2 119899 (13)

(2) radial basis function (RBF)

119870(119909 119910) = exp minus1205741003817100381710038171003817119909 minus 11991010038171003817100381710038172

120574 ge 0 (14)

(3) sigmoid kernel function

119870(119909 119910) = tan [120593 (119909 sdot 119910) + 120579] (15)

where 120572119894 120572lowast119894satisfy 120572

119894120572lowast119894= 0 120572

119894ge 0 and 120572lowast

119894ge 0

Advances in Materials Science and Engineering 7

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

119891119888(MPa) 119891spt (MPa) GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

7790 545 minus008 minus024 minus051 002 minus033 048 032

5710 313 103 133 110 132 115 174 145

7160 490 011 009 minus016 027 minus002 081 051

4410 391 minus052 001 minus019 minus017 minus023 028 minus017

6480 457 004 018 minus006 027 003 073 042

4260 366 minus036 019 000 000 minus005 047 minus004

3040 279 minus026 046 030 013 017 046 000

4680 389 minus034 015 minus006 001 minus008 049 004

8950 591 011 minus033 minus061 009 minus036 073 041

6690 455 019 028 003 039 014 096 059

7610 547 minus021 minus032 minus058 minus008 minus042 041 022

5360 437 minus041 minus005 minus027 minus011 minus025 047 minus002

6480 507 minus046 minus032 minus056 minus023 minus047 023 minus008

4580 402 minus053 minus003 minus023 minus018 minus026 040 minus016

3760 316 minus017 046 027 021 019 063 014

5310 408 minus015 022 000 016 002 078 024

3490 350 minus068 minus001 minus019 minus030 minus029 012 minus040

5390 540 minus142 minus107 minus129 minus112 minus127 minus062 minus100

4650 380 minus027 022 002 008 minus001 060 007

1380 170 minus029 049 038 003 017 minus042 minus017

6710 480 minus005 003 minus021 015 minus010 064 035

2030 340 minus153 minus074 minus088 minus117 minus106 minus126 minus136

2240 210 minus009 069 055 028 038 038 010

6140 430 012 032 009 037 016 079 051

3310 300 minus029 039 022 009 011 053 minus003

7900 560 minus017 minus036 minus062 minus008 minus044 046 021

4360 360 minus024 030 010 012 006 059 010

2560 242 minus020 057 041 019 026 031 002

2620 262 minus036 040 025 003 010 023 minus015

1690 188 minus025 055 042 010 023 minus013 minus010

1490 171 minus022 057 045 011 025 minus038 minus008

2730 281 minus048 027 012 minus009 minus003 019 minus026

2840 287 minus047 027 011 minus008 minus002 022 minus023

1690 191 minus028 052 039 007 020 minus016 minus013

1690 211 minus048 032 019 minus013 000 minus036 minus033

3090 292 minus035 036 019 003 007 031 minus010

3230 308 minus043 027 010 minus004 minus001 036 minus017

1960 222 minus040 039 026 minus004 008 minus010 minus022

1910 208 minus030 050 037 006 018 minus011 minus010

3530 324 minus040 027 009 minus001 minus001 038 minus011

3840 336 minus032 030 011 005 004 057 minus003

2250 247 minus045 033 019 minus008 002 minus005 minus027

2340 257 minus049 028 014 minus012 minus002 minus008 minus029

3100 290 minus033 038 022 006 010 035 minus006

1700 230 minus066 013 001 minus032 minus018 minus060 minus051

4700 400 minus044 004 minus016 minus009 minus018 044 minus009

3700 360 minus065 minus001 minus019 minus027 minus028 021 minus035

4200 400 minus074 minus018 minus037 minus038 minus042 016 minus039

5400 460 minus062 minus026 minus048 minus031 minus046 021 minus024

8 Advances in Materials Science and Engineering

Table 4 Continued

119891119888(MPa) 119891spt (MPa) GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

4200 440 minus114 minus058 minus077 minus078 minus082 minus024 minus079

6200 520 minus075 minus055 minus079 minus050 minus072 003 minus035

6700 425 049 058 033 070 044 115 091

7250 510 minus004 minus008 minus033 012 minus019 056 038

7450 520 minus003 minus011 minus037 011 minus021 059 038

7950 575 minus029 minus049 minus076 minus020 minus057 030 011

7800 515 022 006 minus020 033 minus003 083 060

8500 575 002 minus031 minus059 005 minus036 069 044

8600 480 103 067 039 105 062 167 140

9050 440 168 121 093 165 119 232 204

5480 334 069 103 081 099 083 161 106

3290 285 minus016 053 036 023 025 068 011

3450 292 minus013 055 037 026 027 066 013

5360 348 048 084 062 078 064 136 087

7410 526 minus011 minus018 minus044 003 minus029 058 030

6770 481 minus003 004 minus020 017 minus009 068 037

8420 657 minus085 minus116 minus143 minus081 minus121 minus025 minus046

7530 535 minus013 minus023 minus049 000 minus033 048 028

10230 729 minus055 minus132 minus163 minus073 minus129 052 minus031

4010 330 minus016 044 025 021 018 070 015

4050 340 minus023 035 016 014 010 057 007

9550 570 066 007 minus023 057 006 153 096

3520 317 minus033 033 015 005 006 048 minus005

5750 369 051 078 056 078 060 129 086

3860 338 minus033 029 010 005 003 052 minus004

6150 448 minus005 015 minus009 019 minus002 070 032

4740 376 minus017 030 010 017 008 067 018

4710 368 minus011 037 016 023 014 086 023

4470 326 017 068 048 052 045 109 046

5240 399 minus010 028 006 021 008 079 027

5790 449 minus027 000 minus023 000 minus018 061 012

3950 404 minus093 minus033 minus052 minus056 minus059 000 minus065

3670 403 minus109 minus046 minus064 minus072 minus072 minus029 minus082

3800 365 minus063 minus001 minus020 minus026 minus028 020 minus035

2420 220 minus007 070 055 031 040 038 014

1740 180 minus013 066 054 021 034 minus006 002

2930 280 minus033 039 023 005 010 027 minus009

2700 255 minus023 052 036 015 022 033 minus002

3260 321 minus053 016 minus001 minus015 minus012 026 minus026

4630 520 minus167 minus119 minus139 minus133 minus141 minus076 minus137

3580 430 minus142 minus077 minus095 minus104 minus104 minus057 minus111

4310 418 minus085 minus031 minus050 minus049 minus055 minus004 minus051

3520 317 minus033 033 015 005 006 048 minus005

5750 369 051 078 056 078 060 129 086

4280 490 minus159 minus104 minus124 minus123 minus128 minus076 minus126

6150 448 minus005 015 minus009 019 minus002 070 032

4740 376 minus017 030 010 017 008 067 018

4713 368 minus010 037 016 023 014 086 023

4473 326 017 069 049 052 045 104 050

Advances in Materials Science and Engineering 9

Table 4 Continued

119891119888(MPa) 119891spt (MPa) GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

3460 410 minus130 minus063 minus081 minus091 minus090 minus051 minus102

5970 540 minus108 minus084 minus107 minus082 minus101 minus030 minus069

5190 400 minus014 025 003 017 004 076 021

5000 390 minus015 027 006 017 006 077 022

4850 410 minus044 001 minus020 minus011 minus021 049 minus008

4880 370 minus002 042 021 031 020 080 032

4650 380 minus026 022 002 008 minus001 060 007

4520 400 minus054 minus003 minus024 minus019 minus027 030 minus021

4520 330 016 067 046 051 043 100 049

4260 298 032 087 068 068 063 115 064

8160 440 119 093 066 124 086 180 158

4270 300 031 086 066 066 061 121 064

4190 350 minus024 032 012 012 007 060 009

4120 380 minus059 minus001 minus021 minus022 minus026 031 minus026

3970 270 042 102 083 079 076 124 073

6550 550 minus084 minus073 minus097 minus063 minus087 minus021 minus047

3950 370 minus059 001 minus018 minus022 minus025 034 minus031

1710 190 minus025 054 042 009 022 minus011 minus010

1210 110 018 095 085 048 064 000 031

1290 160 minus026 052 041 005 020 minus048 minus013

1690 150 013 093 080 048 061 025 028

2450 250 minus035 042 027 003 012 016 minus016

3 SVM for Predicting 119891spt of Concrete

As mentioned previously many methods have been pro-posed for the prediction of 119891spt from compressive strengthof concrete This study attempts to utilize SVM for theprediction of concrete of 119891spt SVM-I model is devel-oped for 150 times 300mm cylinder 119891spt prediction from 150 times300mm cylinder 119891

119888of concrete and SVM-IImodel is devel-

oped for 100 times 200mm and 150 times 200mm cylinder 119891sptprediction from 100 times 200mm cylinder 119891

119888of concrete The

experimental data which are taken from studies [9 14 15] areused in this study Among 184 experimental data 138 datawere randomly selected as the training set for SVM-I modeland among 168 cases 126 cases were randomly selected asthe training set for SVM-II model the rest are considered astesting data setThe data are normalized before being used inthe model as follows

119883 =(119883119894minus 119883min)

(119883max minus 119883min) (16)

where 119883max and 119883min are the maximum and minimuminput values data respectively

In case of SVM training two types of kernel func-tions were used namely radical basis function (RBF) andpolynomial function in training process 119862 and 120576 and otherkernel-specific parameters have been chosen by a trial-and-error approach The best simulation performances ofSVM are summarized in Table 1 in order to evaluate theabilities of SVM models and other models mean absolutepercentage error (MAPE) roof-mean-squared error (RMES)

and R-square (1198772) were used as the criteria between theexperimental and predicted values which are according tothe equationsas follow

MAPE =1

119899[sum119899

119894=1

1003816100381610038161003816119905119894 minus 1199001198941003816100381610038161003816

sum119899

119894=1119905119894

times 100]

RMSE = radic1

119899

119899

sum119894=1

(119905119894minus 119900119894)

1198772 =(119899sum 119905

119894119900119894minus sum 119905119894sum119900119894)2

(119899sum 1199052119894minus (sum 1199052

119894)) (119899sum 1199002

119894minus (sum 119900

119894)2

)

(17)

where 119905119894is the experimental value 119900

119894is the predicted value

and 119899 is total number of data

4 Results and Discussion

There is no doubt that the splitting tensile strength 119891spt withan increase in the compression strength of concrete butthere is no agreement on the precise form of the relationshipCodes propose different formulas for prediction cylinder 119891sptof concrete from compressive strength In this paper 119891sptresults are investigated for 150 times 300mm cylinder concreteand 100 times 200mm cylinder concrete separately The errors(predicted values subtract measured values) computed bySVM model other models and measured values of split-ting tensile strengths are shown in Tables 2 and 3 for 150times 300mm cylinder concrete and in Tables 4 and 5 for

10 Advances in Materials Science and Engineering

Table 5 Comparison of experimental results to testing results of RBF polynomial of SVM-II and other models

119891119888(MPa) 119891spt (MPa) GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression RBF [9] Polynomial

3600 349 minus060 005 minus013 minus022 minus022 minus017 minus004

3040 308 minus054 017 001 minus016 minus012 minus010 minus001

2160 200 minus004 074 060 033 043 017 034

2120 270 minus077 002 minus012 minus040 minus030 minus044 minus039

2510 260 minus041 036 021 minus003 005 minus009 006

3190 330 minus067 003 minus014 minus028 minus025 minus026 minus012

3780 310 minus010 053 034 028 026 048 045

2920 250 minus004 069 053 034 040 041 049

3540 310 minus025 041 023 013 014 019 032

2860 340 minus098 minus024 minus041 minus059 minus054 minus051 minus046

2950 310 minus062 010 minus006 minus024 minus019 minus023 minus008

3490 350 minus068 minus001 minus019 minus030 minus029 minus023 minus011

3860 338 minus033 029 010 005 003 015 023

5235 399 minus010 028 006 021 008 042 036

3840 380 minus076 minus014 minus033 minus039 minus040 minus023 minus020

5150 420 minus036 003 minus018 minus005 minus017 016 011

4860 400 minus034 011 minus010 000 minus011 011 016

3950 390 minus079 minus019 minus038 minus042 minus045 minus022 minus022

3040 285 minus031 040 024 007 011 013 022

2950 256 minus008 064 048 030 035 031 046

4280 430 minus099 minus044 minus064 minus063 minus068 minus052 minus046

5190 430 minus044 minus005 minus027 minus013 minus026 009 003

4930 420 minus049 minus006 minus027 minus017 minus027 003 000

4330 360 minus026 028 008 010 004 022 028

4360 338 minus002 052 032 034 028 046 052

4990 457 minus083 minus040 minus061 minus050 minus062 minus034 minus033

5740 420 minus001 027 004 026 009 054 039

5850 490 minus065 minus039 minus062 minus038 minus056 minus015 minus024

3140 279 minus019 052 035 020 023 026 036

3490 304 minus022 045 027 016 017 023 035

4160 384 minus060 minus003 minus023 minus024 minus028 minus011 minus005

4680 400 minus044 004 minus017 minus010 minus019 001 009

5170 420 minus035 004 minus017 minus004 minus016 018 012

4330 390 minus056 minus002 minus022 minus020 minus026 minus008 minus002

6080 410 029 050 027 054 033 076 068

6000 430 004 027 004 030 010 060 044

7190 540 minus037 minus040 minus065 minus021 minus051 016 000

7510 540 minus018 minus029 minus055 minus006 minus039 030 020

5787 449 minus027 000 minus023 000 minus018 027 015

3950 340 minus029 031 012 008 005 028 028

Table 6 Error measurement for 119891spt from 150 times 300mm cylinder 119891119888

GEP-I [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF PolynomialRMSE of training set 04603 05995 05228 05027 04512 04978 04819RMSE of testing set 04271 02735 03152 03754 02874 02748 04176MAPE of training set 97554 13647 114913 106213 95596 102434 105454MAPE of testing set 73022 46139 55434 62059 5286 4987 695171198772 of training set 08282 08225 08224 08267 0826 08115 082271198772 of testing set 09416 09486 09486 09471 09476 09422 09327

Advances in Materials Science and Engineering 11

Table 7 Error measurement for 119891spt from 100 times 200mm cylinder 119891119888

GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF PolynomialRMSE of training set 05886 0539 05244 04987 04911 07287 0562RMSE of testing set 0483 03348 03393 02803 03155 03082 02917MAPE of training set 118729 116879 109223 92976 9472 155902 106658MAPE of testing set 108701 71383 7604 62261 7234 69713 628351198772 of training set 08198 08354 08353 083 08337 0837 082531198772 of testing set 08778 08802 08793 088 08801 08815 08823

0

2

4

6

8

0 20 40 60 80 100 120 140

Split

ting

tens

ile st

reng

th (M

Pa)

Sample number

ExperimentalRBFPolynomial

Figure 1 Comparison of experimental results to training results ofSVM-I

100 times 200mm cylinder concrete As seen from these resultsSVM model is sufficiently close to the experimental dataand is able to predict the 119891spt from the 119891

119888of concrete

For convenient comparison purpose the experimental andpredicted results are plotted in Figures 1 2 3 and 4 It canbe seen that the results from SVM method are in goodagreement with the experimental results

The 119877-square errors in the trained SVM are 08115 and08227 for RBF and polynomial function of SVM-I 08370and 08253 for those of SVM-II respectively 119877-square errorsin the tested SVM are 09422 and 09327 for RBF and polyno-mial function of SVM-I 08815 and 08823 for those of SVM-II respectively The whole results obtained from the SVMmodels show a successful performance of the SVM modelsof predicting 150 times 300mm cylinder119891spt of concrete from thecorresponding 150 times 300mm cylinder compressive strength119891119888and 100 times 200mm cylinder 119891spt from the corresponding

100 times 200mm cylinder compressive strength 119891119888of concrete

for each of the training and testing sets It is also clear thatthere are no major difference on performance between theradical basis function and the polynomial function kernelsthat used in this paper However in generally the radical basisfunction kernel exhibits slightly better performance than thepolynomial kernel

The performance of the trained and tested sets is analyzedby computing MAPE RMSE and 1198772 of experimental 119891spt

0

2

4

6

8

0 5 10 15 20 25 30 35 40 45 50

Split

ting

tens

ile st

reng

th (M

Pa)

Sample number

ExperimentalRBFPolynomial

Figure 2 Comparison of experimental results to testing results ofSVM-I

between the SVM model and other methods The MAPERMSE and 1198772 are calculated for these methods Statisticalparameters of the training and testing sets of all methods arepresented in Tables 6 and 7 The model having the smallestMAPE or RMSE and the highest 1198772 can be regarded to bethe best model with the assumption of analysis Generally1198772 of the SVM models are higher than other methodsand the MAPE and RMSE of SVM are smaller than othermodels The analysis shows that the SVM model is betterthan other models and can predict the splitting tensilestrength of concrete well Furthermore SVMmodel has goodgeneralization capacity to avoid over training and can alwaysbe updated to get better results by presenting new trainingexamples as new data become available Thus SVM modelcan be regarded as a very effective method to predict splittingtensile strength of concrete from their compressive strength

5 Conclusion

The splitting tensile strength of concrete estimations fromcompression strength has been obtained so far in the litera-ture either through regression or other methodsThis presentstudy reports a new and influential approach for predictingthe splitting tensile strength using SVM for the first timein the literature The study conducted in this paper shows

12 Advances in Materials Science and Engineering

Polynomial

0

1

2

3

4

5

6

7

8

9

0 20 40 60 80 100 120 140

Split

ting

tens

ile st

reng

hth

(MPa

)

Sample number

ExperimentalRBF

Figure 3 Comparison of experimental results to training results ofSVM-II

Polynomial

0

2

4

6

0 10 20 30 40 50

Split

ting

tens

ile st

reng

th (M

Pa)

Sample number

ExperimentalRBF

Figure 4 Comparison of experimental results to testing results ofSVM-II

the feasibility of using a simple SVM to estimate the splittingtensile strength of concrete After learning from a set ofselected training data involving compressive strengths ofconcrete collected from the previous literature the SVM canbe utilized to predict the splitting tensile strengths of the testdata

In this paper radical basis function and polynomial areadopted for predicting splitting tensile strength from 150 times300mm cylinder and 100times200mm cylinder 119891

119888of concrete

It is found that the splitting tensile strength obtained fromthe SVM is more accurate than those obtained from designcodes and several researchesrsquo empirical equation when acomparison is made on the basis of the experimental dataSince the SVM is largely characterized by the type of its kernel

function it is necessary to choose the appropriate kernel foreach particular application problem in order to guaranteesatisfactory results The results of radial basis function andpolynomial indicate that RBF and polynomial kernel have theability to predict the splitting tensile strength of concrete fromcompression strength with an acceptable degree of accuracy

The statistical parameters of MAPE RMSE and 1198772 showthat the proposed SVM model results have the best accu-racy and can predict splitting tensile strength very close toexperiment results The use of SVM is very advantageousfor the prediction of the splitting tensile strength of concretefrom compression strength because it can perform nonlinearregression efficiently for high-dimensional data sets Further-more its solution is globalThe satisfactory predictions of thesplitting tensile strength of concrete by the model indicatethat SVM is a useful modeling tool for engineers and researchscientists at concrete construction fields

Acknowledgments

This project was granted financial support from the NationalNatural Science Foundation of China (no 50808077 andno 51278188) and the Fundamental Research Funds for theCentral Universities The authors are grateful for support

References

[1] B W Xu and H S Shi ldquoCorrelations among mechanicalproperties of steel fiber reinforced concreterdquo Construction andBuilding Materials vol 23 no 12 pp 3468ndash3474 2009

[2] M F M Zain H B Mahmud A Ilham and M FaizalldquoPrediction of splitting tensile strength of high-performanceconcreterdquoCement andConcrete Research vol 32 no 8 pp 1251ndash1258 2002

[3] F A Oluokun E G Burdette and J H Deatherage ldquoSplittingtensile strength and compressive strength relationship at earlyagesrdquo ACI Materials Journal vol 88 no 2 pp 115ndash121 1991

[4] Y Choi and R L Yuan ldquoExperimental relationship betweensplitting tensile strength and compressive strength of GFRC andPFRCrdquo Cement and Concrete Research vol 35 no 8 pp 1587ndash1591 2005

[5] F D Larrad and Y Malier ldquoEngineering properties of very highperformance concreterdquo in High Performance Concrete fromMaterial to Structure Y Malier Ed pp 85ndash114 EFN SponLondon UK 1992

[6] J K Kim S H Han Y D Park and J H Noh ldquoMaterialproperties of self-flowing concreterdquo Journal of Materials in CivilEngineering vol 10 no 4 pp 244ndash249 1998

[7] J K Kim S H Han and Y C Song ldquoEffect of temperatureand aging on the mechanical properties of concretemdashpart Iexperimental resultsrdquo Cement and Concrete Research vol 32no 7 pp 1087ndash1094 2002

[8] AM Shaaban andHGesund ldquoSplitting tensile strength of steelfiber reinforced concrete cylinders consolidated by rodding orvibratingrdquo ACI Materials Journal vol 90 no 4 pp 366ndash3691993

[9] M Saridemir ldquoEmpirical modeling of splitting tensile strengthfrom cylinder compressive strength of concrete by geneticprogrammingrdquo Expert Systems with Applications vol 38 no 11pp 14257ndash14268 2011

Advances in Materials Science and Engineering 13

[10] U Thissen R van Brakel A P de Weijer W J Melssen and LM C Buydens ldquoUsing support vector machines for time seriespredictionrdquo Chemometrics and Intelligent Laboratory Systemsvol 69 no 1-2 pp 35ndash49 2003

[11] P Samui ldquoPredicted ultimate capacity of laterally loaded pilesin clay using support vector machinerdquo Geomechanics andGeoengineering vol 3 no 2 pp 113ndash120 2008

[12] VN VapnikTheNature of Statistical LearningTheory SpringerNew York NY USA 1995

[13] V N Vapnik S E Golowich and A J Smola ldquoSupport vectormachine for function approximation regression estimationand signal processingrdquoAdvances in Neural Information Process-ing Systems vol 9 pp 281ndash287 1996

[14] M Khanzadi and A Behnood ldquoMechanical properties ofhigh-strength concrete incorporating copper slag as coarseaggregaterdquo Construction and Building Materials vol 23 no 6pp 2183ndash2188 2009

[15] C K Y Leung and J Pan ldquoEffect of concrete composition onFRPconcrete bond capacityrdquo in Proceedings of the InternationalSymposium on Bond Behaviour of FRP in Structures (BBFS rsquo05)pp 69ndash76 2005

Submit your manuscripts athttpwwwhindawicom

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CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

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Biomaterials

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Advances in

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Nano

materials

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Journal ofNanomaterials

2 Advances in Materials Science and Engineering

waterbinder (WB) Shaaban and Gesund [8] investigatedthe splitting tensile strength and compressive strength of steelfiber reinforced concrete but unfortunately did not analyzethe relationship between the tensile strength and compressivestrength Choi and Yuan [4] investigated the relationshipbetween the tensile strength and compression strength ofglass fiber reinforced concrete and polypropylene fiber rein-forced concrete Xu and Shi [1] developed the formulationbetween the 119891spt and 119891

119888of steel fiber reinforced con-

crete Mathematical regression models are usually imperfectdescription of complex physical phenomena The accuracyof the estimated result depends on the size of available dataSaridemir [9] proposed for 119891spt prediction from cylinder119891119888of concrete or age of specimen (AS) and cylinder 119891

119888

of concrete by gene expression programming (GEP) Thesupport vector machine (SVM) is a new efficient and novelapproach to improve generalization performance and it canattain a global minimum SVM achieves good generalizationability by adopting a structural risk minimization inductionprinciple that aims at minimizing a bound on generalizationerror of a model rather than minimizing the error on thetraining data only It has the ability to avoid overtrainingand has better generalization capability than artificial neuralnetworks (ANN) model Moreover the SVM can alwaysbe updated to get better results by presenting new trainingexamples as new data become available

In this paper support vectormachine is applied to predictthe splitting tensile strength of concrete using the datumcollected by Saridemir [9] Compressive strength used topredict the splitting tensile strength is considered as inputin SVM model that is compared with experimental data andother methods The mean absolute parentage error (MAPE)root-mean-squared error (RMSE) and 119877-square (1198772) areused as the criteria to compare the performance of SVMmodels and other models

2 Support Vector Machine

SVM is a machine-learning algorithm based on statisticallearning theory The main idea of the SVM is to transformthe input space into a high-dimensional space by a nonlineartransformation defined by an inner product function SVMcalculation takes the form of a problem in convex quadraticoptimization ensuring that the solution is optimal It is betterthan the traditional artificial neural networks which is basedon the traditional minimization principle of experience risk[10] The SVM has a good ability to generalize and resolvesome practical problems such as small samples nonlinearityand high-dimensional input space

In this section a brief description of the process ofconstructing a SVM for a regression problem is presentedThere are three distinct characteristics to consider when anSVM is used to solve a regression problem [11] First theSVM estimates the regression by a set of linear functions thatare defined in a high-dimensional space Second the SVMcarries out the regression estimation by risk minimizationwhere the risk is measured using Vapnikrsquos 120576-insensitive lossfunction Third the SVM uses a risk function consisting

Table 1 Parameters of SVM-I and SVM-II

Kernel

SVM-I SVM-IIRadialbasis

functionPolynomial

Radialbasis

functionPolynomial

119862 500000 160000 820000 870000120576 010 010 004 005Parameter 120574 = 001 119889 = 2 120574 = 010 119889 = 3

of the empirical error and a regularization term whichis derived from the structural risk minimization (SRM)principle

Consider a set of training samples 119909119894 119910119894 119894 = 1 2

119899 119909119894isin 119877119889 119910

119894isin 119877 where 119909

119894is an input vector 119910

119894is the

corresponding output value and 119899 is the number of trainingsamples The regression problem is to select a function thatpredicts the actual value of 119910 as closely as possible with aprecision of 120576 Therefore the purpose of the SVM is to seekthe optimum regression function

119891 (119909) = 120596 sdot 119909 + 119887 (4)

where 120596 isin 119877119899 and 119887 isin 119877 120596 is an adjustable weightvector 119887 is the scalar threshold 119877119899 is 119899-dimensional vectorspace and 119909 is one-dimensional vector space

Following statistical theory SVM determines the regres-sion function by minimizing an objective function Theparameters 120596 and 119887 of the regression function are estimatedby minimizing the regularized risk function as follows

Minimize

1

21205962 + 119862

119899

sum119894=1

(120585119894+ 120585lowast119894) (5)

Subject to

119910119894minus [(120596 sdot 119909

119894) + 119887] le 120576 + 120585

119894 (6)

[(120596 sdot 119909119894) + 119887] minus 119910

119894le 120576 + 120585lowast

119894 (7)

120585119894ge 0 120585lowast

119894ge 0 119894 = 1 2 119899 (8)

where 119862 gt 0 is a penalty factor 120585 and 120585lowast are slack variables120576 is the insensitive loss function and can be described in thefollowing way

119871120576(119910) =

1003816100381610038161003816119891 (119909) minus 1199101003816100381610038161003816 minus 120576

1003816100381610038161003816119910 minus 119891 (119909)1003816100381610038161003816 ge 120576

0 otherwise(9)

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

Maximize119899

sum119894=1

119910119894(120572lowast119894minus 120572119894) minus 120576119899

sum119894=1

(120572lowast119894+ 120572119894)

minus1

2

119899

sum119894=1

119899

sum119895=1

(120572lowast119894minus 120572119894) (120572lowast119895minus 120572119895) (119909119894 119909119895)

(10)

Advances in Materials Science and Engineering 3

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

119891119888(MPa) 119891spt (MPa) GEP-I [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

2633 307 minus072 minus004 minus020 minus041 minus052 minus030 minus029

3582 312 minus018 041 023 014 minus002 034 001

3093 321 minus057 007 minus010 minus025 minus039 minus033 minus030

3742 207 097 154 136 129 112 000 114

2910 301 minus049 017 001 minus017 minus029 001 minus017

2803 341 minus095 minus029 minus045 minus064 minus076 minus079 minus053

3698 325 minus024 034 016 008 minus009 037 minus012

3165 339 minus071 minus007 minus024 minus039 minus053 minus107 minus040

3858 316 minus005 050 032 027 009 037 016

8100 660 minus110 minus129 minus156 minus098 minus140 minus006 minus119

6800 590 minus111 minus103 minus128 minus090 minus124 minus176 minus099

6000 500 minus065 minus043 minus066 minus040 minus070 minus022 minus060

5400 470 minus069 minus036 minus058 minus041 minus068 minus015 minus068

3908 311 004 058 039 034 017 076 013

2233 229 minus020 050 036 009 001 025 039

2323 219 minus004 065 051 025 016 032 049

2289 238 minus025 044 030 004 minus005 036 034

2216 217 minus009 061 047 020 011 034 050

2048 215 minus018 052 038 010 002 007 048

2082 199 000 070 057 028 021 002 065

2283 241 minus028 041 027 000 minus008 010 025

2514 262 minus034 034 019 minus005 minus014 012 012

2560 269 minus038 030 014 minus008 minus019 minus015 009

3520 320 minus029 030 012 002 minus013 039 minus010

4440 290 057 103 083 086 065 076 062

3760 240 066 122 103 097 080 000 084

4180 360 minus029 021 002 001 minus018 012 minus020

4200 350 minus017 032 013 012 minus007 047 minus010

3830 270 040 095 077 071 053 041 051

5540 340 071 099 077 096 069 000 078

5400 280 123 154 132 149 122 000 122

4860 280 091 131 110 120 096 000 094

5650 410 007 033 011 032 004 minus006 009

4700 390 minus028 014 minus006 001 minus022 019 minus022

4569 419 minus065 minus020 minus040 minus036 minus057 minus037 minus059

4171 309 022 072 053 052 032 032 033

4249 374 minus038 011 minus009 minus009 minus029 minus053 minus027

3369 293 minus011 049 032 020 005 040 016

3730 314 minus010 046 028 021 004 049 007

2475 262 minus037 032 017 minus007 minus017 minus021 011

2696 268 minus028 038 023 002 minus009 036 007

2214 235 minus027 043 028 002 minus007 016 032

2391 224 minus004 064 050 025 016 003 044

4100 390 minus063 minus012 minus031 minus033 minus052 minus094 minus049

4200 400 minus067 minus018 minus037 minus038 minus057 minus003 minus060

5768 426 minus004 022 minus001 022 minus007 057 minus002

4575 410 minus057 minus011 minus031 minus026 minus048 minus028 minus045

3810 362 minus054 002 minus016 minus022 minus040 minus019 minus037

3195 331 minus061 002 minus014 minus029 minus043 minus027 minus040

4 Advances in Materials Science and Engineering

Table 2 Continued

119891119888(MPa) 119891spt (MPa) GEP-I [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

1820 190 minus010 062 049 018 012 minus024 068

2870 270 minus020 046 030 011 minus001 minus020 023

2390 230 minus011 058 044 019 010 minus003 038

4370 330 011 060 040 042 022 044 021

2460 230 minus007 063 048 024 014 minus001 045

4300 350 minus013 037 017 018 minus002 000 000

1420 180 minus028 042 031 minus004 minus008 minus029 063

3670 320 minus021 037 019 011 minus005 minus048 001

2270 170 041 111 097 071 062 035 095

3530 240 051 111 093 083 067 000 074

1360 130 018 088 077 041 038 079 112

2860 170 079 146 129 111 099 140 114

2660 240 minus004 064 049 027 017 minus001 042

4350 290 050 099 079 081 061 052 057

3090 240 024 088 071 055 042 048 051

4990 380 minus003 037 016 027 003 039 004

3220 300 minus028 035 018 004 minus010 minus009 001

5780 350 072 099 076 098 070 033 081

2680 240 minus002 065 050 029 018 047 044

3650 290 008 066 048 040 024 027 026

6560 440 026 038 014 048 015 014 036

3030 240 020 085 068 052 039 137 054

5280 380 014 049 027 042 017 minus039 023

4060 350 minus027 026 007 004 minus014 015 minus013

5230 420 minus029 007 minus015 000 minus026 010 minus024

4120 320 006 059 039 038 019 065 022

7280 510 minus005 minus007 minus032 013 minus024 003 minus005

4120 320 006 059 039 038 019 065 022

7280 510 minus005 minus007 minus032 013 minus024 003 minus005

7280 510 minus005 minus007 minus032 013 minus024 003 minus005

4180 330 000 051 032 031 012 042 010

6840 410 071 078 053 092 057 038 076

4230 290 043 094 074 074 054 000 057

5910 380 050 074 051 075 046 025 060

4130 370 minus043 009 minus010 minus012 minus031 minus019 minus027

5970 410 023 046 023 048 019 014 030

4230 320 013 064 044 044 024 071 027

7510 510 008 001 minus025 024 minus014 067 minus005

4850 370 minus001 041 020 029 006 008 009

8190 480 075 054 027 086 044 094 053

4130 330 minus003 049 030 028 009 021 013

6710 440 034 043 019 055 022 068 036

4200 320 011 062 043 042 023 077 020

7010 430 061 064 039 080 045 065 058

4480 340 008 055 035 038 017 036 021

6540 400 065 077 053 087 054 067 070

3870 350 minus039 017 minus002 minus007 minus024 040 minus021

5640 430 minus016 013 minus009 011 minus016 minus025 minus014

4240 320 013 064 045 045 025 041 025

Advances in Materials Science and Engineering 5

Table 2 Continued

119891119888(MPa) 119891spt (MPa) GEP-I [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

