Effect of specimen size and shape on compressive strength of concrete containing fly ash: Application of genetic programming for design

Accepted Manuscript Technical Report Effect of specimen size and shape on compressive strength of concrete containing fly ash: Application of genetic programming for design Mustafa Sarıdemir PII: DOI: Reference:

S0261-3069(13)01016-9 http://dx.doi.org/10.1016/j.matdes.2013.10.073 JMAD 5982

To appear in:

Materials and Design

Received Date: Accepted Date:

29 April 2013 27 October 2013

Please cite this article as: Sarıdemir, M., Effect of specimen size and shape on compressive strength of concrete containing fly ash: Application of genetic programming for design, Materials and Design (2013), doi: http:// dx.doi.org/10.1016/j.matdes.2013.10.073

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Effect of specimen size and shape on compressive strength of concrete containing fly ash: Application of genetic programming for design

Mustafa Sarıdemir Department of Civil Engineering, Niğde University, 51240 Niğde, Turkey

The use of fly ash as a mineral admixture in the manufacture of concrete has received considerable attention in recent years. For this reason, several experimental studies are carried out by using fly ash at different proportions replacement of cement in concrete. In the present study, the models are developed in genetic programming for predicting the compressive strength values of cube (100 and 150 mm) and cylinder (100x200 and 150x300 mm) concrete containing fly ash at different proportions. The experimental data of different mixtures are obtained by searching 36 different literatures to predict these models. In the set of the models, the age of specimen, cement, water, sand, aggregate, superplasticizers, fly ash and CaO are entered as input parameters, while the compressive strength values of concrete containing fly ash are used as output parameter. The training, testing and validation set results of the explicit formulations obtained by the genetic programming models show that artificial intelligent methods have strong potential and can be applied for the prediction of the compressive strength of concrete containing fly ash with different specimen size and shape.

Keywords: Fly ash; Compressive strength; Size and shape effects; Genetic programming. Corresponding author. Tel.: +090 388 225 2485

E-mail: [email protected] (Mustafa Sarıdemir)

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1. Introduction

Environmental pollution of industrial waste has become an important problem in recent years which is potentially damaging to both the environment and human health. New trends in environmental regulations related to disposal of wastes such as fly ash (FA), silica fume or granulated blast furnace slag have begun increasing interests in using the wastes as construction materials partially replacing Portland cement in concrete [1]. FA is the byproduct of the burning of pulverized coal and is collected by mechanical and electrostatic separators from the fuel gas of power plants where coal is used as a fuel [2]. A material, which forms of environmental pollution, is evaluated by using FA in concrete production. FA has been commonly used to replace part of cement in concrete, and the percentage of replacement ranges from about 20% (low volume FA) to more than 50% (high volume FA) of the total cementitious materials [2,3]. Furthermore, if the early strength is not an important factor, FA can be used as high as 70%. This replacement rates is determined by a variety of experimental studies.

FA improves the some properties when used in concrete as a replacement of cement. When producing concrete containing FA or building concrete containing FA structures, it is necessary to predict the mechanical properties of concrete containing FA such as compressive strength (fc), flexural strength, splitting tensile strength, modulus of elasticity, creep, durability, shrinkage, etc. Among the mechanical properties used in design, fc is most important since other properties can be predicted on the basis of fc [1].

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Artificial neural networks and fuzzy logic methods are commonly used in the many of civil engineering applications. In addition, gene expression programming (GEP) has also been used for civil engineering applications in recent years. In the present study, four different models have been developed for predicting the fc values of concrete containing FA with different size and shape. The fc values of cube (100 mm and 150 mm) and cylinder (100x200 and 150x300 mm) concrete containing FA at different proportions are predicted by the models developed in GEP method. In GEP method, the models were developed to predict the fc values of (100 and 150 mm) cube concrete, and also (100x200 and 150x300 mm) cylinder concrete, respectively. The developed GEP models were named as GEP-I, GEP-II, GEP-III and GEP-IV, respectively. The age of specimen (AS), cement (C), water (W), sand (S), aggregate (A), superplasticizer (SP) and fly ash (FA) were used as input variables in training, testing and validation sets of the GEP models, while the fc values were used as output in the training set. In addition to these input variables, CaO was used in the fourth model.

