Modeling compressive strength of EPS lightweight concrete using regression, neural network and ANFIS

Construction and Building Materials 42 (2013) 205–216

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Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat

Modeling compressive strength of EPS lightweight concrete using regression, neural network and ANFIS A. Sadrmomtazi a, J. Sobhani b,⇑, M.A. Mirgozar a a b

Faculty of Engineering, University of Guilan, Rasht, Iran Department of Concrete Technology, Road, Housing & Urban Development Research Center (BHRC), Pas Farhangian St., Sheikh Fazlollah Exp. Way, Tehran, P.O. Box 13145-1696, Iran

h i g h l i g h t s " For the first time, models developed for prediction of the strength properties of EPS concrete. " Robust ANN and ANFIS models proposed for predicting the compressive strength of EPS concrete. " The overall performance of trained ANN is more accurate than ANFIS model. " Such robust models could be easily utilized for EPS concrete mix proportioning as a problem with high complexities included. " Higher accuracy of neural network is due to application of Levenberg–Marquardt backpropagation algorithm.

a r t i c l e

i n f o

Article history: Received 1 August 2012 Received in revised form 4 November 2012 Accepted 12 January 2013 Available online 27 February 2013 Keywords: EPS concrete Silica fume Compressive strength Modeling Regression Neural network ANFIS

a b s t r a c t EPS concrete is an especial type of lightweight concrete made by partial replacement of concrete’s stone aggregates with lightweight expanded polystyrene beads (EPSs). This type of concrete is very sensitive to its constituent materials which complicate the modeling process. Considering the involved complexities, this paper dealt with developing and comparing parametric regression, neural network (ANN) and adaptive network-based fuzzy inference system (ANFIS) models for predicting the compressive strength of EPS concrete for possible use in mix-design framework. The results emphasized that the elite ANN model constructed with two hidden layers and comprised of three neurons in each layers, could be effectively used for prediction purposes. Moreover, ANFIS elite model developed by bell-shaped membership function was recognized as a proper model to this means; however, its prediction performances were evaluated to be diluted than ANN model. On the other hand, the prediction results of second-order partial polynomial regression model as elite empirical one showed the weakness of this model comparing ANN and ANFIS models. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction A unique attribute of concrete, which makes it truly versatile, is that it consists of a family of materials with a large range in color, density, strength and durability characteristics [1]. It can be manufactured from a great number of materials, in many ways for different applications. Structural concrete is now available with a density range between 1800 and 3000 kg/m3 as lightweight, normal weight and heavy weight concrete [2]. Lightweight concrete (LWC) is a multi-purpose material for construction, which offers a range of technical, economical and environment-enhancing and preserving advantages and is destined to become a dominant material for construction in the new millennium [1]. ⇑ Corresponding author. Tel.: +98 21 88255942x6; fax: +98 21 88255941. E-mail addresses: [email protected], [email protected] (J. Sobhani). 0950-0618/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.conbuildmat.2013.01.016

The first known use of lightweight concrete dates back over 2000 years. There are several lightweight concrete structures in the Mediterranean region, but the three most notable structures were built during the early Roman Empire and include the Port of Cosa, the Pantheon Dome, and the Coliseum [3]. Lightweight concretes can be produced by replacing the normal aggregates in concrete either partially or fully, depending upon the requirements of density and strength [4]. Lightweight aggregates are broadly classified in two main types: natural (pumice, diatomite, volcanic cinders, etc.) and artificial (perlite, expanded shale, clay, slate, sintered PFA, and expanded polystyrene beads, etc.). Lightweight aggregates can be used to produce low density concretes required for building applications like cladding panels, curtain walls, composite flooring systems, and load-bearing concrete blocks [3,5,6]. When expanded polystyrene beads used for production of LWC, it is typically referred to EPS concrete.

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Nomenclature J

Jacobin matrix learning rate b delay rate (0 < b < 1) x (or y) input of the ANFIS node Ai (or Bj) linguistic label (Fuzzy membership) l(x) (or l(y)) membership function firing strength of a rule wi f1 and f2 fuzzy if–then rules in TSK system f() regression model’s representative function y regression model’s output bi parameters of the regression model xi independent variables RMS root means square