6770 480 minus003 005 minus019 018 minus016 027 minus003

5070 360 022 060 039 051 026 048 026

8110 480 070 051 024 082 041 062 052

4200 320 011 062 043 042 023 077 020

4890 360 011 053 032 041 018 061 021

8530 580 minus007 minus035 minus063 001 minus042 032 minus037

5230 420 minus029 007 minus015 000 minus026 010 minus024

8510 580 minus008 minus036 minus063 000 minus043 minus034 minus042

4970 420 minus044 minus004 minus025 minus014 minus038 minus049 minus036

7120 530 minus033 minus032 minus057 minus015 minus050 minus052 minus033

5190 380 009 045 023 037 012 036 017

8770 610 minus024 minus057 minus086 minus018 minus063 minus005 minus066

5190 380 009 045 023 037 012 036 017

8770 610 minus024 minus057 minus086 minus018 minus063 minus005 minus066

5190 380 009 045 023 037 012 036 017

8770 610 minus024 minus057 minus086 minus018 minus063 minus005 minus066

5380 410 minus010 023 001 018 minus009 minus010 minus006

8530 510 063 035 007 071 028 102 033

5870 440 minus013 012 minus011 013 minus016 minus026 minus009

8710 560 022 minus009 minus037 029 minus015 minus009 minus020

4930 450 minus076 minus036 minus057 minus047 minus070 minus135 minus069

6620 510 minus041 minus030 minus054 minus019 minus052 minus023 minus040

5310 380 016 050 028 044 018 050 033

8430 560 007 minus018 minus046 017 minus026 010 minus015

8670 603 003 minus054 minus080 minus016 minus060 002 minus054

5610 430 minus017 012 minus011 010 minus018 minus058 minus017

5210 380 010 046 024 038 013 011 016

7850 500 036 023 minus004 050 010 087 029

4990 410 minus033 007 minus014 minus003 minus027 009 minus026

8750 510 075 042 014 081 037 048 029

5510 440 minus033 minus002 minus024 minus006 minus033 minus011 minus023

7540 400 122 112 086 135 097 000 105

2467 274 minus050 019 004 minus020 minus029 minus033 002

2301 279 minus066 004 minus010 minus036 minus045 000 minus009

3203 303 minus032 031 014 000 minus014 minus064 minus003

2878 293 minus043 024 007 minus011 minus023 minus043 000

3360 203 077 139 122 109 095 130 106

2330 304 minus089 minus019 minus034 minus059 minus068 minus125 minus031

GEP-I model is developed by Saridemir [9] based on gene expression programming Regression is regression-based formulation results by Saridemir [9]

Subject to119899

sum119894=1

120572119894=119899

sum119894=1

120572lowast119894 0 le 120572lowast

119894le 119888 0 le 120572

119894le 119888 (11)

where the 120572119894 120572lowast119894are called Lagrangian multipliers When

the Lagrangian multipliers (120572119894 120572lowast119894) are equal to zero shows

that the training object is irrelevant to the final solutionotherwise the training samples with nonzero Lagrangianmultipliers are called support vectors

When linear regression is not appropriate then the inputdata have to bemapped into a high-dimensional feature space

through some nonlinear mapping A nonlinear transforma-tion 120601(119909) replaces the input 119909 in (7) and the regressionfunction can be written as

119891 (119909) =nsvsum119894=1

(120572119894minus 120572lowast119894) 119896 (119909119894 119909119895) + 119887 (12)

where nsv is the number of support vectors and 119896(119909119894 119909119895) =

120601(119909119894) sdot 120601(119909

119895) 119896(119909

119894 119909119895) is a kernel function Any function sat-

isfying Mercersquos condition can be used as the kernel function[12 13] Some popular kernel functions are

6 Advances in Materials Science and Engineering

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

119891119888(MPa) 119891spt (MPa) GEP-I [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

4480 380 minus031 015 minus005 minus002 minus023 minus001 minus033

5620 440 minus025 002 minus020 000 minus027 minus041 minus020

5340 360 039 071 049 065 039 033 039

5670 460 minus042 minus016 minus038 minus017 minus045 minus009 minus036

6450 480 minus018 minus006 minus030 002 minus030 minus018 minus012

6650 470 003 011 minus013 022 minus011 010 001

6540 430 037 047 023 057 024 021 043

7380 490 023 017 minus009 038 001 022 014

7400 530 minus016 minus022 minus048 minus001 minus039 minus026 minus015

7250 500 006 002 minus023 022 minus015 024 minus006

7460 490 028 020 minus006 042 004 006 015

7790 550 minus014 minus029 minus056 minus003 minus042 minus036 minus019

7910 520 022 005 minus022 033 minus007 001 015

8420 570 000 minus029 minus056 006 minus037 minus034 minus028

8330 590 minus025 minus052 minus079 minus018 minus060 minus060 minus039

8010 530 015 minus002 minus029 027 minus013 minus022 010

8650 550 032 minus001 minus029 037 minus007 minus027 012

10200 550 115 046 016 105 053 037 034

10100 650 009 minus057 minus087 001 minus051 minus045 minus060

11100 620 092 002 minus030 073 016 040 minus017

9450 580 045 minus006 minus036 042 minus006 008 minus001

11800 620 128 021 minus012 102 041 064 minus014

1900 230 minus043 027 014 minus016 minus023 minus004 minus100

2000 240 minus046 024 010 minus019 minus026 minus013 minus096

2860 290 minus040 026 009 minus009 minus021 011 minus067

2890 300 minus048 017 001 minus017 minus030 008 minus082

3610 330 minus033 024 006 minus002 minus019 minus011 minus048

4450 370 minus023 024 004 007 minus014 007 minus026

2300 250 minus036 033 019 minus007 minus016 minus037 minus082

3300 350 minus072 minus011 minus028 minus041 minus056 minus043 minus102

5800 420 005 029 006 030 001 minus025 020

6900 450 037 040 015 055 020 012 044

3600 330 minus034 024 006 minus003 minus019 minus004 minus045

4280 390 minus053 minus004 minus024 minus023 minus043 minus019 minus063

5930 400 033 054 031 056 027 029 040

6330 450 005 019 minus004 026 minus005 minus018 020

6770 470 010 015 minus009 028 minus006 026 007

7700 510 021 008 minus019 033 minus006 minus005 001

4990 430 minus051 minus013 minus034 minus023 minus047 minus032 minus045

9670 570 063 010 minus019 062 012 005 001

6170 460 minus014 003 minus020 008 minus022 minus036 minus008

7320 510 000 minus005 minus031 015 minus022 017 minus003

8650 550 032 minus001 minus029 037 minus007 minus027 012

9230 590 023 minus023 minus052 023 minus025 010 minus019

(1) polynomial kernel function

119870(119909 119910) = [(119909 + 119910) + 1]119889

119889 = 1 2 119899 (13)

(2) radial basis function (RBF)

119870(119909 119910) = exp minus1205741003817100381710038171003817119909 minus 11991010038171003817100381710038172

120574 ge 0 (14)

(3) sigmoid kernel function

119870(119909 119910) = tan [120593 (119909 sdot 119910) + 120579] (15)

where 120572119894 120572lowast119894satisfy 120572

119894120572lowast119894= 0 120572

119894ge 0 and 120572lowast

119894ge 0

Advances in Materials Science and Engineering 7

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

119891119888(MPa) 119891spt (MPa) GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

7790 545 minus008 minus024 minus051 002 minus033 048 032

5710 313 103 133 110 132 115 174 145

7160 490 011 009 minus016 027 minus002 081 051

4410 391 minus052 001 minus019 minus017 minus023 028 minus017

6480 457 004 018 minus006 027 003 073 042

4260 366 minus036 019 000 000 minus005 047 minus004

3040 279 minus026 046 030 013 017 046 000

4680 389 minus034 015 minus006 001 minus008 049 004

8950 591 011 minus033 minus061 009 minus036 073 041

6690 455 019 028 003 039 014 096 059

7610 547 minus021 minus032 minus058 minus008 minus042 041 022

5360 437 minus041 minus005 minus027 minus011 minus025 047 minus002

6480 507 minus046 minus032 minus056 minus023 minus047 023 minus008

4580 402 minus053 minus003 minus023 minus018 minus026 040 minus016

3760 316 minus017 046 027 021 019 063 014

5310 408 minus015 022 000 016 002 078 024

3490 350 minus068 minus001 minus019 minus030 minus029 012 minus040

5390 540 minus142 minus107 minus129 minus112 minus127 minus062 minus100

4650 380 minus027 022 002 008 minus001 060 007

1380 170 minus029 049 038 003 017 minus042 minus017

6710 480 minus005 003 minus021 015 minus010 064 035

2030 340 minus153 minus074 minus088 minus117 minus106 minus126 minus136

2240 210 minus009 069 055 028 038 038 010

6140 430 012 032 009 037 016 079 051

3310 300 minus029 039 022 009 011 053 minus003

7900 560 minus017 minus036 minus062 minus008 minus044 046 021

4360 360 minus024 030 010 012 006 059 010

2560 242 minus020 057 041 019 026 031 002

2620 262 minus036 040 025 003 010 023 minus015

1690 188 minus025 055 042 010 023 minus013 minus010

1490 171 minus022 057 045 011 025 minus038 minus008

2730 281 minus048 027 012 minus009 minus003 019 minus026

2840 287 minus047 027 011 minus008 minus002 022 minus023

1690 191 minus028 052 039 007 020 minus016 minus013

1690 211 minus048 032 019 minus013 000 minus036 minus033

3090 292 minus035 036 019 003 007 031 minus010

3230 308 minus043 027 010 minus004 minus001 036 minus017

1960 222 minus040 039 026 minus004 008 minus010 minus022

1910 208 minus030 050 037 006 018 minus011 minus010

3530 324 minus040 027 009 minus001 minus001 038 minus011

3840 336 minus032 030 011 005 004 057 minus003

2250 247 minus045 033 019 minus008 002 minus005 minus027

2340 257 minus049 028 014 minus012 minus002 minus008 minus029

3100 290 minus033 038 022 006 010 035 minus006

1700 230 minus066 013 001 minus032 minus018 minus060 minus051

4700 400 minus044 004 minus016 minus009 minus018 044 minus009

3700 360 minus065 minus001 minus019 minus027 minus028 021 minus035

4200 400 minus074 minus018 minus037 minus038 minus042 016 minus039

5400 460 minus062 minus026 minus048 minus031 minus046 021 minus024

8 Advances in Materials Science and Engineering

Table 4 Continued

119891119888(MPa) 119891spt (MPa) GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

4200 440 minus114 minus058 minus077 minus078 minus082 minus024 minus079

6200 520 minus075 minus055 minus079 minus050 minus072 003 minus035

6700 425 049 058 033 070 044 115 091

7250 510 minus004 minus008 minus033 012 minus019 056 038

7450 520 minus003 minus011 minus037 011 minus021 059 038

7950 575 minus029 minus049 minus076 minus020 minus057 030 011

7800 515 022 006 minus020 033 minus003 083 060

8500 575 002 minus031 minus059 005 minus036 069 044

8600 480 103 067 039 105 062 167 140

9050 440 168 121 093 165 119 232 204

5480 334 069 103 081 099 083 161 106

3290 285 minus016 053 036 023 025 068 011

3450 292 minus013 055 037 026 027 066 013

5360 348 048 084 062 078 064 136 087

7410 526 minus011 minus018 minus044 003 minus029 058 030

6770 481 minus003 004 minus020 017 minus009 068 037

8420 657 minus085 minus116 minus143 minus081 minus121 minus025 minus046

7530 535 minus013 minus023 minus049 000 minus033 048 028

10230 729 minus055 minus132 minus163 minus073 minus129 052 minus031

4010 330 minus016 044 025 021 018 070 015

4050 340 minus023 035 016 014 010 057 007

9550 570 066 007 minus023 057 006 153 096

3520 317 minus033 033 015 005 006 048 minus005

5750 369 051 078 056 078 060 129 086

3860 338 minus033 029 010 005 003 052 minus004

6150 448 minus005 015 minus009 019 minus002 070 032

4740 376 minus017 030 010 017 008 067 018

4710 368 minus011 037 016 023 014 086 023

4470 326 017 068 048 052 045 109 046

5240 399 minus010 028 006 021 008 079 027

5790 449 minus027 000 minus023 000 minus018 061 012

3950 404 minus093 minus033 minus052 minus056 minus059 000 minus065

3670 403 minus109 minus046 minus064 minus072 minus072 minus029 minus082

3800 365 minus063 minus001 minus020 minus026 minus028 020 minus035

2420 220 minus007 070 055 031 040 038 014

1740 180 minus013 066 054 021 034 minus006 002

2930 280 minus033 039 023 005 010 027 minus009

2700 255 minus023 052 036 015 022 033 minus002

3260 321 minus053 016 minus001 minus015 minus012 026 minus026

4630 520 minus167 minus119 minus139 minus133 minus141 minus076 minus137

3580 430 minus142 minus077 minus095 minus104 minus104 minus057 minus111

4310 418 minus085 minus031 minus050 minus049 minus055 minus004 minus051

3520 317 minus033 033 015 005 006 048 minus005

5750 369 051 078 056 078 060 129 086

4280 490 minus159 minus104 minus124 minus123 minus128 minus076 minus126

6150 448 minus005 015 minus009 019 minus002 070 032

4740 376 minus017 030 010 017 008 067 018

4713 368 minus010 037 016 023 014 086 023

4473 326 017 069 049 052 045 104 050

Advances in Materials Science and Engineering 9

Table 4 Continued

119891119888(MPa) 119891spt (MPa) GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

3460 410 minus130 minus063 minus081 minus091 minus090 minus051 minus102

5970 540 minus108 minus084 minus107 minus082 minus101 minus030 minus069

5190 400 minus014 025 003 017 004 076 021

5000 390 minus015 027 006 017 006 077 022

4850 410 minus044 001 minus020 minus011 minus021 049 minus008

4880 370 minus002 042 021 031 020 080 032

4650 380 minus026 022 002 008 minus001 060 007

4520 400 minus054 minus003 minus024 minus019 minus027 030 minus021

4520 330 016 067 046 051 043 100 049

4260 298 032 087 068 068 063 115 064

8160 440 119 093 066 124 086 180 158

4270 300 031 086 066 066 061 121 064

4190 350 minus024 032 012 012 007 060 009

4120 380 minus059 minus001 minus021 minus022 minus026 031 minus026

3970 270 042 102 083 079 076 124 073

6550 550 minus084 minus073 minus097 minus063 minus087 minus021 minus047

3950 370 minus059 001 minus018 minus022 minus025 034 minus031

1710 190 minus025 054 042 009 022 minus011 minus010

1210 110 018 095 085 048 064 000 031

1290 160 minus026 052 041 005 020 minus048 minus013

1690 150 013 093 080 048 061 025 028

2450 250 minus035 042 027 003 012 016 minus016

3 SVM for Predicting 119891spt of Concrete

As mentioned previously many methods have been pro-posed for the prediction of 119891spt from compressive strengthof concrete This study attempts to utilize SVM for theprediction of concrete of 119891spt SVM-I model is devel-oped for 150 times 300mm cylinder 119891spt prediction from 150 times300mm cylinder 119891

119888of concrete and SVM-IImodel is devel-

oped for 100 times 200mm and 150 times 200mm cylinder 119891sptprediction from 100 times 200mm cylinder 119891

119888of concrete The

experimental data which are taken from studies [9 14 15] areused in this study Among 184 experimental data 138 datawere randomly selected as the training set for SVM-I modeland among 168 cases 126 cases were randomly selected asthe training set for SVM-II model the rest are considered astesting data setThe data are normalized before being used inthe model as follows

119883 =(119883119894minus 119883min)

(119883max minus 119883min) (16)

where 119883max and 119883min are the maximum and minimuminput values data respectively

In case of SVM training two types of kernel func-tions were used namely radical basis function (RBF) andpolynomial function in training process 119862 and 120576 and otherkernel-specific parameters have been chosen by a trial-and-error approach The best simulation performances ofSVM are summarized in Table 1 in order to evaluate theabilities of SVM models and other models mean absolutepercentage error (MAPE) roof-mean-squared error (RMES)

and R-square (1198772) were used as the criteria between theexperimental and predicted values which are according tothe equationsas follow

MAPE =1

119899[sum119899

119894=1

1003816100381610038161003816119905119894 minus 1199001198941003816100381610038161003816

sum119899

119894=1119905119894

times 100]

RMSE = radic1

119899

119899

sum119894=1

(119905119894minus 119900119894)

1198772 =(119899sum 119905

119894119900119894minus sum 119905119894sum119900119894)2

(119899sum 1199052119894minus (sum 1199052

119894)) (119899sum 1199002

119894minus (sum 119900

119894)2

)

(17)

where 119905119894is the experimental value 119900

119894is the predicted value

and 119899 is total number of data

4 Results and Discussion

There is no doubt that the splitting tensile strength 119891spt withan increase in the compression strength of concrete butthere is no agreement on the precise form of the relationshipCodes propose different formulas for prediction cylinder 119891sptof concrete from compressive strength In this paper 119891sptresults are investigated for 150 times 300mm cylinder concreteand 100 times 200mm cylinder concrete separately The errors(predicted values subtract measured values) computed bySVM model other models and measured values of split-ting tensile strengths are shown in Tables 2 and 3 for 150times 300mm cylinder concrete and in Tables 4 and 5 for

10 Advances in Materials Science and Engineering

Table 5 Comparison of experimental results to testing results of RBF polynomial of SVM-II and other models

119891119888(MPa) 119891spt (MPa) GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression RBF [9] Polynomial

3600 349 minus060 005 minus013 minus022 minus022 minus017 minus004

3040 308 minus054 017 001 minus016 minus012 minus010 minus001

2160 200 minus004 074 060 033 043 017 034

2120 270 minus077 002 minus012 minus040 minus030 minus044 minus039

2510 260 minus041 036 021 minus003 005 minus009 006

3190 330 minus067 003 minus014 minus028 minus025 minus026 minus012

3780 310 minus010 053 034 028 026 048 045

2920 250 minus004 069 053 034 040 041 049

3540 310 minus025 041 023 013 014 019 032

2860 340 minus098 minus024 minus041 minus059 minus054 minus051 minus046

2950 310 minus062 010 minus006 minus024 minus019 minus023 minus008

3490 350 minus068 minus001 minus019 minus030 minus029 minus023 minus011

3860 338 minus033 029 010 005 003 015 023

5235 399 minus010 028 006 021 008 042 036

3840 380 minus076 minus014 minus033 minus039 minus040 minus023 minus020

5150 420 minus036 003 minus018 minus005 minus017 016 011

4860 400 minus034 011 minus010 000 minus011 011 016

3950 390 minus079 minus019 minus038 minus042 minus045 minus022 minus022

3040 285 minus031 040 024 007 011 013 022

2950 256 minus008 064 048 030 035 031 046

4280 430 minus099 minus044 minus064 minus063 minus068 minus052 minus046

5190 430 minus044 minus005 minus027 minus013 minus026 009 003

4930 420 minus049 minus006 minus027 minus017 minus027 003 000

4330 360 minus026 028 008 010 004 022 028

4360 338 minus002 052 032 034 028 046 052

4990 457 minus083 minus040 minus061 minus050 minus062 minus034 minus033

5740 420 minus001 027 004 026 009 054 039

5850 490 minus065 minus039 minus062 minus038 minus056 minus015 minus024

3140 279 minus019 052 035 020 023 026 036

3490 304 minus022 045 027 016 017 023 035

4160 384 minus060 minus003 minus023 minus024 minus028 minus011 minus005

4680 400 minus044 004 minus017 minus010 minus019 001 009

5170 420 minus035 004 minus017 minus004 minus016 018 012

4330 390 minus056 minus002 minus022 minus020 minus026 minus008 minus002

6080 410 029 050 027 054 033 076 068

6000 430 004 027 004 030 010 060 044

7190 540 minus037 minus040 minus065 minus021 minus051 016 000

7510 540 minus018 minus029 minus055 minus006 minus039 030 020

5787 449 minus027 000 minus023 000 minus018 027 015

3950 340 minus029 031 012 008 005 028 028

Table 6 Error measurement for 119891spt from 150 times 300mm cylinder 119891119888

GEP-I [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF PolynomialRMSE of training set 04603 05995 05228 05027 04512 04978 04819RMSE of testing set 04271 02735 03152 03754 02874 02748 04176MAPE of training set 97554 13647 114913 106213 95596 102434 105454MAPE of testing set 73022 46139 55434 62059 5286 4987 695171198772 of training set 08282 08225 08224 08267 0826 08115 082271198772 of testing set 09416 09486 09486 09471 09476 09422 09327

Advances in Materials Science and Engineering 11

Table 7 Error measurement for 119891spt from 100 times 200mm cylinder 119891119888

GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF PolynomialRMSE of training set 05886 0539 05244 04987 04911 07287 0562RMSE of testing set 0483 03348 03393 02803 03155 03082 02917MAPE of training set 118729 116879 109223 92976 9472 155902 106658MAPE of testing set 108701 71383 7604 62261 7234 69713 628351198772 of training set 08198 08354 08353 083 08337 0837 082531198772 of testing set 08778 08802 08793 088 08801 08815 08823

0

2

4

6

8

0 20 40 60 80 100 120 140

Split

ting

tens

ile st

reng

th (M

Pa)

Sample number

ExperimentalRBFPolynomial

Figure 1 Comparison of experimental results to training results ofSVM-I

100 times 200mm cylinder concrete As seen from these resultsSVM model is sufficiently close to the experimental dataand is able to predict the 119891spt from the 119891

119888of concrete

For convenient comparison purpose the experimental andpredicted results are plotted in Figures 1 2 3 and 4 It canbe seen that the results from SVM method are in goodagreement with the experimental results

The 119877-square errors in the trained SVM are 08115 and08227 for RBF and polynomial function of SVM-I 08370and 08253 for those of SVM-II respectively 119877-square errorsin the tested SVM are 09422 and 09327 for RBF and polyno-mial function of SVM-I 08815 and 08823 for those of SVM-II respectively The whole results obtained from the SVMmodels show a successful performance of the SVM modelsof predicting 150 times 300mm cylinder119891spt of concrete from thecorresponding 150 times 300mm cylinder compressive strength119891119888and 100 times 200mm cylinder 119891spt from the corresponding

100 times 200mm cylinder compressive strength 119891119888of concrete

for each of the training and testing sets It is also clear thatthere are no major difference on performance between theradical basis function and the polynomial function kernelsthat used in this paper However in generally the radical basisfunction kernel exhibits slightly better performance than thepolynomial kernel

The performance of the trained and tested sets is analyzedby computing MAPE RMSE and 1198772 of experimental 119891spt

0

2

4

6

8

0 5 10 15 20 25 30 35 40 45 50

Split

ting

tens

ile st

reng

th (M

Pa)

Sample number

ExperimentalRBFPolynomial

Figure 2 Comparison of experimental results to testing results ofSVM-I

between the SVM model and other methods The MAPERMSE and 1198772 are calculated for these methods Statisticalparameters of the training and testing sets of all methods arepresented in Tables 6 and 7 The model having the smallestMAPE or RMSE and the highest 1198772 can be regarded to bethe best model with the assumption of analysis Generally1198772 of the SVM models are higher than other methodsand the MAPE and RMSE of SVM are smaller than othermodels The analysis shows that the SVM model is betterthan other models and can predict the splitting tensilestrength of concrete well Furthermore SVMmodel has goodgeneralization capacity to avoid over training and can alwaysbe updated to get better results by presenting new trainingexamples as new data become available Thus SVM modelcan be regarded as a very effective method to predict splittingtensile strength of concrete from their compressive strength

5 Conclusion

The splitting tensile strength of concrete estimations fromcompression strength has been obtained so far in the litera-ture either through regression or other methodsThis presentstudy reports a new and influential approach for predictingthe splitting tensile strength using SVM for the first timein the literature The study conducted in this paper shows

12 Advances in Materials Science and Engineering

Polynomial

0

1

2

3

4

5

6

7

8

9

0 20 40 60 80 100 120 140

Split

ting

tens

ile st

reng

hth

(MPa

)

Sample number

ExperimentalRBF

Figure 3 Comparison of experimental results to training results ofSVM-II

Polynomial

0

2

4

6

0 10 20 30 40 50

Split

ting

tens

ile st

reng

th (M

Pa)

Sample number

ExperimentalRBF

Figure 4 Comparison of experimental results to testing results ofSVM-II

the feasibility of using a simple SVM to estimate the splittingtensile strength of concrete After learning from a set ofselected training data involving compressive strengths ofconcrete collected from the previous literature the SVM canbe utilized to predict the splitting tensile strengths of the testdata

In this paper radical basis function and polynomial areadopted for predicting splitting tensile strength from 150 times300mm cylinder and 100times200mm cylinder 119891

119888of concrete

It is found that the splitting tensile strength obtained fromthe SVM is more accurate than those obtained from designcodes and several researchesrsquo empirical equation when acomparison is made on the basis of the experimental dataSince the SVM is largely characterized by the type of its kernel

function it is necessary to choose the appropriate kernel foreach particular application problem in order to guaranteesatisfactory results The results of radial basis function andpolynomial indicate that RBF and polynomial kernel have theability to predict the splitting tensile strength of concrete fromcompression strength with an acceptable degree of accuracy

The statistical parameters of MAPE RMSE and 1198772 showthat the proposed SVM model results have the best accu-racy and can predict splitting tensile strength very close toexperiment results The use of SVM is very advantageousfor the prediction of the splitting tensile strength of concretefrom compression strength because it can perform nonlinearregression efficiently for high-dimensional data sets Further-more its solution is globalThe satisfactory predictions of thesplitting tensile strength of concrete by the model indicatethat SVM is a useful modeling tool for engineers and researchscientists at concrete construction fields

Acknowledgments

This project was granted financial support from the NationalNatural Science Foundation of China (no 50808077 andno 51278188) and the Fundamental Research Funds for theCentral Universities The authors are grateful for support

References

[1] B W Xu and H S Shi ldquoCorrelations among mechanicalproperties of steel fiber reinforced concreterdquo Construction andBuilding Materials vol 23 no 12 pp 3468ndash3474 2009

[2] M F M Zain H B Mahmud A Ilham and M FaizalldquoPrediction of splitting tensile strength of high-performanceconcreterdquoCement andConcrete Research vol 32 no 8 pp 1251ndash1258 2002

[3] F A Oluokun E G Burdette and J H Deatherage ldquoSplittingtensile strength and compressive strength relationship at earlyagesrdquo ACI Materials Journal vol 88 no 2 pp 115ndash121 1991

[4] Y Choi and R L Yuan ldquoExperimental relationship betweensplitting tensile strength and compressive strength of GFRC andPFRCrdquo Cement and Concrete Research vol 35 no 8 pp 1587ndash1591 2005

[5] F D Larrad and Y Malier ldquoEngineering properties of very highperformance concreterdquo in High Performance Concrete fromMaterial to Structure Y Malier Ed pp 85ndash114 EFN SponLondon UK 1992

[6] J K Kim S H Han Y D Park and J H Noh ldquoMaterialproperties of self-flowing concreterdquo Journal of Materials in CivilEngineering vol 10 no 4 pp 244ndash249 1998

[7] J K Kim S H Han and Y C Song ldquoEffect of temperatureand aging on the mechanical properties of concretemdashpart Iexperimental resultsrdquo Cement and Concrete Research vol 32no 7 pp 1087ndash1094 2002

[8] AM Shaaban andHGesund ldquoSplitting tensile strength of steelfiber reinforced concrete cylinders consolidated by rodding orvibratingrdquo ACI Materials Journal vol 90 no 4 pp 366ndash3691993

[9] M Saridemir ldquoEmpirical modeling of splitting tensile strengthfrom cylinder compressive strength of concrete by geneticprogrammingrdquo Expert Systems with Applications vol 38 no 11pp 14257ndash14268 2011

Advances in Materials Science and Engineering 13

[10] U Thissen R van Brakel A P de Weijer W J Melssen and LM C Buydens ldquoUsing support vector machines for time seriespredictionrdquo Chemometrics and Intelligent Laboratory Systemsvol 69 no 1-2 pp 35ndash49 2003

[11] P Samui ldquoPredicted ultimate capacity of laterally loaded pilesin clay using support vector machinerdquo Geomechanics andGeoengineering vol 3 no 2 pp 113ndash120 2008

[12] VN VapnikTheNature of Statistical LearningTheory SpringerNew York NY USA 1995

[13] V N Vapnik S E Golowich and A J Smola ldquoSupport vectormachine for function approximation regression estimationand signal processingrdquoAdvances in Neural Information Process-ing Systems vol 9 pp 281ndash287 1996

[14] M Khanzadi and A Behnood ldquoMechanical properties ofhigh-strength concrete incorporating copper slag as coarseaggregaterdquo Construction and Building Materials vol 23 no 6pp 2183ndash2188 2009

[15] C K Y Leung and J Pan ldquoEffect of concrete composition onFRPconcrete bond capacityrdquo in Proceedings of the InternationalSymposium on Bond Behaviour of FRP in Structures (BBFS rsquo05)pp 69ndash76 2005

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

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TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

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BioMed Research International

MaterialsJournal of

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Nano

materials

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Journal ofNanomaterials

Advances in Materials Science and Engineering 3

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

119891119888(MPa) 119891spt (MPa) GEP-I [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

2633 307 minus072 minus004 minus020 minus041 minus052 minus030 minus029

3582 312 minus018 041 023 014 minus002 034 001

3093 321 minus057 007 minus010 minus025 minus039 minus033 minus030

3742 207 097 154 136 129 112 000 114

2910 301 minus049 017 001 minus017 minus029 001 minus017

2803 341 minus095 minus029 minus045 minus064 minus076 minus079 minus053

3698 325 minus024 034 016 008 minus009 037 minus012

3165 339 minus071 minus007 minus024 minus039 minus053 minus107 minus040

3858 316 minus005 050 032 027 009 037 016

8100 660 minus110 minus129 minus156 minus098 minus140 minus006 minus119

6800 590 minus111 minus103 minus128 minus090 minus124 minus176 minus099

6000 500 minus065 minus043 minus066 minus040 minus070 minus022 minus060

5400 470 minus069 minus036 minus058 minus041 minus068 minus015 minus068

3908 311 004 058 039 034 017 076 013

2233 229 minus020 050 036 009 001 025 039

2323 219 minus004 065 051 025 016 032 049

2289 238 minus025 044 030 004 minus005 036 034

2216 217 minus009 061 047 020 011 034 050

2048 215 minus018 052 038 010 002 007 048

2082 199 000 070 057 028 021 002 065

2283 241 minus028 041 027 000 minus008 010 025

2514 262 minus034 034 019 minus005 minus014 012 012

2560 269 minus038 030 014 minus008 minus019 minus015 009

3520 320 minus029 030 012 002 minus013 039 minus010

4440 290 057 103 083 086 065 076 062

3760 240 066 122 103 097 080 000 084

4180 360 minus029 021 002 001 minus018 012 minus020

4200 350 minus017 032 013 012 minus007 047 minus010

3830 270 040 095 077 071 053 041 051

5540 340 071 099 077 096 069 000 078

5400 280 123 154 132 149 122 000 122

4860 280 091 131 110 120 096 000 094

5650 410 007 033 011 032 004 minus006 009

4700 390 minus028 014 minus006 001 minus022 019 minus022

4569 419 minus065 minus020 minus040 minus036 minus057 minus037 minus059

4171 309 022 072 053 052 032 032 033

4249 374 minus038 011 minus009 minus009 minus029 minus053 minus027

3369 293 minus011 049 032 020 005 040 016

3730 314 minus010 046 028 021 004 049 007

2475 262 minus037 032 017 minus007 minus017 minus021 011

2696 268 minus028 038 023 002 minus009 036 007

2214 235 minus027 043 028 002 minus007 016 032

2391 224 minus004 064 050 025 016 003 044

4100 390 minus063 minus012 minus031 minus033 minus052 minus094 minus049

4200 400 minus067 minus018 minus037 minus038 minus057 minus003 minus060

5768 426 minus004 022 minus001 022 minus007 057 minus002

4575 410 minus057 minus011 minus031 minus026 minus048 minus028 minus045

3810 362 minus054 002 minus016 minus022 minus040 minus019 minus037

3195 331 minus061 002 minus014 minus029 minus043 minus027 minus040

4 Advances in Materials Science and Engineering

Table 2 Continued

119891119888(MPa) 119891spt (MPa) GEP-I [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

