INTRODUCTION
In the last 40 years the application of polymer on concrete has significantly
progressed. Although PolymerModified Concrete (PMC) and Polymer Concrete
(PC) came into use in the 1950s, only in the 1970s, after PolymerImpregnated
Concrete (PIC) was developed the PC materials were received fairly publicity
in concrete industry (Fowler, 1999). The use domain of PC in time was
extended from precast components for buildings, bridges panel, waste
components, transportation components to repair of structural members,
waterproofing, anticorrosive and decorative finishes, overlay of pavements,
etc. (Aggarwal et al., 2007). The PC occurred and developed in
construction industry due to its advantages compared with the Portland
cement concrete, such as: quick setting characteristics, high mechanical
strength, chemical resistance and wear resistance (Blaga and Beaudoin,
1985; AbdelFattah and ElHawary, 1999). In the composition of polymeric
concrete are used the same components: aggregates and the binder that
for polymeric concrete is a resin that reacts with a hardener and bind
together the aggregates. Different types, properties and applications
of PC have been reported by Fowler (1999). The performances and the use
domain of PC depend on the polymer binder, type of filler and aggregates.
The mechanical properties and the curing behavior depend on the selection
and the content of the polymer, temperature and aggregate type and dosage.
The presence of filler is also important and the use of Silica Fume (SUF)
in the mix improved the mechanical properties. The relatively high cost
of PC has led to studies for reducing the polymer dosage in the mix, without
diminishing the properties. A limited dosage of polymer, between 12.418.8%
was adopted in this study. The objective of this study is to analyze by
statistical methods the experimental mechanical properties of PC realized
with epoxy resin, SUF and aggregate. The PC mixes were determined on the
basis of mixture design of experiments and based on statistical analysis
properties such as compressive strength, flexural strength, splitting
tensile strength and adhesion stress are analyzed. Each individual response,
such as compressive strength, can be predicted by the regression equation
(Muthukumar et al., 2003; Muthukumar and Mohan, 2004). The chosen
statistical model gives a correlation between the experimental response
and the predicted response.
EXPERIMENTAL PROGRAM
The materials used were: epoxy resin, SUF and crushed aggregates of two
grades 04 mm (Sort I) and 48 mm (Sort II). The epoxy resin in combination
with the hardener forms the binder of the PC.

Fig. 1: 
Test samples of PC 
Table 1: 
Mixture design combinations for PC 

The SUF is a byproduct that results from ferrosilicon production having the following
characteristics:
• 
Particle sizes of 0.01…0.5 μ 
• 
Shape of particles is spherical 
• 
Specific surface is between 13000 and 23000 m^{2} kg^{1} 
• 
Density between 2.1 and 2.25 g cm^{3} (Aggarwal et al.,
2007). 
The aggregates were obtained from river gravel by crushing.. A minimum
resin content of 12.4% was adopted from the workability conditions and
the maximum dosage of 18.8% was imposed by the segregation of the mix.
PC of different compositions as is given in Table 1
was prepared by mixing required quantities of epoxy resin firstly with
aggregates, than with the filler (SUF), that was added slowly in a mechanical
mixer. Than the casting of specimens: cubes of 70.7 mm sides and prism
of 210x70x70 mm sides were prepared for determining the mechanical characteristics
and for the adhesion stress the sample was realized by casting PC circular
tablet on an usual concrete cube surface (Fig. 1).
MIXTURE DESIGN OF EXPERIMENT AND RESPONSE SURFACE METHODOLOGY
Mixture design of experiment: Research in many disciplines frequently
involves blending two or more ingredients together. The design factors
in a mixture experiment (Muthukumar et al., 2003; Muthukumar and
Mohan, 2004; Cornell, 1990; Mayer and Montgomery, 1995; Montgomery and
Douglas, 2000; Marcia et al., 1997) are the proportions of the
components of a blend and the response variables vary as a function of
these proportions making the total and not actual quality of each component.
The total amount of the mixture is normally fixed in a mixture experiment
and the component settings are proportions of the total amount. The component
proportions in a mixture experiment cannot vary independently as in factorial
experiments since they are constrained to sum to a constant (1 or 100%
for standard design). Imposing such constraint on the component proportions
complicates the design and the analysis of mixture experiments. Although
the bestknown constraint in a mixture experiment is to set the sum of
the components to one (100%) additional constraints such as imposing a
maximum or minimum value on each mixture component may also apply.
In the mixture design approach the total of amount of the input variables
was fixed and constrained to sum 100. For each statistical combination,
all properties of interest were measured and empirical models for each
property as function of the input variables were determined from regression
analysis. The advantage of the mixture approach is that the experimental
region of interest is more naturally defined. To simplify calculation
and analysis, the actual variables ranges were transformed to dimensionless
coded variables with a range 0 and 1. Intermediate values were also translated
similarly. Those variables were codified using the following formula:
where, R_{i} = A_{i}/ΣA_{i}, L_{i}
is the lower constraint in real value, L is the sum of lower constraint
in real value, A is the actual value and A_{i} is the total of
actual values.
Response surface methodology: Response Surface Methodology (RSM) consists
of a group of empirical techniques devoted to the evaluation of relations existing
between a cluster of controlled experimental factors and the measured responses,
according to one or more selected criteria (Cornell, 1990; Mayer and Montgomery,
1995; Montgomery, 2000). Prior knowledge and understanding of the process and
the process variables under investigation is necessary for achieving a realistic
model.
RSM provides an approximate relationship between a true response y and
p design variables, which is based on the observed data from the process
or system (Lepadatu et al., 2005, 2006). The response is generally
obtained from real experiments or computer simulations and the true response
y is the expected response. Thus, real experiments are performed in this
study. We suppose that the true response Y_{t} can be written
as:
where, the variables x_{1}, x_{2}, … x_{p}
are expressed in natural units of a measurement, so are called as the
natural variables. The experimentally obtained response Y_{t}
differs from the expected value y due to random error. Because the form
of the true response function F is unknown and perhaps very complicated,
we must approximate it and y can be written as:
where, ε denotes the random error, which includes measurement error
on the response and is inherent in the process or system and the variables
ς_{1}, ς_{2}, . . . ,ς_{n} are
the coded variables of the natural variables. We treat ε as a statistical
error, often assuming it to have a normal distribution with mean zero
and variance σ_{2}.
In many cases, the approximating function F of the true response y is
normally chosen to be either a firstorder or a secondorder polynomial
model, which is based on Taylor series expansion. In general the secondorder
model is:
In order to more accurately predict the response, the secondorder model
is used to fit a curvature response. From the above approximating function,
the estimated response Y_{t} at the n^{th} data point
can be written in matrix form as:
In Eq. 5, X is a matrix of model terms evaluated at the data points.
The regression coefficients of the predictive model are estimated by the
method of the least squares using the general formulation as:
where, X^{T} is the transpose of the matrix X.
The secondorder polynomial relation with special cubic interactions
can approximate the mathematical relationship between the independent
variables x_{i} and the response Y:
where, β_{i} are linear coefficients, β_{ii}
are quadratic coefficients, β_{ij} are crossproduct coefficients,
β_{ijk} are the special cubic coefficients and ε is
the random error which includes measurement error on the response and
is inherent in the process or system. These coefficients are unknown coefficients
usually estimated to minimize the sum of the squares of the error term,
which is a process known as regression.
RESULTS AND DISCUSSION
Out of number of factors identified by their simplified notation (A,
B, C and D), the following ones were considered to be most important and
necessary to control:
• 
Epoxy resin (A) 

