The results obtained from an experiment can be shown by a mathematical model. Here, the primary factors that affected the compressive strength of concrete are the ratios of water, SBR, SF to binder materials and time of curing in water. (In modeling, from effects of superplastysizer on compressive strength is neglected.)

Relationship compressive strength with main effective factors can be determined by regression. But before regression needs to determine each factor how influence in compressive strength.

As per the classical formulation of Abrams’ law, there exists an inverse relationship between the compressive strength and water to cement ratio of concrete (Popovics 1998). Abrams’ equation can be showed by:

f=\frac{A}{{B}^{\left(\frac{\mathit{w}}{\mathit{c}}\right)}}

*A*, *B* are constant coefficients and \left(\frac{\mathit{w}}{\mathit{c}}\right) is the ratio of water to cement.

A lot of researchers introduced relationship between compressive strength and time of curing in water with a logarithmic equation by (Popovics 1998): (*a*, *b* are constant coefficients.)

f=a\times log\left(\mathit{t}\right)+b

Bihanja and Khan (Bhanja and Sengupta 2002; Iqbal khan 2009) proposed power equations for the effect of SF on compressive strength of concrete. Bhanja (Bhanja and Sengupta 2002) proposed a three degree function for prediction compressive strength of SF concrete. When the percentage of replacement of SF is less than 10, the relationship between compressive strength and SF can be considered with a parabola curve. (Base on Bhanja’s relationship, the maximum error will be less than 2.5 %.)

Although, Barleonga used linear approximation to show the relationship between compressive strength and SBR % but, in mixed curing system, a little of increasing of compressive strength is observed in 5 % of SBR and a 2° parabolic can be appropriate to show the effect of SBR in compressive strength. In Figs. 7, 8, and 9 relationship between the compressive strength with each factor is determined.

The relationship between compressive strength with considered variables may be represented by:

{f}_{\text{c}}=\frac{A}{{\text{B}}^{\left(\frac{w}{\mathit{b}}\right)}}\times {\left(11.04\times log\left(t\right)+20.22\right)}^{C}\times {\left(-525.9{\mathit{s}}^{2}+83.81\mathit{s}+30.54\right)}^{D}\times {(-346{p}^{2}+18.72p+34.49)}^{E}

where {f}_{c} is the compressive strength (MPa), *w/b* is the ratio of water to binder materials, *t* is time of curing in water (day), *s* is the ratio of SF to binder materials and *p* is the ratio of SBR polymer to binder materials. *A*, *B*, *C*, *D* and *E* are constant coefficients. The above equation can be written in the following form:

\begin{array}{c}log({f}_{c})=log\left(A\right)-log\phantom{\rule{0.166667em}{0ex}}B\times \frac{w}{b}+C\times log\left(11.04\times log\left(t\right)+20.22\right)+D\times log\left(-525.9{s}^{2}+83.81s+30.54\right)\hfill \\ +E\times log\left(-346{p}^{2}+18.72p+34.49\right)\hfill \end{array}

where *A*, *B*, *C*, *D* and *E* can be determined with multiple linear regressions. The values of these coefficients are shown in Table 3. The value of multiple correlation coefficients (*r*) has been obtained as 0.95.

With replacement of one to coefficients that are near one and some simplification, the following equation is obtained.

{f}_{c}=\frac{49.2}{{10}^{\left(\frac{w}{\mathit{b}}\right)}}\times \left(0.546\times log\left(\mathit{t}\right)+1\right)\left(-17.22{\mathit{s}}^{2}+2.74\mathit{s}+1\right)(-10{\mathit{p}}^{2}+0.54\mathit{p}+1)

For investigation of the accuracy of above equation, diagram of residuals (the difference of observed and fitted values) for compressive strength has been drawn (Fig. 10). The diagram of residuals for compressive strength shows that the maximum error percent of prediction of compressive strength is about 10 %. Although, the distribution residuals does not completely obey the normal distribution but the mean and maximum values of histogram of residuals is located near zero (Fig. 11).