1820 190 minus010 062 049 018 012 minus024 068

2870 270 minus020 046 030 011 minus001 minus020 023

2390 230 minus011 058 044 019 010 minus003 038

4370 330 011 060 040 042 022 044 021

2460 230 minus007 063 048 024 014 minus001 045

4300 350 minus013 037 017 018 minus002 000 000

1420 180 minus028 042 031 minus004 minus008 minus029 063

3670 320 minus021 037 019 011 minus005 minus048 001

2270 170 041 111 097 071 062 035 095

3530 240 051 111 093 083 067 000 074

1360 130 018 088 077 041 038 079 112

2860 170 079 146 129 111 099 140 114

2660 240 minus004 064 049 027 017 minus001 042

4350 290 050 099 079 081 061 052 057

3090 240 024 088 071 055 042 048 051

4990 380 minus003 037 016 027 003 039 004

3220 300 minus028 035 018 004 minus010 minus009 001

5780 350 072 099 076 098 070 033 081

2680 240 minus002 065 050 029 018 047 044

3650 290 008 066 048 040 024 027 026

6560 440 026 038 014 048 015 014 036

3030 240 020 085 068 052 039 137 054

5280 380 014 049 027 042 017 minus039 023

4060 350 minus027 026 007 004 minus014 015 minus013

5230 420 minus029 007 minus015 000 minus026 010 minus024

4120 320 006 059 039 038 019 065 022

7280 510 minus005 minus007 minus032 013 minus024 003 minus005

4120 320 006 059 039 038 019 065 022

7280 510 minus005 minus007 minus032 013 minus024 003 minus005

7280 510 minus005 minus007 minus032 013 minus024 003 minus005

4180 330 000 051 032 031 012 042 010

6840 410 071 078 053 092 057 038 076

4230 290 043 094 074 074 054 000 057

5910 380 050 074 051 075 046 025 060

4130 370 minus043 009 minus010 minus012 minus031 minus019 minus027

5970 410 023 046 023 048 019 014 030

4230 320 013 064 044 044 024 071 027

7510 510 008 001 minus025 024 minus014 067 minus005

4850 370 minus001 041 020 029 006 008 009

8190 480 075 054 027 086 044 094 053

4130 330 minus003 049 030 028 009 021 013

6710 440 034 043 019 055 022 068 036

4200 320 011 062 043 042 023 077 020

7010 430 061 064 039 080 045 065 058

4480 340 008 055 035 038 017 036 021

6540 400 065 077 053 087 054 067 070

3870 350 minus039 017 minus002 minus007 minus024 040 minus021

5640 430 minus016 013 minus009 011 minus016 minus025 minus014

4240 320 013 064 045 045 025 041 025

Advances in Materials Science and Engineering 5

Table 2 Continued

119891119888(MPa) 119891spt (MPa) GEP-I [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

6770 480 minus003 005 minus019 018 minus016 027 minus003

5070 360 022 060 039 051 026 048 026

8110 480 070 051 024 082 041 062 052

4200 320 011 062 043 042 023 077 020

4890 360 011 053 032 041 018 061 021

8530 580 minus007 minus035 minus063 001 minus042 032 minus037

5230 420 minus029 007 minus015 000 minus026 010 minus024

8510 580 minus008 minus036 minus063 000 minus043 minus034 minus042

4970 420 minus044 minus004 minus025 minus014 minus038 minus049 minus036

7120 530 minus033 minus032 minus057 minus015 minus050 minus052 minus033

5190 380 009 045 023 037 012 036 017

8770 610 minus024 minus057 minus086 minus018 minus063 minus005 minus066

5190 380 009 045 023 037 012 036 017

8770 610 minus024 minus057 minus086 minus018 minus063 minus005 minus066

5190 380 009 045 023 037 012 036 017

8770 610 minus024 minus057 minus086 minus018 minus063 minus005 minus066

5380 410 minus010 023 001 018 minus009 minus010 minus006

8530 510 063 035 007 071 028 102 033

5870 440 minus013 012 minus011 013 minus016 minus026 minus009

8710 560 022 minus009 minus037 029 minus015 minus009 minus020

4930 450 minus076 minus036 minus057 minus047 minus070 minus135 minus069

6620 510 minus041 minus030 minus054 minus019 minus052 minus023 minus040

5310 380 016 050 028 044 018 050 033

8430 560 007 minus018 minus046 017 minus026 010 minus015

8670 603 003 minus054 minus080 minus016 minus060 002 minus054

5610 430 minus017 012 minus011 010 minus018 minus058 minus017

5210 380 010 046 024 038 013 011 016

7850 500 036 023 minus004 050 010 087 029

4990 410 minus033 007 minus014 minus003 minus027 009 minus026

8750 510 075 042 014 081 037 048 029

5510 440 minus033 minus002 minus024 minus006 minus033 minus011 minus023

7540 400 122 112 086 135 097 000 105

2467 274 minus050 019 004 minus020 minus029 minus033 002

2301 279 minus066 004 minus010 minus036 minus045 000 minus009

3203 303 minus032 031 014 000 minus014 minus064 minus003

2878 293 minus043 024 007 minus011 minus023 minus043 000

3360 203 077 139 122 109 095 130 106

2330 304 minus089 minus019 minus034 minus059 minus068 minus125 minus031

GEP-I model is developed by Saridemir [9] based on gene expression programming Regression is regression-based formulation results by Saridemir [9]

Subject to119899

sum119894=1

120572119894=119899

sum119894=1

120572lowast119894 0 le 120572lowast

119894le 119888 0 le 120572

119894le 119888 (11)

where the 120572119894 120572lowast119894are called Lagrangian multipliers When

the Lagrangian multipliers (120572119894 120572lowast119894) are equal to zero shows

that the training object is irrelevant to the final solutionotherwise the training samples with nonzero Lagrangianmultipliers are called support vectors

When linear regression is not appropriate then the inputdata have to bemapped into a high-dimensional feature space

through some nonlinear mapping A nonlinear transforma-tion 120601(119909) replaces the input 119909 in (7) and the regressionfunction can be written as

119891 (119909) =nsvsum119894=1

(120572119894minus 120572lowast119894) 119896 (119909119894 119909119895) + 119887 (12)

where nsv is the number of support vectors and 119896(119909119894 119909119895) =

120601(119909119894) sdot 120601(119909

119895) 119896(119909

119894 119909119895) is a kernel function Any function sat-

isfying Mercersquos condition can be used as the kernel function[12 13] Some popular kernel functions are

6 Advances in Materials Science and Engineering

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

119891119888(MPa) 119891spt (MPa) GEP-I [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

4480 380 minus031 015 minus005 minus002 minus023 minus001 minus033

5620 440 minus025 002 minus020 000 minus027 minus041 minus020

5340 360 039 071 049 065 039 033 039

5670 460 minus042 minus016 minus038 minus017 minus045 minus009 minus036

6450 480 minus018 minus006 minus030 002 minus030 minus018 minus012

6650 470 003 011 minus013 022 minus011 010 001

6540 430 037 047 023 057 024 021 043

7380 490 023 017 minus009 038 001 022 014

7400 530 minus016 minus022 minus048 minus001 minus039 minus026 minus015

7250 500 006 002 minus023 022 minus015 024 minus006

7460 490 028 020 minus006 042 004 006 015

7790 550 minus014 minus029 minus056 minus003 minus042 minus036 minus019

7910 520 022 005 minus022 033 minus007 001 015

8420 570 000 minus029 minus056 006 minus037 minus034 minus028

8330 590 minus025 minus052 minus079 minus018 minus060 minus060 minus039

8010 530 015 minus002 minus029 027 minus013 minus022 010

8650 550 032 minus001 minus029 037 minus007 minus027 012

10200 550 115 046 016 105 053 037 034

10100 650 009 minus057 minus087 001 minus051 minus045 minus060

11100 620 092 002 minus030 073 016 040 minus017

9450 580 045 minus006 minus036 042 minus006 008 minus001

11800 620 128 021 minus012 102 041 064 minus014

1900 230 minus043 027 014 minus016 minus023 minus004 minus100

2000 240 minus046 024 010 minus019 minus026 minus013 minus096

2860 290 minus040 026 009 minus009 minus021 011 minus067

2890 300 minus048 017 001 minus017 minus030 008 minus082

3610 330 minus033 024 006 minus002 minus019 minus011 minus048

4450 370 minus023 024 004 007 minus014 007 minus026

2300 250 minus036 033 019 minus007 minus016 minus037 minus082

3300 350 minus072 minus011 minus028 minus041 minus056 minus043 minus102

5800 420 005 029 006 030 001 minus025 020

6900 450 037 040 015 055 020 012 044

3600 330 minus034 024 006 minus003 minus019 minus004 minus045

4280 390 minus053 minus004 minus024 minus023 minus043 minus019 minus063

5930 400 033 054 031 056 027 029 040

6330 450 005 019 minus004 026 minus005 minus018 020

6770 470 010 015 minus009 028 minus006 026 007

7700 510 021 008 minus019 033 minus006 minus005 001

4990 430 minus051 minus013 minus034 minus023 minus047 minus032 minus045

9670 570 063 010 minus019 062 012 005 001

6170 460 minus014 003 minus020 008 minus022 minus036 minus008

7320 510 000 minus005 minus031 015 minus022 017 minus003

8650 550 032 minus001 minus029 037 minus007 minus027 012

9230 590 023 minus023 minus052 023 minus025 010 minus019

(1) polynomial kernel function

119870(119909 119910) = [(119909 + 119910) + 1]119889

119889 = 1 2 119899 (13)

(2) radial basis function (RBF)

119870(119909 119910) = exp minus1205741003817100381710038171003817119909 minus 11991010038171003817100381710038172

120574 ge 0 (14)

(3) sigmoid kernel function

119870(119909 119910) = tan [120593 (119909 sdot 119910) + 120579] (15)

where 120572119894 120572lowast119894satisfy 120572

119894120572lowast119894= 0 120572

119894ge 0 and 120572lowast

119894ge 0

Advances in Materials Science and Engineering 7

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

119891119888(MPa) 119891spt (MPa) GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

7790 545 minus008 minus024 minus051 002 minus033 048 032

5710 313 103 133 110 132 115 174 145

7160 490 011 009 minus016 027 minus002 081 051

4410 391 minus052 001 minus019 minus017 minus023 028 minus017

6480 457 004 018 minus006 027 003 073 042

4260 366 minus036 019 000 000 minus005 047 minus004

3040 279 minus026 046 030 013 017 046 000

4680 389 minus034 015 minus006 001 minus008 049 004

8950 591 011 minus033 minus061 009 minus036 073 041

6690 455 019 028 003 039 014 096 059

7610 547 minus021 minus032 minus058 minus008 minus042 041 022

5360 437 minus041 minus005 minus027 minus011 minus025 047 minus002

6480 507 minus046 minus032 minus056 minus023 minus047 023 minus008

4580 402 minus053 minus003 minus023 minus018 minus026 040 minus016

3760 316 minus017 046 027 021 019 063 014

5310 408 minus015 022 000 016 002 078 024

3490 350 minus068 minus001 minus019 minus030 minus029 012 minus040

5390 540 minus142 minus107 minus129 minus112 minus127 minus062 minus100

4650 380 minus027 022 002 008 minus001 060 007

1380 170 minus029 049 038 003 017 minus042 minus017

6710 480 minus005 003 minus021 015 minus010 064 035

2030 340 minus153 minus074 minus088 minus117 minus106 minus126 minus136

2240 210 minus009 069 055 028 038 038 010

6140 430 012 032 009 037 016 079 051

3310 300 minus029 039 022 009 011 053 minus003

7900 560 minus017 minus036 minus062 minus008 minus044 046 021

4360 360 minus024 030 010 012 006 059 010

2560 242 minus020 057 041 019 026 031 002

2620 262 minus036 040 025 003 010 023 minus015

1690 188 minus025 055 042 010 023 minus013 minus010

1490 171 minus022 057 045 011 025 minus038 minus008

2730 281 minus048 027 012 minus009 minus003 019 minus026

2840 287 minus047 027 011 minus008 minus002 022 minus023

1690 191 minus028 052 039 007 020 minus016 minus013

1690 211 minus048 032 019 minus013 000 minus036 minus033

3090 292 minus035 036 019 003 007 031 minus010

3230 308 minus043 027 010 minus004 minus001 036 minus017

1960 222 minus040 039 026 minus004 008 minus010 minus022

1910 208 minus030 050 037 006 018 minus011 minus010

3530 324 minus040 027 009 minus001 minus001 038 minus011

3840 336 minus032 030 011 005 004 057 minus003

2250 247 minus045 033 019 minus008 002 minus005 minus027

2340 257 minus049 028 014 minus012 minus002 minus008 minus029

3100 290 minus033 038 022 006 010 035 minus006

1700 230 minus066 013 001 minus032 minus018 minus060 minus051

4700 400 minus044 004 minus016 minus009 minus018 044 minus009

3700 360 minus065 minus001 minus019 minus027 minus028 021 minus035

4200 400 minus074 minus018 minus037 minus038 minus042 016 minus039

5400 460 minus062 minus026 minus048 minus031 minus046 021 minus024

8 Advances in Materials Science and Engineering

Table 4 Continued

119891119888(MPa) 119891spt (MPa) GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

4200 440 minus114 minus058 minus077 minus078 minus082 minus024 minus079

6200 520 minus075 minus055 minus079 minus050 minus072 003 minus035

6700 425 049 058 033 070 044 115 091

7250 510 minus004 minus008 minus033 012 minus019 056 038

7450 520 minus003 minus011 minus037 011 minus021 059 038

7950 575 minus029 minus049 minus076 minus020 minus057 030 011

7800 515 022 006 minus020 033 minus003 083 060

8500 575 002 minus031 minus059 005 minus036 069 044

8600 480 103 067 039 105 062 167 140

9050 440 168 121 093 165 119 232 204

5480 334 069 103 081 099 083 161 106

3290 285 minus016 053 036 023 025 068 011

3450 292 minus013 055 037 026 027 066 013

5360 348 048 084 062 078 064 136 087

7410 526 minus011 minus018 minus044 003 minus029 058 030

6770 481 minus003 004 minus020 017 minus009 068 037

8420 657 minus085 minus116 minus143 minus081 minus121 minus025 minus046

7530 535 minus013 minus023 minus049 000 minus033 048 028

10230 729 minus055 minus132 minus163 minus073 minus129 052 minus031

4010 330 minus016 044 025 021 018 070 015

4050 340 minus023 035 016 014 010 057 007

9550 570 066 007 minus023 057 006 153 096

3520 317 minus033 033 015 005 006 048 minus005

5750 369 051 078 056 078 060 129 086

3860 338 minus033 029 010 005 003 052 minus004

6150 448 minus005 015 minus009 019 minus002 070 032

4740 376 minus017 030 010 017 008 067 018

4710 368 minus011 037 016 023 014 086 023

4470 326 017 068 048 052 045 109 046

5240 399 minus010 028 006 021 008 079 027

5790 449 minus027 000 minus023 000 minus018 061 012

3950 404 minus093 minus033 minus052 minus056 minus059 000 minus065

3670 403 minus109 minus046 minus064 minus072 minus072 minus029 minus082

3800 365 minus063 minus001 minus020 minus026 minus028 020 minus035

2420 220 minus007 070 055 031 040 038 014

1740 180 minus013 066 054 021 034 minus006 002

2930 280 minus033 039 023 005 010 027 minus009

2700 255 minus023 052 036 015 022 033 minus002

3260 321 minus053 016 minus001 minus015 minus012 026 minus026

4630 520 minus167 minus119 minus139 minus133 minus141 minus076 minus137

3580 430 minus142 minus077 minus095 minus104 minus104 minus057 minus111

4310 418 minus085 minus031 minus050 minus049 minus055 minus004 minus051

3520 317 minus033 033 015 005 006 048 minus005

5750 369 051 078 056 078 060 129 086

4280 490 minus159 minus104 minus124 minus123 minus128 minus076 minus126

6150 448 minus005 015 minus009 019 minus002 070 032

4740 376 minus017 030 010 017 008 067 018

4713 368 minus010 037 016 023 014 086 023

4473 326 017 069 049 052 045 104 050

Advances in Materials Science and Engineering 9

Table 4 Continued

119891119888(MPa) 119891spt (MPa) GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

3460 410 minus130 minus063 minus081 minus091 minus090 minus051 minus102

5970 540 minus108 minus084 minus107 minus082 minus101 minus030 minus069

5190 400 minus014 025 003 017 004 076 021

5000 390 minus015 027 006 017 006 077 022

4850 410 minus044 001 minus020 minus011 minus021 049 minus008

4880 370 minus002 042 021 031 020 080 032

4650 380 minus026 022 002 008 minus001 060 007

4520 400 minus054 minus003 minus024 minus019 minus027 030 minus021

4520 330 016 067 046 051 043 100 049

4260 298 032 087 068 068 063 115 064

8160 440 119 093 066 124 086 180 158

4270 300 031 086 066 066 061 121 064

4190 350 minus024 032 012 012 007 060 009

4120 380 minus059 minus001 minus021 minus022 minus026 031 minus026

3970 270 042 102 083 079 076 124 073

6550 550 minus084 minus073 minus097 minus063 minus087 minus021 minus047

3950 370 minus059 001 minus018 minus022 minus025 034 minus031

1710 190 minus025 054 042 009 022 minus011 minus010

1210 110 018 095 085 048 064 000 031

1290 160 minus026 052 041 005 020 minus048 minus013

1690 150 013 093 080 048 061 025 028

2450 250 minus035 042 027 003 012 016 minus016

3 SVM for Predicting 119891spt of Concrete

As mentioned previously many methods have been pro-posed for the prediction of 119891spt from compressive strengthof concrete This study attempts to utilize SVM for theprediction of concrete of 119891spt SVM-I model is devel-oped for 150 times 300mm cylinder 119891spt prediction from 150 times300mm cylinder 119891

119888of concrete and SVM-IImodel is devel-

oped for 100 times 200mm and 150 times 200mm cylinder 119891sptprediction from 100 times 200mm cylinder 119891

119888of concrete The

experimental data which are taken from studies [9 14 15] areused in this study Among 184 experimental data 138 datawere randomly selected as the training set for SVM-I modeland among 168 cases 126 cases were randomly selected asthe training set for SVM-II model the rest are considered astesting data setThe data are normalized before being used inthe model as follows

119883 =(119883119894minus 119883min)

(119883max minus 119883min) (16)

where 119883max and 119883min are the maximum and minimuminput values data respectively

In case of SVM training two types of kernel func-tions were used namely radical basis function (RBF) andpolynomial function in training process 119862 and 120576 and otherkernel-specific parameters have been chosen by a trial-and-error approach The best simulation performances ofSVM are summarized in Table 1 in order to evaluate theabilities of SVM models and other models mean absolutepercentage error (MAPE) roof-mean-squared error (RMES)

and R-square (1198772) were used as the criteria between theexperimental and predicted values which are according tothe equationsas follow

MAPE =1

119899[sum119899

119894=1

1003816100381610038161003816119905119894 minus 1199001198941003816100381610038161003816

sum119899

119894=1119905119894

times 100]

RMSE = radic1

119899

119899

sum119894=1

(119905119894minus 119900119894)

1198772 =(119899sum 119905

119894119900119894minus sum 119905119894sum119900119894)2

(119899sum 1199052119894minus (sum 1199052

119894)) (119899sum 1199002

119894minus (sum 119900

119894)2

)

(17)

where 119905119894is the experimental value 119900

119894is the predicted value

and 119899 is total number of data

4 Results and Discussion

There is no doubt that the splitting tensile strength 119891spt withan increase in the compression strength of concrete butthere is no agreement on the precise form of the relationshipCodes propose different formulas for prediction cylinder 119891sptof concrete from compressive strength In this paper 119891sptresults are investigated for 150 times 300mm cylinder concreteand 100 times 200mm cylinder concrete separately The errors(predicted values subtract measured values) computed bySVM model other models and measured values of split-ting tensile strengths are shown in Tables 2 and 3 for 150times 300mm cylinder concrete and in Tables 4 and 5 for

10 Advances in Materials Science and Engineering

Table 5 Comparison of experimental results to testing results of RBF polynomial of SVM-II and other models

119891119888(MPa) 119891spt (MPa) GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression RBF [9] Polynomial

3600 349 minus060 005 minus013 minus022 minus022 minus017 minus004

3040 308 minus054 017 001 minus016 minus012 minus010 minus001

2160 200 minus004 074 060 033 043 017 034

2120 270 minus077 002 minus012 minus040 minus030 minus044 minus039

2510 260 minus041 036 021 minus003 005 minus009 006

3190 330 minus067 003 minus014 minus028 minus025 minus026 minus012

3780 310 minus010 053 034 028 026 048 045

2920 250 minus004 069 053 034 040 041 049

3540 310 minus025 041 023 013 014 019 032

2860 340 minus098 minus024 minus041 minus059 minus054 minus051 minus046

2950 310 minus062 010 minus006 minus024 minus019 minus023 minus008

3490 350 minus068 minus001 minus019 minus030 minus029 minus023 minus011

3860 338 minus033 029 010 005 003 015 023

5235 399 minus010 028 006 021 008 042 036

3840 380 minus076 minus014 minus033 minus039 minus040 minus023 minus020

5150 420 minus036 003 minus018 minus005 minus017 016 011

4860 400 minus034 011 minus010 000 minus011 011 016

3950 390 minus079 minus019 minus038 minus042 minus045 minus022 minus022

3040 285 minus031 040 024 007 011 013 022

2950 256 minus008 064 048 030 035 031 046

4280 430 minus099 minus044 minus064 minus063 minus068 minus052 minus046

5190 430 minus044 minus005 minus027 minus013 minus026 009 003

4930 420 minus049 minus006 minus027 minus017 minus027 003 000

4330 360 minus026 028 008 010 004 022 028

4360 338 minus002 052 032 034 028 046 052

4990 457 minus083 minus040 minus061 minus050 minus062 minus034 minus033

5740 420 minus001 027 004 026 009 054 039

5850 490 minus065 minus039 minus062 minus038 minus056 minus015 minus024

3140 279 minus019 052 035 020 023 026 036

3490 304 minus022 045 027 016 017 023 035

4160 384 minus060 minus003 minus023 minus024 minus028 minus011 minus005

4680 400 minus044 004 minus017 minus010 minus019 001 009

5170 420 minus035 004 minus017 minus004 minus016 018 012

4330 390 minus056 minus002 minus022 minus020 minus026 minus008 minus002

6080 410 029 050 027 054 033 076 068

6000 430 004 027 004 030 010 060 044

7190 540 minus037 minus040 minus065 minus021 minus051 016 000

7510 540 minus018 minus029 minus055 minus006 minus039 030 020

5787 449 minus027 000 minus023 000 minus018 027 015

3950 340 minus029 031 012 008 005 028 028

Table 6 Error measurement for 119891spt from 150 times 300mm cylinder 119891119888

GEP-I [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF PolynomialRMSE of training set 04603 05995 05228 05027 04512 04978 04819RMSE of testing set 04271 02735 03152 03754 02874 02748 04176MAPE of training set 97554 13647 114913 106213 95596 102434 105454MAPE of testing set 73022 46139 55434 62059 5286 4987 695171198772 of training set 08282 08225 08224 08267 0826 08115 082271198772 of testing set 09416 09486 09486 09471 09476 09422 09327

Advances in Materials Science and Engineering 11

Table 7 Error measurement for 119891spt from 100 times 200mm cylinder 119891119888

GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF PolynomialRMSE of training set 05886 0539 05244 04987 04911 07287 0562RMSE of testing set 0483 03348 03393 02803 03155 03082 02917MAPE of training set 118729 116879 109223 92976 9472 155902 106658MAPE of testing set 108701 71383 7604 62261 7234 69713 628351198772 of training set 08198 08354 08353 083 08337 0837 082531198772 of testing set 08778 08802 08793 088 08801 08815 08823

0

2

4

6

8

0 20 40 60 80 100 120 140

Split

ting

tens

ile st

reng

th (M

Pa)

Sample number

ExperimentalRBFPolynomial

Figure 1 Comparison of experimental results to training results ofSVM-I

100 times 200mm cylinder concrete As seen from these resultsSVM model is sufficiently close to the experimental dataand is able to predict the 119891spt from the 119891

119888of concrete

For convenient comparison purpose the experimental andpredicted results are plotted in Figures 1 2 3 and 4 It canbe seen that the results from SVM method are in goodagreement with the experimental results

The 119877-square errors in the trained SVM are 08115 and08227 for RBF and polynomial function of SVM-I 08370and 08253 for those of SVM-II respectively 119877-square errorsin the tested SVM are 09422 and 09327 for RBF and polyno-mial function of SVM-I 08815 and 08823 for those of SVM-II respectively The whole results obtained from the SVMmodels show a successful performance of the SVM modelsof predicting 150 times 300mm cylinder119891spt of concrete from thecorresponding 150 times 300mm cylinder compressive strength119891119888and 100 times 200mm cylinder 119891spt from the corresponding

100 times 200mm cylinder compressive strength 119891119888of concrete

for each of the training and testing sets It is also clear thatthere are no major difference on performance between theradical basis function and the polynomial function kernelsthat used in this paper However in generally the radical basisfunction kernel exhibits slightly better performance than thepolynomial kernel

The performance of the trained and tested sets is analyzedby computing MAPE RMSE and 1198772 of experimental 119891spt

0

2

4

6

8

0 5 10 15 20 25 30 35 40 45 50

Split

ting

tens

ile st

reng

th (M

Pa)

Sample number

ExperimentalRBFPolynomial

Figure 2 Comparison of experimental results to testing results ofSVM-I

between the SVM model and other methods The MAPERMSE and 1198772 are calculated for these methods Statisticalparameters of the training and testing sets of all methods arepresented in Tables 6 and 7 The model having the smallestMAPE or RMSE and the highest 1198772 can be regarded to bethe best model with the assumption of analysis Generally1198772 of the SVM models are higher than other methodsand the MAPE and RMSE of SVM are smaller than othermodels The analysis shows that the SVM model is betterthan other models and can predict the splitting tensilestrength of concrete well Furthermore SVMmodel has goodgeneralization capacity to avoid over training and can alwaysbe updated to get better results by presenting new trainingexamples as new data become available Thus SVM modelcan be regarded as a very effective method to predict splittingtensile strength of concrete from their compressive strength

5 Conclusion

The splitting tensile strength of concrete estimations fromcompression strength has been obtained so far in the litera-ture either through regression or other methodsThis presentstudy reports a new and influential approach for predictingthe splitting tensile strength using SVM for the first timein the literature The study conducted in this paper shows

12 Advances in Materials Science and Engineering

Polynomial

0

1

2

3

4

5

6

7

8

9

0 20 40 60 80 100 120 140

Split

ting

tens

ile st

reng

hth

(MPa

)

Sample number

ExperimentalRBF

Figure 3 Comparison of experimental results to training results ofSVM-II

Polynomial

0

2

4

6

0 10 20 30 40 50

Split

ting

tens

ile st

reng

th (M

Pa)

Sample number

ExperimentalRBF

Figure 4 Comparison of experimental results to testing results ofSVM-II

the feasibility of using a simple SVM to estimate the splittingtensile strength of concrete After learning from a set ofselected training data involving compressive strengths ofconcrete collected from the previous literature the SVM canbe utilized to predict the splitting tensile strengths of the testdata

In this paper radical basis function and polynomial areadopted for predicting splitting tensile strength from 150 times300mm cylinder and 100times200mm cylinder 119891

119888of concrete

It is found that the splitting tensile strength obtained fromthe SVM is more accurate than those obtained from designcodes and several researchesrsquo empirical equation when acomparison is made on the basis of the experimental dataSince the SVM is largely characterized by the type of its kernel

function it is necessary to choose the appropriate kernel foreach particular application problem in order to guaranteesatisfactory results The results of radial basis function andpolynomial indicate that RBF and polynomial kernel have theability to predict the splitting tensile strength of concrete fromcompression strength with an acceptable degree of accuracy

The statistical parameters of MAPE RMSE and 1198772 showthat the proposed SVM model results have the best accu-racy and can predict splitting tensile strength very close toexperiment results The use of SVM is very advantageousfor the prediction of the splitting tensile strength of concretefrom compression strength because it can perform nonlinearregression efficiently for high-dimensional data sets Further-more its solution is globalThe satisfactory predictions of thesplitting tensile strength of concrete by the model indicatethat SVM is a useful modeling tool for engineers and researchscientists at concrete construction fields

Acknowledgments

This project was granted financial support from the NationalNatural Science Foundation of China (no 50808077 andno 51278188) and the Fundamental Research Funds for theCentral Universities The authors are grateful for support

References

[1] B W Xu and H S Shi ldquoCorrelations among mechanicalproperties of steel fiber reinforced concreterdquo Construction andBuilding Materials vol 23 no 12 pp 3468ndash3474 2009

[2] M F M Zain H B Mahmud A Ilham and M FaizalldquoPrediction of splitting tensile strength of high-performanceconcreterdquoCement andConcrete Research vol 32 no 8 pp 1251ndash1258 2002

[3] F A Oluokun E G Burdette and J H Deatherage ldquoSplittingtensile strength and compressive strength relationship at earlyagesrdquo ACI Materials Journal vol 88 no 2 pp 115ndash121 1991

[4] Y Choi and R L Yuan ldquoExperimental relationship betweensplitting tensile strength and compressive strength of GFRC andPFRCrdquo Cement and Concrete Research vol 35 no 8 pp 1587ndash1591 2005

[5] F D Larrad and Y Malier ldquoEngineering properties of very highperformance concreterdquo in High Performance Concrete fromMaterial to Structure Y Malier Ed pp 85ndash114 EFN SponLondon UK 1992

[6] J K Kim S H Han Y D Park and J H Noh ldquoMaterialproperties of self-flowing concreterdquo Journal of Materials in CivilEngineering vol 10 no 4 pp 244ndash249 1998

[7] J K Kim S H Han and Y C Song ldquoEffect of temperatureand aging on the mechanical properties of concretemdashpart Iexperimental resultsrdquo Cement and Concrete Research vol 32no 7 pp 1087ndash1094 2002

[8] AM Shaaban andHGesund ldquoSplitting tensile strength of steelfiber reinforced concrete cylinders consolidated by rodding orvibratingrdquo ACI Materials Journal vol 90 no 4 pp 366ndash3691993

[9] M Saridemir ldquoEmpirical modeling of splitting tensile strengthfrom cylinder compressive strength of concrete by geneticprogrammingrdquo Expert Systems with Applications vol 38 no 11pp 14257ndash14268 2011

Advances in Materials Science and Engineering 13

[10] U Thissen R van Brakel A P de Weijer W J Melssen and LM C Buydens ldquoUsing support vector machines for time seriespredictionrdquo Chemometrics and Intelligent Laboratory Systemsvol 69 no 1-2 pp 35ndash49 2003

[11] P Samui ldquoPredicted ultimate capacity of laterally loaded pilesin clay using support vector machinerdquo Geomechanics andGeoengineering vol 3 no 2 pp 113ndash120 2008

[12] VN VapnikTheNature of Statistical LearningTheory SpringerNew York NY USA 1995

[13] V N Vapnik S E Golowich and A J Smola ldquoSupport vectormachine for function approximation regression estimationand signal processingrdquoAdvances in Neural Information Process-ing Systems vol 9 pp 281ndash287 1996

[14] M Khanzadi and A Behnood ldquoMechanical properties ofhigh-strength concrete incorporating copper slag as coarseaggregaterdquo Construction and Building Materials vol 23 no 6pp 2183ndash2188 2009

[15] C K Y Leung and J Pan ldquoEffect of concrete composition onFRPconcrete bond capacityrdquo in Proceedings of the InternationalSymposium on Bond Behaviour of FRP in Structures (BBFS rsquo05)pp 69ndash76 2005

Submit your manuscripts athttpwwwhindawicom

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Biomaterials

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NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Advances in

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MaterialsJournal of

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Nano

materials

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Journal ofNanomaterials

4 Advances in Materials Science and Engineering

Table 2 Continued

119891119888(MPa) 119891spt (MPa) GEP-I [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

1820 190 minus010 062 049 018 012 minus024 068

2870 270 minus020 046 030 011 minus001 minus020 023

2390 230 minus011 058 044 019 010 minus003 038

4370 330 011 060 040 042 022 044 021

2460 230 minus007 063 048 024 014 minus001 045

4300 350 minus013 037 017 018 minus002 000 000

1420 180 minus028 042 031 minus004 minus008 minus029 063

3670 320 minus021 037 019 011 minus005 minus048 001

2270 170 041 111 097 071 062 035 095

3530 240 051 111 093 083 067 000 074

1360 130 018 088 077 041 038 079 112

2860 170 079 146 129 111 099 140 114

2660 240 minus004 064 049 027 017 minus001 042

4350 290 050 099 079 081 061 052 057

3090 240 024 088 071 055 042 048 051

4990 380 minus003 037 016 027 003 039 004

3220 300 minus028 035 018 004 minus010 minus009 001

5780 350 072 099 076 098 070 033 081

2680 240 minus002 065 050 029 018 047 044

3650 290 008 066 048 040 024 027 026

6560 440 026 038 014 048 015 014 036

3030 240 020 085 068 052 039 137 054

5280 380 014 049 027 042 017 minus039 023

4060 350 minus027 026 007 004 minus014 015 minus013

5230 420 minus029 007 minus015 000 minus026 010 minus024

4120 320 006 059 039 038 019 065 022

7280 510 minus005 minus007 minus032 013 minus024 003 minus005

4120 320 006 059 039 038 019 065 022

7280 510 minus005 minus007 minus032 013 minus024 003 minus005

7280 510 minus005 minus007 minus032 013 minus024 003 minus005

4180 330 000 051 032 031 012 042 010

6840 410 071 078 053 092 057 038 076

4230 290 043 094 074 074 054 000 057

5910 380 050 074 051 075 046 025 060

4130 370 minus043 009 minus010 minus012 minus031 minus019 minus027

5970 410 023 046 023 048 019 014 030

4230 320 013 064 044 044 024 071 027

7510 510 008 001 minus025 024 minus014 067 minus005

4850 370 minus001 041 020 029 006 008 009

8190 480 075 054 027 086 044 094 053

4130 330 minus003 049 030 028 009 021 013

6710 440 034 043 019 055 022 068 036

4200 320 011 062 043 042 023 077 020

7010 430 061 064 039 080 045 065 058

4480 340 008 055 035 038 017 036 021

6540 400 065 077 053 087 054 067 070

3870 350 minus039 017 minus002 minus007 minus024 040 minus021

5640 430 minus016 013 minus009 011 minus016 minus025 minus014

4240 320 013 064 045 045 025 041 025

Advances in Materials Science and Engineering 5

Table 2 Continued

119891119888(MPa) 119891spt (MPa) GEP-I [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

