Empirical modeling of splitting tensile strength from cylinder compressive strength of concrete by genetic programming

Expert Systems with Applications 38 (2011) 14257–14268

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Empirical modeling of splitting tensile strength from cylinder compressive strength of concrete by genetic programming Mustafa Sarıdemir ⇑ Department of Civil Engineering, Nig˘de University, 51100 Nig˘de, Turkey

a r t i c l e

i n f o

Keywords: Compressive strength Splitting tensile strength Genetic programming

a b s t r a c t Compressive strength and splitting tensile strength are both mechanical properties of concrete that are utilized in structural design. This study presents gene expression programming (GEP) as a new tool for the formulations of splitting tensile strength from compressive strength of concrete. For purpose of building the GEP-based formulations, 536 experimental data have been gathered from existing literature. The GEP-based formulations are developed for splitting tensile strength of concrete as a function of age of specimen and cylinder compressive strength. In experimental parts of this study, cylindrical specimens of 150  300 mm and 100  200 mm in dimensions are utilized. Training and testing sets of the GEPbased formulations are randomly separated from the complete experimental data. The GEP-based formulations are also validated with additional 173 data of experimental results other than the data used in training and testing sets of the GEP-based formulations. All of the results obtained from the GEP-based formulations are compared with the results obtained from experimental data, the developed regression-based formulation and formulas given by some national building codes. These comparisons showed that the GEP-based formulations appeared to well agree with the experimental data and found to be quite reliable. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Compressive strength (fc) and splitting tensile strength (fspt) are two significant indexes utilized for characterizing concrete mechanical properties. Generally, fc is necessarily required in structural design (Xu & Shi, 2009). fspt is important for nonreinforced concrete structures such as dam under earthquake excitations. Other concrete structures such as pavement slabs and airfield runway, which are designed based on bending strength, are under the influence of tensile forces. Therefore, in the design of these concrete structures, fspt is more significant than the fc (Xu & Shi, 2009; Zain, Mahmud, Ilham, & Faizal, 2002). Generally, fspt can be determined by direct tension test, splitting tensile test. However, splitting tensile test has been much more popularly carried out, probably because of its easier operation. Moreover, it has been widely reported that fspt can be predicted from fc of concrete through different empirical relations proposed by some national building codes (ACI 363R-92, 1992; ACI 318-99, 1999; CEB-FIP, 1991). Generally, the fc of concrete is the only mechanical property to be considered in the mixture design of the concrete. However the fspt of concrete is a very considerable mechanical property reflecting the ability of the concrete. fspt of concrete is relatively much lower than its fc since it can be developed more quickly with crack ⇑ Tel.: +90 388 225 2348. E-mail address: [email protected] 0957-4174/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2011.04.239

propagation. Usually, fspt of concrete is often assumed proportional to the square root of its fc. But, there has been very few published studies dealing with experimental and analytical researches of the relation of fspt and fc of concretes (Choi & Yuan, 2005). Ideally, the fspt is measured directly on concrete specimens under uniform stresses at the top and bottom across the diameter, but from an experimental point of view, this is not always easy. Therefore, many of the researchers have been developed to formulate between the fspt and fc (Choi & Yuan, 2005; Kim, Han, & Song, 2002; Rashid, Mansur, & Paramasivam, 2004; Xu & Shi, 2009; Zain et al., 2002). In addition, different national building codes propose various formulas for concrete. For example, ACI 363R-92 (1992), ACI 318-99 (1999) and CEB-FIP (1991) propose Eqs. (1)–(3), respectively, for the evaluation of the fspt of concrete. ACI 363R-92 (1992):

fspt ¼ 0:59ðfc Þ1=2

ð1Þ

ACI 318-99 (1999):

fspt ¼ 0:56ðfc Þ1=2

ð2Þ

CEB-FIP (1991):

fspt ¼ 0:3ðfc Þ2=3

ð3Þ

where, fc (MPa) is the compressive strength of concrete and fspt (MPa) is the splitting tensile strength of concrete.

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A lot of researchers pointed out that the relationship between fspt and fc is not a simple one. It depends on the age and strength of concrete, size of specimens, type of curing, type of aggregate, amount of air entrainment and degree of compaction (Xu & Shi, 2009; Zain et al., 2002). In addition, aggregate stiffness, water/binder ratio, kinds and amounts of chemical admixtures and mineral additives of cementitious materials and method of testing of specimens, affect the fspt. In this present study, an alternative method is proposed for fspt prediction from cylinder fc of concrete or age of specimen (AS) and cylinder fc of concrete by following a new approach, which is GEPbased formulations, which have not been carried out in the literature up to now. By means of this new approach, predicting fspt from cylinder fc of concrete or AS and cylinder fc of concrete, GEP-based four formulations, which are named as GEP-I, GEP-II, GEP-III and GEP-IV, were developed. For building these formulations, the 1, 3, 7, 14, 28, 56, 90 and 180 days fspt and cylinder fc results of concretes used in training and testing for GEP-based formulations were obtained from existing literature (Ajdukiewicz & Kliszczewicz, 2002; Choi & Yuan, 2005; Giaccio & Zerbino, 1998; Jerath & Yamane, 1987; Khanzadi & Behnood, 2009; Kim, Han, Park, & Noh, 1998; Lam, Wong, & Poon, 1998; Meddah & Sato, 2010; Mouret, Bascoul, & Escadeillas, 1997; Pul, 2008; Sensale, 2006; Shannag, 2000). The explicit formulations of GEP-based were also presented. In addition, GEP-based formulations were validated with different experimental results taken from the literature (Emiroglu, Kelestemur, & Yıldız, 2007; Leung & Pan, 2005; Li et al., 2004; Pan & Leung, 2009; Rossignolo & Agnesini, 2002; Smaoui, Bérubé, Fournier, Bissonnette, & Durand, 2005; Sofi, van Deventer, Mendis, & Lukey, 2007; Suhaendi & Horiguchi, 2006; Uzal, Turanli, & Mehta, 2007; Yang, Chung, & Ashour, 2008; Zain et al., 2002). The proposed GEP-based formulations results were compared to the formulas results proposed by some national building codes and the developed regression-based formulation results. In the following sections of this study, GEP are shortly described; afterward, the formulations construction for GEP-based are explained together with the comparison and discussion of the obtained results.

2. Gene expression programming Gene expression programming (GEP) is invented by Ferreira (2001a), and is the natural development of genetic algorithms (GAs) and genetic programming (GP). GEP is, like GAs and GP, a GA as it uses populations of individuals, selects them according to fitness, and introduces genetic variation using one or more genetic operators. The fundamental difference between the three algorithms resides in the nature of the individuals: in GAs the individuals are linear strings of fixed length (chromosomes); in GP the individuals are nonlinear entities of different sizes and shapes (parse trees); and in GEP the individuals are encoded as linear strings of fixed length (the genome or chromosomes) which are afterward expressed as nonlinear entities of different sizes and shapes (Ferreira, 2001a; Jedrzejowicz & Ratajczak-Ropel, 2009; Guven & Gunal, 2008). The fundamental steps of GEP are schematically represented in Fig. 1. The process begins with the random generation of the chromosomes of a certain number of individuals (the initial population). Then these chromosomes are expressed and the fitness of each individual is evaluated against a set of fitness cases (also called selection environment). The individuals are then selected according to their fitness (their performance in that particular environment) to reproduce with modification, leaving progeny with new traits. These new individuals are, in their turn, subjected to the same developmental process: expression of the genomes, confrontation of the selection environment, selection, and repro-

Create Chromosomes of Initial Population Express Chromosomes Execute Each Program Evaluate Fitness

Terminate Iterate or Terminate?