l

Lightweight EPS concrete has some distinguished advantages like higher strength to weight ratio, better tensile strain capacity, lower coefficient of thermal expansion, and superior heat and sound insulation characteristics due to inclusion of air voids in the lightweight aggregate. Beside the reduction in of the construction materials led to a remarkable decrease in cross section of concrete structural elements (columns, beams, plates, and foundation), it is also possible to reduce steel reinforcement [5,7,8]. The properties of LWC are very sensitive to its ingredients which intensify the complexities involved in the prediction of its behavior when compared with that of the normal weight concrete. Besides, for the means of concrete mix-design, it is a common practice to produce the trial mixtures based on mix-design and project requirements. To aid in this process and minimize the experimentation tasks, mathematical free models, in the form of regression formulas, are traditionally used to predict the strength behavior of concrete mixes. These models could be applicable in many cases, however; if the problem contains many independent variables, regression methods cannot be used because of a loss of accuracy and increased number of variables in regression form (linear, non-linear, exponential, etc.). In the recent years, artificial intelligent-based modeling techniques like artificial neural networks (ANNs) [5,9], fuzzy systems [10–12], adaptive network-based inference systems (ANFISs) [13,9], neuro-fuzzy systems [14], and genetic fuzzy systems [15], have been utilized to approximate non-linear and complex behavior for various properties of construction materials. Among the aforementioned methods, ANN and ANFIS have been widely applied for various types of concrete. So, this paper aimed to design robust ANN and ANFIS models to predict the compressive strength of and especial type of lightweight concrete named as EPS concrete, incorporating the silica fume, rice husk ash, expanded polypropylene, and waste carpet polypropylene fibers. Moreover, non-linear regression models were also proposed and compared with ANN and ANFIS models.

CF correlation factor P total number of concrete samples xci experimental result for compressive strength ^xci predicted result for compressive strength; b c Þ covariance of experimental and predicted value for covðX c ; X compressive strength um normalized input value wRV rough values of input data C, SF, W, FA, CA, EPS, PP weight per unit volume of concrete respectively for Cement, Silica fume, water, fine aggregates, coarse aggregates, expanded polystyrene beads and waste carpet polypropylene fibers

2.2. Artificial neural network Artificial neural network (ANN) is an artificial intelligence-based method for dealing with recognition of complex phenomenon and solution of problems which could not otherwise be solved through the existing algorithms. ANN is system of simple processing elements called neurons and is connected to a network by a set of weights (see Fig. 1). Generally speaking, the relationship between the elements characterizes the rule of the nets. Nets could be trained to perform a specific job, by setting the relationships between the elements (i.e., the weight and bias terms). Therefore, the trained nets would have a specific output response for a particular input. A neural network, in each setting, compares the value of output outcome with the actual output and hence sets its parameters in such a way that the output response gets closer to the actual value. The trained nets could be utilized to act as an alternative replacement of the complex functions. Every neural network consists of an input, hidden and output layers, respectively. These layers are joined together through connections with different weights. The duty of the hidden layer (HL) is to connect the input to output layers. A hidden layer enables the nets to extract a non-linear correlation from the available dataset. The number of input, hidden and output neuron layers highly depends on the number of input variables, number of output response variables and also to the application of the nets [16–18]. Fig. 1 demonstrates a typical single input neuron model of the nets. In this figure, P stands for the number of inputs, W is the weight, b is the bias which employs the result as the argument for a singular valued function, f the transfer function and a is the output neurons. In neural network, the term backpropagation refers to the procedure in which the gradient is computed for non-linear multiple-layer networks. Furthermore, to assess the performance of the neural network model, an error measure like root mean square error (RMS) might be utilized. The choice of a specific class of networks for the simulation of a non-linear and complex map is dependent upon a variety of parameters. The most popular ANN method is the feed forward multilayer perceptron scheme. In neural network, each neuron is composed of a transfer function which signifies the internal activation level. Generally speaking, the transfer functions are sigmoidal function, hyperbolic tangent and linear function. Amongst the above transfer functions, the log-sigmoidal is the most widely used function for the non-linearity cases. To prevent numerical overflow in the case of a very large or small weights, normalization of the input dataset are unavoidable [16–18]. 2.2.1. Training algorithm Fig. 2 shows ANN model to simulate the experimental results with backpropagation (BP) algorithm. BP algorithm calculates the error, and then used to adjust the weights first in the output layer, and then distributes it backward from the output to hidden and input nodes [9,16]. In this paper Levenberg–Marquardt-type BP (LMBP) algorithm is utilized [9,22]. The weigh update on the base of LMBP is as follows:

2. Modeling methods 2.1. Regression modeling Regression modeling is a statistical tool for the investigation of relationships between variables. Generally, regression is the process of fitting models to data. The process depends on the model. In linear or non-linear parametric regression, it was tried to develop empirical model(s) for system identification and experimental studies purposes. In such systems, estimation is based on search methods from optimization that minimize the norm of a residual vector. In this paper, general parametric regression system proposed in the form of y ¼ f ðbi  xi Þ where f is the model’s representative function, y is the model’s output, bi are the parameters of the model and xi are the independent variables.