6770 480 minus003 005 minus019 018 minus016 027 minus003

5070 360 022 060 039 051 026 048 026

8110 480 070 051 024 082 041 062 052

4200 320 011 062 043 042 023 077 020

4890 360 011 053 032 041 018 061 021

8530 580 minus007 minus035 minus063 001 minus042 032 minus037

5230 420 minus029 007 minus015 000 minus026 010 minus024

8510 580 minus008 minus036 minus063 000 minus043 minus034 minus042

4970 420 minus044 minus004 minus025 minus014 minus038 minus049 minus036

7120 530 minus033 minus032 minus057 minus015 minus050 minus052 minus033

5190 380 009 045 023 037 012 036 017

8770 610 minus024 minus057 minus086 minus018 minus063 minus005 minus066

5190 380 009 045 023 037 012 036 017

8770 610 minus024 minus057 minus086 minus018 minus063 minus005 minus066

5190 380 009 045 023 037 012 036 017

8770 610 minus024 minus057 minus086 minus018 minus063 minus005 minus066

5380 410 minus010 023 001 018 minus009 minus010 minus006

8530 510 063 035 007 071 028 102 033

5870 440 minus013 012 minus011 013 minus016 minus026 minus009

8710 560 022 minus009 minus037 029 minus015 minus009 minus020

4930 450 minus076 minus036 minus057 minus047 minus070 minus135 minus069

6620 510 minus041 minus030 minus054 minus019 minus052 minus023 minus040

5310 380 016 050 028 044 018 050 033

8430 560 007 minus018 minus046 017 minus026 010 minus015

8670 603 003 minus054 minus080 minus016 minus060 002 minus054

5610 430 minus017 012 minus011 010 minus018 minus058 minus017

5210 380 010 046 024 038 013 011 016

7850 500 036 023 minus004 050 010 087 029

4990 410 minus033 007 minus014 minus003 minus027 009 minus026

8750 510 075 042 014 081 037 048 029

5510 440 minus033 minus002 minus024 minus006 minus033 minus011 minus023

7540 400 122 112 086 135 097 000 105

2467 274 minus050 019 004 minus020 minus029 minus033 002

2301 279 minus066 004 minus010 minus036 minus045 000 minus009

3203 303 minus032 031 014 000 minus014 minus064 minus003

2878 293 minus043 024 007 minus011 minus023 minus043 000

3360 203 077 139 122 109 095 130 106

2330 304 minus089 minus019 minus034 minus059 minus068 minus125 minus031

GEP-I model is developed by Saridemir [9] based on gene expression programming Regression is regression-based formulation results by Saridemir [9]

Subject to119899

sum119894=1

120572119894=119899

sum119894=1

120572lowast119894 0 le 120572lowast

119894le 119888 0 le 120572

119894le 119888 (11)

where the 120572119894 120572lowast119894are called Lagrangian multipliers When

the Lagrangian multipliers (120572119894 120572lowast119894) are equal to zero shows

that the training object is irrelevant to the final solutionotherwise the training samples with nonzero Lagrangianmultipliers are called support vectors

When linear regression is not appropriate then the inputdata have to bemapped into a high-dimensional feature space

through some nonlinear mapping A nonlinear transforma-tion 120601(119909) replaces the input 119909 in (7) and the regressionfunction can be written as

119891 (119909) =nsvsum119894=1

(120572119894minus 120572lowast119894) 119896 (119909119894 119909119895) + 119887 (12)

where nsv is the number of support vectors and 119896(119909119894 119909119895) =

120601(119909119894) sdot 120601(119909

119895) 119896(119909

119894 119909119895) is a kernel function Any function sat-

isfying Mercersquos condition can be used as the kernel function[12 13] Some popular kernel functions are

6 Advances in Materials Science and Engineering

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

119891119888(MPa) 119891spt (MPa) GEP-I [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

4480 380 minus031 015 minus005 minus002 minus023 minus001 minus033

5620 440 minus025 002 minus020 000 minus027 minus041 minus020

5340 360 039 071 049 065 039 033 039

5670 460 minus042 minus016 minus038 minus017 minus045 minus009 minus036

6450 480 minus018 minus006 minus030 002 minus030 minus018 minus012

6650 470 003 011 minus013 022 minus011 010 001

6540 430 037 047 023 057 024 021 043

7380 490 023 017 minus009 038 001 022 014

7400 530 minus016 minus022 minus048 minus001 minus039 minus026 minus015

7250 500 006 002 minus023 022 minus015 024 minus006

7460 490 028 020 minus006 042 004 006 015

7790 550 minus014 minus029 minus056 minus003 minus042 minus036 minus019

7910 520 022 005 minus022 033 minus007 001 015

8420 570 000 minus029 minus056 006 minus037 minus034 minus028

8330 590 minus025 minus052 minus079 minus018 minus060 minus060 minus039

8010 530 015 minus002 minus029 027 minus013 minus022 010

8650 550 032 minus001 minus029 037 minus007 minus027 012

10200 550 115 046 016 105 053 037 034

10100 650 009 minus057 minus087 001 minus051 minus045 minus060

11100 620 092 002 minus030 073 016 040 minus017

9450 580 045 minus006 minus036 042 minus006 008 minus001

11800 620 128 021 minus012 102 041 064 minus014

1900 230 minus043 027 014 minus016 minus023 minus004 minus100

2000 240 minus046 024 010 minus019 minus026 minus013 minus096

2860 290 minus040 026 009 minus009 minus021 011 minus067

2890 300 minus048 017 001 minus017 minus030 008 minus082

3610 330 minus033 024 006 minus002 minus019 minus011 minus048

4450 370 minus023 024 004 007 minus014 007 minus026

2300 250 minus036 033 019 minus007 minus016 minus037 minus082

3300 350 minus072 minus011 minus028 minus041 minus056 minus043 minus102

5800 420 005 029 006 030 001 minus025 020

6900 450 037 040 015 055 020 012 044

3600 330 minus034 024 006 minus003 minus019 minus004 minus045

4280 390 minus053 minus004 minus024 minus023 minus043 minus019 minus063

5930 400 033 054 031 056 027 029 040

6330 450 005 019 minus004 026 minus005 minus018 020

6770 470 010 015 minus009 028 minus006 026 007

7700 510 021 008 minus019 033 minus006 minus005 001

4990 430 minus051 minus013 minus034 minus023 minus047 minus032 minus045

9670 570 063 010 minus019 062 012 005 001

6170 460 minus014 003 minus020 008 minus022 minus036 minus008

7320 510 000 minus005 minus031 015 minus022 017 minus003

8650 550 032 minus001 minus029 037 minus007 minus027 012

9230 590 023 minus023 minus052 023 minus025 010 minus019

(1) polynomial kernel function

119870(119909 119910) = [(119909 + 119910) + 1]119889

119889 = 1 2 119899 (13)

(2) radial basis function (RBF)

119870(119909 119910) = exp minus1205741003817100381710038171003817119909 minus 11991010038171003817100381710038172

120574 ge 0 (14)

(3) sigmoid kernel function

119870(119909 119910) = tan [120593 (119909 sdot 119910) + 120579] (15)

where 120572119894 120572lowast119894satisfy 120572

119894120572lowast119894= 0 120572

119894ge 0 and 120572lowast

119894ge 0

Advances in Materials Science and Engineering 7

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

119891119888(MPa) 119891spt (MPa) GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

7790 545 minus008 minus024 minus051 002 minus033 048 032

5710 313 103 133 110 132 115 174 145

7160 490 011 009 minus016 027 minus002 081 051

4410 391 minus052 001 minus019 minus017 minus023 028 minus017

6480 457 004 018 minus006 027 003 073 042

4260 366 minus036 019 000 000 minus005 047 minus004

3040 279 minus026 046 030 013 017 046 000

4680 389 minus034 015 minus006 001 minus008 049 004

8950 591 011 minus033 minus061 009 minus036 073 041

6690 455 019 028 003 039 014 096 059

7610 547 minus021 minus032 minus058 minus008 minus042 041 022

5360 437 minus041 minus005 minus027 minus011 minus025 047 minus002

6480 507 minus046 minus032 minus056 minus023 minus047 023 minus008

4580 402 minus053 minus003 minus023 minus018 minus026 040 minus016

3760 316 minus017 046 027 021 019 063 014

5310 408 minus015 022 000 016 002 078 024

3490 350 minus068 minus001 minus019 minus030 minus029 012 minus040

5390 540 minus142 minus107 minus129 minus112 minus127 minus062 minus100

4650 380 minus027 022 002 008 minus001 060 007

1380 170 minus029 049 038 003 017 minus042 minus017

6710 480 minus005 003 minus021 015 minus010 064 035

2030 340 minus153 minus074 minus088 minus117 minus106 minus126 minus136

2240 210 minus009 069 055 028 038 038 010

6140 430 012 032 009 037 016 079 051

3310 300 minus029 039 022 009 011 053 minus003

7900 560 minus017 minus036 minus062 minus008 minus044 046 021

4360 360 minus024 030 010 012 006 059 010

2560 242 minus020 057 041 019 026 031 002

2620 262 minus036 040 025 003 010 023 minus015

1690 188 minus025 055 042 010 023 minus013 minus010

1490 171 minus022 057 045 011 025 minus038 minus008

2730 281 minus048 027 012 minus009 minus003 019 minus026

2840 287 minus047 027 011 minus008 minus002 022 minus023

1690 191 minus028 052 039 007 020 minus016 minus013

1690 211 minus048 032 019 minus013 000 minus036 minus033

3090 292 minus035 036 019 003 007 031 minus010

3230 308 minus043 027 010 minus004 minus001 036 minus017

1960 222 minus040 039 026 minus004 008 minus010 minus022

1910 208 minus030 050 037 006 018 minus011 minus010

3530 324 minus040 027 009 minus001 minus001 038 minus011

3840 336 minus032 030 011 005 004 057 minus003

2250 247 minus045 033 019 minus008 002 minus005 minus027

2340 257 minus049 028 014 minus012 minus002 minus008 minus029

3100 290 minus033 038 022 006 010 035 minus006

1700 230 minus066 013 001 minus032 minus018 minus060 minus051

4700 400 minus044 004 minus016 minus009 minus018 044 minus009

3700 360 minus065 minus001 minus019 minus027 minus028 021 minus035

4200 400 minus074 minus018 minus037 minus038 minus042 016 minus039

5400 460 minus062 minus026 minus048 minus031 minus046 021 minus024

8 Advances in Materials Science and Engineering

Table 4 Continued

119891119888(MPa) 119891spt (MPa) GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

4200 440 minus114 minus058 minus077 minus078 minus082 minus024 minus079

6200 520 minus075 minus055 minus079 minus050 minus072 003 minus035

6700 425 049 058 033 070 044 115 091

7250 510 minus004 minus008 minus033 012 minus019 056 038

7450 520 minus003 minus011 minus037 011 minus021 059 038

7950 575 minus029 minus049 minus076 minus020 minus057 030 011

7800 515 022 006 minus020 033 minus003 083 060

8500 575 002 minus031 minus059 005 minus036 069 044

8600 480 103 067 039 105 062 167 140

9050 440 168 121 093 165 119 232 204

5480 334 069 103 081 099 083 161 106

3290 285 minus016 053 036 023 025 068 011

3450 292 minus013 055 037 026 027 066 013

5360 348 048 084 062 078 064 136 087

7410 526 minus011 minus018 minus044 003 minus029 058 030

6770 481 minus003 004 minus020 017 minus009 068 037

8420 657 minus085 minus116 minus143 minus081 minus121 minus025 minus046

7530 535 minus013 minus023 minus049 000 minus033 048 028

10230 729 minus055 minus132 minus163 minus073 minus129 052 minus031

4010 330 minus016 044 025 021 018 070 015

4050 340 minus023 035 016 014 010 057 007

9550 570 066 007 minus023 057 006 153 096

3520 317 minus033 033 015 005 006 048 minus005

5750 369 051 078 056 078 060 129 086

3860 338 minus033 029 010 005 003 052 minus004

6150 448 minus005 015 minus009 019 minus002 070 032

4740 376 minus017 030 010 017 008 067 018

4710 368 minus011 037 016 023 014 086 023

4470 326 017 068 048 052 045 109 046

5240 399 minus010 028 006 021 008 079 027

5790 449 minus027 000 minus023 000 minus018 061 012

3950 404 minus093 minus033 minus052 minus056 minus059 000 minus065

3670 403 minus109 minus046 minus064 minus072 minus072 minus029 minus082

3800 365 minus063 minus001 minus020 minus026 minus028 020 minus035

2420 220 minus007 070 055 031 040 038 014

1740 180 minus013 066 054 021 034 minus006 002

2930 280 minus033 039 023 005 010 027 minus009

2700 255 minus023 052 036 015 022 033 minus002

3260 321 minus053 016 minus001 minus015 minus012 026 minus026

4630 520 minus167 minus119 minus139 minus133 minus141 minus076 minus137

3580 430 minus142 minus077 minus095 minus104 minus104 minus057 minus111

4310 418 minus085 minus031 minus050 minus049 minus055 minus004 minus051

3520 317 minus033 033 015 005 006 048 minus005

5750 369 051 078 056 078 060 129 086

4280 490 minus159 minus104 minus124 minus123 minus128 minus076 minus126

6150 448 minus005 015 minus009 019 minus002 070 032

4740 376 minus017 030 010 017 008 067 018

4713 368 minus010 037 016 023 014 086 023

4473 326 017 069 049 052 045 104 050

Advances in Materials Science and Engineering 9

Table 4 Continued

119891119888(MPa) 119891spt (MPa) GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

3460 410 minus130 minus063 minus081 minus091 minus090 minus051 minus102

5970 540 minus108 minus084 minus107 minus082 minus101 minus030 minus069

5190 400 minus014 025 003 017 004 076 021

5000 390 minus015 027 006 017 006 077 022

4850 410 minus044 001 minus020 minus011 minus021 049 minus008

4880 370 minus002 042 021 031 020 080 032

4650 380 minus026 022 002 008 minus001 060 007

4520 400 minus054 minus003 minus024 minus019 minus027 030 minus021

4520 330 016 067 046 051 043 100 049

4260 298 032 087 068 068 063 115 064

8160 440 119 093 066 124 086 180 158

4270 300 031 086 066 066 061 121 064

4190 350 minus024 032 012 012 007 060 009

4120 380 minus059 minus001 minus021 minus022 minus026 031 minus026

3970 270 042 102 083 079 076 124 073

6550 550 minus084 minus073 minus097 minus063 minus087 minus021 minus047

3950 370 minus059 001 minus018 minus022 minus025 034 minus031

1710 190 minus025 054 042 009 022 minus011 minus010

1210 110 018 095 085 048 064 000 031

1290 160 minus026 052 041 005 020 minus048 minus013

1690 150 013 093 080 048 061 025 028

2450 250 minus035 042 027 003 012 016 minus016

3 SVM for Predicting 119891spt of Concrete

As mentioned previously many methods have been pro-posed for the prediction of 119891spt from compressive strengthof concrete This study attempts to utilize SVM for theprediction of concrete of 119891spt SVM-I model is devel-oped for 150 times 300mm cylinder 119891spt prediction from 150 times300mm cylinder 119891

119888of concrete and SVM-IImodel is devel-

oped for 100 times 200mm and 150 times 200mm cylinder 119891sptprediction from 100 times 200mm cylinder 119891

119888of concrete The

experimental data which are taken from studies [9 14 15] areused in this study Among 184 experimental data 138 datawere randomly selected as the training set for SVM-I modeland among 168 cases 126 cases were randomly selected asthe training set for SVM-II model the rest are considered astesting data setThe data are normalized before being used inthe model as follows

119883 =(119883119894minus 119883min)

(119883max minus 119883min) (16)

where 119883max and 119883min are the maximum and minimuminput values data respectively

In case of SVM training two types of kernel func-tions were used namely radical basis function (RBF) andpolynomial function in training process 119862 and 120576 and otherkernel-specific parameters have been chosen by a trial-and-error approach The best simulation performances ofSVM are summarized in Table 1 in order to evaluate theabilities of SVM models and other models mean absolutepercentage error (MAPE) roof-mean-squared error (RMES)

and R-square (1198772) were used as the criteria between theexperimental and predicted values which are according tothe equationsas follow

MAPE =1

119899[sum119899

119894=1

1003816100381610038161003816119905119894 minus 1199001198941003816100381610038161003816

sum119899

119894=1119905119894

times 100]

RMSE = radic1

119899

119899

sum119894=1

(119905119894minus 119900119894)

1198772 =(119899sum 119905

119894119900119894minus sum 119905119894sum119900119894)2

(119899sum 1199052119894minus (sum 1199052

119894)) (119899sum 1199002

119894minus (sum 119900

119894)2

)

(17)

where 119905119894is the experimental value 119900

119894is the predicted value

and 119899 is total number of data

4 Results and Discussion

There is no doubt that the splitting tensile strength 119891spt withan increase in the compression strength of concrete butthere is no agreement on the precise form of the relationshipCodes propose different formulas for prediction cylinder 119891sptof concrete from compressive strength In this paper 119891sptresults are investigated for 150 times 300mm cylinder concreteand 100 times 200mm cylinder concrete separately The errors(predicted values subtract measured values) computed bySVM model other models and measured values of split-ting tensile strengths are shown in Tables 2 and 3 for 150times 300mm cylinder concrete and in Tables 4 and 5 for

10 Advances in Materials Science and Engineering

Table 5 Comparison of experimental results to testing results of RBF polynomial of SVM-II and other models

119891119888(MPa) 119891spt (MPa) GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression RBF [9] Polynomial

3600 349 minus060 005 minus013 minus022 minus022 minus017 minus004

3040 308 minus054 017 001 minus016 minus012 minus010 minus001

2160 200 minus004 074 060 033 043 017 034

2120 270 minus077 002 minus012 minus040 minus030 minus044 minus039

2510 260 minus041 036 021 minus003 005 minus009 006

3190 330 minus067 003 minus014 minus028 minus025 minus026 minus012

3780 310 minus010 053 034 028 026 048 045

2920 250 minus004 069 053 034 040 041 049

3540 310 minus025 041 023 013 014 019 032

2860 340 minus098 minus024 minus041 minus059 minus054 minus051 minus046

2950 310 minus062 010 minus006 minus024 minus019 minus023 minus008

3490 350 minus068 minus001 minus019 minus030 minus029 minus023 minus011

3860 338 minus033 029 010 005 003 015 023

5235 399 minus010 028 006 021 008 042 036

3840 380 minus076 minus014 minus033 minus039 minus040 minus023 minus020

5150 420 minus036 003 minus018 minus005 minus017 016 011

4860 400 minus034 011 minus010 000 minus011 011 016

3950 390 minus079 minus019 minus038 minus042 minus045 minus022 minus022

3040 285 minus031 040 024 007 011 013 022

2950 256 minus008 064 048 030 035 031 046

4280 430 minus099 minus044 minus064 minus063 minus068 minus052 minus046

5190 430 minus044 minus005 minus027 minus013 minus026 009 003

4930 420 minus049 minus006 minus027 minus017 minus027 003 000

4330 360 minus026 028 008 010 004 022 028

4360 338 minus002 052 032 034 028 046 052

4990 457 minus083 minus040 minus061 minus050 minus062 minus034 minus033

5740 420 minus001 027 004 026 009 054 039

5850 490 minus065 minus039 minus062 minus038 minus056 minus015 minus024

3140 279 minus019 052 035 020 023 026 036

3490 304 minus022 045 027 016 017 023 035

4160 384 minus060 minus003 minus023 minus024 minus028 minus011 minus005

4680 400 minus044 004 minus017 minus010 minus019 001 009

5170 420 minus035 004 minus017 minus004 minus016 018 012

4330 390 minus056 minus002 minus022 minus020 minus026 minus008 minus002

6080 410 029 050 027 054 033 076 068

6000 430 004 027 004 030 010 060 044

7190 540 minus037 minus040 minus065 minus021 minus051 016 000

7510 540 minus018 minus029 minus055 minus006 minus039 030 020

5787 449 minus027 000 minus023 000 minus018 027 015

3950 340 minus029 031 012 008 005 028 028

Table 6 Error measurement for 119891spt from 150 times 300mm cylinder 119891119888

GEP-I [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF PolynomialRMSE of training set 04603 05995 05228 05027 04512 04978 04819RMSE of testing set 04271 02735 03152 03754 02874 02748 04176MAPE of training set 97554 13647 114913 106213 95596 102434 105454MAPE of testing set 73022 46139 55434 62059 5286 4987 695171198772 of training set 08282 08225 08224 08267 0826 08115 082271198772 of testing set 09416 09486 09486 09471 09476 09422 09327

Advances in Materials Science and Engineering 11

Table 7 Error measurement for 119891spt from 100 times 200mm cylinder 119891119888

GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF PolynomialRMSE of training set 05886 0539 05244 04987 04911 07287 0562RMSE of testing set 0483 03348 03393 02803 03155 03082 02917MAPE of training set 118729 116879 109223 92976 9472 155902 106658MAPE of testing set 108701 71383 7604 62261 7234 69713 628351198772 of training set 08198 08354 08353 083 08337 0837 082531198772 of testing set 08778 08802 08793 088 08801 08815 08823

0

2

4

6

8

0 20 40 60 80 100 120 140

Split

ting

tens

ile st

reng

th (M

Pa)

Sample number

ExperimentalRBFPolynomial

Figure 1 Comparison of experimental results to training results ofSVM-I

100 times 200mm cylinder concrete As seen from these resultsSVM model is sufficiently close to the experimental dataand is able to predict the 119891spt from the 119891

119888of concrete

For convenient comparison purpose the experimental andpredicted results are plotted in Figures 1 2 3 and 4 It canbe seen that the results from SVM method are in goodagreement with the experimental results

The 119877-square errors in the trained SVM are 08115 and08227 for RBF and polynomial function of SVM-I 08370and 08253 for those of SVM-II respectively 119877-square errorsin the tested SVM are 09422 and 09327 for RBF and polyno-mial function of SVM-I 08815 and 08823 for those of SVM-II respectively The whole results obtained from the SVMmodels show a successful performance of the SVM modelsof predicting 150 times 300mm cylinder119891spt of concrete from thecorresponding 150 times 300mm cylinder compressive strength119891119888and 100 times 200mm cylinder 119891spt from the corresponding

100 times 200mm cylinder compressive strength 119891119888of concrete

for each of the training and testing sets It is also clear thatthere are no major difference on performance between theradical basis function and the polynomial function kernelsthat used in this paper However in generally the radical basisfunction kernel exhibits slightly better performance than thepolynomial kernel

The performance of the trained and tested sets is analyzedby computing MAPE RMSE and 1198772 of experimental 119891spt

0

2

4

6

8

0 5 10 15 20 25 30 35 40 45 50

Split

ting

tens

ile st

reng

th (M

Pa)

Sample number

ExperimentalRBFPolynomial

Figure 2 Comparison of experimental results to testing results ofSVM-I

between the SVM model and other methods The MAPERMSE and 1198772 are calculated for these methods Statisticalparameters of the training and testing sets of all methods arepresented in Tables 6 and 7 The model having the smallestMAPE or RMSE and the highest 1198772 can be regarded to bethe best model with the assumption of analysis Generally1198772 of the SVM models are higher than other methodsand the MAPE and RMSE of SVM are smaller than othermodels The analysis shows that the SVM model is betterthan other models and can predict the splitting tensilestrength of concrete well Furthermore SVMmodel has goodgeneralization capacity to avoid over training and can alwaysbe updated to get better results by presenting new trainingexamples as new data become available Thus SVM modelcan be regarded as a very effective method to predict splittingtensile strength of concrete from their compressive strength

5 Conclusion

The splitting tensile strength of concrete estimations fromcompression strength has been obtained so far in the litera-ture either through regression or other methodsThis presentstudy reports a new and influential approach for predictingthe splitting tensile strength using SVM for the first timein the literature The study conducted in this paper shows

12 Advances in Materials Science and Engineering

Polynomial

0

1

2

3

4

5

6

7

8

9

0 20 40 60 80 100 120 140

Split

ting

tens

ile st

reng

hth

(MPa

)

Sample number

ExperimentalRBF

Figure 3 Comparison of experimental results to training results ofSVM-II

Polynomial

0

2

4

6

0 10 20 30 40 50

Split

ting

tens

ile st

reng

th (M

Pa)

Sample number

ExperimentalRBF

Figure 4 Comparison of experimental results to testing results ofSVM-II

the feasibility of using a simple SVM to estimate the splittingtensile strength of concrete After learning from a set ofselected training data involving compressive strengths ofconcrete collected from the previous literature the SVM canbe utilized to predict the splitting tensile strengths of the testdata

In this paper radical basis function and polynomial areadopted for predicting splitting tensile strength from 150 times300mm cylinder and 100times200mm cylinder 119891

119888of concrete

It is found that the splitting tensile strength obtained fromthe SVM is more accurate than those obtained from designcodes and several researchesrsquo empirical equation when acomparison is made on the basis of the experimental dataSince the SVM is largely characterized by the type of its kernel

function it is necessary to choose the appropriate kernel foreach particular application problem in order to guaranteesatisfactory results The results of radial basis function andpolynomial indicate that RBF and polynomial kernel have theability to predict the splitting tensile strength of concrete fromcompression strength with an acceptable degree of accuracy

The statistical parameters of MAPE RMSE and 1198772 showthat the proposed SVM model results have the best accu-racy and can predict splitting tensile strength very close toexperiment results The use of SVM is very advantageousfor the prediction of the splitting tensile strength of concretefrom compression strength because it can perform nonlinearregression efficiently for high-dimensional data sets Further-more its solution is globalThe satisfactory predictions of thesplitting tensile strength of concrete by the model indicatethat SVM is a useful modeling tool for engineers and researchscientists at concrete construction fields

Acknowledgments

This project was granted financial support from the NationalNatural Science Foundation of China (no 50808077 andno 51278188) and the Fundamental Research Funds for theCentral Universities The authors are grateful for support

References

[1] B W Xu and H S Shi ldquoCorrelations among mechanicalproperties of steel fiber reinforced concreterdquo Construction andBuilding Materials vol 23 no 12 pp 3468ndash3474 2009

[2] M F M Zain H B Mahmud A Ilham and M FaizalldquoPrediction of splitting tensile strength of high-performanceconcreterdquoCement andConcrete Research vol 32 no 8 pp 1251ndash1258 2002

[3] F A Oluokun E G Burdette and J H Deatherage ldquoSplittingtensile strength and compressive strength relationship at earlyagesrdquo ACI Materials Journal vol 88 no 2 pp 115ndash121 1991

[4] Y Choi and R L Yuan ldquoExperimental relationship betweensplitting tensile strength and compressive strength of GFRC andPFRCrdquo Cement and Concrete Research vol 35 no 8 pp 1587ndash1591 2005

[5] F D Larrad and Y Malier ldquoEngineering properties of very highperformance concreterdquo in High Performance Concrete fromMaterial to Structure Y Malier Ed pp 85ndash114 EFN SponLondon UK 1992

[6] J K Kim S H Han Y D Park and J H Noh ldquoMaterialproperties of self-flowing concreterdquo Journal of Materials in CivilEngineering vol 10 no 4 pp 244ndash249 1998

[7] J K Kim S H Han and Y C Song ldquoEffect of temperatureand aging on the mechanical properties of concretemdashpart Iexperimental resultsrdquo Cement and Concrete Research vol 32no 7 pp 1087ndash1094 2002

[8] AM Shaaban andHGesund ldquoSplitting tensile strength of steelfiber reinforced concrete cylinders consolidated by rodding orvibratingrdquo ACI Materials Journal vol 90 no 4 pp 366ndash3691993

[9] M Saridemir ldquoEmpirical modeling of splitting tensile strengthfrom cylinder compressive strength of concrete by geneticprogrammingrdquo Expert Systems with Applications vol 38 no 11pp 14257ndash14268 2011

Advances in Materials Science and Engineering 13

[10] U Thissen R van Brakel A P de Weijer W J Melssen and LM C Buydens ldquoUsing support vector machines for time seriespredictionrdquo Chemometrics and Intelligent Laboratory Systemsvol 69 no 1-2 pp 35ndash49 2003

[11] P Samui ldquoPredicted ultimate capacity of laterally loaded pilesin clay using support vector machinerdquo Geomechanics andGeoengineering vol 3 no 2 pp 113ndash120 2008

[12] VN VapnikTheNature of Statistical LearningTheory SpringerNew York NY USA 1995

[13] V N Vapnik S E Golowich and A J Smola ldquoSupport vectormachine for function approximation regression estimationand signal processingrdquoAdvances in Neural Information Process-ing Systems vol 9 pp 281ndash287 1996

[14] M Khanzadi and A Behnood ldquoMechanical properties ofhigh-strength concrete incorporating copper slag as coarseaggregaterdquo Construction and Building Materials vol 23 no 6pp 2183ndash2188 2009

[15] C K Y Leung and J Pan ldquoEffect of concrete composition onFRPconcrete bond capacityrdquo in Proceedings of the InternationalSymposium on Bond Behaviour of FRP in Structures (BBFS rsquo05)pp 69ndash76 2005

Submit your manuscripts athttpwwwhindawicom

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Advances in

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MaterialsJournal of

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Nano

materials

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Journal ofNanomaterials

Advances in Materials Science and Engineering 5

Table 2 Continued

119891119888(MPa) 119891spt (MPa) GEP-I [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

6770 480 minus003 005 minus019 018 minus016 027 minus003

5070 360 022 060 039 051 026 048 026

8110 480 070 051 024 082 041 062 052

4200 320 011 062 043 042 023 077 020

4890 360 011 053 032 041 018 061 021

8530 580 minus007 minus035 minus063 001 minus042 032 minus037

5230 420 minus029 007 minus015 000 minus026 010 minus024

8510 580 minus008 minus036 minus063 000 minus043 minus034 minus042

4970 420 minus044 minus004 minus025 minus014 minus038 minus049 minus036

7120 530 minus033 minus032 minus057 minus015 minus050 minus052 minus033

5190 380 009 045 023 037 012 036 017

8770 610 minus024 minus057 minus086 minus018 minus063 minus005 minus066

5190 380 009 045 023 037 012 036 017

8770 610 minus024 minus057 minus086 minus018 minus063 minus005 minus066

5190 380 009 045 023 037 012 036 017

8770 610 minus024 minus057 minus086 minus018 minus063 minus005 minus066

5380 410 minus010 023 001 018 minus009 minus010 minus006

8530 510 063 035 007 071 028 102 033

5870 440 minus013 012 minus011 013 minus016 minus026 minus009

8710 560 022 minus009 minus037 029 minus015 minus009 minus020

4930 450 minus076 minus036 minus057 minus047 minus070 minus135 minus069

6620 510 minus041 minus030 minus054 minus019 minus052 minus023 minus040

5310 380 016 050 028 044 018 050 033

8430 560 007 minus018 minus046 017 minus026 010 minus015

8670 603 003 minus054 minus080 minus016 minus060 002 minus054

5610 430 minus017 012 minus011 010 minus018 minus058 minus017

5210 380 010 046 024 038 013 011 016

7850 500 036 023 minus004 050 010 087 029

4990 410 minus033 007 minus014 minus003 minus027 009 minus026

8750 510 075 042 014 081 037 048 029

5510 440 minus033 minus002 minus024 minus006 minus033 minus011 minus023

7540 400 122 112 086 135 097 000 105

2467 274 minus050 019 004 minus020 minus029 minus033 002

2301 279 minus066 004 minus010 minus036 minus045 000 minus009

3203 303 minus032 031 014 000 minus014 minus064 minus003

2878 293 minus043 024 007 minus011 minus023 minus043 000

3360 203 077 139 122 109 095 130 106

2330 304 minus089 minus019 minus034 minus059 minus068 minus125 minus031

GEP-I model is developed by Saridemir [9] based on gene expression programming Regression is regression-based formulation results by Saridemir [9]

Subject to119899

sum119894=1

120572119894=119899

sum119894=1

120572lowast119894 0 le 120572lowast

119894le 119888 0 le 120572

119894le 119888 (11)

where the 120572119894 120572lowast119894are called Lagrangian multipliers When

the Lagrangian multipliers (120572119894 120572lowast119894) are equal to zero shows

that the training object is irrelevant to the final solutionotherwise the training samples with nonzero Lagrangianmultipliers are called support vectors

When linear regression is not appropriate then the inputdata have to bemapped into a high-dimensional feature space

through some nonlinear mapping A nonlinear transforma-tion 120601(119909) replaces the input 119909 in (7) and the regressionfunction can be written as

119891 (119909) =nsvsum119894=1

(120572119894minus 120572lowast119894) 119896 (119909119894 119909119895) + 119887 (12)

where nsv is the number of support vectors and 119896(119909119894 119909119895) =

120601(119909119894) sdot 120601(119909

119895) 119896(119909

119894 119909119895) is a kernel function Any function sat-

isfying Mercersquos condition can be used as the kernel function[12 13] Some popular kernel functions are

6 Advances in Materials Science and Engineering

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

119891119888(MPa) 119891spt (MPa) GEP-I [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

