Analysis of Variance, ANOVA, is the common way to verify the significance of model coefficients. The procedure standard of ANOVA is to evaluate the general observations’ variability in its components. The procedure consists to confirm the hypotheses tests by application of F tests. These tests are applied by relation of F factor tabled and calculated in the characteristic studied. The F factor calculated is obtained by relation between Square Means, corresponding to model characteristics, which would be tested. These Square Means are calculated by relation between square sums by degrees of freedom of characteristic studied. Finally, the square sum is determinate by variability credited for each factor. The sum of individual contributions will constitute the experimental error. The square sum is calculated by the observation deviation around of estimate media of studied characteristic. The ANOVA design with complete randomization of experiments is shown in Vieira et al. [9] .

The statistics model’s parameters are determinate by least squares method. It can be done through the matrix equation: (X^{t}X)^{−1} (X^{t}y), in which X represents the design matrix and the y, the observed responses variable. The estimative coefficients’ errors can be calculated by the equation (X^{t}X)^{−1} s, where the s^{2} term represents the calculated experimental variance [24] .

3. Material and Method

In this topic is presented the materials utilized and the experimental design carried out, in this study.

3.1. Material

This study utilized one type of spherical EPS beads produced for a local industry (Manaus), which recycles its EPS after use. Due to the possibility of manufacturing beads with relative homogeneity, it was chosen from three interval diameter of beads: 1.2 mm to 2.4 mm; 2.4 mm to 4.8 mm and; larger than 4.8 mm (Figure 1). Table 1 shows the data relating to the aggregates and the EPS bead properties. The type of Portland cement used was CP II Z. This cement is produced in Manaus. The sand utilized in this study is from natural river sand.

Figure 1. Expanded polystyrene beads after burning process.

Table 1. Material properties and mix proportioning of control mixtures.

^{*}As described in Table 2. ^{Ä}% of Polystyrene beads that substituted the aggregate.

Development of this research, only were analyzed ﬁne and coarse aggregates that were treated as production variables and mixed with water and cement, constituting the mixture variables. Process variables were chosen, as were the EPS beads size and their percentage which substituted the aggregate in concrete.

3.2. Modeling

The experimental design applied in this study followed the Completely Random Experimental Design (CRED) with a mixture and process variables combined as presented by Vieira [9] .

In this study the set of mixture variables had three compounds: cement, aggregates and water. The process variables were definedas: Size of the polystyrenebeads and proportion of polystyrene beads which substitute aggregates. These two types of variables were simultaneously adjusted.

Table 2 presents the experimental design applied in this study. It was used as analytical responses compression strength (MPa) regarding 28-d for all the experimental conditions, shown as well in Table 3.

Table 2. A 3^{2} factorial design with size EPS beads and proportion of EPS beads that substitute aggregates and mixture design and water proportions for the CRED.

a) standardized variables.

Table 3. Experimental 28-d compressive strength EPS concrete determinations for the duplicated 54 experiment complete random design for two processes and three mixture variables.

The analytical responses are given as compressive strength, in megapascals. a) Formulation number from Table 2. b) Replicate number.

For this experimental design different combined models were tested. These models were obtaine d by com- bination of process variable models [Equations (2) and (3)] with mixture variable models [Equations (4)-(6)]. An example of this procedure is presented in Equation (7).

Mechanical properties of polymerized concrete were investigated through the use of specimens of a cylindrical shape, with dimension of 15 cm in diameter andof 30 cm in height. All experiments were carried out using the leads presented in Table 2, but in a random order.

The polymerized concrete wereproduced fixing the water/cement ratio in 0.5. A proportion of the ﬁne aggregate (sand) was replaced by polystyrene beads. Mass percentage of polystyrene beads used was 5%, 15% and 25%.

The polystyrene beads which were utilized not have and pretreatment were only washed with water.

The effect of surface treatment has to be motivationof a different study. In this study, the polymerized concrete could be produced with the optimized conditions of process variables and mixture compounds.