Fig. 1. Single input neuron model.

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Fig. 2. Architecture of ANN [9].

Fig. 3. Schematic of ANFIS architecture [9].

Dw ¼ ½JT J þ lI1 JT e

ð1Þ

where J is the Jacobin matrix, l is the learning rate which is to be updated using the delay rate b (0 < b < 1) depending on the outcome as lnew = loldb. 2.3. Adaptive network-based fuzzy inference system (ANFIS) ANFIS is the famous hybrid neuro-fuzzy network for modeling the complex systems [19–20]. ANFIS incorporates the human-like reasoning style of fuzzy systems through the use of fuzzy sets and a linguistic model consisting of a set of If-Then fuzzy rules. The main strength of ANFIS models is that they are universal approximators [19] with the ability to solicit interpretable If-Then rules. To illustrate the procedures of an ANFIS, for simplicity, we consider only two inputs x, y and one output fout in this system. The framework of ANFIS is shown in Fig. 3, and the node function in each layer is described below. Layer 1: Every node in this layer is an adaptive node with node function as:

O1;i ¼ lAi ðxÞ for i ¼ 1; 2

ð2Þ

O1;i ¼ lBi2 ðyÞ for i ¼ 3; 4

ð3Þ

where x (or y) is the input of the node, Ai (or Bj) is the linguistic label, l(x) (or l(y)) is the membership function, usually adopting the bell shape with maximum and minimum equal to 1 and 0, respectively. Layer 2: Every node in this layer is a fixed node, marked by a circle and labeled P, with the node function to be multiplied by input signals to serve as output signal

O2;i ¼ lAi ðxÞ  lAi ðxÞ ¼ wi

for i ¼ 1; 2

ð4Þ

The output signal wi represents the firing strength of a rule. Layer 3: Every node in this layer is a fixed node, marked by a circle and labeled N, with the node function to normalize the firing strength by calculating the ratio of the ith node firing strength to the sum of all rules’ firing strength.

wi wi O3;i ¼ P ¼ wi w1 þ w2

for i ¼ 1; 2

ð5Þ

Layer 4: Every node in this layer is an adaptive node, marked by a square, with node function

 i  fi O3;i ¼ w

for i ¼ 1; 2

ð6Þ

where f1 and f2 are the fuzzy if-then rules as follows: Rule 1: if x is A1 and y is B1 then

f 1 ¼ p1 x þ q 1 y þ r 1 Rule 2: if x is A2 and y is B2 then

f 2 ¼ p2 x þ q 2 y þ r 2 where {pi, qi, ri} is the parameters set, referred to as the consequent parameters.

Layer 5: Every node in this layer is a fixed node, marked by a circle and labeled R, with node function to compute the overall output by

O5 ¼

X

 i  fi ¼ fout for i ¼ 1; 2 w

ð7Þ

The basic learning rule of ANFIS is the backpropagation gradient descent, which calculates error signals recursively from the output layer backward to the input nodes. This learning rule is exactly the same as the backpropagation learning rule used in the common feed-forward neural networks [21]. Recently, ANFIS adopted a rapid learning method named as hybrid learning method which utilizes the gradient descent and the least-squares method to find a feasible set of antecedent and consequent parameters [9,13,19,20]. Thus in this paper, the later method is used for constructing the proposed models.

3. Materials and data collection 3.1. Materials Locally available ordinary Portland cement meeting the requirements of ASTM C150 [23], and two types of supplementary cementitious materials (CMs) including silica fume (SF) and rice husk ash (RHA) were used in this investigation. The chemical compositions of these binders are presented in Table 1. The fine aggregate was natural siliceous river sand and the coarse aggregate was crushed limestone aggregate. The physical and mechanical properties of the fine and coarse aggregates are reported in Table 2. The grading of these aggregates is presented in Fig. 4 together with the ASTM C33 limits [24]. In addition to the natural aggregates, EPS beads were utilized as artificial lightweight aggregates in order to decrease the density of concrete and produce different strength grades of EPS concrete. The size of 85% of EPS particles were about 3.5 mm and their density as measure of mass per volume was evaluated through continuous water injecting-withdrawal technique as 0.0257 g/cm3. Moreover,

Table 1 Chemical composition and properties of cement, silica fume and rice husk ash. Chemical composition (%)

Cement

Silica fume (SF)

Rice husk ash (RHA)

SiO2 Al2O3 Fe2O3 CaO MgO SO3 Na2O + 0.685K2O

21 4.6 3.2 64.5 2.0 2.9 1.0

91.1 1.55 2.0 2.42 0.06 0.45 –

91.62 0.49 0.73 2.51 0.88 – 2.39

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Table 2 Aggregate properties.