4480 380 minus031 015 minus005 minus002 minus023 minus001 minus033

5620 440 minus025 002 minus020 000 minus027 minus041 minus020

5340 360 039 071 049 065 039 033 039

5670 460 minus042 minus016 minus038 minus017 minus045 minus009 minus036

6450 480 minus018 minus006 minus030 002 minus030 minus018 minus012

6650 470 003 011 minus013 022 minus011 010 001

6540 430 037 047 023 057 024 021 043

7380 490 023 017 minus009 038 001 022 014

7400 530 minus016 minus022 minus048 minus001 minus039 minus026 minus015

7250 500 006 002 minus023 022 minus015 024 minus006

7460 490 028 020 minus006 042 004 006 015

7790 550 minus014 minus029 minus056 minus003 minus042 minus036 minus019

7910 520 022 005 minus022 033 minus007 001 015

8420 570 000 minus029 minus056 006 minus037 minus034 minus028

8330 590 minus025 minus052 minus079 minus018 minus060 minus060 minus039

8010 530 015 minus002 minus029 027 minus013 minus022 010

8650 550 032 minus001 minus029 037 minus007 minus027 012

10200 550 115 046 016 105 053 037 034

10100 650 009 minus057 minus087 001 minus051 minus045 minus060

11100 620 092 002 minus030 073 016 040 minus017

9450 580 045 minus006 minus036 042 minus006 008 minus001

11800 620 128 021 minus012 102 041 064 minus014

1900 230 minus043 027 014 minus016 minus023 minus004 minus100

2000 240 minus046 024 010 minus019 minus026 minus013 minus096

2860 290 minus040 026 009 minus009 minus021 011 minus067

2890 300 minus048 017 001 minus017 minus030 008 minus082

3610 330 minus033 024 006 minus002 minus019 minus011 minus048

4450 370 minus023 024 004 007 minus014 007 minus026

2300 250 minus036 033 019 minus007 minus016 minus037 minus082

3300 350 minus072 minus011 minus028 minus041 minus056 minus043 minus102

5800 420 005 029 006 030 001 minus025 020

6900 450 037 040 015 055 020 012 044

3600 330 minus034 024 006 minus003 minus019 minus004 minus045

4280 390 minus053 minus004 minus024 minus023 minus043 minus019 minus063

5930 400 033 054 031 056 027 029 040

6330 450 005 019 minus004 026 minus005 minus018 020

6770 470 010 015 minus009 028 minus006 026 007

7700 510 021 008 minus019 033 minus006 minus005 001

4990 430 minus051 minus013 minus034 minus023 minus047 minus032 minus045

9670 570 063 010 minus019 062 012 005 001

6170 460 minus014 003 minus020 008 minus022 minus036 minus008

7320 510 000 minus005 minus031 015 minus022 017 minus003

8650 550 032 minus001 minus029 037 minus007 minus027 012

9230 590 023 minus023 minus052 023 minus025 010 minus019

(1) polynomial kernel function

119870(119909 119910) = [(119909 + 119910) + 1]119889

119889 = 1 2 119899 (13)

(2) radial basis function (RBF)

119870(119909 119910) = exp minus1205741003817100381710038171003817119909 minus 11991010038171003817100381710038172

120574 ge 0 (14)

(3) sigmoid kernel function

119870(119909 119910) = tan [120593 (119909 sdot 119910) + 120579] (15)

where 120572119894 120572lowast119894satisfy 120572

119894120572lowast119894= 0 120572

119894ge 0 and 120572lowast

119894ge 0

Advances in Materials Science and Engineering 7

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

119891119888(MPa) 119891spt (MPa) GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

7790 545 minus008 minus024 minus051 002 minus033 048 032

5710 313 103 133 110 132 115 174 145

7160 490 011 009 minus016 027 minus002 081 051

4410 391 minus052 001 minus019 minus017 minus023 028 minus017

6480 457 004 018 minus006 027 003 073 042

4260 366 minus036 019 000 000 minus005 047 minus004

3040 279 minus026 046 030 013 017 046 000

4680 389 minus034 015 minus006 001 minus008 049 004

8950 591 011 minus033 minus061 009 minus036 073 041

6690 455 019 028 003 039 014 096 059

7610 547 minus021 minus032 minus058 minus008 minus042 041 022

5360 437 minus041 minus005 minus027 minus011 minus025 047 minus002

6480 507 minus046 minus032 minus056 minus023 minus047 023 minus008

4580 402 minus053 minus003 minus023 minus018 minus026 040 minus016

3760 316 minus017 046 027 021 019 063 014

5310 408 minus015 022 000 016 002 078 024

3490 350 minus068 minus001 minus019 minus030 minus029 012 minus040

5390 540 minus142 minus107 minus129 minus112 minus127 minus062 minus100

4650 380 minus027 022 002 008 minus001 060 007

1380 170 minus029 049 038 003 017 minus042 minus017

6710 480 minus005 003 minus021 015 minus010 064 035

2030 340 minus153 minus074 minus088 minus117 minus106 minus126 minus136

2240 210 minus009 069 055 028 038 038 010

6140 430 012 032 009 037 016 079 051

3310 300 minus029 039 022 009 011 053 minus003

7900 560 minus017 minus036 minus062 minus008 minus044 046 021

4360 360 minus024 030 010 012 006 059 010

2560 242 minus020 057 041 019 026 031 002

2620 262 minus036 040 025 003 010 023 minus015

1690 188 minus025 055 042 010 023 minus013 minus010

1490 171 minus022 057 045 011 025 minus038 minus008

2730 281 minus048 027 012 minus009 minus003 019 minus026

2840 287 minus047 027 011 minus008 minus002 022 minus023

1690 191 minus028 052 039 007 020 minus016 minus013

1690 211 minus048 032 019 minus013 000 minus036 minus033

3090 292 minus035 036 019 003 007 031 minus010

3230 308 minus043 027 010 minus004 minus001 036 minus017

1960 222 minus040 039 026 minus004 008 minus010 minus022

1910 208 minus030 050 037 006 018 minus011 minus010

3530 324 minus040 027 009 minus001 minus001 038 minus011

3840 336 minus032 030 011 005 004 057 minus003

2250 247 minus045 033 019 minus008 002 minus005 minus027

2340 257 minus049 028 014 minus012 minus002 minus008 minus029

3100 290 minus033 038 022 006 010 035 minus006

1700 230 minus066 013 001 minus032 minus018 minus060 minus051

4700 400 minus044 004 minus016 minus009 minus018 044 minus009

3700 360 minus065 minus001 minus019 minus027 minus028 021 minus035

4200 400 minus074 minus018 minus037 minus038 minus042 016 minus039

5400 460 minus062 minus026 minus048 minus031 minus046 021 minus024

8 Advances in Materials Science and Engineering

Table 4 Continued

119891119888(MPa) 119891spt (MPa) GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

4200 440 minus114 minus058 minus077 minus078 minus082 minus024 minus079

6200 520 minus075 minus055 minus079 minus050 minus072 003 minus035

6700 425 049 058 033 070 044 115 091

7250 510 minus004 minus008 minus033 012 minus019 056 038

7450 520 minus003 minus011 minus037 011 minus021 059 038

7950 575 minus029 minus049 minus076 minus020 minus057 030 011

7800 515 022 006 minus020 033 minus003 083 060

8500 575 002 minus031 minus059 005 minus036 069 044

8600 480 103 067 039 105 062 167 140

9050 440 168 121 093 165 119 232 204

5480 334 069 103 081 099 083 161 106

3290 285 minus016 053 036 023 025 068 011

3450 292 minus013 055 037 026 027 066 013

5360 348 048 084 062 078 064 136 087

7410 526 minus011 minus018 minus044 003 minus029 058 030

6770 481 minus003 004 minus020 017 minus009 068 037

8420 657 minus085 minus116 minus143 minus081 minus121 minus025 minus046

7530 535 minus013 minus023 minus049 000 minus033 048 028

10230 729 minus055 minus132 minus163 minus073 minus129 052 minus031

4010 330 minus016 044 025 021 018 070 015

4050 340 minus023 035 016 014 010 057 007

9550 570 066 007 minus023 057 006 153 096

3520 317 minus033 033 015 005 006 048 minus005

5750 369 051 078 056 078 060 129 086

3860 338 minus033 029 010 005 003 052 minus004

6150 448 minus005 015 minus009 019 minus002 070 032

4740 376 minus017 030 010 017 008 067 018

4710 368 minus011 037 016 023 014 086 023

4470 326 017 068 048 052 045 109 046

5240 399 minus010 028 006 021 008 079 027

5790 449 minus027 000 minus023 000 minus018 061 012

3950 404 minus093 minus033 minus052 minus056 minus059 000 minus065

3670 403 minus109 minus046 minus064 minus072 minus072 minus029 minus082

3800 365 minus063 minus001 minus020 minus026 minus028 020 minus035

2420 220 minus007 070 055 031 040 038 014

1740 180 minus013 066 054 021 034 minus006 002

2930 280 minus033 039 023 005 010 027 minus009

2700 255 minus023 052 036 015 022 033 minus002

3260 321 minus053 016 minus001 minus015 minus012 026 minus026

4630 520 minus167 minus119 minus139 minus133 minus141 minus076 minus137

3580 430 minus142 minus077 minus095 minus104 minus104 minus057 minus111

4310 418 minus085 minus031 minus050 minus049 minus055 minus004 minus051

3520 317 minus033 033 015 005 006 048 minus005

5750 369 051 078 056 078 060 129 086

4280 490 minus159 minus104 minus124 minus123 minus128 minus076 minus126

6150 448 minus005 015 minus009 019 minus002 070 032

4740 376 minus017 030 010 017 008 067 018

4713 368 minus010 037 016 023 014 086 023

4473 326 017 069 049 052 045 104 050

Advances in Materials Science and Engineering 9

Table 4 Continued

119891119888(MPa) 119891spt (MPa) GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

3460 410 minus130 minus063 minus081 minus091 minus090 minus051 minus102

5970 540 minus108 minus084 minus107 minus082 minus101 minus030 minus069

5190 400 minus014 025 003 017 004 076 021

5000 390 minus015 027 006 017 006 077 022

4850 410 minus044 001 minus020 minus011 minus021 049 minus008

4880 370 minus002 042 021 031 020 080 032

4650 380 minus026 022 002 008 minus001 060 007

4520 400 minus054 minus003 minus024 minus019 minus027 030 minus021

4520 330 016 067 046 051 043 100 049

4260 298 032 087 068 068 063 115 064

8160 440 119 093 066 124 086 180 158

4270 300 031 086 066 066 061 121 064

4190 350 minus024 032 012 012 007 060 009

4120 380 minus059 minus001 minus021 minus022 minus026 031 minus026

3970 270 042 102 083 079 076 124 073

6550 550 minus084 minus073 minus097 minus063 minus087 minus021 minus047

3950 370 minus059 001 minus018 minus022 minus025 034 minus031

1710 190 minus025 054 042 009 022 minus011 minus010

1210 110 018 095 085 048 064 000 031

1290 160 minus026 052 041 005 020 minus048 minus013

1690 150 013 093 080 048 061 025 028

2450 250 minus035 042 027 003 012 016 minus016

3 SVM for Predicting 119891spt of Concrete

As mentioned previously many methods have been pro-posed for the prediction of 119891spt from compressive strengthof concrete This study attempts to utilize SVM for theprediction of concrete of 119891spt SVM-I model is devel-oped for 150 times 300mm cylinder 119891spt prediction from 150 times300mm cylinder 119891

119888of concrete and SVM-IImodel is devel-

oped for 100 times 200mm and 150 times 200mm cylinder 119891sptprediction from 100 times 200mm cylinder 119891

119888of concrete The

experimental data which are taken from studies [9 14 15] areused in this study Among 184 experimental data 138 datawere randomly selected as the training set for SVM-I modeland among 168 cases 126 cases were randomly selected asthe training set for SVM-II model the rest are considered astesting data setThe data are normalized before being used inthe model as follows

119883 =(119883119894minus 119883min)

(119883max minus 119883min) (16)

where 119883max and 119883min are the maximum and minimuminput values data respectively

In case of SVM training two types of kernel func-tions were used namely radical basis function (RBF) andpolynomial function in training process 119862 and 120576 and otherkernel-specific parameters have been chosen by a trial-and-error approach The best simulation performances ofSVM are summarized in Table 1 in order to evaluate theabilities of SVM models and other models mean absolutepercentage error (MAPE) roof-mean-squared error (RMES)

and R-square (1198772) were used as the criteria between theexperimental and predicted values which are according tothe equationsas follow

MAPE =1

119899[sum119899

119894=1

1003816100381610038161003816119905119894 minus 1199001198941003816100381610038161003816

sum119899

119894=1119905119894

times 100]

RMSE = radic1

119899

119899

sum119894=1

(119905119894minus 119900119894)

1198772 =(119899sum 119905

119894119900119894minus sum 119905119894sum119900119894)2

(119899sum 1199052119894minus (sum 1199052

119894)) (119899sum 1199002

119894minus (sum 119900

119894)2

)

(17)

where 119905119894is the experimental value 119900

119894is the predicted value

and 119899 is total number of data

4 Results and Discussion

There is no doubt that the splitting tensile strength 119891spt withan increase in the compression strength of concrete butthere is no agreement on the precise form of the relationshipCodes propose different formulas for prediction cylinder 119891sptof concrete from compressive strength In this paper 119891sptresults are investigated for 150 times 300mm cylinder concreteand 100 times 200mm cylinder concrete separately The errors(predicted values subtract measured values) computed bySVM model other models and measured values of split-ting tensile strengths are shown in Tables 2 and 3 for 150times 300mm cylinder concrete and in Tables 4 and 5 for

10 Advances in Materials Science and Engineering

Table 5 Comparison of experimental results to testing results of RBF polynomial of SVM-II and other models

119891119888(MPa) 119891spt (MPa) GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression RBF [9] Polynomial

3600 349 minus060 005 minus013 minus022 minus022 minus017 minus004

3040 308 minus054 017 001 minus016 minus012 minus010 minus001

2160 200 minus004 074 060 033 043 017 034

2120 270 minus077 002 minus012 minus040 minus030 minus044 minus039

2510 260 minus041 036 021 minus003 005 minus009 006

3190 330 minus067 003 minus014 minus028 minus025 minus026 minus012

3780 310 minus010 053 034 028 026 048 045

2920 250 minus004 069 053 034 040 041 049

3540 310 minus025 041 023 013 014 019 032

2860 340 minus098 minus024 minus041 minus059 minus054 minus051 minus046

2950 310 minus062 010 minus006 minus024 minus019 minus023 minus008

3490 350 minus068 minus001 minus019 minus030 minus029 minus023 minus011

3860 338 minus033 029 010 005 003 015 023

5235 399 minus010 028 006 021 008 042 036

3840 380 minus076 minus014 minus033 minus039 minus040 minus023 minus020

5150 420 minus036 003 minus018 minus005 minus017 016 011

4860 400 minus034 011 minus010 000 minus011 011 016

3950 390 minus079 minus019 minus038 minus042 minus045 minus022 minus022

3040 285 minus031 040 024 007 011 013 022

2950 256 minus008 064 048 030 035 031 046

4280 430 minus099 minus044 minus064 minus063 minus068 minus052 minus046

5190 430 minus044 minus005 minus027 minus013 minus026 009 003

4930 420 minus049 minus006 minus027 minus017 minus027 003 000

4330 360 minus026 028 008 010 004 022 028

4360 338 minus002 052 032 034 028 046 052

4990 457 minus083 minus040 minus061 minus050 minus062 minus034 minus033

5740 420 minus001 027 004 026 009 054 039

5850 490 minus065 minus039 minus062 minus038 minus056 minus015 minus024

3140 279 minus019 052 035 020 023 026 036

3490 304 minus022 045 027 016 017 023 035

4160 384 minus060 minus003 minus023 minus024 minus028 minus011 minus005

4680 400 minus044 004 minus017 minus010 minus019 001 009

5170 420 minus035 004 minus017 minus004 minus016 018 012

4330 390 minus056 minus002 minus022 minus020 minus026 minus008 minus002

6080 410 029 050 027 054 033 076 068

6000 430 004 027 004 030 010 060 044

7190 540 minus037 minus040 minus065 minus021 minus051 016 000

7510 540 minus018 minus029 minus055 minus006 minus039 030 020

5787 449 minus027 000 minus023 000 minus018 027 015

3950 340 minus029 031 012 008 005 028 028

Table 6 Error measurement for 119891spt from 150 times 300mm cylinder 119891119888

GEP-I [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF PolynomialRMSE of training set 04603 05995 05228 05027 04512 04978 04819RMSE of testing set 04271 02735 03152 03754 02874 02748 04176MAPE of training set 97554 13647 114913 106213 95596 102434 105454MAPE of testing set 73022 46139 55434 62059 5286 4987 695171198772 of training set 08282 08225 08224 08267 0826 08115 082271198772 of testing set 09416 09486 09486 09471 09476 09422 09327

Advances in Materials Science and Engineering 11

Table 7 Error measurement for 119891spt from 100 times 200mm cylinder 119891119888

GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF PolynomialRMSE of training set 05886 0539 05244 04987 04911 07287 0562RMSE of testing set 0483 03348 03393 02803 03155 03082 02917MAPE of training set 118729 116879 109223 92976 9472 155902 106658MAPE of testing set 108701 71383 7604 62261 7234 69713 628351198772 of training set 08198 08354 08353 083 08337 0837 082531198772 of testing set 08778 08802 08793 088 08801 08815 08823

0

2

4

6

8

0 20 40 60 80 100 120 140

Split

ting

tens

ile st

reng

th (M

Pa)

Sample number

ExperimentalRBFPolynomial

Figure 1 Comparison of experimental results to training results ofSVM-I

100 times 200mm cylinder concrete As seen from these resultsSVM model is sufficiently close to the experimental dataand is able to predict the 119891spt from the 119891

119888of concrete

For convenient comparison purpose the experimental andpredicted results are plotted in Figures 1 2 3 and 4 It canbe seen that the results from SVM method are in goodagreement with the experimental results

The 119877-square errors in the trained SVM are 08115 and08227 for RBF and polynomial function of SVM-I 08370and 08253 for those of SVM-II respectively 119877-square errorsin the tested SVM are 09422 and 09327 for RBF and polyno-mial function of SVM-I 08815 and 08823 for those of SVM-II respectively The whole results obtained from the SVMmodels show a successful performance of the SVM modelsof predicting 150 times 300mm cylinder119891spt of concrete from thecorresponding 150 times 300mm cylinder compressive strength119891119888and 100 times 200mm cylinder 119891spt from the corresponding

100 times 200mm cylinder compressive strength 119891119888of concrete

for each of the training and testing sets It is also clear thatthere are no major difference on performance between theradical basis function and the polynomial function kernelsthat used in this paper However in generally the radical basisfunction kernel exhibits slightly better performance than thepolynomial kernel

The performance of the trained and tested sets is analyzedby computing MAPE RMSE and 1198772 of experimental 119891spt

0

2

4

6

8

0 5 10 15 20 25 30 35 40 45 50

Split

ting

tens

ile st

reng

th (M

Pa)

Sample number

ExperimentalRBFPolynomial

Figure 2 Comparison of experimental results to testing results ofSVM-I

between the SVM model and other methods The MAPERMSE and 1198772 are calculated for these methods Statisticalparameters of the training and testing sets of all methods arepresented in Tables 6 and 7 The model having the smallestMAPE or RMSE and the highest 1198772 can be regarded to bethe best model with the assumption of analysis Generally1198772 of the SVM models are higher than other methodsand the MAPE and RMSE of SVM are smaller than othermodels The analysis shows that the SVM model is betterthan other models and can predict the splitting tensilestrength of concrete well Furthermore SVMmodel has goodgeneralization capacity to avoid over training and can alwaysbe updated to get better results by presenting new trainingexamples as new data become available Thus SVM modelcan be regarded as a very effective method to predict splittingtensile strength of concrete from their compressive strength

5 Conclusion

The splitting tensile strength of concrete estimations fromcompression strength has been obtained so far in the litera-ture either through regression or other methodsThis presentstudy reports a new and influential approach for predictingthe splitting tensile strength using SVM for the first timein the literature The study conducted in this paper shows

12 Advances in Materials Science and Engineering

Polynomial

0

1

2

3

4

5

6

7

8

9

0 20 40 60 80 100 120 140

Split

ting

tens

ile st

reng

hth

(MPa

)

Sample number

ExperimentalRBF

Figure 3 Comparison of experimental results to training results ofSVM-II

Polynomial

0

2

4

6

0 10 20 30 40 50

Split

ting

tens

ile st

reng

th (M

Pa)

Sample number

ExperimentalRBF

Figure 4 Comparison of experimental results to testing results ofSVM-II

the feasibility of using a simple SVM to estimate the splittingtensile strength of concrete After learning from a set ofselected training data involving compressive strengths ofconcrete collected from the previous literature the SVM canbe utilized to predict the splitting tensile strengths of the testdata

In this paper radical basis function and polynomial areadopted for predicting splitting tensile strength from 150 times300mm cylinder and 100times200mm cylinder 119891

119888of concrete

It is found that the splitting tensile strength obtained fromthe SVM is more accurate than those obtained from designcodes and several researchesrsquo empirical equation when acomparison is made on the basis of the experimental dataSince the SVM is largely characterized by the type of its kernel

function it is necessary to choose the appropriate kernel foreach particular application problem in order to guaranteesatisfactory results The results of radial basis function andpolynomial indicate that RBF and polynomial kernel have theability to predict the splitting tensile strength of concrete fromcompression strength with an acceptable degree of accuracy

The statistical parameters of MAPE RMSE and 1198772 showthat the proposed SVM model results have the best accu-racy and can predict splitting tensile strength very close toexperiment results The use of SVM is very advantageousfor the prediction of the splitting tensile strength of concretefrom compression strength because it can perform nonlinearregression efficiently for high-dimensional data sets Further-more its solution is globalThe satisfactory predictions of thesplitting tensile strength of concrete by the model indicatethat SVM is a useful modeling tool for engineers and researchscientists at concrete construction fields

Acknowledgments

This project was granted financial support from the NationalNatural Science Foundation of China (no 50808077 andno 51278188) and the Fundamental Research Funds for theCentral Universities The authors are grateful for support

References

[1] B W Xu and H S Shi ldquoCorrelations among mechanicalproperties of steel fiber reinforced concreterdquo Construction andBuilding Materials vol 23 no 12 pp 3468ndash3474 2009

[2] M F M Zain H B Mahmud A Ilham and M FaizalldquoPrediction of splitting tensile strength of high-performanceconcreterdquoCement andConcrete Research vol 32 no 8 pp 1251ndash1258 2002

[3] F A Oluokun E G Burdette and J H Deatherage ldquoSplittingtensile strength and compressive strength relationship at earlyagesrdquo ACI Materials Journal vol 88 no 2 pp 115ndash121 1991

[4] Y Choi and R L Yuan ldquoExperimental relationship betweensplitting tensile strength and compressive strength of GFRC andPFRCrdquo Cement and Concrete Research vol 35 no 8 pp 1587ndash1591 2005

[5] F D Larrad and Y Malier ldquoEngineering properties of very highperformance concreterdquo in High Performance Concrete fromMaterial to Structure Y Malier Ed pp 85ndash114 EFN SponLondon UK 1992

[6] J K Kim S H Han Y D Park and J H Noh ldquoMaterialproperties of self-flowing concreterdquo Journal of Materials in CivilEngineering vol 10 no 4 pp 244ndash249 1998

[7] J K Kim S H Han and Y C Song ldquoEffect of temperatureand aging on the mechanical properties of concretemdashpart Iexperimental resultsrdquo Cement and Concrete Research vol 32no 7 pp 1087ndash1094 2002

[8] AM Shaaban andHGesund ldquoSplitting tensile strength of steelfiber reinforced concrete cylinders consolidated by rodding orvibratingrdquo ACI Materials Journal vol 90 no 4 pp 366ndash3691993

[9] M Saridemir ldquoEmpirical modeling of splitting tensile strengthfrom cylinder compressive strength of concrete by geneticprogrammingrdquo Expert Systems with Applications vol 38 no 11pp 14257ndash14268 2011

Advances in Materials Science and Engineering 13

[10] U Thissen R van Brakel A P de Weijer W J Melssen and LM C Buydens ldquoUsing support vector machines for time seriespredictionrdquo Chemometrics and Intelligent Laboratory Systemsvol 69 no 1-2 pp 35ndash49 2003

[11] P Samui ldquoPredicted ultimate capacity of laterally loaded pilesin clay using support vector machinerdquo Geomechanics andGeoengineering vol 3 no 2 pp 113ndash120 2008

[12] VN VapnikTheNature of Statistical LearningTheory SpringerNew York NY USA 1995

[13] V N Vapnik S E Golowich and A J Smola ldquoSupport vectormachine for function approximation regression estimationand signal processingrdquoAdvances in Neural Information Process-ing Systems vol 9 pp 281ndash287 1996

[14] M Khanzadi and A Behnood ldquoMechanical properties ofhigh-strength concrete incorporating copper slag as coarseaggregaterdquo Construction and Building Materials vol 23 no 6pp 2183ndash2188 2009

[15] C K Y Leung and J Pan ldquoEffect of concrete composition onFRPconcrete bond capacityrdquo in Proceedings of the InternationalSymposium on Bond Behaviour of FRP in Structures (BBFS rsquo05)pp 69ndash76 2005

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MaterialsJournal of

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Nano

materials

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Journal ofNanomaterials

6 Advances in Materials Science and Engineering

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

119891119888(MPa) 119891spt (MPa) GEP-I [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

4480 380 minus031 015 minus005 minus002 minus023 minus001 minus033

5620 440 minus025 002 minus020 000 minus027 minus041 minus020

5340 360 039 071 049 065 039 033 039

5670 460 minus042 minus016 minus038 minus017 minus045 minus009 minus036

6450 480 minus018 minus006 minus030 002 minus030 minus018 minus012

6650 470 003 011 minus013 022 minus011 010 001

6540 430 037 047 023 057 024 021 043

7380 490 023 017 minus009 038 001 022 014

7400 530 minus016 minus022 minus048 minus001 minus039 minus026 minus015

7250 500 006 002 minus023 022 minus015 024 minus006

7460 490 028 020 minus006 042 004 006 015

7790 550 minus014 minus029 minus056 minus003 minus042 minus036 minus019

7910 520 022 005 minus022 033 minus007 001 015

8420 570 000 minus029 minus056 006 minus037 minus034 minus028

8330 590 minus025 minus052 minus079 minus018 minus060 minus060 minus039

8010 530 015 minus002 minus029 027 minus013 minus022 010

8650 550 032 minus001 minus029 037 minus007 minus027 012

10200 550 115 046 016 105 053 037 034

10100 650 009 minus057 minus087 001 minus051 minus045 minus060

11100 620 092 002 minus030 073 016 040 minus017

9450 580 045 minus006 minus036 042 minus006 008 minus001

11800 620 128 021 minus012 102 041 064 minus014

1900 230 minus043 027 014 minus016 minus023 minus004 minus100

2000 240 minus046 024 010 minus019 minus026 minus013 minus096

2860 290 minus040 026 009 minus009 minus021 011 minus067

2890 300 minus048 017 001 minus017 minus030 008 minus082

3610 330 minus033 024 006 minus002 minus019 minus011 minus048

4450 370 minus023 024 004 007 minus014 007 minus026

2300 250 minus036 033 019 minus007 minus016 minus037 minus082

3300 350 minus072 minus011 minus028 minus041 minus056 minus043 minus102

5800 420 005 029 006 030 001 minus025 020

6900 450 037 040 015 055 020 012 044

3600 330 minus034 024 006 minus003 minus019 minus004 minus045

4280 390 minus053 minus004 minus024 minus023 minus043 minus019 minus063

5930 400 033 054 031 056 027 029 040

6330 450 005 019 minus004 026 minus005 minus018 020

6770 470 010 015 minus009 028 minus006 026 007

7700 510 021 008 minus019 033 minus006 minus005 001

4990 430 minus051 minus013 minus034 minus023 minus047 minus032 minus045

9670 570 063 010 minus019 062 012 005 001

6170 460 minus014 003 minus020 008 minus022 minus036 minus008

7320 510 000 minus005 minus031 015 minus022 017 minus003

8650 550 032 minus001 minus029 037 minus007 minus027 012

9230 590 023 minus023 minus052 023 minus025 010 minus019

(1) polynomial kernel function

119870(119909 119910) = [(119909 + 119910) + 1]119889

119889 = 1 2 119899 (13)

(2) radial basis function (RBF)

119870(119909 119910) = exp minus1205741003817100381710038171003817119909 minus 11991010038171003817100381710038172

120574 ge 0 (14)

(3) sigmoid kernel function

119870(119909 119910) = tan [120593 (119909 sdot 119910) + 120579] (15)

where 120572119894 120572lowast119894satisfy 120572

119894120572lowast119894= 0 120572

119894ge 0 and 120572lowast

119894ge 0

Advances in Materials Science and Engineering 7

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

119891119888(MPa) 119891spt (MPa) GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

7790 545 minus008 minus024 minus051 002 minus033 048 032

5710 313 103 133 110 132 115 174 145

7160 490 011 009 minus016 027 minus002 081 051

4410 391 minus052 001 minus019 minus017 minus023 028 minus017

6480 457 004 018 minus006 027 003 073 042

4260 366 minus036 019 000 000 minus005 047 minus004

3040 279 minus026 046 030 013 017 046 000

4680 389 minus034 015 minus006 001 minus008 049 004

8950 591 011 minus033 minus061 009 minus036 073 041

6690 455 019 028 003 039 014 096 059

7610 547 minus021 minus032 minus058 minus008 minus042 041 022

5360 437 minus041 minus005 minus027 minus011 minus025 047 minus002

6480 507 minus046 minus032 minus056 minus023 minus047 023 minus008

4580 402 minus053 minus003 minus023 minus018 minus026 040 minus016

3760 316 minus017 046 027 021 019 063 014

5310 408 minus015 022 000 016 002 078 024

3490 350 minus068 minus001 minus019 minus030 minus029 012 minus040

5390 540 minus142 minus107 minus129 minus112 minus127 minus062 minus100

4650 380 minus027 022 002 008 minus001 060 007

1380 170 minus029 049 038 003 017 minus042 minus017

6710 480 minus005 003 minus021 015 minus010 064 035

2030 340 minus153 minus074 minus088 minus117 minus106 minus126 minus136

2240 210 minus009 069 055 028 038 038 010

6140 430 012 032 009 037 016 079 051

3310 300 minus029 039 022 009 011 053 minus003

7900 560 minus017 minus036 minus062 minus008 minus044 046 021

4360 360 minus024 030 010 012 006 059 010

2560 242 minus020 057 041 019 026 031 002

2620 262 minus036 040 025 003 010 023 minus015

1690 188 minus025 055 042 010 023 minus013 minus010

1490 171 minus022 057 045 011 025 minus038 minus008

2730 281 minus048 027 012 minus009 minus003 019 minus026

2840 287 minus047 027 011 minus008 minus002 022 minus023

1690 191 minus028 052 039 007 020 minus016 minus013

1690 211 minus048 032 019 minus013 000 minus036 minus033

3090 292 minus035 036 019 003 007 031 minus010

3230 308 minus043 027 010 minus004 minus001 036 minus017

1960 222 minus040 039 026 minus004 008 minus010 minus022

1910 208 minus030 050 037 006 018 minus011 minus010

3530 324 minus040 027 009 minus001 minus001 038 minus011

3840 336 minus032 030 011 005 004 057 minus003

2250 247 minus045 033 019 minus008 002 minus005 minus027

2340 257 minus049 028 014 minus012 minus002 minus008 minus029

3100 290 minus033 038 022 006 010 035 minus006

1700 230 minus066 013 001 minus032 minus018 minus060 minus051

4700 400 minus044 004 minus016 minus009 minus018 044 minus009

3700 360 minus065 minus001 minus019 minus027 minus028 021 minus035

4200 400 minus074 minus018 minus037 minus038 minus042 016 minus039

5400 460 minus062 minus026 minus048 minus031 minus046 021 minus024

8 Advances in Materials Science and Engineering

Table 4 Continued

119891119888(MPa) 119891spt (MPa) GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

4200 440 minus114 minus058 minus077 minus078 minus082 minus024 minus079

6200 520 minus075 minus055 minus079 minus050 minus072 003 minus035

6700 425 049 058 033 070 044 115 091

7250 510 minus004 minus008 minus033 012 minus019 056 038

7450 520 minus003 minus011 minus037 011 minus021 059 038

7950 575 minus029 minus049 minus076 minus020 minus057 030 011

7800 515 022 006 minus020 033 minus003 083 060

8500 575 002 minus031 minus059 005 minus036 069 044

8600 480 103 067 039 105 062 167 140

9050 440 168 121 093 165 119 232 204

5480 334 069 103 081 099 083 161 106

3290 285 minus016 053 036 023 025 068 011

3450 292 minus013 055 037 026 027 066 013

5360 348 048 084 062 078 064 136 087

7410 526 minus011 minus018 minus044 003 minus029 058 030

6770 481 minus003 004 minus020 017 minus009 068 037

8420 657 minus085 minus116 minus143 minus081 minus121 minus025 minus046

7530 535 minus013 minus023 minus049 000 minus033 048 028

10230 729 minus055 minus132 minus163 minus073 minus129 052 minus031

4010 330 minus016 044 025 021 018 070 015

4050 340 minus023 035 016 014 010 057 007

9550 570 066 007 minus023 057 006 153 096

3520 317 minus033 033 015 005 006 048 minus005

5750 369 051 078 056 078 060 129 086

3860 338 minus033 029 010 005 003 052 minus004

6150 448 minus005 015 minus009 019 minus002 070 032

4740 376 minus017 030 010 017 008 067 018

4710 368 minus011 037 016 023 014 086 023

4470 326 017 068 048 052 045 109 046

5240 399 minus010 028 006 021 008 079 027

5790 449 minus027 000 minus023 000 minus018 061 012

3950 404 minus093 minus033 minus052 minus056 minus059 000 minus065

3670 403 minus109 minus046 minus064 minus072 minus072 minus029 minus082

3800 365 minus063 minus001 minus020 minus026 minus028 020 minus035

2420 220 minus007 070 055 031 040 038 014

1740 180 minus013 066 054 021 034 minus006 002

2930 280 minus033 039 023 005 010 027 minus009

2700 255 minus023 052 036 015 022 033 minus002

3260 321 minus053 016 minus001 minus015 minus012 026 minus026

4630 520 minus167 minus119 minus139 minus133 minus141 minus076 minus137

3580 430 minus142 minus077 minus095 minus104 minus104 minus057 minus111

4310 418 minus085 minus031 minus050 minus049 minus055 minus004 minus051

3520 317 minus033 033 015 005 006 048 minus005

5750 369 051 078 056 078 060 129 086

4280 490 minus159 minus104 minus124 minus123 minus128 minus076 minus126

6150 448 minus005 015 minus009 019 minus002 070 032

4740 376 minus017 030 010 017 008 067 018

4713 368 minus010 037 016 023 014 086 023

4473 326 017 069 049 052 045 104 050

Advances in Materials Science and Engineering 9

Table 4 Continued

119891119888(MPa) 119891spt (MPa) GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