The reason of the development four different models in this study is that the size and shape of the specimens significantly affect the results of fc test which is commonly used in the quality control procedures of concrete. Because of the size and shape effect, the relative strength of concrete shows variability for different dimensions and forms of the specimens. This variability is clearly observed in the made experimental studies and the references used for create models. Besides, these models are developed for eliminating loss

of time and materials by predicting the fc values of concrete containing FA without experimental study.

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2. Gene expression programming

Gene expression programming (GEP) is, like genetic algorithms (GAs) and genetic programming (GP), a GA as it uses populations of individuals, selects them according to fitness, and initiates genetic diversity using one or more genetic operators [4]. GEP algorithm is a wide range of functions scans as a combination of GA and GP algorithms. The flowchart of a gene expression algorithm is shown in Fig. 1. A large portion of the flowchart in GEP algorithm constitutes the genetic operators.

In GEP algorithm, all the problems are represented by ETs that compose of operators, functions, constants and variables. An algebraic expression [((a × b)-c) + d-e ] can be represented by a two genes chromosome or an ET, as shown in Fig. 2. This figure shows how a chromosome with two genes is encoded as a linear string and also how it is expressed as an ET [5].

2.1. Development of GEP model

The main task of the present article is to model the fc values of (100 and 150 mm) cube

concrete, and also (100x200 and 150x300 cm) cylinder concrete containing FA at different proportions based on experimental studies by using GEP. For this purpose, the fc values of concrete were obtained by using the four GEP models in connection with AS, C, W, S, A, SP, FA and CaO (CaO was only used in the fourth model) values of concrete mixtures. The four GEP models were named as GEP-I (for compressive strength (fc10) of 100 mm cube concrete), GEP-II (for compressive strength (fc15) of 150 mm cube

concrete), GEP-III (for compressive strength (fc10x20) of 100x200 mm cylinder concrete),

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and GEP-IV (for compressive strength (fc15x30) of 150x300 mm cylinder concrete), respectively.

The GEP-I and GEP-II models were developed for predicting the fc10 and fc15 values of cube concrete from the experimental results of 572 and 274 specimens of concrete containing

FA which were obtained from the references [3,6-15] and [2,16-22], respectively. These models were trained with 270 and 107 of the experimental data, respectively, and they were also tested with 136 and 53 of them which not used in the training set, respectively. These experimental data, relating to the input and output variables of training and testing sets, have been entered into computer program for creating models in the GEP. After the models were trained and tested, formulations depending on the input variables were obtained. These formulations were verified with using the 166 and 114 experimental data obtained from the references [12-15] and [21,22], respectively. Similarly, for predicting the fc10x20 and fc15x30 values of cylinder concrete, the GEP-III and GEP-IV models were developed by using the experimental results of 660 and 542 of

concrete containing which was obtained from the references [1,3,23-31] and [32-38], respectively. While these models were trained with 278 and 312 of the experimental data, respectively, they were also tested with 140 and 154 of them, respectively. Afterwards formulations depending on the input variables were obtained. These formulations were verified with using the 166 and 80 experimental data obtained from the references [3,27-31] and [38], respectively. So, the formulations obtained from the models were used to predict the fc values of concrete containing FA with different specimen size and shape. The data sets of the proposed GEP models were given in Table 1.

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For the GEP-based formulations, firstly, the fitness, fi , of an individual program, i, is measured

in the following equation:

Ct

(

fi = ∑ M − C(i , j ) − T( j ) j =i

)

(1)

where “M” is the range of selection, “C(i,j)” 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 ) − T( j ) (the precision) is less than or equal to 0.01, then the precision is equal to zero, and fi = fmax = Ct.M. 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 by itself [4, 5]. Afterwards, the set of terminals “T” and the set of functions “F” to create the chromosomes are chosen, namely, “T”={AS, C, W, S, A, SP, FA, CaO} (CaO was used in the GEP-IV model) and four basic arithmetic operators (+, -, *, /) and some basic mathematical functions (Sqrt, 3Rt, 4Rt, Sub3, Exp, x3, Add3, 1/x, Ln) were utilized as shown Table 2. Another major step is to choose the chromosomal architecture, i.e., the length of the head and the number of genes. After several trials, length of the head, h = 10, and the genes per chromosome were found to give the best results. The sub-ETs (genes) were linked by multiplication. Finally, a combination of all genetic operators (mutation, transposition and recombination) was used as set of genetic operators. The training parameters of the GEP models are given in Table 2.