Table 3 Characteristics of waste carpet polypropylene fibers.

Aggregate type

Specific gravity

Absorption (%)

Fineness modulus

Properties

Description

Fine (0–4.75 mm) Coarse (4.75–12 mm)

2.51 2.54

3.40 2.57

2.82 –

Morphology Specific gravity (g/cm3) Diameter (lm) Modulus of elasticity (GPa) Tensile strength (MPa) Ultimate strain (%) Elongation of fracture (%) Melting point (°C) Bonding with cement Stability in cement Aspect ratio (L/d)

Fibrillated or mono filament 0.95 50 5 450 5–15 20 160 Good Good 120

polypropylene (PP) fibers obtained from waste carpets were utilized in this study to improve the toughness of EPS concrete. The properties of these fibers are presented in Table 3. The water used for mixing and curing of all concrete mixes and specimens was clean, fresh, and free from any impurities. Also, owing to the necessity of lowering water to cement ratio for obtaining enough compressive strength and desired fluidity, a polycarboxylate based superplasticizer was incorporated in all mixtures. 3.2. Specimen preparation As homogeneity is a main issue in EPS concrete, to prepare humongous specimens, the following steps adopted: Step 1: EPS beads were wetted with a part of the mixing water and superplasticizer, before adding the remaining materials. Step 2: The remaining materials were added to the mixer and the remaining water was gradually added while the mixing was in progress. Step 3: Mixing was continued until a uniform and flowing mixture was obtained. The observation on specimens as depicted in Fig. 5 showed a uniforms distribution of EPS beads in matrix. 3.3. Data collection The mix proportions of the study are summarized in Table 4. As seen, three percentages of EPS beads and four percentages of PP fiber (0.1%, 0.3%, 0.5% and 1%) were used in preparation of concrete specimens. Respectively 10% and 20% of silica fume and rice husk were replaced by weight of cement as supplementary cementitious material. Based on the mix proportions, EPS concrete specimens were prepared in the standard condition. To gather the database for training and testing pairs of neural network or ANFIS models, cube specimens based on the mix designs reported in Table 4 were made, cured for 28 days. Afterwards the compressive strength of these specimens determined according to ASTM C39 [25]. The average compressive strength for EPS mixtures without PP could be seen in this table. A total number of 75 records of EPS concrete compressive strength at 28 days were gathered based on the aforementioned mix design to construct the training–testing database. For training and testing of the proposed models, 64 and 11 samples were randomly chosen respectively. Moreover, to monitor the training

Fig. 5. Distribution of EPS beads in concrete mixture. process, 10 checking data were selected randomly from both training and testing data. This idea might be utilized to avoid miss-training (over training, saturation problem) [14,16]. The structure of the input–output of the modeler systems were schematically shown in Fig. 6. In this figure, the input parameters are (i) cement (C), (ii) silica fume (SF), (iii) water (W), (iv) fine aggregates (FA), (v) coarse aggregates (CA), (vi)

Fig. 4. Grading of fine and coarse aggregates and limits of ASTM C33.

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A. Sadrmomtazi et al. / Construction and Building Materials 42 (2013) 205–216 Table 4 Mix proportion of the specimens. Mixture

1 2 3 4 5 6 7 8 9 10 11 12 a

Cement (kg/m3)

S.F (%)

400 400 400 400 360 360 360 360 320 320 320 320

– – – – 10 10 10 10 – – – –

R.H (%)

– – – – – – – – 20% 20% 20% 20%

EPS (kg/m3)

PP

– 15 25 40 – 15 25 40 – 15 25 40

0%, 0.1%, 0.3%, 0.5%, 1% By volume

Water (kg/m3)

W/(C + CM)

180 170 165 160 190 175 175 170 210 205 205 200

0.45 0.43 0.41 0.4 0.48 0.44 0.44 0.43 0.52 0.51 0.51 0.5

Aggregate (size mm) 0–3 (kg/m3)

3–6 (kg/m3)

6–12 (kg/m3)

666 540 431 294 652 524 422 282 620 470 385 245

118 95 76 52 115 93 75 50 110 80 68 43

957 777 620 423 940 755 607 406 895 670 555 352

Compressive strengtha (MPa)

43 33 16.7 9.8 47.6 27.8 24.4 10.2 29.5 22.4 10.6 6.7

Mixture without PP fiber.

and

C

b c Þ ¼ E½ðX c  l Þ  ð X bc  l ^ c Þ covðX c ; X c

SF

ð11Þ

where

W

lc ¼ EðX c Þ; l^ c ¼ Eð Xb c Þ

Neural network/ANFIS Model

FA CA

Compressive Strength (CS)

ð12Þ

where E is the mathematical expectation. 3.5. Data preprocessing

EPS PP Fig. 6. Schematic structure of modeler systems.

expanded polystyrene beads (EPS) and (vii) waste carpet polypropylene fibers (PP) by weight per unit volume of concrete. Moreover Table 5 summarizes the ranges of input and output of total data used for modeling purposes.