3460 410 minus130 minus063 minus081 minus091 minus090 minus051 minus102

5970 540 minus108 minus084 minus107 minus082 minus101 minus030 minus069

5190 400 minus014 025 003 017 004 076 021

5000 390 minus015 027 006 017 006 077 022

4850 410 minus044 001 minus020 minus011 minus021 049 minus008

4880 370 minus002 042 021 031 020 080 032

4650 380 minus026 022 002 008 minus001 060 007

4520 400 minus054 minus003 minus024 minus019 minus027 030 minus021

4520 330 016 067 046 051 043 100 049

4260 298 032 087 068 068 063 115 064

8160 440 119 093 066 124 086 180 158

4270 300 031 086 066 066 061 121 064

4190 350 minus024 032 012 012 007 060 009

4120 380 minus059 minus001 minus021 minus022 minus026 031 minus026

3970 270 042 102 083 079 076 124 073

6550 550 minus084 minus073 minus097 minus063 minus087 minus021 minus047

3950 370 minus059 001 minus018 minus022 minus025 034 minus031

1710 190 minus025 054 042 009 022 minus011 minus010

1210 110 018 095 085 048 064 000 031

1290 160 minus026 052 041 005 020 minus048 minus013

1690 150 013 093 080 048 061 025 028

2450 250 minus035 042 027 003 012 016 minus016

3 SVM for Predicting 119891spt of Concrete

As mentioned previously many methods have been pro-posed for the prediction of 119891spt from compressive strengthof concrete This study attempts to utilize SVM for theprediction of concrete of 119891spt SVM-I model is devel-oped for 150 times 300mm cylinder 119891spt prediction from 150 times300mm cylinder 119891

119888of concrete and SVM-IImodel is devel-

oped for 100 times 200mm and 150 times 200mm cylinder 119891sptprediction from 100 times 200mm cylinder 119891

119888of concrete The

experimental data which are taken from studies [9 14 15] areused in this study Among 184 experimental data 138 datawere randomly selected as the training set for SVM-I modeland among 168 cases 126 cases were randomly selected asthe training set for SVM-II model the rest are considered astesting data setThe data are normalized before being used inthe model as follows

119883 =(119883119894minus 119883min)

(119883max minus 119883min) (16)

where 119883max and 119883min are the maximum and minimuminput values data respectively

In case of SVM training two types of kernel func-tions were used namely radical basis function (RBF) andpolynomial function in training process 119862 and 120576 and otherkernel-specific parameters have been chosen by a trial-and-error approach The best simulation performances ofSVM are summarized in Table 1 in order to evaluate theabilities of SVM models and other models mean absolutepercentage error (MAPE) roof-mean-squared error (RMES)

and R-square (1198772) were used as the criteria between theexperimental and predicted values which are according tothe equationsas follow

MAPE =1

119899[sum119899

119894=1

1003816100381610038161003816119905119894 minus 1199001198941003816100381610038161003816

sum119899

119894=1119905119894

times 100]

RMSE = radic1

119899

119899

sum119894=1

(119905119894minus 119900119894)

1198772 =(119899sum 119905

119894119900119894minus sum 119905119894sum119900119894)2

(119899sum 1199052119894minus (sum 1199052

119894)) (119899sum 1199002

119894minus (sum 119900

119894)2

)

(17)

where 119905119894is the experimental value 119900

119894is the predicted value

and 119899 is total number of data

4 Results and Discussion

There is no doubt that the splitting tensile strength 119891spt withan increase in the compression strength of concrete butthere is no agreement on the precise form of the relationshipCodes propose different formulas for prediction cylinder 119891sptof concrete from compressive strength In this paper 119891sptresults are investigated for 150 times 300mm cylinder concreteand 100 times 200mm cylinder concrete separately The errors(predicted values subtract measured values) computed bySVM model other models and measured values of split-ting tensile strengths are shown in Tables 2 and 3 for 150times 300mm cylinder concrete and in Tables 4 and 5 for

10 Advances in Materials Science and Engineering

Table 5 Comparison of experimental results to testing results of RBF polynomial of SVM-II and other models

119891119888(MPa) 119891spt (MPa) GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression RBF [9] Polynomial

3600 349 minus060 005 minus013 minus022 minus022 minus017 minus004

3040 308 minus054 017 001 minus016 minus012 minus010 minus001

2160 200 minus004 074 060 033 043 017 034

2120 270 minus077 002 minus012 minus040 minus030 minus044 minus039

2510 260 minus041 036 021 minus003 005 minus009 006

3190 330 minus067 003 minus014 minus028 minus025 minus026 minus012

3780 310 minus010 053 034 028 026 048 045

2920 250 minus004 069 053 034 040 041 049

3540 310 minus025 041 023 013 014 019 032

2860 340 minus098 minus024 minus041 minus059 minus054 minus051 minus046

2950 310 minus062 010 minus006 minus024 minus019 minus023 minus008

3490 350 minus068 minus001 minus019 minus030 minus029 minus023 minus011

3860 338 minus033 029 010 005 003 015 023

5235 399 minus010 028 006 021 008 042 036

3840 380 minus076 minus014 minus033 minus039 minus040 minus023 minus020

5150 420 minus036 003 minus018 minus005 minus017 016 011

4860 400 minus034 011 minus010 000 minus011 011 016

3950 390 minus079 minus019 minus038 minus042 minus045 minus022 minus022

3040 285 minus031 040 024 007 011 013 022

2950 256 minus008 064 048 030 035 031 046

4280 430 minus099 minus044 minus064 minus063 minus068 minus052 minus046

5190 430 minus044 minus005 minus027 minus013 minus026 009 003

4930 420 minus049 minus006 minus027 minus017 minus027 003 000

4330 360 minus026 028 008 010 004 022 028

4360 338 minus002 052 032 034 028 046 052

4990 457 minus083 minus040 minus061 minus050 minus062 minus034 minus033

5740 420 minus001 027 004 026 009 054 039

5850 490 minus065 minus039 minus062 minus038 minus056 minus015 minus024

3140 279 minus019 052 035 020 023 026 036

3490 304 minus022 045 027 016 017 023 035

4160 384 minus060 minus003 minus023 minus024 minus028 minus011 minus005

4680 400 minus044 004 minus017 minus010 minus019 001 009

5170 420 minus035 004 minus017 minus004 minus016 018 012

4330 390 minus056 minus002 minus022 minus020 minus026 minus008 minus002

6080 410 029 050 027 054 033 076 068

6000 430 004 027 004 030 010 060 044

7190 540 minus037 minus040 minus065 minus021 minus051 016 000

7510 540 minus018 minus029 minus055 minus006 minus039 030 020

5787 449 minus027 000 minus023 000 minus018 027 015

3950 340 minus029 031 012 008 005 028 028

Table 6 Error measurement for 119891spt from 150 times 300mm cylinder 119891119888

GEP-I [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF PolynomialRMSE of training set 04603 05995 05228 05027 04512 04978 04819RMSE of testing set 04271 02735 03152 03754 02874 02748 04176MAPE of training set 97554 13647 114913 106213 95596 102434 105454MAPE of testing set 73022 46139 55434 62059 5286 4987 695171198772 of training set 08282 08225 08224 08267 0826 08115 082271198772 of testing set 09416 09486 09486 09471 09476 09422 09327

Advances in Materials Science and Engineering 11

Table 7 Error measurement for 119891spt from 100 times 200mm cylinder 119891119888

GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF PolynomialRMSE of training set 05886 0539 05244 04987 04911 07287 0562RMSE of testing set 0483 03348 03393 02803 03155 03082 02917MAPE of training set 118729 116879 109223 92976 9472 155902 106658MAPE of testing set 108701 71383 7604 62261 7234 69713 628351198772 of training set 08198 08354 08353 083 08337 0837 082531198772 of testing set 08778 08802 08793 088 08801 08815 08823

0

2

4

6

8

0 20 40 60 80 100 120 140

Split

ting

tens

ile st

reng

th (M

Pa)

Sample number

ExperimentalRBFPolynomial

Figure 1 Comparison of experimental results to training results ofSVM-I

100 times 200mm cylinder concrete As seen from these resultsSVM model is sufficiently close to the experimental dataand is able to predict the 119891spt from the 119891

119888of concrete

For convenient comparison purpose the experimental andpredicted results are plotted in Figures 1 2 3 and 4 It canbe seen that the results from SVM method are in goodagreement with the experimental results

The 119877-square errors in the trained SVM are 08115 and08227 for RBF and polynomial function of SVM-I 08370and 08253 for those of SVM-II respectively 119877-square errorsin the tested SVM are 09422 and 09327 for RBF and polyno-mial function of SVM-I 08815 and 08823 for those of SVM-II respectively The whole results obtained from the SVMmodels show a successful performance of the SVM modelsof predicting 150 times 300mm cylinder119891spt of concrete from thecorresponding 150 times 300mm cylinder compressive strength119891119888and 100 times 200mm cylinder 119891spt from the corresponding

100 times 200mm cylinder compressive strength 119891119888of concrete

for each of the training and testing sets It is also clear thatthere are no major difference on performance between theradical basis function and the polynomial function kernelsthat used in this paper However in generally the radical basisfunction kernel exhibits slightly better performance than thepolynomial kernel

The performance of the trained and tested sets is analyzedby computing MAPE RMSE and 1198772 of experimental 119891spt

0

2

4

6

8

0 5 10 15 20 25 30 35 40 45 50

Split

ting

tens

ile st

reng

th (M

Pa)

Sample number

ExperimentalRBFPolynomial

Figure 2 Comparison of experimental results to testing results ofSVM-I

between the SVM model and other methods The MAPERMSE and 1198772 are calculated for these methods Statisticalparameters of the training and testing sets of all methods arepresented in Tables 6 and 7 The model having the smallestMAPE or RMSE and the highest 1198772 can be regarded to bethe best model with the assumption of analysis Generally1198772 of the SVM models are higher than other methodsand the MAPE and RMSE of SVM are smaller than othermodels The analysis shows that the SVM model is betterthan other models and can predict the splitting tensilestrength of concrete well Furthermore SVMmodel has goodgeneralization capacity to avoid over training and can alwaysbe updated to get better results by presenting new trainingexamples as new data become available Thus SVM modelcan be regarded as a very effective method to predict splittingtensile strength of concrete from their compressive strength

5 Conclusion

The splitting tensile strength of concrete estimations fromcompression strength has been obtained so far in the litera-ture either through regression or other methodsThis presentstudy reports a new and influential approach for predictingthe splitting tensile strength using SVM for the first timein the literature The study conducted in this paper shows

12 Advances in Materials Science and Engineering

Polynomial

0

1

2

3

4

5

6

7

8

9

0 20 40 60 80 100 120 140

Split

ting

tens

ile st

reng

hth

(MPa

)

Sample number

ExperimentalRBF

Figure 3 Comparison of experimental results to training results ofSVM-II

Polynomial

0

2

4

6

0 10 20 30 40 50

Split

ting

tens

ile st

reng

th (M

Pa)

Sample number

ExperimentalRBF

Figure 4 Comparison of experimental results to testing results ofSVM-II

the feasibility of using a simple SVM to estimate the splittingtensile strength of concrete After learning from a set ofselected training data involving compressive strengths ofconcrete collected from the previous literature the SVM canbe utilized to predict the splitting tensile strengths of the testdata

In this paper radical basis function and polynomial areadopted for predicting splitting tensile strength from 150 times300mm cylinder and 100times200mm cylinder 119891

119888of concrete

It is found that the splitting tensile strength obtained fromthe SVM is more accurate than those obtained from designcodes and several researchesrsquo empirical equation when acomparison is made on the basis of the experimental dataSince the SVM is largely characterized by the type of its kernel

function it is necessary to choose the appropriate kernel foreach particular application problem in order to guaranteesatisfactory results The results of radial basis function andpolynomial indicate that RBF and polynomial kernel have theability to predict the splitting tensile strength of concrete fromcompression strength with an acceptable degree of accuracy

The statistical parameters of MAPE RMSE and 1198772 showthat the proposed SVM model results have the best accu-racy and can predict splitting tensile strength very close toexperiment results The use of SVM is very advantageousfor the prediction of the splitting tensile strength of concretefrom compression strength because it can perform nonlinearregression efficiently for high-dimensional data sets Further-more its solution is globalThe satisfactory predictions of thesplitting tensile strength of concrete by the model indicatethat SVM is a useful modeling tool for engineers and researchscientists at concrete construction fields

Acknowledgments

This project was granted financial support from the NationalNatural Science Foundation of China (no 50808077 andno 51278188) and the Fundamental Research Funds for theCentral Universities The authors are grateful for support

References

[1] B W Xu and H S Shi ldquoCorrelations among mechanicalproperties of steel fiber reinforced concreterdquo Construction andBuilding Materials vol 23 no 12 pp 3468ndash3474 2009

[2] M F M Zain H B Mahmud A Ilham and M FaizalldquoPrediction of splitting tensile strength of high-performanceconcreterdquoCement andConcrete Research vol 32 no 8 pp 1251ndash1258 2002

[3] F A Oluokun E G Burdette and J H Deatherage ldquoSplittingtensile strength and compressive strength relationship at earlyagesrdquo ACI Materials Journal vol 88 no 2 pp 115ndash121 1991

[4] Y Choi and R L Yuan ldquoExperimental relationship betweensplitting tensile strength and compressive strength of GFRC andPFRCrdquo Cement and Concrete Research vol 35 no 8 pp 1587ndash1591 2005

[5] F D Larrad and Y Malier ldquoEngineering properties of very highperformance concreterdquo in High Performance Concrete fromMaterial to Structure Y Malier Ed pp 85ndash114 EFN SponLondon UK 1992

[6] J K Kim S H Han Y D Park and J H Noh ldquoMaterialproperties of self-flowing concreterdquo Journal of Materials in CivilEngineering vol 10 no 4 pp 244ndash249 1998

[7] J K Kim S H Han and Y C Song ldquoEffect of temperatureand aging on the mechanical properties of concretemdashpart Iexperimental resultsrdquo Cement and Concrete Research vol 32no 7 pp 1087ndash1094 2002

[8] AM Shaaban andHGesund ldquoSplitting tensile strength of steelfiber reinforced concrete cylinders consolidated by rodding orvibratingrdquo ACI Materials Journal vol 90 no 4 pp 366ndash3691993

[9] M Saridemir ldquoEmpirical modeling of splitting tensile strengthfrom cylinder compressive strength of concrete by geneticprogrammingrdquo Expert Systems with Applications vol 38 no 11pp 14257ndash14268 2011

Advances in Materials Science and Engineering 13

[10] U Thissen R van Brakel A P de Weijer W J Melssen and LM C Buydens ldquoUsing support vector machines for time seriespredictionrdquo Chemometrics and Intelligent Laboratory Systemsvol 69 no 1-2 pp 35ndash49 2003

[11] P Samui ldquoPredicted ultimate capacity of laterally loaded pilesin clay using support vector machinerdquo Geomechanics andGeoengineering vol 3 no 2 pp 113ndash120 2008

[12] VN VapnikTheNature of Statistical LearningTheory SpringerNew York NY USA 1995

[13] V N Vapnik S E Golowich and A J Smola ldquoSupport vectormachine for function approximation regression estimationand signal processingrdquoAdvances in Neural Information Process-ing Systems vol 9 pp 281ndash287 1996

[14] M Khanzadi and A Behnood ldquoMechanical properties ofhigh-strength concrete incorporating copper slag as coarseaggregaterdquo Construction and Building Materials vol 23 no 6pp 2183ndash2188 2009

[15] C K Y Leung and J Pan ldquoEffect of concrete composition onFRPconcrete bond capacityrdquo in Proceedings of the InternationalSymposium on Bond Behaviour of FRP in Structures (BBFS rsquo05)pp 69ndash76 2005

Submit your manuscripts athttpwwwhindawicom

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Nano

materials

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Journal ofNanomaterials

Advances in Materials Science and Engineering 7

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

119891119888(MPa) 119891spt (MPa) GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

7790 545 minus008 minus024 minus051 002 minus033 048 032

5710 313 103 133 110 132 115 174 145

7160 490 011 009 minus016 027 minus002 081 051

4410 391 minus052 001 minus019 minus017 minus023 028 minus017

6480 457 004 018 minus006 027 003 073 042

4260 366 minus036 019 000 000 minus005 047 minus004

3040 279 minus026 046 030 013 017 046 000

4680 389 minus034 015 minus006 001 minus008 049 004

8950 591 011 minus033 minus061 009 minus036 073 041

6690 455 019 028 003 039 014 096 059

7610 547 minus021 minus032 minus058 minus008 minus042 041 022

5360 437 minus041 minus005 minus027 minus011 minus025 047 minus002

6480 507 minus046 minus032 minus056 minus023 minus047 023 minus008

4580 402 minus053 minus003 minus023 minus018 minus026 040 minus016

3760 316 minus017 046 027 021 019 063 014

5310 408 minus015 022 000 016 002 078 024

3490 350 minus068 minus001 minus019 minus030 minus029 012 minus040

5390 540 minus142 minus107 minus129 minus112 minus127 minus062 minus100

4650 380 minus027 022 002 008 minus001 060 007

1380 170 minus029 049 038 003 017 minus042 minus017

6710 480 minus005 003 minus021 015 minus010 064 035

2030 340 minus153 minus074 minus088 minus117 minus106 minus126 minus136

2240 210 minus009 069 055 028 038 038 010

6140 430 012 032 009 037 016 079 051

3310 300 minus029 039 022 009 011 053 minus003

7900 560 minus017 minus036 minus062 minus008 minus044 046 021

4360 360 minus024 030 010 012 006 059 010

2560 242 minus020 057 041 019 026 031 002

2620 262 minus036 040 025 003 010 023 minus015

1690 188 minus025 055 042 010 023 minus013 minus010

1490 171 minus022 057 045 011 025 minus038 minus008

2730 281 minus048 027 012 minus009 minus003 019 minus026

2840 287 minus047 027 011 minus008 minus002 022 minus023

1690 191 minus028 052 039 007 020 minus016 minus013

1690 211 minus048 032 019 minus013 000 minus036 minus033

3090 292 minus035 036 019 003 007 031 minus010

3230 308 minus043 027 010 minus004 minus001 036 minus017

1960 222 minus040 039 026 minus004 008 minus010 minus022

1910 208 minus030 050 037 006 018 minus011 minus010

3530 324 minus040 027 009 minus001 minus001 038 minus011

3840 336 minus032 030 011 005 004 057 minus003

2250 247 minus045 033 019 minus008 002 minus005 minus027

2340 257 minus049 028 014 minus012 minus002 minus008 minus029

3100 290 minus033 038 022 006 010 035 minus006

1700 230 minus066 013 001 minus032 minus018 minus060 minus051

4700 400 minus044 004 minus016 minus009 minus018 044 minus009

3700 360 minus065 minus001 minus019 minus027 minus028 021 minus035

4200 400 minus074 minus018 minus037 minus038 minus042 016 minus039

5400 460 minus062 minus026 minus048 minus031 minus046 021 minus024

8 Advances in Materials Science and Engineering

Table 4 Continued

119891119888(MPa) 119891spt (MPa) GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

4200 440 minus114 minus058 minus077 minus078 minus082 minus024 minus079

6200 520 minus075 minus055 minus079 minus050 minus072 003 minus035

6700 425 049 058 033 070 044 115 091

7250 510 minus004 minus008 minus033 012 minus019 056 038

7450 520 minus003 minus011 minus037 011 minus021 059 038

7950 575 minus029 minus049 minus076 minus020 minus057 030 011

7800 515 022 006 minus020 033 minus003 083 060

8500 575 002 minus031 minus059 005 minus036 069 044

8600 480 103 067 039 105 062 167 140

9050 440 168 121 093 165 119 232 204

5480 334 069 103 081 099 083 161 106

3290 285 minus016 053 036 023 025 068 011

3450 292 minus013 055 037 026 027 066 013

5360 348 048 084 062 078 064 136 087

7410 526 minus011 minus018 minus044 003 minus029 058 030

6770 481 minus003 004 minus020 017 minus009 068 037

8420 657 minus085 minus116 minus143 minus081 minus121 minus025 minus046

7530 535 minus013 minus023 minus049 000 minus033 048 028

10230 729 minus055 minus132 minus163 minus073 minus129 052 minus031

4010 330 minus016 044 025 021 018 070 015

4050 340 minus023 035 016 014 010 057 007

9550 570 066 007 minus023 057 006 153 096

3520 317 minus033 033 015 005 006 048 minus005

5750 369 051 078 056 078 060 129 086

3860 338 minus033 029 010 005 003 052 minus004

6150 448 minus005 015 minus009 019 minus002 070 032

4740 376 minus017 030 010 017 008 067 018

4710 368 minus011 037 016 023 014 086 023

4470 326 017 068 048 052 045 109 046

5240 399 minus010 028 006 021 008 079 027

5790 449 minus027 000 minus023 000 minus018 061 012

3950 404 minus093 minus033 minus052 minus056 minus059 000 minus065

3670 403 minus109 minus046 minus064 minus072 minus072 minus029 minus082

3800 365 minus063 minus001 minus020 minus026 minus028 020 minus035

2420 220 minus007 070 055 031 040 038 014

1740 180 minus013 066 054 021 034 minus006 002

2930 280 minus033 039 023 005 010 027 minus009

2700 255 minus023 052 036 015 022 033 minus002

3260 321 minus053 016 minus001 minus015 minus012 026 minus026

4630 520 minus167 minus119 minus139 minus133 minus141 minus076 minus137

3580 430 minus142 minus077 minus095 minus104 minus104 minus057 minus111

4310 418 minus085 minus031 minus050 minus049 minus055 minus004 minus051

3520 317 minus033 033 015 005 006 048 minus005

5750 369 051 078 056 078 060 129 086

4280 490 minus159 minus104 minus124 minus123 minus128 minus076 minus126

6150 448 minus005 015 minus009 019 minus002 070 032

4740 376 minus017 030 010 017 008 067 018

4713 368 minus010 037 016 023 014 086 023

4473 326 017 069 049 052 045 104 050

Advances in Materials Science and Engineering 9

Table 4 Continued

119891119888(MPa) 119891spt (MPa) GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

3460 410 minus130 minus063 minus081 minus091 minus090 minus051 minus102

5970 540 minus108 minus084 minus107 minus082 minus101 minus030 minus069

5190 400 minus014 025 003 017 004 076 021

5000 390 minus015 027 006 017 006 077 022

4850 410 minus044 001 minus020 minus011 minus021 049 minus008

4880 370 minus002 042 021 031 020 080 032

4650 380 minus026 022 002 008 minus001 060 007

4520 400 minus054 minus003 minus024 minus019 minus027 030 minus021

4520 330 016 067 046 051 043 100 049

4260 298 032 087 068 068 063 115 064

8160 440 119 093 066 124 086 180 158

4270 300 031 086 066 066 061 121 064

4190 350 minus024 032 012 012 007 060 009

4120 380 minus059 minus001 minus021 minus022 minus026 031 minus026

3970 270 042 102 083 079 076 124 073

6550 550 minus084 minus073 minus097 minus063 minus087 minus021 minus047

3950 370 minus059 001 minus018 minus022 minus025 034 minus031

1710 190 minus025 054 042 009 022 minus011 minus010

1210 110 018 095 085 048 064 000 031

1290 160 minus026 052 041 005 020 minus048 minus013

1690 150 013 093 080 048 061 025 028

2450 250 minus035 042 027 003 012 016 minus016

3 SVM for Predicting 119891spt of Concrete

As mentioned previously many methods have been pro-posed for the prediction of 119891spt from compressive strengthof concrete This study attempts to utilize SVM for theprediction of concrete of 119891spt SVM-I model is devel-oped for 150 times 300mm cylinder 119891spt prediction from 150 times300mm cylinder 119891

119888of concrete and SVM-IImodel is devel-

oped for 100 times 200mm and 150 times 200mm cylinder 119891sptprediction from 100 times 200mm cylinder 119891

119888of concrete The

experimental data which are taken from studies [9 14 15] areused in this study Among 184 experimental data 138 datawere randomly selected as the training set for SVM-I modeland among 168 cases 126 cases were randomly selected asthe training set for SVM-II model the rest are considered astesting data setThe data are normalized before being used inthe model as follows

119883 =(119883119894minus 119883min)

(119883max minus 119883min) (16)

where 119883max and 119883min are the maximum and minimuminput values data respectively

In case of SVM training two types of kernel func-tions were used namely radical basis function (RBF) andpolynomial function in training process 119862 and 120576 and otherkernel-specific parameters have been chosen by a trial-and-error approach The best simulation performances ofSVM are summarized in Table 1 in order to evaluate theabilities of SVM models and other models mean absolutepercentage error (MAPE) roof-mean-squared error (RMES)

and R-square (1198772) were used as the criteria between theexperimental and predicted values which are according tothe equationsas follow

MAPE =1

119899[sum119899

119894=1

1003816100381610038161003816119905119894 minus 1199001198941003816100381610038161003816

sum119899

119894=1119905119894

times 100]

RMSE = radic1

119899

119899

sum119894=1

(119905119894minus 119900119894)

1198772 =(119899sum 119905

119894119900119894minus sum 119905119894sum119900119894)2

(119899sum 1199052119894minus (sum 1199052

119894)) (119899sum 1199002

119894minus (sum 119900

119894)2

)

(17)

where 119905119894is the experimental value 119900

119894is the predicted value

and 119899 is total number of data

4 Results and Discussion

There is no doubt that the splitting tensile strength 119891spt withan increase in the compression strength of concrete butthere is no agreement on the precise form of the relationshipCodes propose different formulas for prediction cylinder 119891sptof concrete from compressive strength In this paper 119891sptresults are investigated for 150 times 300mm cylinder concreteand 100 times 200mm cylinder concrete separately The errors(predicted values subtract measured values) computed bySVM model other models and measured values of split-ting tensile strengths are shown in Tables 2 and 3 for 150times 300mm cylinder concrete and in Tables 4 and 5 for

10 Advances in Materials Science and Engineering

Table 5 Comparison of experimental results to testing results of RBF polynomial of SVM-II and other models

119891119888(MPa) 119891spt (MPa) GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression RBF [9] Polynomial

3600 349 minus060 005 minus013 minus022 minus022 minus017 minus004

3040 308 minus054 017 001 minus016 minus012 minus010 minus001

2160 200 minus004 074 060 033 043 017 034

2120 270 minus077 002 minus012 minus040 minus030 minus044 minus039

2510 260 minus041 036 021 minus003 005 minus009 006

3190 330 minus067 003 minus014 minus028 minus025 minus026 minus012

3780 310 minus010 053 034 028 026 048 045

2920 250 minus004 069 053 034 040 041 049

3540 310 minus025 041 023 013 014 019 032

2860 340 minus098 minus024 minus041 minus059 minus054 minus051 minus046

2950 310 minus062 010 minus006 minus024 minus019 minus023 minus008

3490 350 minus068 minus001 minus019 minus030 minus029 minus023 minus011

3860 338 minus033 029 010 005 003 015 023

5235 399 minus010 028 006 021 008 042 036

3840 380 minus076 minus014 minus033 minus039 minus040 minus023 minus020

5150 420 minus036 003 minus018 minus005 minus017 016 011

4860 400 minus034 011 minus010 000 minus011 011 016

3950 390 minus079 minus019 minus038 minus042 minus045 minus022 minus022

3040 285 minus031 040 024 007 011 013 022

2950 256 minus008 064 048 030 035 031 046

4280 430 minus099 minus044 minus064 minus063 minus068 minus052 minus046

5190 430 minus044 minus005 minus027 minus013 minus026 009 003

4930 420 minus049 minus006 minus027 minus017 minus027 003 000

4330 360 minus026 028 008 010 004 022 028

4360 338 minus002 052 032 034 028 046 052

4990 457 minus083 minus040 minus061 minus050 minus062 minus034 minus033

5740 420 minus001 027 004 026 009 054 039

5850 490 minus065 minus039 minus062 minus038 minus056 minus015 minus024

3140 279 minus019 052 035 020 023 026 036

3490 304 minus022 045 027 016 017 023 035

4160 384 minus060 minus003 minus023 minus024 minus028 minus011 minus005

4680 400 minus044 004 minus017 minus010 minus019 001 009

5170 420 minus035 004 minus017 minus004 minus016 018 012

4330 390 minus056 minus002 minus022 minus020 minus026 minus008 minus002

6080 410 029 050 027 054 033 076 068

6000 430 004 027 004 030 010 060 044

7190 540 minus037 minus040 minus065 minus021 minus051 016 000

7510 540 minus018 minus029 minus055 minus006 minus039 030 020

5787 449 minus027 000 minus023 000 minus018 027 015

3950 340 minus029 031 012 008 005 028 028

Table 6 Error measurement for 119891spt from 150 times 300mm cylinder 119891119888

GEP-I [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF PolynomialRMSE of training set 04603 05995 05228 05027 04512 04978 04819RMSE of testing set 04271 02735 03152 03754 02874 02748 04176MAPE of training set 97554 13647 114913 106213 95596 102434 105454MAPE of testing set 73022 46139 55434 62059 5286 4987 695171198772 of training set 08282 08225 08224 08267 0826 08115 082271198772 of testing set 09416 09486 09486 09471 09476 09422 09327

Advances in Materials Science and Engineering 11

Table 7 Error measurement for 119891spt from 100 times 200mm cylinder 119891119888

GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF PolynomialRMSE of training set 05886 0539 05244 04987 04911 07287 0562RMSE of testing set 0483 03348 03393 02803 03155 03082 02917MAPE of training set 118729 116879 109223 92976 9472 155902 106658MAPE of testing set 108701 71383 7604 62261 7234 69713 628351198772 of training set 08198 08354 08353 083 08337 0837 082531198772 of testing set 08778 08802 08793 088 08801 08815 08823

0

2

4

6

8

0 20 40 60 80 100 120 140

Split

ting

tens

ile st

reng

th (M

Pa)

Sample number

ExperimentalRBFPolynomial

Figure 1 Comparison of experimental results to training results ofSVM-I

100 times 200mm cylinder concrete As seen from these resultsSVM model is sufficiently close to the experimental dataand is able to predict the 119891spt from the 119891

119888of concrete

For convenient comparison purpose the experimental andpredicted results are plotted in Figures 1 2 3 and 4 It canbe seen that the results from SVM method are in goodagreement with the experimental results

The 119877-square errors in the trained SVM are 08115 and08227 for RBF and polynomial function of SVM-I 08370and 08253 for those of SVM-II respectively 119877-square errorsin the tested SVM are 09422 and 09327 for RBF and polyno-mial function of SVM-I 08815 and 08823 for those of SVM-II respectively The whole results obtained from the SVMmodels show a successful performance of the SVM modelsof predicting 150 times 300mm cylinder119891spt of concrete from thecorresponding 150 times 300mm cylinder compressive strength119891119888and 100 times 200mm cylinder 119891spt from the corresponding

100 times 200mm cylinder compressive strength 119891119888of concrete

for each of the training and testing sets It is also clear thatthere are no major difference on performance between theradical basis function and the polynomial function kernelsthat used in this paper However in generally the radical basisfunction kernel exhibits slightly better performance than thepolynomial kernel

The performance of the trained and tested sets is analyzedby computing MAPE RMSE and 1198772 of experimental 119891spt

0

2

4

6

8

0 5 10 15 20 25 30 35 40 45 50

Split

ting

tens

ile st

reng

th (M

Pa)

Sample number

ExperimentalRBFPolynomial

Figure 2 Comparison of experimental results to testing results ofSVM-I

between the SVM model and other methods The MAPERMSE and 1198772 are calculated for these methods Statisticalparameters of the training and testing sets of all methods arepresented in Tables 6 and 7 The model having the smallestMAPE or RMSE and the highest 1198772 can be regarded to bethe best model with the assumption of analysis Generally1198772 of the SVM models are higher than other methodsand the MAPE and RMSE of SVM are smaller than othermodels The analysis shows that the SVM model is betterthan other models and can predict the splitting tensilestrength of concrete well Furthermore SVMmodel has goodgeneralization capacity to avoid over training and can alwaysbe updated to get better results by presenting new trainingexamples as new data become available Thus SVM modelcan be regarded as a very effective method to predict splittingtensile strength of concrete from their compressive strength