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The explicit formulations of the GEP-I, GEP-II, GEP-III, and GEP-IV for the prediction of fc were obtained by Eq. (2). The ETs of the formulations in the Eqs. (3)-(6) are shown in Figs. 3-6 for the GEP-I, GEP-II, GEP-III and GEP-IV models, respectively.

f c = f (AS, C, W, S, A, SP, FA, CaO)

(2)

⎡ ⎛ d4 − 2 d 2 ⎞ ⎤ ⎡ ⎤ ⎢18 ⎜⎜⎝ 3 ( d 6 + d03 ) ⎟⎟⎠ ⎥ d6 f c10 = ⎢ 4 3c1 + − d1 + d 4 − d 6 ⎥ × ⎢ e ⎥ c2 ⎣ ⎦ ⎢ ⎥⎦ ⎣ 1 d ⎤ ⎡ d 0 + 2d1 + c1 xd 5 − − 3 ⎥ ⎢ ⎡2 ⎤ c2 d 0 1 ⎥ ×⎢ + 3 ⎥×⎢ d5 + d 2 ⎥ ⎣ c2 d 6 + c2 + 2 d3 − c0 − d 4 ⎦ ⎢ ⎢⎣ ⎥⎦

× ⎡ Ln ⎢⎣

f c10

3

)

d4 − 2d 0 + d5 × Ln(d 0 ) ⎤ ⎥⎦

A− 2 S ⎞ ⎤ ⎡ ⎛ ⎡ ⎤ ⎢18 ⎜⎜⎝ 3 (UK + G3 ) ⎟⎟⎠ ⎥ UK = ⎢ 4 −29.991 − − C + A − UK ⎥ × e ⎢ ⎥ 4.046 ⎣ ⎦ ⎢ ⎥⎦ ⎣ K⎤ ⎡ G + 2C + 6.480 xSA + 5.780 − ⎥ ⎢ 1 ⎡ ⎤ G × ⎢1.236 + ×⎢ ⎥ 3 ⎥ UK + 1.618 + 2 K − 5.951 − A ⎦ ⎢ SA + S ⎣ ⎥ ⎣ ⎦

× ⎡ Ln ⎢⎣

f c15

(

(

3

)

A − 2G + SA × Ln(G) ⎤ ⎥⎦

⎡⎛ d +d = ⎢⎜ 12 d 0 + 4 1 ⎜ d2 ⎢⎝ ⎣

⎞ ⎟⎟ ⎠

2

(3)

⎤ ⎡ ⎤ ⎥ × ⎢ c1 + 2d1 − d 4 + d 6 + c1 ⎥ c2 ⎥ ⎣ ⎦ ⎦

7

⎡ d ⎤ ⎡ ( 4 d −d ) ⎤ × ⎢ 4 3d ⎥ × ⎢ 3 e 1 5 × c0 + c1 + d 6 ⎥ 2 ⎦ ⎣ e ⎦ ⎣

f c15

⎡⎛ = ⎢⎜⎜ 12 G + ⎢⎝ ⎣

A+G S

⎞ ⎟⎟ ⎠

2

⎤ ⎡ ⎤ A ⎥ × ⎢ −9.837 + 2C − + UK − 9.837 ⎥ 6.290 ⎥ ⎣ ⎦ ⎦

⎡ K ⎤ ⎡ 3 ( 4 C − SA) ⎤ ×⎢4 S ⎥×⎢ e × 9.172 + 4.770 + UK ⎥ ⎦ ⎣ e ⎦ ⎣

f c10×20

(4)

⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ c0 c1 ⎥×⎢ ⎥ =⎢ ⎢ d1 × d 6 ⎥ ⎢ ⎥ ⎛ d3 c2 ⎞ + d2 + d5 ⎥ ⎢ d4 + ⎜ × ⎟ × ( d6 + 2c0 ) ⎥ ⎢ ⎢⎣ d0 − d4 ⎥⎦ ⎢⎣ ⎥⎦ ⎝ d0 c1 ⎠

⎡ ⎤ ⎢ ⎥ ⎡d ⎤ d4 ×⎢ + d1 + d 6 ⎥ × ⎢ 0 xd5 xc2 + d 4 + d 0 × d1 − d5 ⎥ d ⎢(d × c × d 2 × d ) − 1 ⎥ ⎣ d2 ⎦ 0 ⎢⎣ 6 1 5 ⎥⎦ c0

f c10×20

⎤ ⎡ ⎤ ⎡ ⎢ ⎥ ⎢ ⎥ 2.974 7.082 ⎥ =⎢ ⎥×⎢ ⎢ C × UK ⎥ ⎢ A + ⎛ 0.045 × K ⎞ × (UK + 19.968 ) ⎥ ⎜ ⎟ ⎥⎦ ⎢⎣ G − A + S + SA ⎥⎦ ⎢⎣ G⎠ ⎝

⎡ ⎤ ⎢ ⎥ A ×⎢ + C + UK ⎥ C ⎢ (UK × (−0.052) × SA2 × G ) − ⎥ 9.951 ⎣ ⎦ ⎡G ⎤ × ⎢ × SA × ( −6128) + A + G × C − SA⎥ ⎣S ⎦

(5)

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f c15×30

⎤ ⎡ ⎤ ⎡⎛ ⎛ ⎞⎞ d4 ⎢ ⎥ ⎜ ⎟ 4 ⎜ ⎟ ⎢ ⎥ d1 ⎥ × ⎢ ⎜ 4 ⎝ d5 + d 0 + d 2 ⎠ ⎟ × c ⎥ =⎢ ⎟ 1⎥ ⎢8 ⎛ d ⎞ ⎥ ⎢⎜ d3 6 ⎢ ⎥ ⎜ ⎟ ⎢ ⎜ ⎟ + c1 ⎥ ⎢⎜ ⎥ ⎟ ⎢⎣ ⎝ c2 × d o ⎠ ⎥⎦ ⎝ ⎠ ⎣ ⎦

⎡ ×⎢4 ⎣

4

((

⎤ d 6 × c22 × d 5 × ( d5 + d 7 ) × d13 + d1 ⎥ ⎦

)

)

× ⎡ 4 c22 + d7 + d1 + c1 + d0 − d6 × d5 ⎤ ⎣⎢ ⎦⎥

f c15×30

⎡⎛ ⎤ ⎞⎞ A ⎡ ⎤ ⎢⎜ ⎛ 4 ⎥ ⎟ ⎜ ⎟ ⎢ ⎥ ⎢⎜ 4 S A G S + + ⎟ C ⎠ × 8.612 ⎥ ⎥ × ⎢⎜ ⎝ =⎢ ⎥ ⎟ K ⎢ 8 ⎛ UK ⎞ + 7.404 ⎥ ⎢⎜ ⎥ ⎟ ⎢ ⎜⎝ 1.421× G ⎟⎠ ⎥ ⎢⎜ ⎥ ⎟ ⎣ ⎦ ⎠ ⎣⎝ ⎦

⎡ ×⎢4 ⎣

4

((

⎤ UK × 89.506 × SA × ( SA + CaO ) × C 3 + C ⎥ ⎦

)

)

× ⎡ 4 0.632 + CaO + C − 6.401 + G − UK × SA ⎤ ⎣⎢ ⎦⎥

(6)

The ETs of the GEP-I, GEP-II, GEP-III, and GEP-IV for predicting the fc values of cube (100 and 150 mm) and cylinder (100x200 and 150x300 mm) concrete containing FA at different proportions are given in Figs. 3-6, respectively. In all the GEP models, multiplication was used as the linking functions and the numbers of genes (Sub-ETs) are given in Table 2. The ETs of all formulations used for predicting fc are shown in Figs. 3-6 where d0, d1, d2, d3, d4, d5, d6, and d7 refer to AS, C, W, S, A, SP, FA and CaO, respectively. The constants in all formulations are given in Table 3.

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3. Results and discussion

In the evaluation of the model results, it is important to define the criteria, which is the performance of the model and its prediction accuracy. Several statistical parameters were used to evaluate the performance of model. The mean absolute percentage error (MAPE), root-mean-squared error (RMSE) and R-square (R2) were used as the criteria between the experimental (target) and predicted values, according to the Eqs. (7)-(9), respectively.

⎡ n ⎤ ti − oi ∑ ⎢ ⎥ 1 MAPE = ⎢ i =1 n × 100 ⎥ n⎢ ⎥ ti ⎢⎣ ∑ ⎥⎦ i =1

RMSE =

R2 =

(7)

1 n (ti − oi )2 ∑ n i =1

(8)

(n∑ ti oi − ∑ ti ∑ oi )2

(9)

(n∑ t i2 − (∑ ti )2 )(n∑ oi2 − (∑ oi )2

where “t” is the experimental value, “o” is the predicted value, and “n” is total number of data. If the RMS statistical value is smaller, the results obtained from the models are

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closer to the experimental (target) results. On the other hand, if the RMS statistical value is higher, the results obtained by the models are far from the experimental (target) results. If R2 values are above 0.7, and also closer to 1, the results obtained from the models are very close to the experimental (target) results. The total errors between the experimental results (target) and model results are also evaluated by using the MAPE statistical value. In addition, the absolute errors between each experimental result (target) and model results are determined by using the MAPE statistical value.

Figs. 7-10 show the results obtained from experimental studies and predicted by using the training, testing and validation sets of the GEP-I, GEP-II, GEP-III and GEP-IV models, respectively. The linear least square fit line and the R2 values are shown in these figures for the training, testing and validation sets of the models. As can be clearly seen in Figs. 7-10, the fc values obtained from the training, testing and validation sets in GEP-I, GEP-II, GEP-III and GEP-IV models are very close to the experimental results. R2 values given on the figures for all sets confirm this situation. Also, the statistic parameter results of all the GEP models are given in Tables 4. The statistical analysis results of the training, testing and validation sets of the GEP models indicate a good correlation between the input parameters and the fc values of cube and cylinder concrete containing FA at different proportions. As seen in Table 4, R2 of the models for the training, testing and validation sets are higher than 0.82. The best value of R2 is 0.9386 for training set in the GEP-II model, while the minimum value of R2 is 0.8223 for validation set in the GEP-IV model. All of the statistical parameter values seen in Table 4 indicate that the proposed GEP models predict the fc values of cube and cylinder concrete containing FA at different ages with good accuracy.

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Although benefiting from the experimental studies conducted by researchers in different parts of the world, the results of this study indicate that the formulations obtained by

using the GEP are available in wide range of predicting the fc values of concrete containing FA. Moreover, these formulations are used for designing new concrete mixtures containing FA. Therefore, GEP can provide an effective alternative approach to the experimental studies or artificial neural networks and fuzzy logic used in modeling the fc of concrete mixtures containing FA.

4. Conclusions

This paper evaluates GEP as an alternative tool for modeling the compressive strength values of (100 and 150 mm) cube and (100x200 and 150x300 mm) cylinder concrete containing fly ash at different proportions. Four different models which are named GEP-I, GEP-II, GEP-III and GEP-IV, are proposed for predicting the compressive strength values of concrete. The experimental results, which are obtained by a widely dispersed database of previously published compressive strength values of concrete containing fly ash, are used for developing the models. The models have been found to have a high prediction capability of the compressive strength values of concrete in connection with age of specimen, cement, water, sand, aggregate, superplasticizer, fly ash and CaO. The statistical parameters of MAPE, RMS and R2 indicate that the proposed GEP models can predict the compressive strength values of concrete containing fly ash for each of training, testing and validation sets. The validation set results of four GEP models confirms the efficiency of the models for its general

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application to the compressive strength values prediction of concrete containing fly ash. As a result, this paper suggests that GEP is an alternative approach for the prediction of the compressive strength of concrete containing fly ash with different specimen size and shape. Acknowledgement

This study has partially been supported by Niğde University, The Scientific Research Project Unit with the project number of FEB-2011/21. I am grateful for support.

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Fig. 1. The flowchart of a gene expression algorithm [4]. Fig. 2. Chromosome with two genes and its decoding in GEP. Fig. 3. Expression tree of GEP-I model. Fig. 4. Expression tree of GEP-II model. Fig. 5. Expression tree of GEP-III model. Fig. 6. Expression tree of GEP-IV model. Fig. 7. Comparison of the experimental results of 10 cm cube fc with GEP-I. Fig. 8. Comparison of the experimental results of 15 cm cube fc with GEP-II. Fig. 9. Comparison of the experimental results of 10x20 cm cylinder fc with GEP-III. Fig. 10. Comparison of the experimental results of 15x30 cm cylinder fc with GEP-IV.