To avoid the saturation problem and consequently the low rate of the training [9,14,16], in particular to avoid the saturation region of log-sigmoid activation function, normally used in the neural networks with backpropagation algorithm, it is necessary to normalize the real rough data into a suitable range. In this paper, a linear normalization adopted to map the rough data range to the range of [0.1, 0.95] as follows:

um ¼ 0:1 þ 0:85 

wRV  wRV min RV wRV min  wmax

ð13Þ

To evaluate the performance of models, root means square (RMS), and correlation factor (CF) are utilized as follows:

RV where um is the normalized value, wRV is the rough values of data, wRV max and wmin are the maximum and minimum of rough values respectively. Obviously inverse mapping could be then applied to draw the real values for the purposes of representation and possible practical applications.

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u P . uX RMS ¼ t ðxci  ^xci Þ2 P

4. Results and discussion

3.4. Measures for evaluation of models

ð8Þ

i¼1

where P is the total number of concrete samples, xci is experimental result and ^xci the predicted result:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bcÞ bc; X b c Þ  covðX c ; X c Þ CFðxc ; ^xc Þ ¼ covðX c ; X covð X

ð9Þ

where

X c ¼ ðxc1 ; xc2 ; :::; xcp Þ;

b c ¼ ð^xc1 ; ^xc2 ; . . . ; ^xcp Þ X

ð10Þ

Table 5 Ranges of input/output variables. Variable

Range Min

Max

Input Cement (kg/m3) Silica fume (kg/m3) Water (kg/m3) Fine aggregate (kg/m3) Coarse aggregate (kg/m3) Expanded polystyrene beads (kg/m3) Waste carper propylene fiber (kg/m3)

320 0 160 97 118 0 0

400 40 230 784 958 55 9.1

Output Compressive Strength (MPa)

0.9

47.6

Modeling with regression analysis, neural network and ANFIS, consist of three stages: (a) preprocessing of data, (b) designing the model (architecture), (c) training, and (d) testing of regression, neural network or ANFIS models. For implementing the proposed models, MATLAB software is utilized. 4.1. Regression models Table 6 summarizes the proposed regression models for predicting the compressive strength of EPS concrete. Table 7 shows the b-parameters of the proposed models. The performance of proposed models were evaluated and presented in Table 8 in terms of correlation factor and root means square. Moreover, Fig. 7 demonstrates a comparison between results of prediction of EPS concrete with experimental observation. Considering results of model’s performance summarized in Table 8 and illustrated in Fig. 7, it could be deduced that second order polynomial model (NRM2) is the best model for the purpose of compressive strength prediction. This model predicts the compressive strength of EPS concrete with CF of 0.9663 and 0.9879 respectively for training and testing pairs and RMS of 21.4631 and 21.7042 for training and testing pairs respectively. As seen this model has the minimum value of RMS for testing data set with an acceptable correlation factor. In justifi-

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Table 6 Proposed regression models for predicting compressive strength of EPS concrete. Model

Type

Equation

NRM1 NRM2

1st Polynomial Partial 2nd Polynomial

b0 + b1C + b1SF + b1W + b1FA + b1CA + b1EPS + b1PP

NRM3

Power-fractional

NRM4

Power-fractional

b0 þ b1 C þ b2 SF þ b3 W þ b4 FA þ b5 CA þ b6 EPS þ b7 PP þ b8 C 2 þ b9 SF 2 þ b10 W 2 þ b11 FA2 þ b12 CA2 þ b13 EPS2 þ b14 PP2  2  2 b1 W 4 Cþb5 SF þ b FAþbbCAþb b0 þ b Cþb 2 3 SF 6 7 8 EPSþb9 PP  b2  b4   b6 W CþSF W b0 þ b1 CþSF þ b3 FAþCAþEPSþPP þ b5 EPSþPP

Table 7 b-Parameters of the proposed models. Model

b0

b1

b2

b3

b4

b5

b6

b7

b8

b9

b10

b11

b12

b13

b14

NRM1 NRM2 NRM3 NRM4

.246 .341 .136 43.637

.065 .176 .079 2.349

.099 .238 1.669 .205

.057 .288 .807 38.737

2.07 4.198 .021 .012

1.53 3.541 .01 84.756

.2641 .493 .031 .005

.09 .038 .049 –

– .16 .033 –

– .284 .003 –

– .176 – –

– 5.641 – –

– 4.368 – –

– .587 – –

– .054 – –

Table 8 Performance of proposed regression models. Model

NRM1 NRM2 NRM3 NRM4

Training (Interpolation)