End

Iterate Keep Best Program Select Programs Reproduction Prepare New Chromosomes of Next Generation Fig. 1. The flowchart of gene expression programming (Ferreira, 2004).

duction with modification. The process is repeated for a certain number of generations or until a good solution has been found (Ferreira, 2004). The fundamental players in GEP are only two: the chromosomes and the expression trees (ETs), being the latter the expression of the genetic information encoded in the chromosomes. As in nature, the process of information decoding is called translation, and this translation implies obviously a kind of code and a set of rules. The genetic code is very simple: a one-to-one relationship between the symbols of the chromosome and the functions or terminals they represent. The rules are also very simple: they determine the spatial organization of the functions and terminals in the ETs and the type of interaction between sub-ETs (Ferreira, 2004). Therefore, there are two languages in GEP: the language of the genes and the language of ETs, and knowing the sequence or structure of one, knowing the other. In GEP, thanks to the simple rules that determine the structure of ETs and their interactions, it is possible to infer immediately the phenotype given the sequence of a gene, and vice versa. This unequivocal bilingual notation is called Karva language (Ferreira, 2001a, 2001b, 2004). For example, a mathematical expression [(a  c)  b] + [a  (b  c)] can

Linking function

+ Gene 1

Gene 2

-

b

*

a

*

c

a

-

b

Fig. 2. Example of GEP expression tree.

c

M. Sarıdemir / Expert Systems with Applications 38 (2011) 14257–14268

be represented by a two gene chromosome or an ET, as shown in Fig. 2. This figure shows how two genes are encoded as a linear string and how it is expressed as an ET.

2.1. Gene expression programming operators Genetic operators are the core of all GAs, and two of them are common to all evolutionary systems: selection and replication. Despite the center of the storm, these operators, by themselves, do nothing in terms of evolution. In fact, they can only cause genetic drift, making populations less and less diverse with time until all of the individuals are exactly the same. So, the touch stone of all evolutionary systems is modification, or more specifically, the genetic operators that cause variation, and different algorithms create this modification differently (Ferreira, 2001b, 2004). There are four main operators in GEP: mutation, inversion, transposition, and crossover (recombination). These operators modify some chromosome for next generation.

2.1.1. Mutation In GEP, mutations can take place anywhere in the chromosome. However, the structural organization of chromosomes must remain undamaged, that is, in the heads of genes any symbol can change into another (function or terminal), while in the tails terminals can only change into terminals. This way, the structural organization of chromosomes is preserved, and all of the new individuals produced by mutation are structurally correct programs (Ferreira, 2001b, 2004; Guven & Aytek, 2009; Kayadelen, Gunaydın, Fener, Demir, & Ozvan, 2009).

2.1.2. Inversion In GEP, the inversion operator is restricted to the heads of genes, where any sequence might be randomly selected and inverted. In GEP, the inversion operator randomly chooses the chromosome, the gene to be modified, and the start and termination points of the sequence to be inverted. It is worth pointing out that this is the first time the inversion operator is described in GEP (Ferreira, 2004; Guven & Aytek, 2009).

2.1.3. Transposition The transposable elements of GEP are fragments of the genome that can be activated afterward leap to another place in the chromosome. In GEP, there are three kinds of transposable elements: (i) short fragments with a function or terminal in the first position that transpose to the head of genes exclude the root; ii) short fragments with a function in the first position that transpose to the start position of genes; (iii) and entire genes that transpose to the beginning of chromosomes (Ferreira, 2001b, 2004).

2.1.4. Crossover (recombination) In GEP, there are three kinds of crossover: one-point crossover, two-point crossover and gene crossover. In all types of crossover, two chromosomes are randomly selected and paired to exchange some material between them, creating two new daughter chromosomes (Ferreira, 2004). The genetic operators are implemented by operator rate showing a certain probability of a chromosome. Operator rate is specified at the beginning of the model construction. The mutation rate is proposed to be small values between the 0.001 and 0.1. The transposition operator and crossover operator are proposed to be 0.1 and 0.4, respectively (Kayadelen et al., 2009; Teodorescu & Sherwood, 2008).

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2.2. Gene expression programming formulation The main aim of development of GEP-based formulations is to generate the mathematical functions for the prediction of fspt. For that purpose, fspt is modeled in connection with cylinder fc of concrete or AS and cylinder fc of concrete using the GEP-based four formulations, which are named as GEP-I, GEP-II, GEP-III and GEP-IV. GEP-I formulation was developed for 150  300 mm cylinder fspt-I prediction from 150  300 mm cylinder fc-I of concrete at the ages of 1, 3, 7, 14, 28, 56 and 90 days. Besides, GEP-II formulation was developed for 150x300 mm cylinder fspt-II prediction from AS and 150  300 mm cylinder fc-II of concrete at the same ages. The proposed GEP-based formulations were trained and tested with experimental results taken from four different experimental studies (Ajdukiewicz & Kliszczewicz, 2002; Choi & Yuan, 2005; Jerath & Yamane, 1987; Pul, 2008). The number of experimental data used for training and testing in the formulations are 209 and 105, respectively. In addition, the proposed GEP-based formulations were validated with 79 data of experimental results taken from four different experimental studies (Emiroglu et al., 2007; Li et al., 2004; Sofi et al., 2007; Zain et al., 2002). GEP-III formulation was developed for 100  200 mm and 150  200 mm cylinder fspt-III prediction from 100  200 mm cylinder fc-III of concrete at the ages of 1, 3, 7, 14, 28, 56, 90 and 180 days. In addition, GEP-IV formulation was developed for 100  200 mm and 150  200 mm cylinder fspt-IV prediction from AS and 100  200 mm cylinder fc-IV of concrete at the same ages. The proposed GEP-based formulations were trained and tested with experimental results taken from eight different experimental studies (Giaccio & Zerbino, 1998; Khanzadi & Behnood, 2009; Kim et al., 1998; Lam et al., 1998; Meddah & Sato, 2010; Mouret et al., 1997; Sensale, 2006; Shannag, 2000). The number of experimental data used for training and testing in the GEP-based formulations are 148 and 74, respectively. Moreover, the proposed GEP-based formulations were validated with 94 data of experimental results taken from six different experimental studies (Leung & Pan, 2005; Pan & Leung, 2009; Rossignolo & Agnesini, 2002; Smaoui et al., 2005; Uzal et al., 2007; Yang et al., 2008). For the GEP-based formulations, the first is to choose the fitness function. For this problem, firstly, the fitness, fi, of an individual program, i, is measured by Eq. (4)

fi ¼

Ct X ðM  jC ði;jÞ  T j jÞ

ð4Þ

j¼1

where M is the range of selection, C(i,j) is the value returned by the individual chromosome i for fitness case j (out of Ct fitness cases) and Tj is the target value for fitness case j. If |C(i,j)  Tj| (the precision) is less than or equal to 0.01, then the precision is equal to zero, and fi = fmax = CtM. In this case, M = 100 was used, therefore, fmax = 1000. The advantage of this kind of fitness functions is that the system can find the optimal solution for itself (Ferreira, 2001a, 2001b). The second important step makes up by choosing the set of terminals T and the set of functions F to create the chromosomes. In this problem, the terminal set consists obviously of the independent variable, i.e., T = {fc} or T = {AS, fc}. The choice of the appropriate function set is not so obvious, but a good guess can always be done in order to include all the necessary functions (Ferreira, 2001b). In this situation, four basic arithmetic operators (+, , , /) and some basic functions (Sqrt, x2, Power, Exp). The third important step is to choose the chromosomal tree, i.e., the length of the head and the number of genes. GEP-based formulations initially used single gene and 2 lengths of heads, and increased the number of genes and heads, one after another during each run, and monitored the training and testing performance of each formulation. In this present study, after several trials, for all

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Table 1 GEP parameters used for proposed formulations. GEP-I

p1 p2

Number of generation Function set

p3 p4 p5 p6 p7 p8 p9

Number of chromosomes Head size Number of genes Linking function Mutation rate Inversion rate One-point recombination rate Two-point recombination rate Gene recombination rate Gene transposition rate

706 708 +, /, +, /, Sqrt Sqrt 20 20 3 4 2 2 Multiplication 0.044 0.1 0.3

p10 p11 p12

Sub-ET 2

Sub-ET 1 c2

Parameter definition

GEP-II

GEP-III

GEP-IV

664 +, , Sqrt 20 3 2

672 +, /, , Sqrt 10 4 2

+

Sqrt

/ d0

c0

d1

Fig. 4. Expression tree of the proposed GEP-II formulation.