4. Results

Due to the degree freedom number, between the others combined models possible only two models were examined. These models were: the linear [Equation (2)] and bilinear [Equation (3)] process variable models combined with the linear mixture model [Equation (4)].The others combined models embodying the others mixture models [(Equations (5) and (6)], require bigger numbers of degree freedom, which the experimental design carried out, in this study, cannot provide.

Table 3 presents the results of ANOVA for the combination between the modelsof linear process variable and linear mixture (Combined model L-L). The result for the combined model embodying the linear process model and bilinear mixture model was unnoticed, because it presented worse values of regression, when compared to the simplest model. The combined model L-L presented no significant Lack of fit. Once, the Lack of fit is obtained by division between the means square/main-plot pure error and means square ratio (MSLOF/MSPE) of 8.60/5.75 = 1.50 is less than the 95% confidence F18, 27 tabled value of 2.00. It indicates that the variance results are explained only by the uncertainty of the experimental errors, it is not evidence which the model cannot fit the data at a confidence level of 95%.

Table 5 presents data for the L-L combined model: Coefficients, standard errors and t-test values. The standard errors were obtained by the data showed in Table 4.

There is no specific method to be applied in setting the t-test for combined model [16] . One way to obtain the critical t-values is by application of a normal probability graph in the t-tests values presented in Table 5. Figure 2, shows this plotting, where values of cutoff are between two lines, and the model combined L-L after this adjustment is shown in the Equation (9):

(9)

Figure 3 presents the graph of relation between experimental values and predicted values obtained from Equation (9). The variations in the graph keep up a correspondence to a root average square error of 20.48. The distribution of the points can considered random about the line. It confirms the no evidence of Lack of fit for the

Table 4. CRED ANOVA table for combined linear-linear model experimental.

Table 5. Parameters, estimated standard errors and t-test ratios for the linear-linear combined model.

Figure 2. Graph of the values predicted by Equation (9) against the experimental values. The line representing exact agreement is also shown expanded.

Figure 3. Normal probability graph of the t-test values in Table 5. The points between the two lines were discarded.

L-L combined model. Another graph, which relates the residuals versus the predicted values also presents a random behavior, this graph is not presented in this study.

By analyzing the Equation (9), it is possible to note, which the term with greater values is associated to x_{3}, this result is initially unexpected, because plastic concrete follows the Abrams law [25] . These concretes utilize high consumption of water beyond the amount needed to cement hydration. In this case, the increases of water/cement ratio will provide a decrease of compressive strength. Its occurs because the amount of water consumed, in this study, is lower. For this reason, the behavior of this concrete it is contrary to that suggested by Abrams law. For this kind of concretes, the greater the amount of water used in the mixture, the bigger the concrete compressive strength. It is because the water used provides a better plasticity of the mixture, i.e., the water serves as a “lubricant”, reducing the internal friction between grains, thus making it easier to accommodate grains and minimizing the formation of voids [26] .

The significance of crossing terms xz indicates which the levels of the size and proportion of additional Polystyrene beads interact with mixture variables and affect surface response. By analysing of each set of mixture separately for the least-square fit of process variables data, it will resuts the following set of equations:

(10)

(11)

(12)

When, each value of set mixture is substituted by values of x_{1}, x_{2}, and x_{3} into Equation (9), the coefficients resultants are consistent that presented by the Equations (10)-(12).

In consequence of significance of xz crossing terms, it is expected, which each set of mixture condition or different level of process variable will impact the variable response. This fact is evidenced by different coefficients obtained from Equations (10)-(12). In the Figure 4, is possible to verify the results of the data application presented in Table 3, in the Equations (10)-(12).

4.1. Polystyrene Beads Percentage Effect Substituting Aggregates in Concrete

Through the analysis of the Figure 4, it can be verified,which the compressive strength is inversely proportional to the percentage of Polystyrene beads that replace the aggregates. This performance was expected because the difference between the chemical composition of Polystyrene beads and the conventional concrete. Their compounds are usually formed by mineral products, while the Polystyrene beads are composed of polymer organic origin. Furthermore, this behavior is already described in the literature [27] - [29] , where the greater the replacement of Polystyrene beads by aggregate, the lower will be concrete compression strength. Thus, considering this level of variable studied, the best % of Polystyrene in concrete is 5.0%, and the optimum mixture condition is 18.3% cement, 73.4% aggregate and 8.3% water. When, are considered all the set of mixtures, it was observed which it is possible to produce a polymerized concrete with compressive strength above 15 MPa for 28 days. It allows many possibilities for the use in this of civil construction where the use of non-structural concrete is recommended.