• 
Silica fume (B) 

• 
Aggregate sort I, 04 (C) 

• 
Aggregate sort II, 48 (D) 

The input variables, range chosen for the study, their coded value and
mixture design combination are given in Table 2 and
3.
Each PC mixture was used for casting specimens that were tested under
identical conditions according to European Standard (EN 12390/2001).
Table 2: 
Range of variables and their coded form 

Table 3: 
Mixture design combination for PC 

The
Compressive Strength (CS), Flexural Strength (FS), Splitting Tensile Strength (STS)
and Adherence Stress (AS), at 14 days were determined experimentally,
adopting standard techniques (EN 12390/2001) for all combinations given
in Table 3. The adhesion stress was determined with
the following relation.
Where:
P 
= 
Split force (N) 
d 
= 
Diameter of PC tablet (mm) 
h 
= 
Depth of polymeric concrete tablet (mm) 
Table 4 summarizes mixture design and their experimental
responsesCS, FS, STS and AS for each PC combination based on the concept
of design of experiments. Mixture designs (110 runs) are sometimes augmented
by adding interior points (1115 runs). A center points will be added
to the design data (Table 4) with 5 runs making 15 runs
total. This addition will change the design from simplexlattice to simplexcentroid
design. The experimentally studied response based on the results observed
at the 14 days was analyzed statistically used Statstica software. In
Table 4, each individual response (CS, FS, FTS, AS)
can be predicted by the regression equation, which expresses the relationship
between the input variables and the concerned response.
For example the regression equation for CS can be predicted by the following
equation:
In the same case can be determined the other predicted responses. The
predicted responses obtained by the regression equation (Eq. 9) are compared
with the experimental values and are given in Table 4.
The coefficients of the individual variable in each equation give a measure
of the effect of variable on the predicted response. For variable having
coefficient of large magnitudes, even a marginal increment will give a
significant change in the response. However, for variables having coefficients
of lower magnitudes, even a large increase will result in only a small
change in the response. Thus significant and less significant variable
can be identified from the equation. The above equations are based on
the special cubic model, because this model fitted well with the experimental
data.
A standard statistical technique to carry it out is the analysis of varianceANOVA
(Cornell, 1990; Mayer and Montgomery Douglas, 1995; Montgomer Douglas,
2000; Goupy, 1999), it is routinely used to provide a measure of confidence.
ANOVA results for 14 days strength are shown in Table 5.
By this way we can observe the importance of interaction effect of the
three leading factors (ABC), which is expressed by the coefficient Rsqr
 0.99987 (Table 5). This coefficient shows an adequate
fit for the predictive response surface model of CS.
Table 6 shows the ANOVA results for 28 days for Adherence
Stress.
ANOVA for each model (Table 58)
gives the sum of squares and degrees of freedom for the model terms from
which mean squares values of the individual model terms are calculated.
The lackoffit test compares the residual error to the pure error from
replication and gives Fvalues for all the models. The Fvalues must be
lower if a particular model is to be significant. From the Ftest, it
was found that only the model passed the Ftest.
Analyses of variance (ANOVA) for the four responses that were studied
clearly shows that the cubic modulation is adequate for this study and
is explained by the fact that only by interaction of three factors (is
a case of a mixture) we have a modeling corresponding to the physical
phenomenon.
Pareto charts obtained from the statistical analysis are shown in Fig.
2 and 3 and show the importance order of the variables.
The charts show the variables effects on each analyzed response variation
of the PC.
Table 4: 
Experimental and predicted value for mixture design
combinations of PC 