0.3

Sub-ET 1

Sub-ET 2 c2

+

0.1 0.1

Sqrt

*

fspt-i ¼ f ðfc-i Þ ði ¼ I; IIIÞ

ð5Þ

fspt-j ¼ f ðAS;fc-j Þ ði ¼ II; IVÞ     pffiffiffiffi 2:091 fc-I  þ fc fspt-I ¼ 8:243 6:393 pffiffiffiffiffiffi fspt-I ¼ 0:04f c-I þ 0:254 fc-I pffiffiffiffiffiffiffi   1:109 fc-II  fspt-II ¼ ð0:560Þ  AS pffiffiffiffiffiffiffi 0:621 fspt-II ¼ 0:560 fc-II  AS pffiffiffiffiffiffiffiffi fspt-III ¼ ð0:212f c-III þ fc-III Þ  0:212 pffiffiffiffiffiffiffiffi fspt-III ¼ ð0:045f c-III þ 0:212 fc-III Þ pffiffiffiffiffiffi!  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi AS  AS  fcIV þ AS fsptIV ¼ 2AS pffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi AS  ASfcIV fspt-IV ¼ þ 0:5 2AS

ð6Þ

d0

c2

of the GEP-based formulations, the number of genes and length of head are determined as given in Table 1. The fourth important step is to select the linking function. In this present study, all of the GEP-based formulations used the linking function multiplication. Finally, the fifth important step is to select the set of genetic operators that give rise to variation and their rates. The combinations of all genetic operators (mutation, transposition and crossover) are given in Table 1. The explicit formulations of the GEP-I and GEP-III for fspt prediction from cylinder fc of concrete were obtained by Eq. (5). Moreover, the explicit formulations of the GEP-II and GEP-IV for fspt prediction from AS and cylinder fc of concrete were obtained by Eq. (6). For all of the GEP-I, GEP-II, GEP-III and GEP-IV formulations, the expression tree of the Eqs. (7)–(10) formulations are shown in Figs. 3–6, respectively

d0

Fig. 5. Expression tree of the proposed GEP-III formulation.

Sub-ET 1

Sub-ET 2

/

Sqrt

+ d0

d0

d0

+

Sqrt

Sqrt

*

d0 d0

d1

Fig. 6. Expression tree of the proposed GEP-IV formulation.

9 fc:150x300 mm cylinder

ð7Þ

ð8Þ

ð9Þ

Splitting tensile strength, MPa

8

fc:100x200 mm cylinder

7 6 5

f c :100x200 mm cylinder

4

f spt = 0.4079(f c )0.5809 R2 = 0.84

3 2

f c :150x300 mm cylinder f spt = 0.3191(f c )0.6353

1

R2 = 0.8131

0

ð10Þ

0

10

20

30

40 50 60 70 80 90 100 110 120 130 Compressive strength, MPa

Fig. 7. Scatter diagram and the regression curve of training set.

Sub-ET 1 c2

Sub-ET 2

/ c1

+ Sqrt

/ d0

c1

d0

Fig. 3. Expression tree of the proposed GEP-I formulation.

The actual parameter in the GEP-I formulation is d0 = fc-I and the constant in the formulation are; in the Sub-ET 1 c1 = 8.243 and c2 = 2.091, in the Sub-ET 2 c1 = 8.243. For the GEP-II formulation is d0 = AS and d1 = fc-II, the constant in the formulation are; in the Sub-ET 1 c2 = 0.560, in the Sub-ET 2 c0 = 1.109. In addition, the actual parameter in the GEP-III formulation is d0 = fc-III and the constants in the formulation are: in the Sub-ET 1 c2=0.212, in the Sub-ET 2 c1 = 0.212. For the GEP-IV formulation is d0 = AS and d1 = fc-IV.

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8 GEP-I training GEP-II training ACI 363R-92 ACI 318-99 CEB-FIP Regression training

6

5

4

Predicted splitting tensile strength, MPa

Predicted splitting tensile strength, MPa

7

ACI 318-99 R2 = 0.8324

3

2

Regression R2 = 0.8388

GEP-II R2 = 0.8409

ACI 363R-92 R2 = 0.8324

GEP-I R2 = 0.845

CEB-FIP R2 = 0.84

1 1

2 3 4 5 6 Experimental splitting tensile strength, MPa

ACI 318-99 R2 = 0.8581

3

GEP-I ACI 363R-92 R2 = 0.8627 R2 = 0.8581 Regression GEP-II R2 = 0.8613 R2 = 0.8629

CEB-FIP R2 = 0.8618

1 1

2

3

4

ACI 318-99 R2 = 0.868

3

ACI 363R-92 GEP-III 2 R2 = 0.8628 R = 0.868 Regression R2 = 0.868

2

CEB-FIP R2 = 0.8672

2

3 4 5 6 7 Experimental splitting tensile strength, MPa

8

4

5

6

GEP-I validating GEP-II validating ACI 363R-92 ACI 318-99 CEB-FIP Regression validating

7 6 5

ACI 363R-92 2 R = 0.8849

4 3

ACI 318-99 R = 0.8849

Regression GEP-II 2 R = 0.8856 R2 = 0.881

2

2

CEB-FIP 2

R = 0.8855

1

7

Fig. 9. Comparison of experimental results to testing results of GEP-I, GEP-II, regression and codes.

GEP-I R = 0.8826

2

1

Experimental splitting tensile strength, MPa

2

3 4 5 6 7 Experimental splitting tensile strength, MPa

8

Fig. 12. Comparison of experimental results to validating results of GEP-I, GEP-II, regression and codes.

10

6 GEP-III training GEP-IV training ACI 363R-92 ACI 318-99 CEB-FIP Regression training

9 8 7 6

Predicted splitting tensile strength, MPa

Predicted splitting tensile strength, MPa

GEP-IV R2 = 0.868

Fig. 11. Comparison of experimental results to testing results of GEP-III, GEP-IV, regression and codes.

Predicted splitting tensile strength, MPa

Predicted splitting tensile strength, MPa

4

2

5

8

GEP-I testing GEP-II testing ACI 363R-92 ACI 318-99 CEB-FIP Regression testing

5

6

1

7

6

7

1

7

Fig. 8. Comparison of experimental results to training results of GEP-I, GEP-II, regression and codes.

GEP-III testing GEP-IV testing ACI 363R-92 ACI 318-99 CEB-FIP Regression testing

ACI 318-99 R2 = 0.818

5 4

GEP-III R2 = 0.823

ACI 363R-92 R2 = 0.818

GEP-IV R2 = 0.818

CEB-FIP R2 = 0.8219

3 Regression R2 = 0.8203

2 1

GEP-III validating GEP-IV validating ACI 363R-92 ACI 318-99 CEB-FIP Regression validating

5

4 ACI 318-99 2 R = 0.7939

3 GEP-III 2 R = 0.7711

2

Regression GEP-IV 2 R = 0.7899 R2 = 0.7939

ACI 363R-92 2

R = 0.7939 CEB-FIP 2

R = 0.7849

1 1

2

3 4 5 6 7 8 Experimental splitting tensile strength, MPa

9

10

Fig. 10. Comparison of experimental results to training results of GEP-III, GEP-IV, regression and codes.

1

2 3 4 5 Experimental splitting tensile strength, MPa

6

Fig. 13. Comparison of experimental results to validating results of GEP-III, GEP-IV, regression and codes.