4.2. Mixture Composition Effect

By analyzing the mixture sets, it is possible to identify the optimal mixture composition: x_{1} = 0.183; x_{2} = 0.734; x_{3} = 0.083. Each mixture compositionis presented different water/cement ratio, however this mixture set presents the greatest relationship. Considering what was previously discussed, the amount of water available for the mixture showed the major influence and contributed significantly to the compressive strength of concrete. This fact was determinant to impose this set as being the best.

4.3. Effect Regarding the Polystyrene Size

Finalizing, when considering the particle size of polystyrene beads, all mixtures composition shows a tendency

Figure 4. Process response surfaces for variation of size and proportion of EPS beads for each set of mixture variable conditions. Figures 4(a)-(c) correspond to each set of mixture and (d) compressive strength against of EPS beads size, with two levels of % EPS highlighted.

directly proportional between the size of polystyrene beads and the concrete compressive strength. In consequence of this, and considering this studied level of limit imposed on the variable studied, the best size of beads is 4.8 mm. This observation is obvious when are compared the compressive strength results versus the size of Polystyrene beads. It can be done by observation in Figures 4(a)-(c), in addition to Figure 4(d). This last figure, it ratifies this finding, because it shows the influence of increase in beads’ size and compressive strength. It is also related to proportion of polystyrene added, as already correlated by the set of mixtures presented in the other three previous figures (Figures 4(a)-(c)). The results found in the literature utilized a different size of beads, which becomes difficult to do any comparison.

The substitution of only 5.0% of Polystyrene beads seems a value without importance. However, it presents as environmental advance, considering the great amounts waste Polystyrene produced annually such as packing and construction residue. The percentage of replacement develops into more significant by the possibility of utilization of this blocks as paving without the need for pretreatment or additives.

5. Conclusions

The optimization method presented adequate performance to simplify operational procedures that require many experiments.

The combined process-mixture variable model has not presented evidence of Lack of fit. It can explain the changes that occur with the isolated changes in each variable (processes and mixtures) as well as the interactions that each has on the other. Considering this level of variable studied, the optimum condition in the polymerized concrete is: 4.8 mm sized polystyrene beads, 5.0% polystyrene beads that substituted the aggregate, 18.3% cement, 73.4% aggregate and 8.3% water.

The results indicate that it is possible to produce a polymerized concrete with compressive strength above 15 MPa for 28 days. Thus, the goals of this works were achieved and the material obtained could be used as block for paving or in walls.

This level of strength was obtained without and surface treatment in beads used or addition of additives. It allows many possibilities for the use in this of civil construction where the use of non-structural concrete. For future works, it is recommended its analysis as an element for thermal and acoustic insulation.

Acknowledgements

R. K. V. thanks the CAPES and the program PROAP.

Cite this paper

Vieira, R. , Vasconcelos, R. , Marangoni, D. and Vieira, A. (2016) Optimization of Expanded Polystyrene Lightweight Aggregate in Pre-Cast Concrete Blocks by a Completely Random Experimental Design (CRED) with Mixture and Process Variables.*Open Journal of Statistics*, **6**, 594-604. doi: 10.4236/ojs.2016.64050.

Vieira, R. , Vasconcelos, R. , Marangoni, D. and Vieira, A. (2016) Optimization of Expanded Polystyrene Lightweight Aggregate in Pre-Cast Concrete Blocks by a Completely Random Experimental Design (CRED) with Mixture and Process Variables.

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[12] Piepel, G., Pasquini, B., Cooley, S., Heredia-Langner, A., Orlandini, S. and Furlanetto, S. (2012) Mixture-Process Variable Approach to Optimize a Microemulsion Electrokinetic Chromatography Method for the Quality Control of a Nutraceutical Based on Coenzyme Q10. Talanta, 97, 73-82.

http://dx.doi.org/10.1016/j.talanta.2012.03.064

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http://dx.doi.org/10.1007/978-3-319-00218-7_6

[16] Hosseini, S.A., Niaei, A., Salari, D. and Nabavi, S.R. (2014) Modeling and Optimization of Combustion Process of 2-Propanol over Perovskite-Type LaMnyCo1-yO3 Nanocatalysts by an Unreplicated Experimental Design with Mixture-Process Variables and Genetic Algorithm Methodology. Journal of the Taiwan Institute of Chemical Engineers, 45, 85-91.

http://dx.doi.org/10.1016/j.jtice.2013.04.010

[17] Alexandrino, G.L. and Poppi, R.J. (2014) Study of the Homogeneity of Drug Loaded in Polymeric Films Using Near-Infrared Chemical Imaging and Split-Plot Design. Journal of Pharmaceutical Sciences, 103, 2356-2365.

http://dx.doi.org/10.1002/jps.24051

[18] Hinkelmann, K. and Kempthorne, O. (2012) Design and Analysis of Experiments, Special Designs and Applications. Vol. 3, John Wiley & Sons, Hoboken.

http://dx.doi.org/10.1002/9781118147634

[19] Shrivastava, A.K. (2013) Efficient Construction of Split-Plot Design Catalogs Using Graphs. IIE Transactions, 45, 1137-1152.

http://dx.doi.org/10.1080/0740817X.2012.723840

[20] Fang, L. and Loughin, T.M. (2013) Analyzing Binomial Data in a Split-Plot Design: Classical Approach or Modern Techniques? Communications in Statistics-Simulation and Computation, 42, 727-740

http://dx.doi.org/10.1080/03610918.2011.650264

[21] Kristensen, K. (2012) Incomplete Split-Plot Designs Based on α-Designs: A Compromise between Traditional Split-Plot Designs and Randomised Complete Block Design. Euphytica, 183, 401-413.

http://dx.doi.org/10.1007/s10681-010-0318-5

[22] Ambro?y, K. and Mejza, I. (2011) Statistical Properties of Some Supplemented Split-Split-Plot Design. Colloquium Biometricum, 41, 65-174.

[23] Montgomery, D.C. (2008) Design and Analysis of Experiments. 8th Edition, John Wiley & Sons, Hoboken.

[24] Benício B.N., Scarminio I.S. and Bruns R.E. (1995) Planejamento e Otimiza??o de experimentos. 2nd Edition, Editora da Unicamp, Campinas.

[25] Abrams, D.A. (1919) Design of Concrete Mixtures. Vol. 1, Structural Materials Research Laboratory, Lewis Institute, Chicago.

[26] Pirola, F.C. (2011) Contribui??o para o estudo de concreto seco utilizadona fabrica??o de pe?as de concreto para pavimenta??o de 50 MPa. MSc Thesis, Universidade Federal de Santa Catarina, Florianópolis.

[27] Chen, B. and Liu, J. (2007) Mechanical Properties of Polymer-Modified Concretes Containing Expanded Polystyrene Beads. Construction and Building Materials, 21, 7-11.

http://dx.doi.org/10.1016/j.conbuildmat.2005.08.001

[28] Saradhi Babu, D., Ganesh Babu, K. and Wee, T.H. (2005) Properties of Lightweight Expanded Polystyrene Aggregate Concretes Containing Fly Ash. Cement and Concrete Research, 35, 1218-1223.

http://dx.doi.org/10.1016/j.cemconres.2004.11.015

[29] Xu, Y., Jiang, L., Xu, J. and Li, Y. (2012) Mechanical Properties of Expanded Polystyrene Lightweight Aggregate Concrete and Brick. Construction and Building Materials, 27, 32-38.

http://dx.doi.org/10.1016/j.conbuildmat.2011.08.030

[1] Demirel, B. (2013) Optimization of the Composite Brick Composed of Expanded Polystyrene and Pumice Blocks. Construction and Building Materials, 40, 306-313.

http://dx.doi.org/10.1016/j.conbuildmat.2012.11.008

[2] Poon, C.S. and Lam, C.S. (2008) The Effect of Aggregate-to-Cement Ratio and Types of Aggregates on the Properties of Pre-Cast Concrete Blocks. Cement & Concrete Composites, 30, 283-289.

http://dx.doi.org/10.1016/j.cemconcomp.2007.10.005

[3] Poon, C.S., Kou, S.C. and Lam, L. (2002) Use of Recycled Aggregates in Molded Concrete Bricks and Blocks. Construction and Building Materials, 16, 281-289.

http://dx.doi.org/10.1016/S0950-0618(02)00019-3

[4] Raut, S.P., Ralegaonkar, R.V. and Mandavgane, S.A. (2011) Development of Sustainable Construction Material Using Industrial and Agricultural Solid Waste: A Review of Waste-Create Bricks. Construction and Building Materials, 10, 4037-4042.

http://dx.doi.org/10.1016/j.conbuildmat.2011.04.038

[5] Kinuthia, J.M. and Nidzam, R.M. (2011) Towards Zero Industrial Waste: Utilisation of Brick Dust Waste in Sustainable Construction. Waste Management, 31, 1867-1878.

http://dx.doi.org/10.1016/j.wasman.2011.03.020

[6] Kushwaha, M., Akhtar, S. and Rajput, S. (2013) Development of the Self Compacting Concrete by Industrial Waste (Red Mud). Development, 3, 539-542.

[7] Hassan, I.O., Ismail, M., Noruzman, A.H., Yusuf, T.O., Mehmannavaz, T. and Usman, J. (2013) Characterization of Some Key Industrial Waste Products for Sustainable Concrete Production. Advanced Materials Research, 690, 1091-1094.

http://dx.doi.org/10.4028/www.scientific.net/AMR.690-693.1091

[8] Ribas, L.F., Cordeiro, G.C., Vieira, A.K. and Vieira, R.K. (2012) A Methodology for Collection of Residential Construction Residue in Manaus. Journal of Environmental Science and Engineering, 1, 1149-1163

[9] Vieira, R.K., Soares, R.C., Pinheiro, S.C., Paiva, O.A., Eleutério, J.O. and Vasconcelos, R.P. (2010) Completely Random Experimental Design with Mixture and Process Variables for Optimization of Rubberized Concrete. Construction and Building Materials, 24, 1754-1760.

http://dx.doi.org/10.1016/j.conbuildmat.2010.02.013

[10] Antony, J. (2014) Design of Experiments for Engineers and Scientists. 2nd Edition, Elsevier, London.

[11] Hosseini, S.A., Niaei, A., Salari, D., Vieira, R.K., Sadigov, S. and Nabavi, S.R. (2013) Optimization and Statistical Modeling of Catalytic Oxidation of 2-Propanol over CuMnmC2?mO4 Nanospinels by Unreplicated Split Design Methodology. Journal of Industrial and Engineering Chemistry, 19, 166-171.

http://dx.doi.org/10.1016/j.jiec.2012.07.020

[12] Piepel, G., Pasquini, B., Cooley, S., Heredia-Langner, A., Orlandini, S. and Furlanetto, S. (2012) Mixture-Process Variable Approach to Optimize a Microemulsion Electrokinetic Chromatography Method for the Quality Control of a Nutraceutical Based on Coenzyme Q10. Talanta, 97, 73-82.

http://dx.doi.org/10.1016/j.talanta.2012.03.064

[13] Kowalski, S.M., Cornell, J.A. and Vining, G.G. (2002) Split-Plot Designs and Estimation Methods for Mixture Experiments with Process Variables. Technometrics, 44, 72-79.

http://dx.doi.org/10.1198/004017002753398344

[14] Bortoloti, J.A., Bruns, R.E., de Andrade, J.C. and Vieira, R.K. (2004) Split-Plot Design Optimization for Trace Determination of Lead by Anodic Stripping Voltammetry in a Homogeneous Ternary Solvent System. Chemometrics and intelligent laboratory systems, 70, 113-121.