Table 5: 
Summary of ANOVA for CS 

Table 6: 
Summary of ANOVA for AS 

Table 7: 
Summary of ANOVA for FTS 

Table 8: 
Summary of ANOVA for FS 


Fig. 2: 
Pareto charts of standardized effect of a) CS b) AS 

Fig. 3: 
Pareto charts of standardized effect of (a) FTS (b) FS 

Fig. 4: 
Effect of Silica fume. Epoxy resin and Aggregate sort I a) CS b)
FS 
The Pareto diagrams shows the four leading factors, epoxy resin (A), silica fume
(B) and crushed aggregates sort I and II (C and D) in the inverse important
order.
Also it can observe that for two responses that are analyzed there are
not significant effects in interaction of factors and for the other two
responses almost all interactions among factors are important. It must
remark that for these responses also there are important interactions
of third degree (ABC interaction for CS Fig. 4a, BCD
interaction for FS Fig. 4b).
Analyzing these charts it can conclude that for all four responses the
factors that are studied have a very important influence. For example
analysis of variance of CS and FS gives the nonlinear response surface
with the significant interactions: ABC (epoxy resin (A), silica fume (B)
and crushed aggregates sort I (C) Fig. 4a, b)
The results from the designed experiment indicate that the factors have
very significant effects. A00ll secondorder interactions have an important
role. The special cubic interaction ABC has very significant effects and
all others interactions have very weak effects whereas the other interactions
are barely noticeable at all. As a consequence, these interactions effects
can be neglected. Analysis of variance (ANOVA) and the responses surfaces for desirability
effects of variables interactions show significant nonlinear effects
(all the response surface are curve Fig. 5) for AS of
PC for example, that means that are not enough known the influences of
each factor on the behavior of this type of PC and on the contact elements.
Figure 6 shows the variation of mechanical characteristics
with resin percentage in the mix.
Analyzing the experimental results we can observe the following:
• 
The values of CS varied between 65.32 N mm^{2}
(for concrete type PC13 with 13.2% polymer) and 43.47 N mm^{2}
(for concrete PC7 with 15.6 % polymer)  that characterizes the PC
as a high strength concrete. 
• 
The values of FS varied from 17.57 N mm^{2} (for concrete
type PC6 with 15.6% polymer) to 12.29 N mm^{2} (for concrete
type PC2 with 12.4% polymer) 
• 
The values of FTS varied from 7.67 N mm^{2} (for concrete
type PC3 with 12.4% polymer) to 5.59 N mm^{2} (for concrete
type PC7 with 15.6% polymer) 
Regarding the mechanical characteristics compressive, flexural and splitting
tensile strengths the PC with good behavior for all these properties is
PC with a percentage of polymer of 15.6%.

Fig. 5: 
Response surface for desirability effects of variables
interactions 

Fig. 6: 
Variation of mechanical characteristics with resin percentage
in the mix 
For compressive strength CS the percentage
can decrease to 13.2% and for split strength the polymer percentage can
decrease to 12.4%.
• 
The values of AS varied from 10.25 N mm^{2}
(for concrete type PC6 with 15.6% polymer) to 5.18 N mm^{2}
(for concrete type PC2 with 12.4% polymer) 
From the predicted values it can observe that the maximum values are
not the same in the case of CS that has the maximum value for concrete
type PC5 (with 15.6% polymer). For FS, FTS and AS the maximum values are
the same as in the case of experimental results. The optimum polymer content
that satisfies all mechanical characteristics is situated between 12.4
and 15.6%.
CONCLUSIONS
Polymer concretes were made with epoxy resin in a reduced dosage, SUF
and aggregates. Response surface method has been used for a better understanding
of the influence of the deviation of the PC parameters on the mechanical
characteristic evolution. The mechanical characteristics of PC were determined
experimentally and compared with predicted values. A statistical analysis
has been carried out using a mixture design of experiments, which allows
for the prediction of the mechanical characteristic sensitivity versus
the process leading factors. From statistical analysis it resulted that
all factors have an important influence on the mechanical characteristics
of PC. The polymer percentage obtained from the statistical analysis satisfies
the requirement of low cost and high strength.