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Table 2 Comparison of experimental results to testing results of GEP-I, GEP-II, regression and codes. AS (day)

fc (MPa)

fspt (MPa)

GEP-I

GEP-II

ACI 363R

ACI 318

CEB-FIP

Regression

28 28 28 28 1 1 1 1 1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 3 3 3 7 7 7 7 7 7 7 7 7 7 7 7 14 14 14 14 14 14 14 14 14 14 14 14 28 28 28 28 28 28 28 28 28 28 28 56 56 56 56 56 56 56 56 56 56 56 56

57.68 45.75 38.10 31.95 18.20 28.70 23.90 43.70 24.60 43.00 14.20 36.70 22.70 35.30 13.60 28.60 26.60 43.50 30.90 49.90 32.20 57.80 26.80 36.50 65.60 30.30 52.80 40.60 52.30 41.20 72.80 41.20 72.80 41.20 72.80 41.80 68.40 42.30 59.10 41.30 59.70 42.30 75.10 48.50 81.90 41.30 67.10 42.00 70.10 44.80 65.40 38.70 56.40 42.40 67.70 50.70 81.10 42.00 48.90 85.30 52.30 85.10 49.70 71.20 51.90 87.70 51.90 87.70 51.90 87.70 53.80 85.30 58.70 87.10

4.26 4.10 3.62 3.31 1.90 2.70 2.30 3.30 2.30 3.50 1.80 3.20 1.70 2.40 1.30 1.70 2.40 2.90 2.40 3.80 3.00 3.50 2.40 2.90 4.40 2.40 3.80 3.50 4.20 3.20 5.10 3.20 5.10 3.20 5.10 3.30 4.10 2.90 3.80 3.70 4.10 3.20 5.10 3.70 4.80 3.30 4.40 3.20 4.30 3.40 4.00 3.50 4.30 3.20 4.80 3.60 4.80 3.20 3.60 5.80 4.20 5.80 4.20 5.30 3.80 6.10 3.80 6.10 3.80 6.10 4.10 5.10 4.40 5.60

4.22 3.53 3.08 2.70 1.80 2.50 2.19 3.41 2.23 3.37 1.52 2.99 2.11 2.91 1.48 2.49 2.36 3.40 2.64 3.77 2.72 4.22 2.38 2.98 4.66 2.60 3.94 3.23 3.91 3.26 5.05 3.26 5.05 3.26 5.05 3.30 4.81 3.33 4.30 3.27 4.33 3.33 5.18 3.69 5.55 3.27 4.74 3.31 4.91 3.48 4.65 3.11 4.14 3.33 4.77 3.82 5.50 3.31 3.71 5.73 3.91 5.72 3.76 4.97 3.89 5.86 3.89 5.86 3.89 5.86 4.00 5.73 4.27 5.82

4.23 3.77 3.43 3.14 1.77 2.38 2.12 3.08 2.16 3.05 1.49 2.77 2.05 2.71 1.44 2.37 2.68 3.49 2.91 3.75 2.97 4.05 2.69 3.18 4.33 2.88 3.86 3.48 3.96 3.51 4.69 3.51 4.69 3.51 4.69 3.53 4.54 3.55 4.22 3.55 4.28 3.60 4.81 3.86 5.02 3.55 4.54 3.58 4.64 3.70 4.48 3.46 4.18 3.62 4.58 3.96 5.02 3.61 3.89 5.15 4.03 5.14 3.94 4.71 4.02 5.23 4.02 5.23 4.02 5.23 4.10 5.16 4.28 5.21

4.48 3.99 3.64 3.33 2.52 3.16 2.88 3.90 2.93 3.87 2.22 3.57 2.81 3.51 2.18 3.16 3.04 3.89 3.28 4.17 3.35 4.49 3.05 3.56 4.78 3.25 4.29 3.76 4.27 3.79 5.03 3.79 5.03 3.79 5.03 3.81 4.88 3.84 4.54 3.79 4.56 3.84 5.11 4.11 5.34 3.79 4.83 3.82 4.94 3.95 4.77 3.67 4.43 3.84 4.85 4.20 5.31 3.82 4.13 5.45 4.27 5.44 4.16 4.98 4.25 5.53 4.25 5.53 4.25 5.53 4.33 5.45 4.52 5.51

4.25 3.79 3.46 3.17 2.39 3.00 2.74 3.70 2.78 3.67 2.11 3.39 2.67 3.33 2.07 2.99 2.89 3.69 3.11 3.96 3.18 4.26 2.90 3.38 4.54 3.08 4.07 3.57 4.05 3.59 4.78 3.59 4.78 3.59 4.78 3.62 4.63 3.64 4.31 3.60 4.33 3.64 4.85 3.90 5.07 3.60 4.59 3.63 4.69 3.75 4.53 3.48 4.21 3.65 4.61 3.99 5.04 3.63 3.92 5.17 4.05 5.17 3.95 4.73 4.03 5.24 4.03 5.24 4.03 5.24 4.11 5.17 4.29 5.23

4.48 3.84 3.40 3.02 2.08 2.81 2.49 3.72 2.54 3.68 1.76 3.31 2.41 3.23 1.71 2.81 2.67 3.71 2.95 4.07 3.04 4.48 2.69 3.30 4.88 2.92 4.22 3.54 4.20 3.58 5.23 3.58 5.23 3.58 5.23 3.61 5.02 3.64 4.55 3.58 4.58 3.64 5.34 3.99 5.66 3.58 4.95 3.62 5.10 3.78 4.87 3.43 4.41 3.65 4.98 4.11 5.62 3.62 4.01 5.81 4.20 5.80 4.06 5.15 4.17 5.92 4.17 5.92 4.17 5.92 4.28 5.81 4.53 5.89

4.19 3.62 3.22 2.88 2.02 2.69 2.40 3.52 2.44 3.48 1.72 3.15 2.32 3.07 1.68 2.69 2.57 3.51 2.82 3.83 2.90 4.20 2.58 3.14 4.55 2.79 3.97 3.36 3.94 3.39 4.86 3.39 4.86 3.39 4.86 3.42 4.67 3.44 4.26 3.39 4.29 3.44 4.96 3.76 5.24 3.39 4.62 3.43 4.75 3.57 4.54 3.26 4.14 3.45 4.64 3.86 5.21 3.43 3.78 5.38 3.94 5.37 3.82 4.80 3.92 5.47 3.92 5.47 3.92 5.47 4.01 5.38 4.24 5.45

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M. Sarıdemir / Expert Systems with Applications 38 (2011) 14257–14268 Table 2 (continued) AS (day)

fc (MPa)

fspt (MPa)

GEP-I

GEP-II

ACI 363R

ACI 318

CEB-FIP

Regression

90 90 90 90 90 90 90 90 90 90 90 90 7 7 7 7 7 28 28 28 28 28 90 90 90 90 90 28 28 28 28

49.30 66.20 53.10 84.30 56.10 96.70 52.10 78.50 49.90 87.50 55.10 80.10 24.67 23.01 32.03 28.78 33.60 23.30 26.33 35.82 30.93 37.42 29.10 28.03 36.98 31.65 38.58 81.00 68.00 60.00 54.00

4.50 5.10 3.80 5.60 4.30 5.70 3.80 5.00 4.10 5.10 4.40 5.30 2.74 2.79 3.03 2.93 2.03 3.04 3.07 3.12 3.21 2.07 3.01 3.41 3.25 3.39 3.16 6.60 5.90 5.00 4.70

3.74 4.69 3.96 5.67 4.13 6.33 3.90 5.36 3.77 5.85 4.07 5.45 2.24 2.13 2.71 2.50 2.80 2.15 2.35 2.94 2.64 3.04 2.52 2.46 3.01 2.68 3.11 5.50 4.79 4.35 4.01

3.92 4.55 4.07 5.13 4.19 5.50 4.03 4.95 3.95 5.23 4.15 5.00 2.69 2.60 3.08 2.92 3.16 2.68 2.85 3.33 3.09 3.40 3.01 2.96 3.40 3.14 3.47 5.02 4.60 4.32 4.09

4.14 4.80 4.30 5.42 4.42 5.80 4.26 5.23 4.17 5.52 4.38 5.28 2.93 2.83 3.34 3.17 3.42 2.85 3.03 3.53 3.28 3.61 3.18 3.12 3.59 3.32 3.66 5.31 4.87 4.57 4.34