5 Conclusion

The splitting tensile strength of concrete estimations fromcompression strength has been obtained so far in the litera-ture either through regression or other methodsThis presentstudy reports a new and influential approach for predictingthe splitting tensile strength using SVM for the first timein the literature The study conducted in this paper shows

12 Advances in Materials Science and Engineering

Polynomial

0

1

2

3

4

5

6

7

8

9

0 20 40 60 80 100 120 140

Split

ting

tens

ile st

reng

hth

(MPa

)

Sample number

ExperimentalRBF

Figure 3 Comparison of experimental results to training results ofSVM-II

Polynomial

0

2

4

6

0 10 20 30 40 50

Split

ting

tens

ile st

reng

th (M

Pa)

Sample number

ExperimentalRBF

Figure 4 Comparison of experimental results to testing results ofSVM-II

the feasibility of using a simple SVM to estimate the splittingtensile strength of concrete After learning from a set ofselected training data involving compressive strengths ofconcrete collected from the previous literature the SVM canbe utilized to predict the splitting tensile strengths of the testdata

In this paper radical basis function and polynomial areadopted for predicting splitting tensile strength from 150 times300mm cylinder and 100times200mm cylinder 119891

119888of concrete

It is found that the splitting tensile strength obtained fromthe SVM is more accurate than those obtained from designcodes and several researchesrsquo empirical equation when acomparison is made on the basis of the experimental dataSince the SVM is largely characterized by the type of its kernel

function it is necessary to choose the appropriate kernel foreach particular application problem in order to guaranteesatisfactory results The results of radial basis function andpolynomial indicate that RBF and polynomial kernel have theability to predict the splitting tensile strength of concrete fromcompression strength with an acceptable degree of accuracy

The statistical parameters of MAPE RMSE and 1198772 showthat the proposed SVM model results have the best accu-racy and can predict splitting tensile strength very close toexperiment results The use of SVM is very advantageousfor the prediction of the splitting tensile strength of concretefrom compression strength because it can perform nonlinearregression efficiently for high-dimensional data sets Further-more its solution is globalThe satisfactory predictions of thesplitting tensile strength of concrete by the model indicatethat SVM is a useful modeling tool for engineers and researchscientists at concrete construction fields

Acknowledgments

This project was granted financial support from the NationalNatural Science Foundation of China (no 50808077 andno 51278188) and the Fundamental Research Funds for theCentral Universities The authors are grateful for support

References

[1] B W Xu and H S Shi ldquoCorrelations among mechanicalproperties of steel fiber reinforced concreterdquo Construction andBuilding Materials vol 23 no 12 pp 3468ndash3474 2009

[2] M F M Zain H B Mahmud A Ilham and M FaizalldquoPrediction of splitting tensile strength of high-performanceconcreterdquoCement andConcrete Research vol 32 no 8 pp 1251ndash1258 2002

[3] F A Oluokun E G Burdette and J H Deatherage ldquoSplittingtensile strength and compressive strength relationship at earlyagesrdquo ACI Materials Journal vol 88 no 2 pp 115ndash121 1991

[4] Y Choi and R L Yuan ldquoExperimental relationship betweensplitting tensile strength and compressive strength of GFRC andPFRCrdquo Cement and Concrete Research vol 35 no 8 pp 1587ndash1591 2005

[5] F D Larrad and Y Malier ldquoEngineering properties of very highperformance concreterdquo in High Performance Concrete fromMaterial to Structure Y Malier Ed pp 85ndash114 EFN SponLondon UK 1992

[6] J K Kim S H Han Y D Park and J H Noh ldquoMaterialproperties of self-flowing concreterdquo Journal of Materials in CivilEngineering vol 10 no 4 pp 244ndash249 1998

[7] J K Kim S H Han and Y C Song ldquoEffect of temperatureand aging on the mechanical properties of concretemdashpart Iexperimental resultsrdquo Cement and Concrete Research vol 32no 7 pp 1087ndash1094 2002

[8] AM Shaaban andHGesund ldquoSplitting tensile strength of steelfiber reinforced concrete cylinders consolidated by rodding orvibratingrdquo ACI Materials Journal vol 90 no 4 pp 366ndash3691993

[9] M Saridemir ldquoEmpirical modeling of splitting tensile strengthfrom cylinder compressive strength of concrete by geneticprogrammingrdquo Expert Systems with Applications vol 38 no 11pp 14257ndash14268 2011

Advances in Materials Science and Engineering 13

[10] U Thissen R van Brakel A P de Weijer W J Melssen and LM C Buydens ldquoUsing support vector machines for time seriespredictionrdquo Chemometrics and Intelligent Laboratory Systemsvol 69 no 1-2 pp 35ndash49 2003

[11] P Samui ldquoPredicted ultimate capacity of laterally loaded pilesin clay using support vector machinerdquo Geomechanics andGeoengineering vol 3 no 2 pp 113ndash120 2008

[12] VN VapnikTheNature of Statistical LearningTheory SpringerNew York NY USA 1995

[13] V N Vapnik S E Golowich and A J Smola ldquoSupport vectormachine for function approximation regression estimationand signal processingrdquoAdvances in Neural Information Process-ing Systems vol 9 pp 281ndash287 1996

[14] M Khanzadi and A Behnood ldquoMechanical properties ofhigh-strength concrete incorporating copper slag as coarseaggregaterdquo Construction and Building Materials vol 23 no 6pp 2183ndash2188 2009

[15] C K Y Leung and J Pan ldquoEffect of concrete composition onFRPconcrete bond capacityrdquo in Proceedings of the InternationalSymposium on Bond Behaviour of FRP in Structures (BBFS rsquo05)pp 69ndash76 2005

Submit your manuscripts athttpwwwhindawicom

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CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

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International Journal of

Biomaterials

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NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CrystallographyJournal of

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

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BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

8 Advances in Materials Science and Engineering

Table 4 Continued

119891119888(MPa) 119891spt (MPa) GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

4200 440 minus114 minus058 minus077 minus078 minus082 minus024 minus079

6200 520 minus075 minus055 minus079 minus050 minus072 003 minus035

6700 425 049 058 033 070 044 115 091

7250 510 minus004 minus008 minus033 012 minus019 056 038

7450 520 minus003 minus011 minus037 011 minus021 059 038

7950 575 minus029 minus049 minus076 minus020 minus057 030 011

7800 515 022 006 minus020 033 minus003 083 060

8500 575 002 minus031 minus059 005 minus036 069 044

8600 480 103 067 039 105 062 167 140

9050 440 168 121 093 165 119 232 204

5480 334 069 103 081 099 083 161 106

3290 285 minus016 053 036 023 025 068 011

3450 292 minus013 055 037 026 027 066 013

5360 348 048 084 062 078 064 136 087

7410 526 minus011 minus018 minus044 003 minus029 058 030

6770 481 minus003 004 minus020 017 minus009 068 037

8420 657 minus085 minus116 minus143 minus081 minus121 minus025 minus046

7530 535 minus013 minus023 minus049 000 minus033 048 028

10230 729 minus055 minus132 minus163 minus073 minus129 052 minus031

4010 330 minus016 044 025 021 018 070 015

4050 340 minus023 035 016 014 010 057 007

9550 570 066 007 minus023 057 006 153 096

3520 317 minus033 033 015 005 006 048 minus005

5750 369 051 078 056 078 060 129 086

3860 338 minus033 029 010 005 003 052 minus004

6150 448 minus005 015 minus009 019 minus002 070 032

4740 376 minus017 030 010 017 008 067 018

4710 368 minus011 037 016 023 014 086 023

4470 326 017 068 048 052 045 109 046

5240 399 minus010 028 006 021 008 079 027

5790 449 minus027 000 minus023 000 minus018 061 012

3950 404 minus093 minus033 minus052 minus056 minus059 000 minus065

3670 403 minus109 minus046 minus064 minus072 minus072 minus029 minus082

3800 365 minus063 minus001 minus020 minus026 minus028 020 minus035

2420 220 minus007 070 055 031 040 038 014

1740 180 minus013 066 054 021 034 minus006 002

2930 280 minus033 039 023 005 010 027 minus009

2700 255 minus023 052 036 015 022 033 minus002

3260 321 minus053 016 minus001 minus015 minus012 026 minus026

4630 520 minus167 minus119 minus139 minus133 minus141 minus076 minus137

3580 430 minus142 minus077 minus095 minus104 minus104 minus057 minus111

4310 418 minus085 minus031 minus050 minus049 minus055 minus004 minus051

3520 317 minus033 033 015 005 006 048 minus005

5750 369 051 078 056 078 060 129 086

4280 490 minus159 minus104 minus124 minus123 minus128 minus076 minus126

6150 448 minus005 015 minus009 019 minus002 070 032

4740 376 minus017 030 010 017 008 067 018

4713 368 minus010 037 016 023 014 086 023

4473 326 017 069 049 052 045 104 050

Advances in Materials Science and Engineering 9

Table 4 Continued

119891119888(MPa) 119891spt (MPa) GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

3460 410 minus130 minus063 minus081 minus091 minus090 minus051 minus102

5970 540 minus108 minus084 minus107 minus082 minus101 minus030 minus069

5190 400 minus014 025 003 017 004 076 021

5000 390 minus015 027 006 017 006 077 022

4850 410 minus044 001 minus020 minus011 minus021 049 minus008

4880 370 minus002 042 021 031 020 080 032

4650 380 minus026 022 002 008 minus001 060 007

4520 400 minus054 minus003 minus024 minus019 minus027 030 minus021

4520 330 016 067 046 051 043 100 049

4260 298 032 087 068 068 063 115 064

8160 440 119 093 066 124 086 180 158

4270 300 031 086 066 066 061 121 064

4190 350 minus024 032 012 012 007 060 009

4120 380 minus059 minus001 minus021 minus022 minus026 031 minus026

3970 270 042 102 083 079 076 124 073

6550 550 minus084 minus073 minus097 minus063 minus087 minus021 minus047

3950 370 minus059 001 minus018 minus022 minus025 034 minus031

1710 190 minus025 054 042 009 022 minus011 minus010

1210 110 018 095 085 048 064 000 031

1290 160 minus026 052 041 005 020 minus048 minus013

1690 150 013 093 080 048 061 025 028

2450 250 minus035 042 027 003 012 016 minus016

3 SVM for Predicting 119891spt of Concrete

As mentioned previously many methods have been pro-posed for the prediction of 119891spt from compressive strengthof concrete This study attempts to utilize SVM for theprediction of concrete of 119891spt SVM-I model is devel-oped for 150 times 300mm cylinder 119891spt prediction from 150 times300mm cylinder 119891

119888of concrete and SVM-IImodel is devel-

oped for 100 times 200mm and 150 times 200mm cylinder 119891sptprediction from 100 times 200mm cylinder 119891

119888of concrete The

experimental data which are taken from studies [9 14 15] areused in this study Among 184 experimental data 138 datawere randomly selected as the training set for SVM-I modeland among 168 cases 126 cases were randomly selected asthe training set for SVM-II model the rest are considered astesting data setThe data are normalized before being used inthe model as follows

119883 =(119883119894minus 119883min)

(119883max minus 119883min) (16)

where 119883max and 119883min are the maximum and minimuminput values data respectively

In case of SVM training two types of kernel func-tions were used namely radical basis function (RBF) andpolynomial function in training process 119862 and 120576 and otherkernel-specific parameters have been chosen by a trial-and-error approach The best simulation performances ofSVM are summarized in Table 1 in order to evaluate theabilities of SVM models and other models mean absolutepercentage error (MAPE) roof-mean-squared error (RMES)

and R-square (1198772) were used as the criteria between theexperimental and predicted values which are according tothe equationsas follow

MAPE =1

119899[sum119899

119894=1

1003816100381610038161003816119905119894 minus 1199001198941003816100381610038161003816

sum119899

119894=1119905119894

times 100]

RMSE = radic1

119899

119899

sum119894=1

(119905119894minus 119900119894)

1198772 =(119899sum 119905

119894119900119894minus sum 119905119894sum119900119894)2

(119899sum 1199052119894minus (sum 1199052

119894)) (119899sum 1199002

119894minus (sum 119900

119894)2

)

(17)

where 119905119894is the experimental value 119900

119894is the predicted value

and 119899 is total number of data

4 Results and Discussion

There is no doubt that the splitting tensile strength 119891spt withan increase in the compression strength of concrete butthere is no agreement on the precise form of the relationshipCodes propose different formulas for prediction cylinder 119891sptof concrete from compressive strength In this paper 119891sptresults are investigated for 150 times 300mm cylinder concreteand 100 times 200mm cylinder concrete separately The errors(predicted values subtract measured values) computed bySVM model other models and measured values of split-ting tensile strengths are shown in Tables 2 and 3 for 150times 300mm cylinder concrete and in Tables 4 and 5 for

10 Advances in Materials Science and Engineering

Table 5 Comparison of experimental results to testing results of RBF polynomial of SVM-II and other models

119891119888(MPa) 119891spt (MPa) GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression RBF [9] Polynomial

3600 349 minus060 005 minus013 minus022 minus022 minus017 minus004

3040 308 minus054 017 001 minus016 minus012 minus010 minus001

2160 200 minus004 074 060 033 043 017 034

2120 270 minus077 002 minus012 minus040 minus030 minus044 minus039

2510 260 minus041 036 021 minus003 005 minus009 006

3190 330 minus067 003 minus014 minus028 minus025 minus026 minus012

3780 310 minus010 053 034 028 026 048 045

2920 250 minus004 069 053 034 040 041 049

3540 310 minus025 041 023 013 014 019 032

2860 340 minus098 minus024 minus041 minus059 minus054 minus051 minus046

2950 310 minus062 010 minus006 minus024 minus019 minus023 minus008

3490 350 minus068 minus001 minus019 minus030 minus029 minus023 minus011

3860 338 minus033 029 010 005 003 015 023

5235 399 minus010 028 006 021 008 042 036

3840 380 minus076 minus014 minus033 minus039 minus040 minus023 minus020

5150 420 minus036 003 minus018 minus005 minus017 016 011

4860 400 minus034 011 minus010 000 minus011 011 016

3950 390 minus079 minus019 minus038 minus042 minus045 minus022 minus022

3040 285 minus031 040 024 007 011 013 022

2950 256 minus008 064 048 030 035 031 046

4280 430 minus099 minus044 minus064 minus063 minus068 minus052 minus046

5190 430 minus044 minus005 minus027 minus013 minus026 009 003

4930 420 minus049 minus006 minus027 minus017 minus027 003 000

4330 360 minus026 028 008 010 004 022 028

4360 338 minus002 052 032 034 028 046 052

4990 457 minus083 minus040 minus061 minus050 minus062 minus034 minus033

5740 420 minus001 027 004 026 009 054 039

5850 490 minus065 minus039 minus062 minus038 minus056 minus015 minus024

3140 279 minus019 052 035 020 023 026 036

3490 304 minus022 045 027 016 017 023 035

4160 384 minus060 minus003 minus023 minus024 minus028 minus011 minus005

4680 400 minus044 004 minus017 minus010 minus019 001 009

5170 420 minus035 004 minus017 minus004 minus016 018 012

4330 390 minus056 minus002 minus022 minus020 minus026 minus008 minus002

6080 410 029 050 027 054 033 076 068

6000 430 004 027 004 030 010 060 044

7190 540 minus037 minus040 minus065 minus021 minus051 016 000

7510 540 minus018 minus029 minus055 minus006 minus039 030 020

5787 449 minus027 000 minus023 000 minus018 027 015

3950 340 minus029 031 012 008 005 028 028

Table 6 Error measurement for 119891spt from 150 times 300mm cylinder 119891119888

GEP-I [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF PolynomialRMSE of training set 04603 05995 05228 05027 04512 04978 04819RMSE of testing set 04271 02735 03152 03754 02874 02748 04176MAPE of training set 97554 13647 114913 106213 95596 102434 105454MAPE of testing set 73022 46139 55434 62059 5286 4987 695171198772 of training set 08282 08225 08224 08267 0826 08115 082271198772 of testing set 09416 09486 09486 09471 09476 09422 09327

Advances in Materials Science and Engineering 11

Table 7 Error measurement for 119891spt from 100 times 200mm cylinder 119891119888

GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF PolynomialRMSE of training set 05886 0539 05244 04987 04911 07287 0562RMSE of testing set 0483 03348 03393 02803 03155 03082 02917MAPE of training set 118729 116879 109223 92976 9472 155902 106658MAPE of testing set 108701 71383 7604 62261 7234 69713 628351198772 of training set 08198 08354 08353 083 08337 0837 082531198772 of testing set 08778 08802 08793 088 08801 08815 08823

0

2

4

6

8

0 20 40 60 80 100 120 140

Split

ting

tens

ile st

reng

th (M

Pa)

Sample number

ExperimentalRBFPolynomial

Figure 1 Comparison of experimental results to training results ofSVM-I

100 times 200mm cylinder concrete As seen from these resultsSVM model is sufficiently close to the experimental dataand is able to predict the 119891spt from the 119891

119888of concrete

For convenient comparison purpose the experimental andpredicted results are plotted in Figures 1 2 3 and 4 It canbe seen that the results from SVM method are in goodagreement with the experimental results

The 119877-square errors in the trained SVM are 08115 and08227 for RBF and polynomial function of SVM-I 08370and 08253 for those of SVM-II respectively 119877-square errorsin the tested SVM are 09422 and 09327 for RBF and polyno-mial function of SVM-I 08815 and 08823 for those of SVM-II respectively The whole results obtained from the SVMmodels show a successful performance of the SVM modelsof predicting 150 times 300mm cylinder119891spt of concrete from thecorresponding 150 times 300mm cylinder compressive strength119891119888and 100 times 200mm cylinder 119891spt from the corresponding

100 times 200mm cylinder compressive strength 119891119888of concrete

for each of the training and testing sets It is also clear thatthere are no major difference on performance between theradical basis function and the polynomial function kernelsthat used in this paper However in generally the radical basisfunction kernel exhibits slightly better performance than thepolynomial kernel

The performance of the trained and tested sets is analyzedby computing MAPE RMSE and 1198772 of experimental 119891spt

0

2

4

6

8

0 5 10 15 20 25 30 35 40 45 50

Split

ting

tens

ile st

reng

th (M

Pa)

Sample number

ExperimentalRBFPolynomial

Figure 2 Comparison of experimental results to testing results ofSVM-I

between the SVM model and other methods The MAPERMSE and 1198772 are calculated for these methods Statisticalparameters of the training and testing sets of all methods arepresented in Tables 6 and 7 The model having the smallestMAPE or RMSE and the highest 1198772 can be regarded to bethe best model with the assumption of analysis Generally1198772 of the SVM models are higher than other methodsand the MAPE and RMSE of SVM are smaller than othermodels The analysis shows that the SVM model is betterthan other models and can predict the splitting tensilestrength of concrete well Furthermore SVMmodel has goodgeneralization capacity to avoid over training and can alwaysbe updated to get better results by presenting new trainingexamples as new data become available Thus SVM modelcan be regarded as a very effective method to predict splittingtensile strength of concrete from their compressive strength

5 Conclusion

The splitting tensile strength of concrete estimations fromcompression strength has been obtained so far in the litera-ture either through regression or other methodsThis presentstudy reports a new and influential approach for predictingthe splitting tensile strength using SVM for the first timein the literature The study conducted in this paper shows

12 Advances in Materials Science and Engineering

Polynomial

0

1

2

3

4

5

6

7

8

9

0 20 40 60 80 100 120 140

Split

ting

tens

ile st

reng

hth

(MPa

)

Sample number

ExperimentalRBF

Figure 3 Comparison of experimental results to training results ofSVM-II

Polynomial

0

2

4

6

0 10 20 30 40 50

Split

ting

tens

ile st

reng

th (M

Pa)

Sample number

ExperimentalRBF

Figure 4 Comparison of experimental results to testing results ofSVM-II

the feasibility of using a simple SVM to estimate the splittingtensile strength of concrete After learning from a set ofselected training data involving compressive strengths ofconcrete collected from the previous literature the SVM canbe utilized to predict the splitting tensile strengths of the testdata

In this paper radical basis function and polynomial areadopted for predicting splitting tensile strength from 150 times300mm cylinder and 100times200mm cylinder 119891

119888of concrete

It is found that the splitting tensile strength obtained fromthe SVM is more accurate than those obtained from designcodes and several researchesrsquo empirical equation when acomparison is made on the basis of the experimental dataSince the SVM is largely characterized by the type of its kernel

function it is necessary to choose the appropriate kernel foreach particular application problem in order to guaranteesatisfactory results The results of radial basis function andpolynomial indicate that RBF and polynomial kernel have theability to predict the splitting tensile strength of concrete fromcompression strength with an acceptable degree of accuracy

The statistical parameters of MAPE RMSE and 1198772 showthat the proposed SVM model results have the best accu-racy and can predict splitting tensile strength very close toexperiment results The use of SVM is very advantageousfor the prediction of the splitting tensile strength of concretefrom compression strength because it can perform nonlinearregression efficiently for high-dimensional data sets Further-more its solution is globalThe satisfactory predictions of thesplitting tensile strength of concrete by the model indicatethat SVM is a useful modeling tool for engineers and researchscientists at concrete construction fields

Acknowledgments

This project was granted financial support from the NationalNatural Science Foundation of China (no 50808077 andno 51278188) and the Fundamental Research Funds for theCentral Universities The authors are grateful for support

References

[1] B W Xu and H S Shi ldquoCorrelations among mechanicalproperties of steel fiber reinforced concreterdquo Construction andBuilding Materials vol 23 no 12 pp 3468ndash3474 2009

[2] M F M Zain H B Mahmud A Ilham and M FaizalldquoPrediction of splitting tensile strength of high-performanceconcreterdquoCement andConcrete Research vol 32 no 8 pp 1251ndash1258 2002

[3] F A Oluokun E G Burdette and J H Deatherage ldquoSplittingtensile strength and compressive strength relationship at earlyagesrdquo ACI Materials Journal vol 88 no 2 pp 115ndash121 1991

[4] Y Choi and R L Yuan ldquoExperimental relationship betweensplitting tensile strength and compressive strength of GFRC andPFRCrdquo Cement and Concrete Research vol 35 no 8 pp 1587ndash1591 2005

[5] F D Larrad and Y Malier ldquoEngineering properties of very highperformance concreterdquo in High Performance Concrete fromMaterial to Structure Y Malier Ed pp 85ndash114 EFN SponLondon UK 1992

[6] J K Kim S H Han Y D Park and J H Noh ldquoMaterialproperties of self-flowing concreterdquo Journal of Materials in CivilEngineering vol 10 no 4 pp 244ndash249 1998

[7] J K Kim S H Han and Y C Song ldquoEffect of temperatureand aging on the mechanical properties of concretemdashpart Iexperimental resultsrdquo Cement and Concrete Research vol 32no 7 pp 1087ndash1094 2002

[8] AM Shaaban andHGesund ldquoSplitting tensile strength of steelfiber reinforced concrete cylinders consolidated by rodding orvibratingrdquo ACI Materials Journal vol 90 no 4 pp 366ndash3691993

[9] M Saridemir ldquoEmpirical modeling of splitting tensile strengthfrom cylinder compressive strength of concrete by geneticprogrammingrdquo Expert Systems with Applications vol 38 no 11pp 14257ndash14268 2011

Advances in Materials Science and Engineering 13

[10] U Thissen R van Brakel A P de Weijer W J Melssen and LM C Buydens ldquoUsing support vector machines for time seriespredictionrdquo Chemometrics and Intelligent Laboratory Systemsvol 69 no 1-2 pp 35ndash49 2003

[11] P Samui ldquoPredicted ultimate capacity of laterally loaded pilesin clay using support vector machinerdquo Geomechanics andGeoengineering vol 3 no 2 pp 113ndash120 2008

[12] VN VapnikTheNature of Statistical LearningTheory SpringerNew York NY USA 1995

[13] V N Vapnik S E Golowich and A J Smola ldquoSupport vectormachine for function approximation regression estimationand signal processingrdquoAdvances in Neural Information Process-ing Systems vol 9 pp 281ndash287 1996

[14] M Khanzadi and A Behnood ldquoMechanical properties ofhigh-strength concrete incorporating copper slag as coarseaggregaterdquo Construction and Building Materials vol 23 no 6pp 2183ndash2188 2009

[15] C K Y Leung and J Pan ldquoEffect of concrete composition onFRPconcrete bond capacityrdquo in Proceedings of the InternationalSymposium on Bond Behaviour of FRP in Structures (BBFS rsquo05)pp 69ndash76 2005

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

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Advances in

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Nano

materials

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Journal ofNanomaterials

Advances in Materials Science and Engineering 9

Table 4 Continued

119891119888(MPa) 119891spt (MPa) GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF Polynomial

3460 410 minus130 minus063 minus081 minus091 minus090 minus051 minus102

5970 540 minus108 minus084 minus107 minus082 minus101 minus030 minus069

5190 400 minus014 025 003 017 004 076 021

5000 390 minus015 027 006 017 006 077 022

4850 410 minus044 001 minus020 minus011 minus021 049 minus008

4880 370 minus002 042 021 031 020 080 032

4650 380 minus026 022 002 008 minus001 060 007

4520 400 minus054 minus003 minus024 minus019 minus027 030 minus021

4520 330 016 067 046 051 043 100 049

4260 298 032 087 068 068 063 115 064

8160 440 119 093 066 124 086 180 158

4270 300 031 086 066 066 061 121 064

4190 350 minus024 032 012 012 007 060 009

4120 380 minus059 minus001 minus021 minus022 minus026 031 minus026

3970 270 042 102 083 079 076 124 073

6550 550 minus084 minus073 minus097 minus063 minus087 minus021 minus047

3950 370 minus059 001 minus018 minus022 minus025 034 minus031

1710 190 minus025 054 042 009 022 minus011 minus010

1210 110 018 095 085 048 064 000 031

1290 160 minus026 052 041 005 020 minus048 minus013

1690 150 013 093 080 048 061 025 028

2450 250 minus035 042 027 003 012 016 minus016

3 SVM for Predicting 119891spt of Concrete

As mentioned previously many methods have been pro-posed for the prediction of 119891spt from compressive strengthof concrete This study attempts to utilize SVM for theprediction of concrete of 119891spt SVM-I model is devel-oped for 150 times 300mm cylinder 119891spt prediction from 150 times300mm cylinder 119891

119888of concrete and SVM-IImodel is devel-

oped for 100 times 200mm and 150 times 200mm cylinder 119891sptprediction from 100 times 200mm cylinder 119891

119888of concrete The

experimental data which are taken from studies [9 14 15] areused in this study Among 184 experimental data 138 datawere randomly selected as the training set for SVM-I modeland among 168 cases 126 cases were randomly selected asthe training set for SVM-II model the rest are considered astesting data setThe data are normalized before being used inthe model as follows

119883 =(119883119894minus 119883min)

(119883max minus 119883min) (16)

where 119883max and 119883min are the maximum and minimuminput values data respectively

In case of SVM training two types of kernel func-tions were used namely radical basis function (RBF) andpolynomial function in training process 119862 and 120576 and otherkernel-specific parameters have been chosen by a trial-and-error approach The best simulation performances ofSVM are summarized in Table 1 in order to evaluate theabilities of SVM models and other models mean absolutepercentage error (MAPE) roof-mean-squared error (RMES)

and R-square (1198772) were used as the criteria between theexperimental and predicted values which are according tothe equationsas follow

MAPE =1

119899[sum119899

119894=1

1003816100381610038161003816119905119894 minus 1199001198941003816100381610038161003816

sum119899

119894=1119905119894

times 100]

RMSE = radic1

119899

119899

sum119894=1

(119905119894minus 119900119894)

1198772 =(119899sum 119905

119894119900119894minus sum 119905119894sum119900119894)2

(119899sum 1199052119894minus (sum 1199052

119894)) (119899sum 1199002

119894minus (sum 119900

119894)2

)

(17)

where 119905119894is the experimental value 119900

119894is the predicted value

and 119899 is total number of data

4 Results and Discussion

There is no doubt that the splitting tensile strength 119891spt withan increase in the compression strength of concrete butthere is no agreement on the precise form of the relationshipCodes propose different formulas for prediction cylinder 119891sptof concrete from compressive strength In this paper 119891sptresults are investigated for 150 times 300mm cylinder concreteand 100 times 200mm cylinder concrete separately The errors(predicted values subtract measured values) computed bySVM model other models and measured values of split-ting tensile strengths are shown in Tables 2 and 3 for 150times 300mm cylinder concrete and in Tables 4 and 5 for

10 Advances in Materials Science and Engineering

Table 5 Comparison of experimental results to testing results of RBF polynomial of SVM-II and other models

119891119888(MPa) 119891spt (MPa) GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression RBF [9] Polynomial

3600 349 minus060 005 minus013 minus022 minus022 minus017 minus004

3040 308 minus054 017 001 minus016 minus012 minus010 minus001

2160 200 minus004 074 060 033 043 017 034

2120 270 minus077 002 minus012 minus040 minus030 minus044 minus039

2510 260 minus041 036 021 minus003 005 minus009 006

3190 330 minus067 003 minus014 minus028 minus025 minus026 minus012

3780 310 minus010 053 034 028 026 048 045

2920 250 minus004 069 053 034 040 041 049

3540 310 minus025 041 023 013 014 019 032

2860 340 minus098 minus024 minus041 minus059 minus054 minus051 minus046

2950 310 minus062 010 minus006 minus024 minus019 minus023 minus008

3490 350 minus068 minus001 minus019 minus030 minus029 minus023 minus011

3860 338 minus033 029 010 005 003 015 023

5235 399 minus010 028 006 021 008 042 036

3840 380 minus076 minus014 minus033 minus039 minus040 minus023 minus020

5150 420 minus036 003 minus018 minus005 minus017 016 011

4860 400 minus034 011 minus010 000 minus011 011 016

3950 390 minus079 minus019 minus038 minus042 minus045 minus022 minus022

3040 285 minus031 040 024 007 011 013 022

2950 256 minus008 064 048 030 035 031 046

4280 430 minus099 minus044 minus064 minus063 minus068 minus052 minus046

5190 430 minus044 minus005 minus027 minus013 minus026 009 003

4930 420 minus049 minus006 minus027 minus017 minus027 003 000

4330 360 minus026 028 008 010 004 022 028

4360 338 minus002 052 032 034 028 046 052

4990 457 minus083 minus040 minus061 minus050 minus062 minus034 minus033

5740 420 minus001 027 004 026 009 054 039

5850 490 minus065 minus039 minus062 minus038 minus056 minus015 minus024

3140 279 minus019 052 035 020 023 026 036

3490 304 minus022 045 027 016 017 023 035

4160 384 minus060 minus003 minus023 minus024 minus028 minus011 minus005

4680 400 minus044 004 minus017 minus010 minus019 001 009

5170 420 minus035 004 minus017 minus004 minus016 018 012

4330 390 minus056 minus002 minus022 minus020 minus026 minus008 minus002

6080 410 029 050 027 054 033 076 068

6000 430 004 027 004 030 010 060 044

7190 540 minus037 minus040 minus065 minus021 minus051 016 000

7510 540 minus018 minus029 minus055 minus006 minus039 030 020

5787 449 minus027 000 minus023 000 minus018 027 015

3950 340 minus029 031 012 008 005 028 028

Table 6 Error measurement for 119891spt from 150 times 300mm cylinder 119891119888

GEP-I [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF PolynomialRMSE of training set 04603 05995 05228 05027 04512 04978 04819RMSE of testing set 04271 02735 03152 03754 02874 02748 04176MAPE of training set 97554 13647 114913 106213 95596 102434 105454MAPE of testing set 73022 46139 55434 62059 5286 4987 695171198772 of training set 08282 08225 08224 08267 0826 08115 082271198772 of testing set 09416 09486 09486 09471 09476 09422 09327

Advances in Materials Science and Engineering 11

Table 7 Error measurement for 119891spt from 100 times 200mm cylinder 119891119888

GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF PolynomialRMSE of training set 05886 0539 05244 04987 04911 07287 0562RMSE of testing set 0483 03348 03393 02803 03155 03082 02917MAPE of training set 118729 116879 109223 92976 9472 155902 106658MAPE of testing set 108701 71383 7604 62261 7234 69713 628351198772 of training set 08198 08354 08353 083 08337 0837 082531198772 of testing set 08778 08802 08793 088 08801 08815 08823

0

2

4

6

8

0 20 40 60 80 100 120 140

Split

ting

tens

ile st

reng

th (M

Pa)

Sample number

ExperimentalRBFPolynomial

Figure 1 Comparison of experimental results to training results ofSVM-I

100 times 200mm cylinder concrete As seen from these resultsSVM model is sufficiently close to the experimental dataand is able to predict the 119891spt from the 119891

119888of concrete

For convenient comparison purpose the experimental andpredicted results are plotted in Figures 1 2 3 and 4 It canbe seen that the results from SVM method are in goodagreement with the experimental results

The 119877-square errors in the trained SVM are 08115 and08227 for RBF and polynomial function of SVM-I 08370and 08253 for those of SVM-II respectively 119877-square errorsin the tested SVM are 09422 and 09327 for RBF and polyno-mial function of SVM-I 08815 and 08823 for those of SVM-II respectively The whole results obtained from the SVMmodels show a successful performance of the SVM modelsof predicting 150 times 300mm cylinder119891spt of concrete from thecorresponding 150 times 300mm cylinder compressive strength119891119888and 100 times 200mm cylinder 119891spt from the corresponding

100 times 200mm cylinder compressive strength 119891119888of concrete

for each of the training and testing sets It is also clear thatthere are no major difference on performance between theradical basis function and the polynomial function kernelsthat used in this paper However in generally the radical basisfunction kernel exhibits slightly better performance than thepolynomial kernel