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

Terminate Iterate or Terminate?

End

Iterate Keep Best Program Select Programs

Genetic Modification

Replication

Replication

Prepare New Chromosomes of Next Generation

Fig. 1. The flowchart of a gene expression algorithm [4].

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ORF 012 3 456789

+-Q*c - abde Linking function

+ Gene 1

Gene 2

×

a

Q c

b

d

e

Expression tree

((a × b)-c) + d-e Mathematical expression

Fig. 2. Chromosome with two genes and its decoding in GEP.

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Sub-ET 1

Sqrt

Sub-ET 2

4Rt

3Rt + 3Rt -

c1

Exp +

d6 3Rt

-

d4 +

/

Sub3 d6 c2

d6

d1

c1

/

c1 x3

Sub3

Sub-ET 3

d4

Inv

Inv

Inv

c2

c2

Add3 +

x3 d6

Sub3

c2

d3

d3

Sub-ET 4

d4

c0 /

Sub3 +

d1

× d1

+

c2

Sqrt

/

Inv

Add3 d0

d2

d2

Add3

d3

d5

d0

d5

c1

Sub-ET 5

Ln ×

3Rt

Sqrt

+

Ln

Sqrt

d5

d0

Sub3 d4

d0

d0

Fig. 3. Expression tree of GEP-I model.

20

d2

d0

Sub-ET 1

x2

Sub-ET 2

+

Sqrt

3Rt

d0

/

4Rt

Sqrt

×

+

/

d1

d4

c1

+ d2

+

d1

d1

d6 /

+ c1

d1

Sub-ET 3

+

d1

d4

c2

Sub-ET 4

4Rt

3Rt +

/ d3

+ Exp

× Exp

Sqrt

c1 c0

-

d2 4Rt

d5

d1

Fig. 4. Expression tree of GEP-II model.

21

d6

Sub-ET 1

Sub-ET 2

/

+ c1

+

c0

+ ×

d4

d5

+

×

/

d2 ×

/

-

Sqrt

d1

+

c2

d0

d3

d4

d0

/

+ c1

d6

c0 c0

d6

Sub-ET 3

Sub-ET 4

-

+ +

d5

+

d6

Sqrt

+ d1

/ d0

×

× Mul3 Mul3 d6

c1

d0

/

/ d5

d1

d4

d0

c0

× d2

d5

d0 c2

d5

Fig. 5. Expression tree of GEP-III model.

22

× d1

Sub-ET 1

x3

Sub-ET 1

Sqrt

4Rt

Sqrt

4Rt

/

c1

4Rt

/

/

c1

/

Add3

d4

d0

d6

d3

/

+

d1

d5

d0

d2

c2

Sub-ET 3

Sub-ET 4

4Rt

4Rt -

+ Add3

4Rt

d1

Mul3 Sqrt

Mul3 c2

c2

+ d5

x3

x2

d6

c2

d7

d1

d5

Fig. 6. Expression tree of GEP-IV model.

23

4Rt

Sqrt

+

×

d3

×

×

+ c1

d0

d6

d5

Predicted compressive strength results, MPa

130 120

GEP-I training results

110

GEP-I testing results

100

GEP-I validation results

90 80 70 Training R² = 0.9049

60 50

Testing R² = 0.9047

40 30

Validation R² = 0.8993

20 10 0 0

10

20

30 40 50 60 70 80 90 100 110 120 130 Experimental compressive strength, MPa

Fig. 7. Comparison of the experimental results of 100 mm cube fc with GEP-I.

24

Predicted compressive strength results, MPa

120 GEP-II training results

110

GEP-II testing results

100

GEP-II validation results

90 80 70

Training R² = 0.9386

60 50

Testing R² = 0.908

40 30

Validation R² = 0.8885

20 10 0 0

10

20

30

40

50

60

70

80

90

100 110 120

Experimental compressive strength results, MPa

Fig. 8. Comparison of the experimental results of 150 mm cube fc with GEP-II.