3.93 4.56 4.08 5.14 4.19 5.51 4.04 4.96 3.96 5.24 4.16 5.01 2.78 2.69 3.17 3.00 3.25 2.70 2.87 3.35 3.11 3.43 3.02 2.96 3.41 3.15 3.48 5.04 4.62 4.34 4.12

4.03 4.91 4.24 5.77 4.40 6.32 4.18 5.50 4.07 5.91 4.34 5.57 2.54 2.43 3.03 2.82 3.12 2.45 2.66 3.26 2.96 3.36 2.84 2.77 3.33 3.00 3.43 5.62 5.00 4.60 4.29

3.80 4.58 3.98 5.34 4.12 5.82 3.93 5.10 3.83 5.47 4.07 5.17 2.45 2.34 2.89 2.70 2.98 2.36 2.55 3.10 2.82 3.19 2.72 2.65 3.16 2.86 3.25 5.20 4.66 4.30 4.02

3. Results and discussion In this present study, in order to evaluate the capabilities of GEP-based formulations, formulas given by some national building codes and the developed regression-based formulation, mean absolute percentage error (MAPE), root-mean-squared error (RMSE) and R-square (R2) were used as the criteria between the experimental and predicted values which are according to the Eqs. (13)–(15), respectively

 Pn  1 i¼1 jt i  oi j Pn  100 n i¼1 t i vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n u1 X ðti  oi Þ2 RMSE ¼ t n i¼1 P P P ðn ti oi  ti oi Þ2 R2 ¼ P 2 P 2 P P ðn ti  ð t i Þ Þðn o2i  ð oi Þ2 Þ

MAPE ¼

ð13Þ

ð14Þ

ð15Þ

where t is the experimental value, o is the predicted value and n is total number of data. In order to build GEP-based formulations and to show the generalization capability of the GEP, the database made up in the experimental part is subdivided into three sets, namely training, testing and validating sets. Among 314 experimental data (Ajdukiewicz & Kliszczewicz, 2002; Choi & Yuan, 2005; Jerath & Yamane, 1987; Pul, 2008), 209 data were randomly selected as the training set for the GEP-I and GEP-II modeling and the remaining 105 data which are not used in training set were utilized to test the proposed GEP-based formulations. In addition, 79 experimental data (Emiroglu et al., 2007; Li et al., 2004; Sofi et al., 2007; Zain et al., 2002) which are not used in training and testing sets were utilized to validate the generalization capacity of the proposed GEP-I and GEP-II formulations. Moreover, In the GEP-III and GEP-IV formulations, while 148 data taken from experimental studies (Giaccio &

Zerbino, 1998; Khanzadi & Behnood, 2009; Kim et al., 1998; Lam et al., 1998; Meddah & Sato, 2010; Mouret et al., 1997; Sensale, 2006; Shannag, 2000) were utilized for training set, 74 data which are not used in training set were employed for testing set. Besides, 94 experimental data (Leung & Pan, 2005; Pan & Leung, 2009; Rossignolo & Agnesini, 2002; Smaoui et al., 2005; Uzal et al., 2007; Yang et al., 2008) which are not used in training and testing sets were utilized to validate the generalization capacity of the proposed GEP-III and GEP-IV formulations. For fspt prediction from 150  300 mm and 100  200 mm cylinder fc of concrete, the developed regression-based formulations and regression curve of experimental results used in the training set are given in Fig. 7. All of the results obtained from experimental studies and predicted by using the training and testing sets results of the GEP-based formulations, the developed regression-based and formulas given by some national building codes results for fspt are given in Figs. 8–11, respectively. The GEP-based formulations are also validated with additional experimental data which the data not used in training and testing sets of the GEP-based formulations as shown in Figs. 12 and 13. All of the R2 values of the GEPbased formulations are shown in these figures for the training, testing and validating sets. The experimental is plotted against the corresponding predicted fspt in Figs. 8–13. Figs. 8 and 10 show how well GEP-I, GEP-II, GEP-III and GEP-IV learned the nonlinear relation between parameters, while Figs. 9 and 11 illustrate the test results for the accuracy in generalization of the proposed GEP-based formulations and the developed regression-based formulas. In addition, Figs. 12 and 13 show the generalization capacity of the proposed all of the GEP-based formulations. The whole of results obtained from the GEP-based formulations show a successful performance of the GEP-based formulations for predicting 150  300 mm cylinder fspt from the corresponding 150  300 mm cylinder fc of concrete, and 100  200 mm and 150  200 mm cylinder fspt from the corresponding 100  200 mm cylinder fc of concrete for the each of

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M. Sarıdemir / Expert Systems with Applications 38 (2011) 14257–14268

Table 3 Comparison of experimental results to testing results of GEP-III, GEP-IV, regression and codes. AS (day)

fc (MPa)

fspt (MPa)

GEP-III

GEP-IV

ACI 363R

ACI 318

CEB-FIP

Regression

28 28 28 28 28 28 28 28 56 56 56 56 56 56 56 56 1 3 3 3 7 7 7 28 28 28 180 180 180 7 7 7 7 7 7 7 7 28 28 28 28 28 28 28 28 7 7 28 28 28 90 90 90 7 7 14 14 28 28 56 56 28 28 28 7 7 28 28 91 91 1 1 7 7

77.90 57.10 71.60 44.10 64.80 42.60 30.40 46.80 89.50 66.90 76.10 53.60 64.80 45.80 37.60 53.10 34.90 53.90 46.50 13.80 67.10 20.30 22.40 61.40 33.10 79.00 81.30 38.70 43.60 25.60 26.20 16.90 14.90 27.30 28.40 16.90 16.90 30.90 32.30 19.60 19.10 35.30 38.40 22.50 23.40 31.00 17.00 47.00 37.00 42.00 54.00 42.00 62.00 67.00 72.50 74.50 79.50 78.00 85.00 86.00 90.50 54.80 32.90 34.50 53.60 74.10 67.70 84.20 75.30 102.30 40.10 40.50 95.50 92.40

5.45 3.13 4.90 3.91 4.57 3.66 2.79 3.89 5.91 4.55 5.47 4.37 5.07 4.02 3.16 4.08 3.50 5.40 3.80 1.70 4.80 3.40 2.10 4.30 3.00 5.60 5.30 4.00 3.60 2.42 2.62 1.88 1.71 2.81 2.87 1.91 2.11 2.92 3.08 2.22 2.08 3.24 3.36 2.47 2.57 2.90 2.30 4.00 3.60 4.00 4.60 4.40 5.20 4.25 5.10 5.20 5.75 5.15 5.75 4.80 4.40 3.34 2.85 2.92 3.48 5.26 4.81 6.57 5.35 7.29 3.30 3.40 5.70 4.70

5.37 4.16 5.01 3.39 4.61 3.30 2.53 3.55 6.02 4.74 5.26 3.96 4.61 3.49 2.99 3.93 2.82 3.98 3.53 1.41 4.75 1.87 2.01 4.42 2.71 5.43 5.56 3.06 3.36 2.22 2.26 1.63 1.49 2.33 2.40 1.63 1.63 2.57 2.65 1.82 1.78 2.84 3.04 2.02 2.08 2.57 1.64 3.56 2.95 3.26 3.98 3.26 4.45 4.74 5.06 5.17 5.46 5.37 5.77 5.83 6.08 4.03 2.69 2.79 3.96 5.15 4.78 5.72 5.22 6.74 3.14 3.17 6.36 6.18

4.91 4.28 4.73 3.82 4.52 3.76 3.26 3.92 5.23 4.59 4.86 4.16 4.52 3.88 3.57 4.14 3.45 4.17 3.91 2.36 4.60 2.75 2.87 4.42 3.38 4.94 5.01 3.61 3.80 3.03 3.06 2.56 2.43 3.11 3.16 2.56 2.56 3.28 3.34 2.71 2.69 3.47 3.60 2.87 2.92 3.28 2.56 3.93 3.54 3.74 4.17 3.74 4.44 4.59 4.76 4.82 4.96 4.92 5.11 5.14 5.26 4.20 3.37 3.44 4.16 4.80 4.61 5.09 4.84 5.56 3.67 3.68 5.39 5.31