The performance of the trained and tested sets is analyzedby computing MAPE RMSE and 1198772 of experimental 119891spt

0

2

4

6

8

0 5 10 15 20 25 30 35 40 45 50

Split

ting

tens

ile st

reng

th (M

Pa)

Sample number

ExperimentalRBFPolynomial

Figure 2 Comparison of experimental results to testing results ofSVM-I

between the SVM model and other methods The MAPERMSE and 1198772 are calculated for these methods Statisticalparameters of the training and testing sets of all methods arepresented in Tables 6 and 7 The model having the smallestMAPE or RMSE and the highest 1198772 can be regarded to bethe best model with the assumption of analysis Generally1198772 of the SVM models are higher than other methodsand the MAPE and RMSE of SVM are smaller than othermodels The analysis shows that the SVM model is betterthan other models and can predict the splitting tensilestrength of concrete well Furthermore SVMmodel has goodgeneralization capacity to avoid over training and can alwaysbe updated to get better results by presenting new trainingexamples as new data become available Thus SVM modelcan be regarded as a very effective method to predict splittingtensile strength of concrete from their compressive strength

5 Conclusion

The splitting tensile strength of concrete estimations fromcompression strength has been obtained so far in the litera-ture either through regression or other methodsThis presentstudy reports a new and influential approach for predictingthe splitting tensile strength using SVM for the first timein the literature The study conducted in this paper shows

12 Advances in Materials Science and Engineering

Polynomial

0

1

2

3

4

5

6

7

8

9

0 20 40 60 80 100 120 140

Split

ting

tens

ile st

reng

hth

(MPa

)

Sample number

ExperimentalRBF

Figure 3 Comparison of experimental results to training results ofSVM-II

Polynomial

0

2

4

6

0 10 20 30 40 50

Split

ting

tens

ile st

reng

th (M

Pa)

Sample number

ExperimentalRBF

Figure 4 Comparison of experimental results to testing results ofSVM-II

the feasibility of using a simple SVM to estimate the splittingtensile strength of concrete After learning from a set ofselected training data involving compressive strengths ofconcrete collected from the previous literature the SVM canbe utilized to predict the splitting tensile strengths of the testdata

In this paper radical basis function and polynomial areadopted for predicting splitting tensile strength from 150 times300mm cylinder and 100times200mm cylinder 119891

119888of concrete

It is found that the splitting tensile strength obtained fromthe SVM is more accurate than those obtained from designcodes and several researchesrsquo empirical equation when acomparison is made on the basis of the experimental dataSince the SVM is largely characterized by the type of its kernel

function it is necessary to choose the appropriate kernel foreach particular application problem in order to guaranteesatisfactory results The results of radial basis function andpolynomial indicate that RBF and polynomial kernel have theability to predict the splitting tensile strength of concrete fromcompression strength with an acceptable degree of accuracy

The statistical parameters of MAPE RMSE and 1198772 showthat the proposed SVM model results have the best accu-racy and can predict splitting tensile strength very close toexperiment results The use of SVM is very advantageousfor the prediction of the splitting tensile strength of concretefrom compression strength because it can perform nonlinearregression efficiently for high-dimensional data sets Further-more its solution is globalThe satisfactory predictions of thesplitting tensile strength of concrete by the model indicatethat SVM is a useful modeling tool for engineers and researchscientists at concrete construction fields

Acknowledgments

This project was granted financial support from the NationalNatural Science Foundation of China (no 50808077 andno 51278188) and the Fundamental Research Funds for theCentral Universities The authors are grateful for support

References

[1] B W Xu and H S Shi ldquoCorrelations among mechanicalproperties of steel fiber reinforced concreterdquo Construction andBuilding Materials vol 23 no 12 pp 3468ndash3474 2009

[2] M F M Zain H B Mahmud A Ilham and M FaizalldquoPrediction of splitting tensile strength of high-performanceconcreterdquoCement andConcrete Research vol 32 no 8 pp 1251ndash1258 2002

[3] F A Oluokun E G Burdette and J H Deatherage ldquoSplittingtensile strength and compressive strength relationship at earlyagesrdquo ACI Materials Journal vol 88 no 2 pp 115ndash121 1991

[4] Y Choi and R L Yuan ldquoExperimental relationship betweensplitting tensile strength and compressive strength of GFRC andPFRCrdquo Cement and Concrete Research vol 35 no 8 pp 1587ndash1591 2005

[5] F D Larrad and Y Malier ldquoEngineering properties of very highperformance concreterdquo in High Performance Concrete fromMaterial to Structure Y Malier Ed pp 85ndash114 EFN SponLondon UK 1992

[6] J K Kim S H Han Y D Park and J H Noh ldquoMaterialproperties of self-flowing concreterdquo Journal of Materials in CivilEngineering vol 10 no 4 pp 244ndash249 1998

[7] J K Kim S H Han and Y C Song ldquoEffect of temperatureand aging on the mechanical properties of concretemdashpart Iexperimental resultsrdquo Cement and Concrete Research vol 32no 7 pp 1087ndash1094 2002

[8] AM Shaaban andHGesund ldquoSplitting tensile strength of steelfiber reinforced concrete cylinders consolidated by rodding orvibratingrdquo ACI Materials Journal vol 90 no 4 pp 366ndash3691993

[9] M Saridemir ldquoEmpirical modeling of splitting tensile strengthfrom cylinder compressive strength of concrete by geneticprogrammingrdquo Expert Systems with Applications vol 38 no 11pp 14257ndash14268 2011

Advances in Materials Science and Engineering 13

[10] U Thissen R van Brakel A P de Weijer W J Melssen and LM C Buydens ldquoUsing support vector machines for time seriespredictionrdquo Chemometrics and Intelligent Laboratory Systemsvol 69 no 1-2 pp 35ndash49 2003

[11] P Samui ldquoPredicted ultimate capacity of laterally loaded pilesin clay using support vector machinerdquo Geomechanics andGeoengineering vol 3 no 2 pp 113ndash120 2008

[12] VN VapnikTheNature of Statistical LearningTheory SpringerNew York NY USA 1995

[13] V N Vapnik S E Golowich and A J Smola ldquoSupport vectormachine for function approximation regression estimationand signal processingrdquoAdvances in Neural Information Process-ing Systems vol 9 pp 281ndash287 1996

[14] M Khanzadi and A Behnood ldquoMechanical properties ofhigh-strength concrete incorporating copper slag as coarseaggregaterdquo Construction and Building Materials vol 23 no 6pp 2183ndash2188 2009

[15] C K Y Leung and J Pan ldquoEffect of concrete composition onFRPconcrete bond capacityrdquo in Proceedings of the InternationalSymposium on Bond Behaviour of FRP in Structures (BBFS rsquo05)pp 69ndash76 2005

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

10 Advances in Materials Science and Engineering

Table 5 Comparison of experimental results to testing results of RBF polynomial of SVM-II and other models

119891119888(MPa) 119891spt (MPa) GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression RBF [9] Polynomial

3600 349 minus060 005 minus013 minus022 minus022 minus017 minus004

3040 308 minus054 017 001 minus016 minus012 minus010 minus001

2160 200 minus004 074 060 033 043 017 034

2120 270 minus077 002 minus012 minus040 minus030 minus044 minus039

2510 260 minus041 036 021 minus003 005 minus009 006

3190 330 minus067 003 minus014 minus028 minus025 minus026 minus012

3780 310 minus010 053 034 028 026 048 045

2920 250 minus004 069 053 034 040 041 049

3540 310 minus025 041 023 013 014 019 032

2860 340 minus098 minus024 minus041 minus059 minus054 minus051 minus046

2950 310 minus062 010 minus006 minus024 minus019 minus023 minus008

3490 350 minus068 minus001 minus019 minus030 minus029 minus023 minus011

3860 338 minus033 029 010 005 003 015 023

5235 399 minus010 028 006 021 008 042 036

3840 380 minus076 minus014 minus033 minus039 minus040 minus023 minus020

5150 420 minus036 003 minus018 minus005 minus017 016 011

4860 400 minus034 011 minus010 000 minus011 011 016

3950 390 minus079 minus019 minus038 minus042 minus045 minus022 minus022

3040 285 minus031 040 024 007 011 013 022

2950 256 minus008 064 048 030 035 031 046

4280 430 minus099 minus044 minus064 minus063 minus068 minus052 minus046

5190 430 minus044 minus005 minus027 minus013 minus026 009 003

4930 420 minus049 minus006 minus027 minus017 minus027 003 000

4330 360 minus026 028 008 010 004 022 028

4360 338 minus002 052 032 034 028 046 052

4990 457 minus083 minus040 minus061 minus050 minus062 minus034 minus033

5740 420 minus001 027 004 026 009 054 039

5850 490 minus065 minus039 minus062 minus038 minus056 minus015 minus024

3140 279 minus019 052 035 020 023 026 036

3490 304 minus022 045 027 016 017 023 035

4160 384 minus060 minus003 minus023 minus024 minus028 minus011 minus005

4680 400 minus044 004 minus017 minus010 minus019 001 009

5170 420 minus035 004 minus017 minus004 minus016 018 012

4330 390 minus056 minus002 minus022 minus020 minus026 minus008 minus002

6080 410 029 050 027 054 033 076 068

6000 430 004 027 004 030 010 060 044

7190 540 minus037 minus040 minus065 minus021 minus051 016 000

7510 540 minus018 minus029 minus055 minus006 minus039 030 020

5787 449 minus027 000 minus023 000 minus018 027 015

3950 340 minus029 031 012 008 005 028 028

Table 6 Error measurement for 119891spt from 150 times 300mm cylinder 119891119888

GEP-I [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF PolynomialRMSE of training set 04603 05995 05228 05027 04512 04978 04819RMSE of testing set 04271 02735 03152 03754 02874 02748 04176MAPE of training set 97554 13647 114913 106213 95596 102434 105454MAPE of testing set 73022 46139 55434 62059 5286 4987 695171198772 of training set 08282 08225 08224 08267 0826 08115 082271198772 of testing set 09416 09486 09486 09471 09476 09422 09327

Advances in Materials Science and Engineering 11

Table 7 Error measurement for 119891spt from 100 times 200mm cylinder 119891119888

GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF PolynomialRMSE of training set 05886 0539 05244 04987 04911 07287 0562RMSE of testing set 0483 03348 03393 02803 03155 03082 02917MAPE of training set 118729 116879 109223 92976 9472 155902 106658MAPE of testing set 108701 71383 7604 62261 7234 69713 628351198772 of training set 08198 08354 08353 083 08337 0837 082531198772 of testing set 08778 08802 08793 088 08801 08815 08823

0

2

4

6

8

0 20 40 60 80 100 120 140

Split

ting

tens

ile st

reng

th (M

Pa)

Sample number

ExperimentalRBFPolynomial

Figure 1 Comparison of experimental results to training results ofSVM-I

100 times 200mm cylinder concrete As seen from these resultsSVM model is sufficiently close to the experimental dataand is able to predict the 119891spt from the 119891

119888of concrete

For convenient comparison purpose the experimental andpredicted results are plotted in Figures 1 2 3 and 4 It canbe seen that the results from SVM method are in goodagreement with the experimental results

The 119877-square errors in the trained SVM are 08115 and08227 for RBF and polynomial function of SVM-I 08370and 08253 for those of SVM-II respectively 119877-square errorsin the tested SVM are 09422 and 09327 for RBF and polyno-mial function of SVM-I 08815 and 08823 for those of SVM-II respectively The whole results obtained from the SVMmodels show a successful performance of the SVM modelsof predicting 150 times 300mm cylinder119891spt of concrete from thecorresponding 150 times 300mm cylinder compressive strength119891119888and 100 times 200mm cylinder 119891spt from the corresponding

100 times 200mm cylinder compressive strength 119891119888of concrete

for each of the training and testing sets It is also clear thatthere are no major difference on performance between theradical basis function and the polynomial function kernelsthat used in this paper However in generally the radical basisfunction kernel exhibits slightly better performance than thepolynomial kernel

The performance of the trained and tested sets is analyzedby computing MAPE RMSE and 1198772 of experimental 119891spt

0

2

4

6

8

0 5 10 15 20 25 30 35 40 45 50

Split

ting

tens

ile st

reng

th (M

Pa)

Sample number

ExperimentalRBFPolynomial

Figure 2 Comparison of experimental results to testing results ofSVM-I

between the SVM model and other methods The MAPERMSE and 1198772 are calculated for these methods Statisticalparameters of the training and testing sets of all methods arepresented in Tables 6 and 7 The model having the smallestMAPE or RMSE and the highest 1198772 can be regarded to bethe best model with the assumption of analysis Generally1198772 of the SVM models are higher than other methodsand the MAPE and RMSE of SVM are smaller than othermodels The analysis shows that the SVM model is betterthan other models and can predict the splitting tensilestrength of concrete well Furthermore SVMmodel has goodgeneralization capacity to avoid over training and can alwaysbe updated to get better results by presenting new trainingexamples as new data become available Thus SVM modelcan be regarded as a very effective method to predict splittingtensile strength of concrete from their compressive strength

5 Conclusion

The splitting tensile strength of concrete estimations fromcompression strength has been obtained so far in the litera-ture either through regression or other methodsThis presentstudy reports a new and influential approach for predictingthe splitting tensile strength using SVM for the first timein the literature The study conducted in this paper shows

12 Advances in Materials Science and Engineering

Polynomial

0

1

2

3

4

5

6

7

8

9

0 20 40 60 80 100 120 140

Split

ting

tens

ile st

reng

hth

(MPa

)

Sample number

ExperimentalRBF

Figure 3 Comparison of experimental results to training results ofSVM-II

Polynomial

0

2

4

6

0 10 20 30 40 50

Split

ting

tens

ile st

reng

th (M

Pa)

Sample number

ExperimentalRBF

Figure 4 Comparison of experimental results to testing results ofSVM-II

the feasibility of using a simple SVM to estimate the splittingtensile strength of concrete After learning from a set ofselected training data involving compressive strengths ofconcrete collected from the previous literature the SVM canbe utilized to predict the splitting tensile strengths of the testdata

In this paper radical basis function and polynomial areadopted for predicting splitting tensile strength from 150 times300mm cylinder and 100times200mm cylinder 119891

119888of concrete

It is found that the splitting tensile strength obtained fromthe SVM is more accurate than those obtained from designcodes and several researchesrsquo empirical equation when acomparison is made on the basis of the experimental dataSince the SVM is largely characterized by the type of its kernel

function it is necessary to choose the appropriate kernel foreach particular application problem in order to guaranteesatisfactory results The results of radial basis function andpolynomial indicate that RBF and polynomial kernel have theability to predict the splitting tensile strength of concrete fromcompression strength with an acceptable degree of accuracy

The statistical parameters of MAPE RMSE and 1198772 showthat the proposed SVM model results have the best accu-racy and can predict splitting tensile strength very close toexperiment results The use of SVM is very advantageousfor the prediction of the splitting tensile strength of concretefrom compression strength because it can perform nonlinearregression efficiently for high-dimensional data sets Further-more its solution is globalThe satisfactory predictions of thesplitting tensile strength of concrete by the model indicatethat SVM is a useful modeling tool for engineers and researchscientists at concrete construction fields

Acknowledgments

This project was granted financial support from the NationalNatural Science Foundation of China (no 50808077 andno 51278188) and the Fundamental Research Funds for theCentral Universities The authors are grateful for support

References

[1] B W Xu and H S Shi ldquoCorrelations among mechanicalproperties of steel fiber reinforced concreterdquo Construction andBuilding Materials vol 23 no 12 pp 3468ndash3474 2009

[2] M F M Zain H B Mahmud A Ilham and M FaizalldquoPrediction of splitting tensile strength of high-performanceconcreterdquoCement andConcrete Research vol 32 no 8 pp 1251ndash1258 2002

[3] F A Oluokun E G Burdette and J H Deatherage ldquoSplittingtensile strength and compressive strength relationship at earlyagesrdquo ACI Materials Journal vol 88 no 2 pp 115ndash121 1991

[4] Y Choi and R L Yuan ldquoExperimental relationship betweensplitting tensile strength and compressive strength of GFRC andPFRCrdquo Cement and Concrete Research vol 35 no 8 pp 1587ndash1591 2005

[5] F D Larrad and Y Malier ldquoEngineering properties of very highperformance concreterdquo in High Performance Concrete fromMaterial to Structure Y Malier Ed pp 85ndash114 EFN SponLondon UK 1992

[6] J K Kim S H Han Y D Park and J H Noh ldquoMaterialproperties of self-flowing concreterdquo Journal of Materials in CivilEngineering vol 10 no 4 pp 244ndash249 1998

[7] J K Kim S H Han and Y C Song ldquoEffect of temperatureand aging on the mechanical properties of concretemdashpart Iexperimental resultsrdquo Cement and Concrete Research vol 32no 7 pp 1087ndash1094 2002

[8] AM Shaaban andHGesund ldquoSplitting tensile strength of steelfiber reinforced concrete cylinders consolidated by rodding orvibratingrdquo ACI Materials Journal vol 90 no 4 pp 366ndash3691993

[9] M Saridemir ldquoEmpirical modeling of splitting tensile strengthfrom cylinder compressive strength of concrete by geneticprogrammingrdquo Expert Systems with Applications vol 38 no 11pp 14257ndash14268 2011

Advances in Materials Science and Engineering 13

[10] U Thissen R van Brakel A P de Weijer W J Melssen and LM C Buydens ldquoUsing support vector machines for time seriespredictionrdquo Chemometrics and Intelligent Laboratory Systemsvol 69 no 1-2 pp 35ndash49 2003

[11] P Samui ldquoPredicted ultimate capacity of laterally loaded pilesin clay using support vector machinerdquo Geomechanics andGeoengineering vol 3 no 2 pp 113ndash120 2008

[12] VN VapnikTheNature of Statistical LearningTheory SpringerNew York NY USA 1995

[13] V N Vapnik S E Golowich and A J Smola ldquoSupport vectormachine for function approximation regression estimationand signal processingrdquoAdvances in Neural Information Process-ing Systems vol 9 pp 281ndash287 1996

[14] M Khanzadi and A Behnood ldquoMechanical properties ofhigh-strength concrete incorporating copper slag as coarseaggregaterdquo Construction and Building Materials vol 23 no 6pp 2183ndash2188 2009

[15] C K Y Leung and J Pan ldquoEffect of concrete composition onFRPconcrete bond capacityrdquo in Proceedings of the InternationalSymposium on Bond Behaviour of FRP in Structures (BBFS rsquo05)pp 69ndash76 2005

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Advances in Materials Science and Engineering 11

Table 7 Error measurement for 119891spt from 100 times 200mm cylinder 119891119888

GEP-III [9] ACI 363R ACI 318 CEB-FIP Regression [9] RBF PolynomialRMSE of training set 05886 0539 05244 04987 04911 07287 0562RMSE of testing set 0483 03348 03393 02803 03155 03082 02917MAPE of training set 118729 116879 109223 92976 9472 155902 106658MAPE of testing set 108701 71383 7604 62261 7234 69713 628351198772 of training set 08198 08354 08353 083 08337 0837 082531198772 of testing set 08778 08802 08793 088 08801 08815 08823

0

2

4

6

8

0 20 40 60 80 100 120 140

Split

ting

tens

ile st

reng

th (M

Pa)

Sample number

ExperimentalRBFPolynomial

Figure 1 Comparison of experimental results to training results ofSVM-I

100 times 200mm cylinder concrete As seen from these resultsSVM model is sufficiently close to the experimental dataand is able to predict the 119891spt from the 119891

119888of concrete

For convenient comparison purpose the experimental andpredicted results are plotted in Figures 1 2 3 and 4 It canbe seen that the results from SVM method are in goodagreement with the experimental results

The 119877-square errors in the trained SVM are 08115 and08227 for RBF and polynomial function of SVM-I 08370and 08253 for those of SVM-II respectively 119877-square errorsin the tested SVM are 09422 and 09327 for RBF and polyno-mial function of SVM-I 08815 and 08823 for those of SVM-II respectively The whole results obtained from the SVMmodels show a successful performance of the SVM modelsof predicting 150 times 300mm cylinder119891spt of concrete from thecorresponding 150 times 300mm cylinder compressive strength119891119888and 100 times 200mm cylinder 119891spt from the corresponding

100 times 200mm cylinder compressive strength 119891119888of concrete

for each of the training and testing sets It is also clear thatthere are no major difference on performance between theradical basis function and the polynomial function kernelsthat used in this paper However in generally the radical basisfunction kernel exhibits slightly better performance than thepolynomial kernel

The performance of the trained and tested sets is analyzedby computing MAPE RMSE and 1198772 of experimental 119891spt

0

2

4

6

8

0 5 10 15 20 25 30 35 40 45 50

Split

ting

tens

ile st

reng

th (M

Pa)

Sample number

ExperimentalRBFPolynomial

Figure 2 Comparison of experimental results to testing results ofSVM-I

between the SVM model and other methods The MAPERMSE and 1198772 are calculated for these methods Statisticalparameters of the training and testing sets of all methods arepresented in Tables 6 and 7 The model having the smallestMAPE or RMSE and the highest 1198772 can be regarded to bethe best model with the assumption of analysis Generally1198772 of the SVM models are higher than other methodsand the MAPE and RMSE of SVM are smaller than othermodels The analysis shows that the SVM model is betterthan other models and can predict the splitting tensilestrength of concrete well Furthermore SVMmodel has goodgeneralization capacity to avoid over training and can alwaysbe updated to get better results by presenting new trainingexamples as new data become available Thus SVM modelcan be regarded as a very effective method to predict splittingtensile strength of concrete from their compressive strength

5 Conclusion

The splitting tensile strength of concrete estimations fromcompression strength has been obtained so far in the litera-ture either through regression or other methodsThis presentstudy reports a new and influential approach for predictingthe splitting tensile strength using SVM for the first timein the literature The study conducted in this paper shows

12 Advances in Materials Science and Engineering

Polynomial

0

1

2

3

4

5

6

7

8

9

0 20 40 60 80 100 120 140

Split

ting

tens

ile st

reng

hth

(MPa

)

Sample number

ExperimentalRBF

Figure 3 Comparison of experimental results to training results ofSVM-II

Polynomial

0

2

4

6

0 10 20 30 40 50

Split

ting

tens

ile st

reng

th (M

Pa)

Sample number

ExperimentalRBF

Figure 4 Comparison of experimental results to testing results ofSVM-II

the feasibility of using a simple SVM to estimate the splittingtensile strength of concrete After learning from a set ofselected training data involving compressive strengths ofconcrete collected from the previous literature the SVM canbe utilized to predict the splitting tensile strengths of the testdata

In this paper radical basis function and polynomial areadopted for predicting splitting tensile strength from 150 times300mm cylinder and 100times200mm cylinder 119891

119888of concrete

It is found that the splitting tensile strength obtained fromthe SVM is more accurate than those obtained from designcodes and several researchesrsquo empirical equation when acomparison is made on the basis of the experimental dataSince the SVM is largely characterized by the type of its kernel

function it is necessary to choose the appropriate kernel foreach particular application problem in order to guaranteesatisfactory results The results of radial basis function andpolynomial indicate that RBF and polynomial kernel have theability to predict the splitting tensile strength of concrete fromcompression strength with an acceptable degree of accuracy

The statistical parameters of MAPE RMSE and 1198772 showthat the proposed SVM model results have the best accu-racy and can predict splitting tensile strength very close toexperiment results The use of SVM is very advantageousfor the prediction of the splitting tensile strength of concretefrom compression strength because it can perform nonlinearregression efficiently for high-dimensional data sets Further-more its solution is globalThe satisfactory predictions of thesplitting tensile strength of concrete by the model indicatethat SVM is a useful modeling tool for engineers and researchscientists at concrete construction fields

Acknowledgments

This project was granted financial support from the NationalNatural Science Foundation of China (no 50808077 andno 51278188) and the Fundamental Research Funds for theCentral Universities The authors are grateful for support

References

[1] B W Xu and H S Shi ldquoCorrelations among mechanicalproperties of steel fiber reinforced concreterdquo Construction andBuilding Materials vol 23 no 12 pp 3468ndash3474 2009

[2] M F M Zain H B Mahmud A Ilham and M FaizalldquoPrediction of splitting tensile strength of high-performanceconcreterdquoCement andConcrete Research vol 32 no 8 pp 1251ndash1258 2002

[3] F A Oluokun E G Burdette and J H Deatherage ldquoSplittingtensile strength and compressive strength relationship at earlyagesrdquo ACI Materials Journal vol 88 no 2 pp 115ndash121 1991

[4] Y Choi and R L Yuan ldquoExperimental relationship betweensplitting tensile strength and compressive strength of GFRC andPFRCrdquo Cement and Concrete Research vol 35 no 8 pp 1587ndash1591 2005

[5] F D Larrad and Y Malier ldquoEngineering properties of very highperformance concreterdquo in High Performance Concrete fromMaterial to Structure Y Malier Ed pp 85ndash114 EFN SponLondon UK 1992

[6] J K Kim S H Han Y D Park and J H Noh ldquoMaterialproperties of self-flowing concreterdquo Journal of Materials in CivilEngineering vol 10 no 4 pp 244ndash249 1998

[7] J K Kim S H Han and Y C Song ldquoEffect of temperatureand aging on the mechanical properties of concretemdashpart Iexperimental resultsrdquo Cement and Concrete Research vol 32no 7 pp 1087ndash1094 2002

[8] AM Shaaban andHGesund ldquoSplitting tensile strength of steelfiber reinforced concrete cylinders consolidated by rodding orvibratingrdquo ACI Materials Journal vol 90 no 4 pp 366ndash3691993

[9] M Saridemir ldquoEmpirical modeling of splitting tensile strengthfrom cylinder compressive strength of concrete by geneticprogrammingrdquo Expert Systems with Applications vol 38 no 11pp 14257ndash14268 2011

Advances in Materials Science and Engineering 13

[10] U Thissen R van Brakel A P de Weijer W J Melssen and LM C Buydens ldquoUsing support vector machines for time seriespredictionrdquo Chemometrics and Intelligent Laboratory Systemsvol 69 no 1-2 pp 35ndash49 2003

[11] P Samui ldquoPredicted ultimate capacity of laterally loaded pilesin clay using support vector machinerdquo Geomechanics andGeoengineering vol 3 no 2 pp 113ndash120 2008

[12] VN VapnikTheNature of Statistical LearningTheory SpringerNew York NY USA 1995

[13] V N Vapnik S E Golowich and A J Smola ldquoSupport vectormachine for function approximation regression estimationand signal processingrdquoAdvances in Neural Information Process-ing Systems vol 9 pp 281ndash287 1996

[14] M Khanzadi and A Behnood ldquoMechanical properties ofhigh-strength concrete incorporating copper slag as coarseaggregaterdquo Construction and Building Materials vol 23 no 6pp 2183ndash2188 2009

[15] C K Y Leung and J Pan ldquoEffect of concrete composition onFRPconcrete bond capacityrdquo in Proceedings of the InternationalSymposium on Bond Behaviour of FRP in Structures (BBFS rsquo05)pp 69ndash76 2005

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

12 Advances in Materials Science and Engineering

Polynomial

0

1

2

3

4

5

6

7

8

9

0 20 40 60 80 100 120 140

Split

ting

tens

ile st

reng

hth

(MPa

)

Sample number

ExperimentalRBF

Figure 3 Comparison of experimental results to training results ofSVM-II

Polynomial

0

2

4

6

0 10 20 30 40 50

Split

ting

tens

ile st

reng

th (M

Pa)

Sample number

ExperimentalRBF

Figure 4 Comparison of experimental results to testing results ofSVM-II

the feasibility of using a simple SVM to estimate the splittingtensile strength of concrete After learning from a set ofselected training data involving compressive strengths ofconcrete collected from the previous literature the SVM canbe utilized to predict the splitting tensile strengths of the testdata

In this paper radical basis function and polynomial areadopted for predicting splitting tensile strength from 150 times300mm cylinder and 100times200mm cylinder 119891

119888of concrete

It is found that the splitting tensile strength obtained fromthe SVM is more accurate than those obtained from designcodes and several researchesrsquo empirical equation when acomparison is made on the basis of the experimental dataSince the SVM is largely characterized by the type of its kernel

function it is necessary to choose the appropriate kernel foreach particular application problem in order to guaranteesatisfactory results The results of radial basis function andpolynomial indicate that RBF and polynomial kernel have theability to predict the splitting tensile strength of concrete fromcompression strength with an acceptable degree of accuracy

The statistical parameters of MAPE RMSE and 1198772 showthat the proposed SVM model results have the best accu-racy and can predict splitting tensile strength very close toexperiment results The use of SVM is very advantageousfor the prediction of the splitting tensile strength of concretefrom compression strength because it can perform nonlinearregression efficiently for high-dimensional data sets Further-more its solution is globalThe satisfactory predictions of thesplitting tensile strength of concrete by the model indicatethat SVM is a useful modeling tool for engineers and researchscientists at concrete construction fields

Acknowledgments

This project was granted financial support from the NationalNatural Science Foundation of China (no 50808077 andno 51278188) and the Fundamental Research Funds for theCentral Universities The authors are grateful for support

References

[1] B W Xu and H S Shi ldquoCorrelations among mechanicalproperties of steel fiber reinforced concreterdquo Construction andBuilding Materials vol 23 no 12 pp 3468ndash3474 2009

[2] M F M Zain H B Mahmud A Ilham and M FaizalldquoPrediction of splitting tensile strength of high-performanceconcreterdquoCement andConcrete Research vol 32 no 8 pp 1251ndash1258 2002

[3] F A Oluokun E G Burdette and J H Deatherage ldquoSplittingtensile strength and compressive strength relationship at earlyagesrdquo ACI Materials Journal vol 88 no 2 pp 115ndash121 1991

[4] Y Choi and R L Yuan ldquoExperimental relationship betweensplitting tensile strength and compressive strength of GFRC andPFRCrdquo Cement and Concrete Research vol 35 no 8 pp 1587ndash1591 2005

[5] F D Larrad and Y Malier ldquoEngineering properties of very highperformance concreterdquo in High Performance Concrete fromMaterial to Structure Y Malier Ed pp 85ndash114 EFN SponLondon UK 1992

[6] J K Kim S H Han Y D Park and J H Noh ldquoMaterialproperties of self-flowing concreterdquo Journal of Materials in CivilEngineering vol 10 no 4 pp 244ndash249 1998

[7] J K Kim S H Han and Y C Song ldquoEffect of temperatureand aging on the mechanical properties of concretemdashpart Iexperimental resultsrdquo Cement and Concrete Research vol 32no 7 pp 1087ndash1094 2002

[8] AM Shaaban andHGesund ldquoSplitting tensile strength of steelfiber reinforced concrete cylinders consolidated by rodding orvibratingrdquo ACI Materials Journal vol 90 no 4 pp 366ndash3691993

[9] M Saridemir ldquoEmpirical modeling of splitting tensile strengthfrom cylinder compressive strength of concrete by geneticprogrammingrdquo Expert Systems with Applications vol 38 no 11pp 14257ndash14268 2011

Advances in Materials Science and Engineering 13

[10] U Thissen R van Brakel A P de Weijer W J Melssen and LM C Buydens ldquoUsing support vector machines for time seriespredictionrdquo Chemometrics and Intelligent Laboratory Systemsvol 69 no 1-2 pp 35ndash49 2003

[11] P Samui ldquoPredicted ultimate capacity of laterally loaded pilesin clay using support vector machinerdquo Geomechanics andGeoengineering vol 3 no 2 pp 113ndash120 2008

[12] VN VapnikTheNature of Statistical LearningTheory SpringerNew York NY USA 1995

[13] V N Vapnik S E Golowich and A J Smola ldquoSupport vectormachine for function approximation regression estimationand signal processingrdquoAdvances in Neural Information Process-ing Systems vol 9 pp 281ndash287 1996

[14] M Khanzadi and A Behnood ldquoMechanical properties ofhigh-strength concrete incorporating copper slag as coarseaggregaterdquo Construction and Building Materials vol 23 no 6pp 2183ndash2188 2009

[15] C K Y Leung and J Pan ldquoEffect of concrete composition onFRPconcrete bond capacityrdquo in Proceedings of the InternationalSymposium on Bond Behaviour of FRP in Structures (BBFS rsquo05)pp 69ndash76 2005

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Advances in Materials Science and Engineering 13

[10] U Thissen R van Brakel A P de Weijer W J Melssen and LM C Buydens ldquoUsing support vector machines for time seriespredictionrdquo Chemometrics and Intelligent Laboratory Systemsvol 69 no 1-2 pp 35ndash49 2003

[11] P Samui ldquoPredicted ultimate capacity of laterally loaded pilesin clay using support vector machinerdquo Geomechanics andGeoengineering vol 3 no 2 pp 113ndash120 2008

[12] VN VapnikTheNature of Statistical LearningTheory SpringerNew York NY USA 1995

[13] V N Vapnik S E Golowich and A J Smola ldquoSupport vectormachine for function approximation regression estimationand signal processingrdquoAdvances in Neural Information Process-ing Systems vol 9 pp 281ndash287 1996

[14] M Khanzadi and A Behnood ldquoMechanical properties ofhigh-strength concrete incorporating copper slag as coarseaggregaterdquo Construction and Building Materials vol 23 no 6pp 2183ndash2188 2009

[15] C K Y Leung and J Pan ldquoEffect of concrete composition onFRPconcrete bond capacityrdquo in Proceedings of the InternationalSymposium on Bond Behaviour of FRP in Structures (BBFS rsquo05)pp 69ndash76 2005

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials