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The aim of this study was to determine the optimum design mix to produce pre-cast concrete blocks by a completely random experimental design (CRED) with mixture and process variables. The polymerized concrete was studied its composition: Cement, and water defined as the mixture compounds. To choose the best model, all the possible models were assessed through the ANOVA, which tested each possible model. The linear-linear model was preferred, since that do not present evidence of lack of fit, and it is capable of relating how to react the process variables, when are changed the variable mixture condition levels. The optimum experimental condition, obtained for the polymerized concrete, was: The size of the polystyrene beads was 4.8 mm sized polystyrene beads, 5.0% polystyrene that replaced the aggregate, 18.3% cement, 73.4% aggregate and 8.3% water. In this condition, the blocks made with polymerized concrete show a compressive strength above 15 Mpa, allowing its utilization in paving.

The use of raw materials in the civil construction industry has been very intense in recent years. This increase is very consistent in developing countries, which is motivated primarily by economic growth. It conveys many concerns, because the supply of types of raw materials used, is non-renewable resource. Although, the same economic growth expands the consumption of durable goods throughout their supply chain, it is also responsible for generating an amount of industrial waste.

The use of these residues, in partial or total substitution, of raw material in the civil construction industry, is a constant source of research [

In the major cities of the Amazon region, such as Manaus, this problem is much more critical due to its geological formation. It has rocks, which are easily accessible for use in construction, such as coarse aggregates to produce concrete blocks.

Alternatively, there is used gravel as coarse aggregates, which is obtained through dredging of riverbeds. The replacement of them with industrial waste, even minimally, already implies a direct improvement to the ecosystem of the region.

Also, the Industrial Pole of Manaus, where the electro-electronic companies are the principal industrial sector, generates much waste. This residue comes from packaging, which is based on expanded polystyrene, and does not have any final landfill destination.

In this context, this study presents an alternative technology for the use of waste polystyrene from packaging. This residue can be used as the aggregate in concrete for the production of blocks, after a burning process. These blocks could be used for construction walls or pavement. Thus, the purpose of this study was to obtain the most advantageous percentage and size of the polystyrenebeads that can be integrated to concrete as a substitute for aggregate, with minimum loss of compressive resistance with a CRED.

Vieira [

The design of experiment employs multivariate optimization; this method has been applied more and more in many area of Engineering [

Mixtures―in these variables, their properties not depend on their absolute magnitude, but the percentage of each constituent in the blend. The proportion of each constituent of the system has to be considered as a variable of mixture. These variables, which are dependent of others (i.e., when the quantity of one varies, the percentage of the others also varies). This definition could be better understood by a mathematical description [

where, q is the number of constituent of the mixture and x_{i} are the components.

Process variables―these are factors which can changes its levels without change the levels of others factors. For this reason they are independent and change without limitation [

Linear:

Bilinear:

The models often applied to describe the effect of the mixture variables are: [

Linear:

Quadratic:

Special Cubic:

where, a is the process model factor (the effects); b is the mixture model factor, z is the parameter of the process variable, x is the mixture variable parameter and

Nevertheless, these variables (process and mixture) can fit several systems where one or another has decisive influence. There are other types of systems where the response variable depends on both magnitude of the mixture constituents (mixture variables) and the effects of the process conditions [

where, d_{ij} is the arrangement of a_{i} and b_{j} factors.

Situations where mixture and process variables are studied concomitant and its experiments are planned to be done in a completely random mode are encountered in the literature [

Due to the specificity of this type of design, they not involve necessary a great number of variables; however, the experimental set will require, generally, a large number of experiments, it will depend of the system size in study. In this case, it is recommended to do a preliminary study only with mixture or process variables to exclude the variables that are not influence in the system. The planning suggested is a factorial.

An alternative to this method will be the perform the experiment in blocks by application of a split-plot design, as used by several authors [

The aims of this study was to determine the optimum design mix to produce pre-cast concrete blocks by a completely random experimental design (CRED) with mixture and process variables. The use this procedure will generate a fast response without the need to use complex mathematical or statistic models.

This paper is organized as follows: In introduction (Section 1) are shown the motivation and justification, as well is initiating a short Literature Review about application of experimental design. This review is completed with the topic Analysis of variance (Section 2). Section 3 presents the methodology applied in the study. In Sections 4 are presented three subitems: In the first one, it relates there place effect of polystyreneasa substitute of aggregates in concrete; in the second one, it presents the results relate to mixture effect in concrete; and in the thirst one, it analyzes the effect of polystyrene. The last section presents the conclusions of the article.

Analysis of variance depends of type model analyzed, for a factorial planning with two factors, for example: Z_{1} and Z_{2}, with levels “f” and “g”, and “n” replicates. In this case,the variance model analysis will correspond to Equation (8), according Montgomery, [

where:

_{i} represents the effects of i-^{th}A factor treatment; β_{j} the effects of j-^{th}B factor treatment; (d)_{ij} interactions effects between A and B and e_{ijk} represents a random error.

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. [

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 [

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

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 (

Material | Specific gravity | Set of mixture^{*} | ||
---|---|---|---|---|

1 | 2 | 3 | ||

Cement (kg/m^{3}) | 2305 | 422.24 | 398.21 | 490.24 |

Polystyrene beads (kg/m^{3})^{Ä} | 1006 | 5% | 15% | 25% |

Grit | 2650 | 1083.05 | 1015.43 | 551.52 |

Sand (kg/m^{3}) | 2620 | 548.92 | 398.21 | 735.36 |

Water (kg/m^{3}) | 1000 | 190.01 | 159.28 | 171.58 |

^{*}As described in ^{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.

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 [

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.

Process | Size EPS (mm) | % EPS substitute | ||||||
---|---|---|---|---|---|---|---|---|

1 | −1 | −1 | 1.20 | 5.0 | ||||

2 | 0 | −1 | 2.40 | 5.0 | ||||

3 | 1 | −1 | 4.80 | 5.0 | ||||

4 | −1 | 0 | 1.20 | 15.0 | ||||

5 | 0 | 0 | 2.40 | 15.0 | ||||

6 | 1 | 0 | 4.80 | 15.0 | ||||

7 | −1 | 1 | 1.20 | 25.0 | ||||

8 | 0 | 1 | 2.40 | 25.0 | ||||

9 | 1 | 1 | 4.80 | 25.0 | ||||

Mixtures | Cement (X_{1}) | Aggregate (X_{2}) | Water (X_{3}) | |||||

1 | 0.183 | 0.734 | 0.083 | |||||

2 | 0.185 | 0.741 | 0.074 | |||||

3 | 0.230 | 0.690 | 0.080 | |||||

a) standardized variables.

Formulation Number^{a} | 1 | 2 | 3 | |||
---|---|---|---|---|---|---|

R_{2} | R_{1} | R_{2} | R_{1} | R_{2} | ||

z_{1} = −1; z_{2} = −1 | 16.40 | 11.12 | 6.41 | 12.15 | 6.52 | 07.12 |

z_{1} = 0; z_{2} = −1 | 16.65 | 17.68 | 21.10 | 10.38 | 10.88 | 10.70 |

z_{1} = 1; z_{2} = −1 | 15.59 | 11.02 | 9.21 | 12.29 | 15.23 | 10.63 |

z_{1} = −1; z_{2} = 0 | 5.17 | 4.57 | 3.86 | 4.04 | 3.54 | 0.00 |

z_{1} = 0; z_{2} = 0 | 7.72 | 5.35 | 3.01 | 4.43 | 3.97 | 4.61 |

z_{1} = 1; z_{2} = 0 | 11.05 | 8.89 | 5.46 | 6.02 | 6.09 | 4.99 |

z_{1} = −1; z_{2} = 1 | 3.15 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

z_{1} = 0; z_{2} = 1 | 2.94 | 9.53 | 2.59 | 8.15 | 0.00 | 10.66 |

z_{1} = 1; z_{2} = 1 | 4.00 | 8.61 | 2.69 | 10.91 | 0.00 | 15.41 |

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

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

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.

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.

There is no specific method to be applied in setting the t-test for combined model [

Model | Source | SS | DF | MS |
---|---|---|---|---|

Linear-linear | Regression | 1118.75 | 8 | 139.84 |

Residues | 309.94 | 45 | 6.89 | |

Lack of fit | 154.82 | 18 | 8.60 | |

Pure error | 155.12 | 27 | 5.75 | |

Total | 1428.69 | 53 | ||

% var. explained | 78.31 | |||

% max. explainable | 89.14 |

Variables | Parameters | Standard error | t-test |
---|---|---|---|

x_{1} | −56.65 | 16.43 | −3.45 |

x_{2} | −0.75 | 7.87 | −0.10 |

x_{3} | 232.76 | 89.52 | 2.60 |

x_{1}z_{1} | 11.37 | 20.02 | 0.57 |

x_{2}z_{1} | 17.46 | 20.02 | 0.87 |

x_{3}z_{1} | −4.35 | 9.64 | −0.45 |

x_{1}z_{2} | −3.37 | 9.64 | −0.35 |

x_{2}z_{2} | 29.79 | 109.64 | 0.27 |

x_{3}z_{2} | −77.35 | 109.64 | −0.71 |

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 [

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:

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

Through the analysis of the

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.

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

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

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.

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.

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

Raimundo Kennedy Vieira,Raimundo Pereira de Vasconcelos,Douglas Marangoni,Adalena Kennedy Vieira, (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,06,594-604. doi: 10.4236/ojs.2016.64050