5.21 4.46 4.99 3.92 4.75 3.85 3.25 4.04 5.58 4.83 5.15 4.32 4.75 3.99 3.62 4.30 3.49 4.33 4.02 2.19 4.83 2.66 2.79 4.62 3.39 5.24 5.32 3.67 3.90 2.99 3.02 2.43 2.28 3.08 3.14 2.43 2.43 3.28 3.35 2.61 2.58 3.51 3.66 2.80 2.85 3.28 2.43 4.04 3.59 3.82 4.34 3.82 4.65 4.83 5.02 5.09 5.26 5.21 5.44 5.47 5.61 4.37 3.38 3.47 4.32 5.08 4.85 5.41 5.12 5.97 3.74 3.75 5.77 5.67

4.94 4.23 4.74 3.72 4.51 3.66 3.09 3.83 5.30 4.58 4.89 4.10 4.51 3.79 3.43 4.08 3.31 4.11 3.82 2.08 4.59 2.52 2.65 4.39 3.22 4.98 5.05 3.48 3.70 2.83 2.87 2.30 2.16 2.93 2.98 2.30 2.30 3.11 3.18 2.48 2.45 3.33 3.47 2.66 2.71 3.12 2.31 3.84 3.41 3.63 4.12 3.63 4.41 4.58 4.77 4.83 4.99 4.95 5.16 5.19 5.33 4.15 3.21 3.29 4.10 4.82 4.61 5.14 4.86 5.66 3.55 3.56 5.47 5.38

5.47 4.45 5.17 3.74 4.84 3.66 2.92 3.90 6.00 4.94 5.39 4.26 4.84 3.84 3.37 4.24 3.20 4.28 3.88 1.73 4.95 2.23 2.38 4.67 3.09 5.52 5.63 3.43 3.72 2.61 2.65 1.98 1.82 2.72 2.79 1.98 1.98 2.95 3.04 2.18 2.14 3.23 3.41 2.39 2.45 2.96 1.98 3.91 3.33 3.62 4.29 3.62 4.70 4.95 5.22 5.31 5.55 5.48 5.80 5.85 6.05 4.33 3.08 3.18 4.26 5.29 4.98 5.76 5.35 6.56 3.51 3.54 6.27 6.13

5.12 4.28 4.88 3.68 4.60 3.61 2.96 3.81 5.55 4.69 5.05 4.12 4.60 3.76 3.35 4.10 3.21 4.13 3.79 1.87 4.70 2.34 2.48 4.46 3.11 5.16 5.25 3.41 3.66 2.68 2.72 2.11 1.96 2.78 2.85 2.11 2.11 2.99 3.07 2.30 2.26 3.23 3.40 2.49 2.55 3.00 2.12 3.82 3.32 3.58 4.14 3.58 4.48 4.69 4.91 4.99 5.18 5.12 5.39 5.42 5.59 4.17 3.10 3.19 4.12 4.97 4.72 5.36 5.02 6.00 3.48 3.50 5.76 5.65

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M. Sarıdemir / Expert Systems with Applications 38 (2011) 14257–14268 Table 4 Comparison of experimental results to validating results of GEP-I, GEP-II, regression and codes. AS (day)

fc (MPa)

fspt (MPa)

GEP-I

GEP-II

ACI 363R

ACI 318

CEB-FIP

Regression

28 28 28 28 28 28 28 28 28 28 7 7 7 7 7 7 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 3 3 3 3 3 3 7 7 7 7 7 7 14 14 14 14 14 14 56

39.08 22.33 23.23 22.89 22.16 20.48 20.82 22.83 25.14 25.60 35.20 44.40 37.60 41.80 42.00 38.30 55.40 54.00 48.60 56.50 47.00 52.80 45.69 41.71 42.49 33.69 37.30 24.75 26.96 22.14 23.91 41.00 42.00 44.80 56.20 53.40 56.70 64.50 66.50 65.40 73.80 74.00 72.50 74.60 77.90 79.10 84.20 83.30 75.40 86.50 102.00 101.00 111.00 94.50 118.00 19.00 20.00 28.60 28.90 36.10 44.50 23.00 33.00 40.50 48.20 58.00 69.00 36.00 42.80 59.30 63.30 67.70 77.00 49.90

3.11 2.29 2.19 2.38 2.17 2.15 1.99 2.41 2.62 2.69 3.20 2.90 2.40 3.60 3.50 2.70 3.40 2.80 2.80 4.10 3.90 3.30 4.19 3.09 3.74 2.93 3.14 2.62 2.68 2.35 2.24 3.90 4.00 3.80 4.40 3.60 4.60 4.80 4.70 4.30 4.90 5.30 5.00 4.90 5.50 5.20 5.70 5.90 4.00 5.50 5.50 6.50 6.20 5.80 6.20 2.30 2.40 2.90 3.00 3.30 3.70 2.50 3.50 3.20 4.10 4.20 4.50 3.30 3.90 4.00 4.50 4.70 5.10 4.30

3.15 2.09 2.15 2.13 2.08 1.97 1.99 2.13 2.28 2.31 2.91 3.47 3.06 3.31 3.33 3.10 4.11 4.03 3.71 4.17 3.62 3.96 3.54 3.31 3.36 2.82 3.04 2.25 2.40 2.08 2.20 3.27 3.33 3.49 4.15 3.99 4.18 4.62 4.73 4.67 5.13 5.14 5.06 5.18 5.36 5.42 5.70 5.65 5.22 5.82 6.65 6.59 7.12 6.25 7.48 1.87 1.94 2.50 2.52 2.97 3.47 2.14 2.78 3.24 3.69 4.25 4.87 2.96 3.37 4.33 4.55 4.80 5.31 3.79

3.48 2.62 2.68 2.66 2.61 2.51 2.53 2.65 2.79 2.81 3.23 3.64 3.35 3.53 3.54 3.38 4.15 4.09 3.88 4.19 3.82 4.05 3.76 3.59 3.63 3.23 3.40 2.76 2.89 2.61 2.72 3.56 3.61 3.73 4.18 4.07 4.19 4.48 4.54 4.51 4.79 4.80 4.75 4.81 4.92 4.96 5.12 5.09 4.84 5.19 5.63 5.61 5.88 5.42 6.06 2.23 2.30 2.79 2.80 3.16 3.53 2.60 3.13 3.48 3.80 4.18 4.56 3.32 3.62 4.27 4.41 4.56 4.87 3.94

3.69 2.79 2.84 2.82 2.78 2.67 2.69 2.82 2.96 2.99 3.50 3.93 3.62 3.81 3.82 3.65 4.39 4.34 4.11 4.43 4.04 4.29 3.99 3.81 3.85 3.42 3.60 2.94 3.06 2.78 2.88 3.78 3.82 3.95 4.42 4.31 4.44 4.74 4.81 4.77 5.07 5.08 5.02 5.10 5.21 5.25 5.41 5.38 5.12 5.49 5.96 5.93 6.22 5.74 6.41 2.57 2.64 3.16 3.17 3.54 3.94 2.83 3.39 3.75 4.10 4.49 4.90 3.54 3.86 4.54 4.69 4.85 5.18 4.17

3.50 2.65 2.70 2.68 2.64 2.53 2.56 2.68 2.81 2.83 3.32 3.73 3.43 3.62 3.63 3.47 4.17 4.12 3.90 4.21 3.84 4.07 3.79 3.62 3.65 3.25 3.42 2.79 2.91 2.63 2.74 3.59 3.63 3.75 4.20 4.09 4.22 4.50 4.57 4.53 4.81 4.82 4.77 4.84 4.94 4.98 5.14 5.11 4.86 5.21 5.66 5.63 5.90 5.44 6.08 2.44 2.50 2.99 3.01 3.36 3.74 2.69 3.22 3.56 3.89 4.26 4.65 3.36 3.66 4.31 4.46 4.61 4.91 3.96

3.45 2.38 2.44 2.42 2.37 2.25 2.27 2.41 2.57 2.61 3.22 3.76 3.37 3.61 3.62 3.41 4.36 4.29 4.00 4.42 3.91 4.22 3.83 3.61 3.65 3.13 3.35 2.55 2.70 2.37 2.49 3.57 3.62 3.78 4.40 4.25 4.43 4.82 4.92 4.87 5.28 5.29 5.22 5.32 5.47 5.53 5.76 5.72 5.35 5.87 6.55 6.51 6.93 6.22 7.22 2.14 2.21 2.81 2.83 3.28 3.77 2.43 3.09 3.54 3.97 4.50 5.05 3.27 3.67 4.56 4.76 4.98 5.43 4.07

3.28 2.30 2.35 2.33 2.28 2.17 2.20 2.33 2.48 2.50 3.07 3.55 3.20 3.42 3.43 3.23 4.09 4.02 3.76 4.14 3.68 3.97 3.62 3.41 3.45 2.98 3.18 2.45 2.59 2.28 2.40 3.38 3.43 3.57 4.13 3.99 4.15 4.50 4.59 4.54 4.91 4.91 4.85 4.94 5.08 5.13 5.33 5.30 4.97 5.43 6.03 5.99 6.36 5.74 6.61 2.07 2.14 2.69 2.70 3.11 3.56 2.34 2.94 3.35 3.74 4.21 4.70 3.11 3.47 4.27 4.45 4.64 5.04 3.83 (continued on next page)

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M. Sarıdemir / Expert Systems with Applications 38 (2011) 14257–14268

Table 4 (continued) AS (day) 56 56 56 56 56

fc (MPa) 53.00 61.70 73.20 86.50 92.30

fspt (MPa)

GEP-I

GEP-II

ACI 363R

ACI 318

CEB-FIP

Regression

5.40 4.60 5.10 5.50 5.90

3.97 4.46 5.10 5.82 6.13

4.07 4.39 4.78 5.20 5.37

4.30 4.63 5.05 5.49 5.67

4.08 4.40 4.79 5.21 5.38

4.23 4.68 5.25 5.87 6.13

3.98 4.38 4.88 5.43 5.65

Table 5 Comparison of experimental results to validating results of GEP-III, GEP-IV, regression and codes. AS (day) 1 1 3 5 7 14 1 1 3 5 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 28 3 3 3 3 3 3 7 7 7 7 7 7 28 28 28 28 28 28

fc (MPa)

fspt (MPa)

GEP-III

GEP-IV

ACI 363R

ACI 318

CEB-FIP

Regression

41.30 35.20 57.50 38.60 61.50 47.40 47.10 44.70 52.40 57.90 39.50 36.70 38.00 36.00 30.40 29.30 27.00 32.60 30.4 29.5 43.10 35.20 57.50 38.60 61.50 47.40 47.13 44.73 52.35 57.87 51.90 50.00 48.50 48.80 46.50 45.20 45.20 43.30 43.30 42.70 41.90 41.20 39.70 39.50 39.50 17.1 13.30 13.60 12.10 12.90 16.90 24.50 24.20 17.40 21.60 21.20 25.10 31.90 37.80 29.20 35.40 28.60 29.50

4.18 3.17 3.69 3.38 4.48 3.76 3.68 3.26 3.99 4.49 4.04 4.03 3.65 3.49 3.08 2.80 2.55 3.21 2.85 2.56 4.18 3.17 3.69 3.38 4.48 3.76 3.68 3.26 3.99 4.49 4.00 3.90 4.10 3.70 3.80 4.00 3.30 3.60 3.90 3.00 3.50 3.80 2.70 3.40 3.70 1.9 1.20 1.50 1.10 1.60 1.50 2.50 2.20 1.80 2.00 2.70 2.60 3.30 3.10 2.50 3.10 3.40 3.10

3.22 2.84 4.20 3.05 4.43 3.59 3.57 3.43 3.89 4.22 3.11 2.94 3.02 2.89 2.54 2.47 2.32 2.68 2.54 2.48 3.33 2.84 4.20 3.05 4.43 3.59 3.58 3.43 3.89 4.22 3.86 3.75 3.66 3.68 3.54 3.46 3.46 3.34 3.34 3.31 3.26 3.21 3.12 3.11 3.11 1.65 1.37 1.39 1.28 1.34 1.63 2.15 2.13 1.67 1.96 1.93 2.19 2.63 3.00 2.46 2.85 2.42 2.48

3.71 3.47 4.29 3.61 4.42 3.94 3.93 3.84 4.12 4.30 3.64 3.53 3.58 3.50 3.26 3.21 3.10 3.35 3.26 3.22 3.78 3.47 4.29 3.61 4.42 3.94 3.93 3.84 4.12 4.30 4.10 4.04 3.98 3.99 3.91 3.86 3.86 3.79 3.79 3.77 3.74 3.71 3.65 3.64 3.64 2.57 2.32 2.34 2.24 2.30 2.56 2.97 2.96 2.59 2.82 2.80 3.00 3.32 3.57 3.20 3.47 3.17 3.22

3.79 3.50 4.47 3.67 4.63 4.06 4.05 3.94 4.27 4.49 3.71 3.57 3.64 3.54 3.25 3.19 3.07 3.37 3.25 3.20 3.87 3.50 4.47 3.67 4.63 4.06 4.05 3.95 4.27 4.49 4.25 4.17 4.11 4.12 4.02 3.97 3.97 3.88 3.88 3.86 3.82 3.79 3.72 3.71 3.71 2.44 2.15 2.18 2.05 2.12 2.43 2.92 2.90 2.46 2.74 2.72 2.96 3.33 3.63 3.19 3.51 3.16 3.20

3.60 3.32 4.25 3.48 4.39 3.86 3.84 3.74 4.05 4.26 3.52 3.39 3.45 3.36 3.09 3.03 2.91 3.20 3.09 3.04 3.68 3.32 4.25 3.48 4.39 3.86 3.84 3.75 4.05 4.26 4.03 3.96 3.90 3.91 3.82 3.76 3.76 3.68 3.68 3.66 3.62 3.59 3.53 3.52 3.52 2.32 2.04 2.07 1.95 2.01 2.30 2.77 2.75 2.34 2.60 2.58 2.81 3.16 3.44 3.03 3.33 2.99 3.04

3.58 3.22 4.47 3.43 4.67 3.93 3.91 3.78 4.20 4.49 3.48 3.31 3.39 3.27 2.92 2.85 2.70 3.06 2.92 2.86 3.69 3.22 4.47 3.43 4.67 3.93 3.91 3.78 4.20 4.49 4.17 4.07 3.99 4.01 3.88 3.81 3.81 3.70 3.70 3.66 3.62 3.58 3.49 3.48 3.48 1.99 1.68 1.71 1.58 1.65 1.98 2.53 2.51 2.01 2.33 2.30 2.57 3.02 3.38 2.84 3.23 2.81 2.86

3.54 3.23 4.29 3.41 4.46 3.84 3.82 3.71 4.07 4.31 3.45 3.31 3.37 3.27 2.96 2.90 2.77 3.09 2.96 2.91 3.63 3.23 4.29 3.41 4.46 3.84 3.82 3.71 4.07 4.31 4.04 3.96 3.89 3.90 3.79 3.73 3.73 3.64 3.64 3.61 3.57 3.54 3.46 3.45 3.45 2.12 1.83 1.86 1.74 1.80 2.11 2.62 2.60 2.14 2.43 2.40 2.65 3.05 3.36 2.90 3.24 2.86 2.91

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M. Sarıdemir / Expert Systems with Applications 38 (2011) 14257–14268 Table 5 (continued) AS (day)

fc (MPa)

fspt (MPa)

GEP-III

GEP-IV

ACI 363R

ACI 318

CEB-FIP

Regression

91 91 91 91 91 91 180 180 180 180 180 180 3 7 28 90 180 3 7 28 90 180 28 28 28 28 28 28 28 28 28

34.90 42.80 34.60 38.40 51.50 48.60 39.50 46.30 35.80 42.80 51.90 49.30 42.60 43.60 49.90 57.40 58.50 31.40 34.90 41.60 46.80 51.70 81.60 60.80 60.00 71.90 59.40 68.00 75.10 59.70 65.50

3.50 4.90 4.10 3.80 4.20 4.00 3.90 5.20 4.30 4.30 4.30 4.20 2.98 3.38 4.57 4.20 4.90 2.79 3.04 3.84 4.00 4.20 4.40 4.10 4.30 5.40 4.70 5.20 5.40 5.40 5.50

2.82 3.31 2.80 3.04 3.84 3.66 3.11 3.53 2.88 3.31 3.86 3.71 3.30 3.36 3.74 4.19 4.25 2.60 2.82 3.24 3.56 3.85 5.59 4.39 4.34 5.03 4.31 4.81 5.22 4.32 4.66

3.45 3.77 3.44 3.60 4.09 3.99 3.64 3.90 3.49 3.77 4.10 4.01 3.76 3.80 4.03 4.29 4.32 3.30 3.45 3.72 3.92 4.10 5.02 4.40 4.37 4.74 4.35 4.62 4.83 4.36 4.55

3.49 3.86 3.47 3.66 4.23 4.11 3.71 4.01 3.53 3.86 4.25 4.14 3.85 3.90 4.17 4.47 4.51 3.31 3.49 3.81 4.04 4.24 5.33 4.60 4.57 5.00 4.55 4.87 5.11 4.56 4.77

3.31 3.66 3.29 3.47 4.02 3.90 3.52 3.81 3.35 3.66 4.03 3.93 3.66 3.70 3.96 4.24 4.28 3.14 3.31 3.61 3.83 4.03 5.06 4.37 4.34 4.75 4.32 4.62 4.85 4.33 4.53

3.20 3.67 3.19 3.41 4.15 4.00 3.48 3.87 3.26 3.67 4.17 4.03 3.66 3.72 4.07 4.46 4.52 2.99 3.20 3.60 3.90 4.16 5.64 4.64 4.60 5.19 4.57 5.00 5.34 4.58 4.87

3.21 3.62 3.20 3.40 4.03 3.89 3.45 3.79 3.26 3.62 4.04 3.93 3.61 3.66 3.95 4.29 4.34 3.02 3.21 3.56 3.81 4.04 5.26 4.43 4.40 4.89 4.37 4.73 5.01 4.39 4.63

Table 6 Statistical parameters for predicting fspt from 150  300 mm cylinder fc. Statistical parameters Training MAPE RMSE R2 Testing MAPE RMSE R2 Validating MAPE RMSE R2

GEP-I

GEP-II

ACI 363R

ACI 318

CEB-FIP

Regression

9.2896 0.4495 0.8450

9.5097 0.4462 0.8409

14.5727 0.5579 0.8324

11.9258 0.4965 0.8324

11.878 0.4872 0.8400

9.6911 0.4371 0.8388

9.5570 0.4299 0.8627

9.6155 0.4388 0.8629

14.6788 0.5576 0.8581

12.2167 0.4984 0.8581

11.4680 0.4713 0.8618

9.7688 0.4251 0.8613

10.1023 0.4803 0.8826

9.9451 0.4463 0.8810

12.2108 0.5085 0.8849

10.1541 0.4530 0.8849

9.1134 0.4782 0.8855

8.1762 0.4040 0.8856

Table 7 Statistical parameters for predicting fspt from 100  200 mm cylinder fc. Statistical parameters

GEP-III

GEP-IV

ACI 363R

ACI 318

CEB-FIP

Training MAPE RMSE R2

13.7879 0.6278 0.8230

12.6839 0.6039 0.8180

12.1895 0.5726 0.8180

10.7277 0.5618 0.8180

10.3462 0.5528 0.8219

9.6760 0.5292 0.8203

Testing MAPE RMSE R2

11.9746 0.5655 0.8628

12.9058 0.5524 0.8680

11.5534 0.5028 0.8680

10.1670 0.5049 0.8680

7.7888 0.4780 0.8672

8.0314 0.4631 0.8680

Validating MAPE RMSE R2

11.8543 0.5628 0.7711

14.3742 0.5117 0.7939

13.6486 0.4899 0.7939

12.0395 0.4677 0.7939

9.7524 0.4339 0.7849

10.5474 0.4463 0.7899

training, testing and validating sets. Besides, input values, experimental results, testing and validating sets of results obtained from

Regression

the GEP-based formulations and the comparison of these data with the results obtained from different national building codes formu-

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las, the developed regression-based formulation and the other formulations are given in Tables 2–5. The performance of the trained, tested and validated sets was analyzed by computing MAPE, RMSE and R2 of experimental fspt between the GEP-based formulations and other methods. Statistical parameters of the training, testing and validating sets of all GEPbased formulations are presented in Tables 6 and 7. All of these statistical parameters show a successful performance of the GEPbased formulations for predicting fspt from 150  300 mm cylinder and 100  200 mm cylinder fc in training, testing and validating sets. Generally, R2 of the GEP-based formulations are higher than or equal to R2 of ACI 363R-92 (1992), ACI 318-99 (1999) and CEB-FIP (1991) and regression. 4. Conclusions This present study reports a new and influential approach for the formulations of splitting tensile strength using GEP for the first time in the literature. Four different formulations are proposed for predicting splitting tensile strength from cylinder compressive strength of concrete or age of specimen and cylinder compressive strength of concrete. The proposed GEP-based formulations are empirical and based on experimental data gathered from existing literatures. To predict the splitting tensile strength of concrete, two equations based on only cylinder compressive strength as given Eqs. (7) and (9) are sufficiently accurate. In addition, to predict the splitting tensile strength of concrete, two equations based on age of specimen and cylinder compressive strength as given Eqs. (8) and (10) are sufficiently accurate. The proposed GEP-based equations are so simple that they can be utilized by anybody not absolutely familiar with GEP. The statistical parameters of MAPE, RMS and R2 demonstrate that the proposed GEP-based formulations results has best accuracy and can predict splitting tensile strength very close to experiment results. Besides, comparison of MAPE, RMS and R2 calculations reveal that the GEP-based formulations and the developed regression-based formulation, according to formulas given by some national building codes, have generally the smallest value for the training, testing and validating sets. The results of this present study will give some useful information to civil engineers and structural designers and this methodology can be utilized as a new and influential tool to support the decision process at concrete construction fields. References ACI Committee 318 (1999). Building code requirements for structural concrete (ACI 318-99) and commentary (318R-99). Farmington Hills, MI: American Concrete Institute. ACI Committee 363 (1992). State-of-the art report on high strength concrete (ACI 363R-92). Farmington Hills, MI: American Concrete Institute. Ajdukiewicz, A., & Kliszczewicz, A. (2002). Influence of recycled aggregates on mechanical properties of HS/HPC. Cement and Concrete Composites, 24(2), 269–279. CEB-FIP model code for concrete structures (1991). Evaluation of the time dependent behavior of concrete. Bulletin d’ information No. 199, Comite Europe du Beton/ Federation Internationale de Precontrainte, Lausanne. Choi, Y., & Yuan, R. L. (2005). Experimental relationship between splitting tensile strength and compressive strength of GFRC and PFRC. Cement and Concrete Research, 35(8), 1587–1591. Emiroglu, M., Kelestemur, M. H., Yıldız, S. (2007). An investigation on ITZ microstructure of the concrete containing waste vehicle tire. In Proceedings of 8th international fracture conference (pp. 453–459). Ferreira, C. (2001a). Gene expression programming: A new adaptive algorithm for solving problems. Complex Systems, 13(2), 87–129. Ferreira, C. (2001b). Gene expression programming in problem solving. In Sixth online world conference on soft computing in industrial applications (pp. 1–22).

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