http://dx.doi.org/10.1016/j.chemolab.2003.09.004

[15] Coetzer, R.L. and Haines, L.M. (2013) Optimal Designs for Multiple-Mixture by Process Variable Experiments. In: Ucinski, D., Atkinson, A.C. and Patan, M., Eds., mODa 10—Advances in Model-Oriented Design and Analysis, Springer International Publishing, Cham, 45-53.

http://dx.doi.org/10.1007/978-3-319-00218-7_6

[16] Hosseini, S.A., Niaei, A., Salari, D. and Nabavi, S.R. (2014) Modeling and Optimization of Combustion Process of 2-Propanol over Perovskite-Type LaMnyCo1-yO3 Nanocatalysts by an Unreplicated Experimental Design with Mixture-Process Variables and Genetic Algorithm Methodology. Journal of the Taiwan Institute of Chemical Engineers, 45, 85-91.

http://dx.doi.org/10.1016/j.jtice.2013.04.010

[17] Alexandrino, G.L. and Poppi, R.J. (2014) Study of the Homogeneity of Drug Loaded in Polymeric Films Using Near-Infrared Chemical Imaging and Split-Plot Design. Journal of Pharmaceutical Sciences, 103, 2356-2365.

http://dx.doi.org/10.1002/jps.24051

[18] Hinkelmann, K. and Kempthorne, O. (2012) Design and Analysis of Experiments, Special Designs and Applications. Vol. 3, John Wiley & Sons, Hoboken.

http://dx.doi.org/10.1002/9781118147634

[19] Shrivastava, A.K. (2013) Efficient Construction of Split-Plot Design Catalogs Using Graphs. IIE Transactions, 45, 1137-1152.

http://dx.doi.org/10.1080/0740817X.2012.723840

[20] Fang, L. and Loughin, T.M. (2013) Analyzing Binomial Data in a Split-Plot Design: Classical Approach or Modern Techniques? Communications in Statistics-Simulation and Computation, 42, 727-740

http://dx.doi.org/10.1080/03610918.2011.650264

[21] Kristensen, K. (2012) Incomplete Split-Plot Designs Based on α-Designs: A Compromise between Traditional Split-Plot Designs and Randomised Complete Block Design. Euphytica, 183, 401-413.

http://dx.doi.org/10.1007/s10681-010-0318-5

[22] Ambro?y, K. and Mejza, I. (2011) Statistical Properties of Some Supplemented Split-Split-Plot Design. Colloquium Biometricum, 41, 65-174.

[23] Montgomery, D.C. (2008) Design and Analysis of Experiments. 8th Edition, John Wiley & Sons, Hoboken.

[24] Benício B.N., Scarminio I.S. and Bruns R.E. (1995) Planejamento e Otimiza??o de experimentos. 2nd Edition, Editora da Unicamp, Campinas.

[25] Abrams, D.A. (1919) Design of Concrete Mixtures. Vol. 1, Structural Materials Research Laboratory, Lewis Institute, Chicago.

[26] Pirola, F.C. (2011) Contribui??o para o estudo de concreto seco utilizadona fabrica??o de pe?as de concreto para pavimenta??o de 50 MPa. MSc Thesis, Universidade Federal de Santa Catarina, Florianópolis.

[27] Chen, B. and Liu, J. (2007) Mechanical Properties of Polymer-Modified Concretes Containing Expanded Polystyrene Beads. Construction and Building Materials, 21, 7-11.

http://dx.doi.org/10.1016/j.conbuildmat.2005.08.001

[28] Saradhi Babu, D., Ganesh Babu, K. and Wee, T.H. (2005) Properties of Lightweight Expanded Polystyrene Aggregate Concretes Containing Fly Ash. Cement and Concrete Research, 35, 1218-1223.

http://dx.doi.org/10.1016/j.cemconres.2004.11.015

[29] Xu, Y., Jiang, L., Xu, J. and Li, Y. (2012) Mechanical Properties of Expanded Polystyrene Lightweight Aggregate Concrete and Brick. Construction and Building Materials, 27, 32-38.

http://dx.doi.org/10.1016/j.conbuildmat.2011.08.030