Testing (Extrapolation)

CF

RMS

CF

RMS

0.9990 0.9663 0.9051 0.9287

21.7081 21.4631 20.8990 21.1673

0.9937 0.9879 0.9165 0.9186

22.8693 21.7042 25.2815 23.0578

cation on the goodness of NRM2 as the best model, comparisons made with the experimental observation depicted in Fig. 7a and b are of great importance. These figures show unfeasible results of NRM1, NRM3 and NRM4. It should be note that Fig. 7a and b are related to the prediction results of training and testing data set (or formally interpolation and extrapolation in terms of regression problem) respectively. These figures confirm our justification on proposing NRM2 as elite regression model.

Fig. 7. Comparison of regression models with experimental observation: (a) Training set (interpolation) and (b) testing set (extrapolation).

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Fig. 8. Schematic of NNM architecture.

Table 9 General properties of ANN models. Name

Type

Training method/ algorithm

Activation function in HLs

Activation function in output layer

No. of PE in HL

Layers number

HLs number

NLMBP

Feed-forward backpropagation network

Supervised/LMBP

Log-sigmoid

Linear transfer function

Variable

4

2

Table 10 Summary of ANN models for prediction of 28-days compressive strength. Name

NLMBP11 NLMBP22 NLMBP33 NLMBP44

No. of neurons in

Training set

Testing set

Checking set

HL1

HL2

CF

RMS

CF

RMS

CF

RMS

1 2 3 4

1 2 3 4

0.9804 0.9976 0.9990 0.9998

2.5754 0.9069 0.5718 0.2272

0.9828 0.9798 0.9937 0.9805

2.6113 3.6138 1.9302 3.1545

0.9745 0.9968 0.9979 0.9953

2.7121 1.2211 1.0782 1.1763

4.2. Neural network models The schematic structure and general properties of used ANN are shown in Fig. 8 and Table 9 respectively. Different topologies could be utilized for developing ANN models. As discussed earlier, the architecture of ANNs are composed of input/output/hidden layers and some neurons in each layers. The numbers of neurons in input and output layers are constant and constrained to the number of input and output parameters respectively. However, number of hidden layers and neurons in corresponding layers are variable and these numbers are governing factor of ANN model’s performance. If too few hidden neurons are used, the network will be unable to model complex data; resulting in a poor performance. If too many hidden neurons are used; then training will become excessively long and the network may over fit. Over fitting occurs when the network begins to model random noise contained within the data, resulting in a failure to converge on a generalized solution. So, one practical method to deal with this issue is to exploit the structured trial and error approach. The process generally used to determine the number of hidden layers and number of neurons. In this approach, the goal is try to quickly find the network that converges and simulate the goal of ANN. In this paper, three-hidden-layer structure is supposed for the final ANN model with variable number of neurons in each layer. Moreover, equal number of neurons is supposed to be in each hidden layers. Afterwards, ANN models constructed by this method are trained and their prediction measures were evaluated as correlation factor and root means square. To decide on the suitable model, it is tried to find an architecture which yield the best

correlation factor and root means square concurrently occurred for training and testing pairs. Eventually, this approach proposes ANN model with higher performance for both estimation (interpolation) and generalization (extrapolation) properties concurrently. In this regard, Table 10 summarizes the performance of 6 elite neural network models trained via the aforementioned procedure. Fig. 9a–c illustrates the performances of trained networks for training, testing and checking data sets respectively. In these figures, the horizontal axis represents the number of neurons in hidden layers while two vertical axes dedicated to evaluated performance indexes. Left vertical axis presents the correlation factor and the right vertical axis shows the root means square. As depicted in Fig. 9a the performance indexes (correlation factor and root means square) for training data set were improved by increasing the number of neurons in hidden layers showing the effectiveness of higher order neural networks for the prediction purposes. Fig. 9b shows the performance indexes for testing pairs which confirms the best number of neurons in hidden layers as three neurons for generalization purposes. These findings could be verified by the results gained for checking data sets as seen in Fig. 9c. Thus NLMBP33 which contains three neurons in its layers considered to be the suitable ANN model for prediction of the compressive strength of EPS concrete. 4.3. ANFIS models ANFIS models could be composed of different architectures which govern the output and prediction performances. Numbers of input/output units are constant and basically limited to the

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Formula  xa cx  max min ba ;0 ;  xa cbdx  max min ba ; 1; dc ; 0