25

Predicted compressive strength results, MPa

130 120

GEP-III training results

110

GEP-III testing results

100

GEP-III validation results

90 80 70 Training R² = 0.9335

60 50

Testing R² = 0.9224

40 30

Validation R² = 0.8956

20 10 0 0

10

20 30 40 50 60 70 80 90 100 110 120 130 Experimental compressive strength results, MPa

Fig. 9. Comparison of the experimental results of 100x200 mm cylinder fc with GEP-III.

26

Predicted compressive strength results, MPa

100 GEP-IV training results

90

GEP-IV testing results GEP-IV validation results

80 70 60

Training R² = 0.9035

50 40

Testing R² = 0.896

30

Validation R² = 0.8223

20 10 0 0

10 20 30 40 50 60 70 80 90 Experimental compressive strength results, MPa

100

Fig. 10. Comparison of the experimental results of 150x300 mm cylinder fc with GEP-IV.

27

Table 1

The data set of the proposed GEP models. The model information Number of data set References Age of specimens GEP-II Number of data set References Age of specimens GEP-III Number of data set References Age of specimens GEP-IV Number of data set References Age of specimens GEP-I

Training set Testing set Validation set 270 136 166 [3, 6-11] [3, 6-11] [12-15] 3, 7, 14, 28, 56, 60, 90, 180, 365 107 53 114 [2,16-20] [2,16-20] [21,22] 7, 14, 28, 56, 90, 91, 112, 180, 360, 365, 720 278 140 166 [1,23-26] [1,23-26] [3,27-31] 1, 3, 7, 14, 21, 28, 56, 90, 91, 180, 182, 364, 365 312 154 80 [32-37] [32-37] [38] 3, 7, 14, 28, 56, 90, 91, 365

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Table 2

GEP parameters used for proposed formulas. Parameter Definition

GEP-I

GEP-II

p1 Number of generation p2 Arithmetic operators P3 P4 P5 p6 p7 p8 p9 p10 p11 p12 p13

GEP-III

GEP-IV

764 801 795 763 +, -, *, / +, -, *, / +, -, *, / +, -, *, / Sqrt, 3Rt, 4Rt, Sub3, Sqrt, x2, Exp, x2, x3, Sqrt, Mathematical functions Sqrt, Mul3 3 Exp, x , Add3, 1/x, Ln 3Rt, 4Rt, Mul3, Add3, 4Rt Number of chromosomes 20 30 40 30 Head size 10 10 10 10 Number of genes 5 4 4 4 Linking function Multiplication Mutation rate 0.044 Inversion rate 0.1 One-point recombination rate 0.3 Two-point recombination rate 0.3 Gene recombination rate 0.1 Gene transposition rate 0.1

29

Table 3

The constants in the proposed GEP models. Sub-expression trees c0 GEP-I c1 c2 GEP-II c0 c1 c2 c0 GEP-III c1 c2 GEP-IV c1 c2

Sub-ET 1

Sub-ET 2

-9.997 -4.046

2.974

7.404 1.421

Sub-ET 3 5.951 1.618

-9.837 6.290 9.984 7.082 0.315 8.612

30

Sub-ET 4 6.480 -0.173 9.172 4.770

9.951 -0.052

-6.128

9.461

-6.401 0.795

Table 4

Statistical parameters of proposed GEP models. Statistical parameters Training set GEP-I Testing set Validation set GEP-II Training set Testing set Validation set Training set GEP-III Testing set Validation set GEP-IV Training set Testing set Validation set

MAPE 17.7721 16.0322 12.1387 11.1759 11.6023 17.3961 13.4958 16.9603 38.2951 16.5128 17.0508 17.1245

31

RMSE 7.4657 7.5731 7.5707 5.4727 8.0779 8.2033 5.8904 6.2895 9.6132 5.6464 5.9783 7.6187

R2 0.9049 0.9047 0.8993 0.9386 0.9080 0.8885 0.9335 0.9224 0.8956 0.9035 0.8960 0.8223

HIGHLIGHTS

¾ GEP was used for the prediction of the fc of concrete containing FA.

¾ After the GEP models were trained, the formulations were obtained.

¾ These formulations were verified with independent data from training and

testing sets.

¾ The formulations’ results are in good agreement with experimental results.

32