Triangular Trapezoidal Bell-shape

1 2b ð1þjxc a j Þ

Gaussian

Fig. 9. Performance of ANN models with different number of neurons in hidden layers: (a) Training, (b) testing and (c) checking data set.

e

ðxcÞ2 2r

space of concerned problem. Similar to the ANN models, in ANFIS case, number of input/output units is constant. Units specified in layer I (schematically represented in Fig. 3) are related to the number of fuzzy sets (labels or membership function) utilized for decomposing the input space of each input variable. Other important option of this architecture is the type of fuzzy membership function (MF) used for such decompositions. It was practically studied by author that such membership function is an important factor affect the prediction performance [13]. So, considering the constraints of utilized computer’s CPU, three fuzzy sets used for decomposition of each input space with variable fuzzy label types. In this regard, similar to that of neural network models, the structure of proposed ANFIS networks was consisted of seven input variables (i.e., C, SF, W, FA, CA, EPS and PP) and CS as its output variable. As said, the input space is decomposed by three fuzzy labels. In this study, for comparison purposes, four types of MFs including the triangular, trapezoidal, bell-shape, and Gaussian functions (see Table 11) were utilized to construct the suggested models. ANFIS models were then trained in a similar manner for neural network. 100 epochs were specified for training process to assure gaining the minimum error tolerance. To identify the parameters of Sugeno-type fuzzy inference system, a hybrid learning algorithm was utilized [22]. Fig. 10 compares the error trends of ANFIS models during 100 epochs regarding checking data set which underscore reaching the equilibrium state after completion of training process. The performance of ANFIS models are examined by RMS and CF and the results summarized in Table 12. As seen, the correlation factors of training data set for all ANFIS models are near 1 as

Fig. 10. Training error trend for ANFIS models.

Table 12 Summary of ANFIS predictions. ANFIS model

ANFTRI ANFTRA ANFBEL ANFGUS

MF

Triangular Trapezoidal Bell-shape Gaussian

Training set

Testing set

Checking set

CF

RMS

CF

RMS

CF

RMS

1.0000 0.9998 1.0000 1.0000

0.0079 0.2335 3.9  104 7.6  104

0.8333 0.9505 0.9783 0.9098

12.0251 5.0968 3.4053 8.3303

0.7033 0.9634 0.9836 0.8546

12.2271 3.5216 2.4327 7.6508

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the optimum value; however this norm for testing data set demonstrated different values. Of all, ANFBEL exhibited the best correlation of 0.9783. Correlation factor gained for ANFTRI, ANFTRA, and ANFGUS are as 0.8333, 0.9505 and 0.9098 respectively. RMS values for ANFTRI, ANFTRA, ANFBEL, and ANFGUS calculated as 0.0079, 0.2335, 3.9  104, and 7.6  104 for training data set and 12.0251, 5.0968, 3.4053, and 8.3303 for testing pairs respectively. As seen, the RMS values for all of models are satisfactory for training set; however, this norm is elevated for testing pairs. Again, ANFBEL exhibited the best performance; however, other models have a great amount of error in comparison. It should be noted that these findings could be confirmed by performance norms of checking data set. Thus ANFBEL which consisted of bell-shaped fuzzy

Table 13 Parameters of bell-shaped convex fuzzy sets as vector of [a,b,c] (see Fig. 11 and Table 11).

a

Variablea

Small

Medium

Big

NSF NW NFA NCA NEPS NPP NCS

[0.137,2,0.07] [0.20,2,0.097] [0.174,2,0.063] [0.12,2,0.065] [0.09,2,0.029] [0.094,2,0.043] [0.13,1.99,0.012]

[0.126,2,0.518] [0.01,2,0.526] [0.23,2,0.47] [0.058,2,0.42] [0.115,2,0.435] [0.127,2,0.458] [0.142,2,0.429]

[0.138,2,0.979] [0.2,2,0.952] [0.19,2,0.94] [0.224,2,0.9] [0.16,2,0.927] [0.174,2,0.942] [0.318,2,0.837]

Prefix N stands for normalized data.

Fig. 11. Fuzzy domain decomposition using bell-shaped linguistic variables.

Fig. 12. Comparison of NLMBP33 and ANFBEL with experimental observations: (a) Training set and (b) testing set.

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membership functions is nominated as the best model for prediction of the compressive of EPS concrete. Fig. 11, with respect to Table 13, shows the membership functions for input variables of ANFBEL after completion of training process.

4.4. Comparison of non-linear regression, ANN and ANFIS Fig. 12a and b represents the prediction results of NLMBP33 and ANFBEL as elite models of ANN and ANFIS models respectively in comparison with the experimental observations. In these figures, the horizontal axis is representative of experimental results and the vertical one is related to the results of model’s prediction for compressive strength of EPS concrete. Error lines of +25% and 25% are also plotted to visualize the prediction performances. Based on these figures, it is evident that the proposed models are capable to predict the compressive strength of EPS concrete for training data set. As seen in Fig. 12b, using ANFBEL, two samples of testing data were predicted beyond 25% error line; however, all of samples could be successfully predicted using NLMPB33. NRM2 as elite regression model is justified to be the worst model due to wrong predictions made by this model depicted in Fig. 12a and also numerous predictions beyond 25% error line shown in Fig. 12b. It should be noted that the ANFIS model might be improved by introduction of more fuzzy sets (or fuzzy labels) to each of input

variables. For example in the current modeling system with ANFIS, input space is decomposed by three labels defined here as Small, Medium and Big (Table 13). By adding up the number of fuzzy sets for decomposing the input space, for example four fuzzy sets, the number of rules for estimation procedures would be increased and thus a better estimation performance might be gained. However, such task increase the complexity of the system and accordingly more memory needed to complete the training process. This issue is unfeasible in computers with limited memory and CPU speed for problems comprising numerous input variables. On the other hand, the neural network operates in the parallel processing strategy and needed less memory for training process. One another reason might be related to the utilized training algorithm of ANN and ANFIS. It should be noted that ANN trained by so-called Levenberg–Marquardt algorithm while ANFIS trained by gradient descent backpropagation combined with least-squares approach. It is well established that prior one is more rapid and robust than other later method utilized in ANFIS training and optimization process.

4.5. Effects of PP and EPS content on compressive strength To evaluate the effects of EPS and PP content on the compressive strength of EPS concrete, trained models of ANN (NLMBP33) and ANFIS (ANFBEL) were utilized. Figs. 13a and 10b demonstrate

Fig. 13. Effect of PP content on compressive strength: (a) NLMBP33 estimation and (b) ANFBEL estimation.

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the interactive effects of PP fiber content on the compressive strength of EPS concrete regarding the cement content estimated by elite model of ANN and ANFIS respectively in view of 3-D plots. X, Y, and Z axis of these plots are PP fibers content, cement content and compressive strength respectively. As seen, by increasing the cement content, both model proposed an increase of compressive strength while increasing the PP content lead to a reductive effect on the compressive strength of EPS concretes. It should be noted that increasing effects of cement content on compressive strength is more evident for PP content below 0.8%. Moreover, Fig. 14a and b illustrate the interactive effects of EPS and cement contents on the compressive strength of EPS concrete in a similar manner molded by ANN and ANFIS elite models. It should be noted that, in these plots, X-axis are the EPS content. Considering the response surface, it could be concluded that the more cement content used, the more compressive strength could be gained and vice versa, by application of higher amount of EPS, the compressive strength of such concretes reduced substantially. Meanwhile, the 3-D surface of interactive effects of PP, EPS and cement content might be assisted in the EPS-concrete mix designer to have a general view on how these parameters affects the compressive strength and accordingly these surfaces might be utilized in a framework of concrete mixdesign system with capability of optimization task. Another point could be drawn is that the effect of PP fibers is more tangible in the mixtures with EPS content of below 20%.

215

5. Conclusion Concrete is a highly non-homogenous material, so modeling its behavior is a difficult task. The artificial intelligent-based models are known to be robust tools to model complex systems. In this paper, the application of two types of such systems including ANN and ANFIS in the estimation of 28 days compressive strength of an especial type of LWC mixtures made by EPS beads have been outlined. Based on an experimental program, a database was collected for strength of EPS concretes and then four regression models, ANN and ANFIS models were designed and trained by training data set randomly chosen from the whole database. The performances of the proposed models were evaluated by correlation factor and root means square to assess the best regression, ANN and ANFIS models. The results showed that NRM2 as elite regression model in the form of partial second order polynomial, NLMBP33 as elite ANN model constructed by two hidden layers having three neurons in each and ANFBEL as elite ANFIS one constructed by bell-shaped fuzzy membership functions are proposed as the suitable models. Promising results were obtained using both ANN and ANFIS models; however, comparing the prediction results of elite ANN and ANFIS models for training and testing data sets revealed the higher accuracy and generalization capabilities of NLMBP33 as elite neural network model. The regression model was found to be unable to predict the compressive strength. In general, materials and civil engineers

Fig. 14. Effect of EPS content on compressive strength: (a) NLMBP33 estimation and (b) ANFBEL estimation.

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may use the proposed model to predict the stability of EPS concrete mixtures and avoid conducting costly experimental tests that require specialized equipments and expertise.

[12] [13]

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