Introduction

Nowadays, research on cement matrix materials is focused on the inclusion of additives, admixtures and short fibers, to improve certain physical and mechanical properties. Adherence, permeability, ductility and strength are some of the main research lines on binder materials [1].

The flexural strength or modulus of rupture of concrete is determined with simple rectangular beams without any reinforcement submitted to increasing bending until failure occurs. The modulus of rupture is determined from the moment as failures. This knowledge is useful in the design of pavement slabs and airfield runways as flexural tension is critical in these cases.

A most frequently used polymer is Styrene-Butadiene Rubber (SBR). The previous researches showed that SBR modified mortars and concretes had good mechanical properties, with respect to polymer-free materials. Their workability, adhesion to substrate, durability and abrasion resistance can be significantly improved [2–6]. Polymer can form another network in the material, which fills up pores in cement matrix and improves the bonding between aggregates and cement paste [7].

Also, the use of silica fume increases the resistance of concrete against acid and sulfate attack and improves durability of concrete by reducing porosity and permeability of cement paste matrix, it makes the concrete more resistant against abrasive forces and reduces the expansion generated by alkali-aggregate properties [8].

A recent report said the highest values of the thickness reduction of the Interfacial Transition Zone (ITZ) were obtained for the concretes which both silica fume and SBR latex, besides the thickness reduction of the ITZ [9], it is expected some properties of the concrete, such as mechanical strength, elastic modules and permeability are improved.

In this study, the flexural strength is tried to improve with addition of supplementary materials (SBR and silica fume) to concrete. In this research, the combination of wet and dry curing is applied to the polymer-modified silica fume concretes. The benefits of this curing system, completion of hydration of cement matrix in the early curing period and solidification of SBR latex in the dry curing during the following curing period [10]. Therefore, the adhesion and strength of both cement matrix and SBR film are developed effective.

A lot of researches investigated the isolated effects of silica fume or SBR on flexural strength [1–4, 7, 10–14] but it seems that few reports on the relationship between flexural strength and ratios of SBR or silica fume to binder materials are found until now. The current empirical equations presented are based on tests of concrete without supplementary cementations materials. The validity of these relationships for concrete with supplementary materials (silica fume, SBR, etc.) should be investigated.

Most of the empirical models used to study the behavior of concrete mixes consist of mathematical rules and expressions that capture relationship between components of concrete mixes. In the present work, the flexural strength is modeled as a function of four variables: the ratios of polymer, silica fume, water to binder materials and time of curing in water. The proposed model is determined based on regression methods. Linear or nonlinear regression methods are conventional methods of predicting the strength of concrete.

Materials

The materials used in this study were ordinary Portland cement, river sand with a fines modulus of 3.01 and coarse aggregate with a specific gravity of 2.25. A locally produced ordinary silica fume was used in this investigation. The binder content weighted keep constant 400 kg/m3.

The polymer dispersion used was styrene-butadiene rubber (SBR) latex from BASF. The SBR latex is in a liquid state of low viscosity, having total solids content of 40.0, a pH value of 10.5, a density of 1.01.

Preparation of Specimens

In this study, prisms 150 × 150 × 600 mm over span of 450 mm were cast for flexural strength test. Before casting, coarse aggregate, sand, silica fume and water blend was mixed. Then cement, SBR latex, water put in the mixer. Finally, supper-plasticizer was added. The mixed concrete cast in molds and full compacted. The concrete mixture parameters were two different water-binder ratios (0.35, 0.45), four different SBR dosages (0, 5, 10, and 15 %) and four partial replacement ratios of silica fume (0, 5, 7.5, and 10 %). The specimens were subjected to the following three different curing systems:

  1. 1.

    7-day water curing at 20 °C and 53-day dry curing at 20 °C (7W53D).

  2. 2.

    14-day water curing at 20 °C and 46-day dry curing at 20 °C (14W46D).

  3. 3.

    28-day water curing at 20 °C and 32-day dry curing at 20 °C (28W32D).

The cement hydration and polymer film in the modified mortars develop with prolongation cured age, which results in enhanced strength. The coherent polymer film reduces weakness in the mortars, prevents extending of tiny cracks [1].

The specimens were tested for flexural strength in the manner described in BS standard 1881-118.

Test Results and Discussion

Figures (1, 2, 3, 4) show flexural strength of samples.

Fig. 1
figure 1

Effect of polymer-cement ratio on flexural strength of concrete (w/c = 0.35 and 0 % silica fume)

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Fig. 2
figure 2

Effect of silica fume-binder ratio on flexural strength of concrete (w/b = 0.35 and 0 % SBR)

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Fig. 3
figure 3

Effect of polymer–binder materials ratio on flexural strength of polymer modified concrete (w/b = 0.35 and 7.5 % silica fume)

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Fig. 4
figure 4

Effect of polymer–binder materials ratio on flexural strength of polymer modified concrete (w/b = 0.45 and 7.5 % silica fume)

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Effects of SBR on Flexural Strength of Concrete

In polymer-cement ratio [P/C] (5 %), the flexural strength of modified samples increases remarkably and maximizes, but with increasing polymer ratio, flexural strength decreases (Fig. 1). It seems at p/c ratio 5 %; the continuity is only present through small tiny bridges on a limited number of spots. The bridging and tensile performance of polymer induces such increasing. At p/c ratio 15 %, positive effect of polymer in increasing flexural strength is less than decreasing effect of polymer due to increasing porosity of concrete and therefore flexural strength decreases.

Effects of Silica Fume on Flexural Strength

A significant improvement in the flexural strength of concrete is observed due to high pozzolanic activity and its void filling ability of silica fume [14, 15]. The flexural strength of silica fume concrete continuously increases and maximizes at 7.5 % replacement of silica fume (Fig. 2).

Interfacial Effects of Silica Fume and SBR on Flexural Strength of Concrete

The strength of concrete is affected by the ratio of SBR and silica fume to binder materials. When the ratio of polymer/binder is certain, Silica fume increases flexural strength of concrete. Percentage of silica fume that optimizes flexural strength remains 7.5 % constant (Fig. 3).

Interfacial Effects of Silica Fume and SBR on Flexural Strength of Concrete

The flexural strength of concrete is affected by the ratio of SBR and silica fume to binder materials (Fig. 3). When the ratio of polymer/binder is certain, the amount of silica fume affects strength of concrete. The percentage of silica fume that optimizes flexural strength remains 7.5 %.

In a few samples, a local decrease in flexural strength is observed in 5 % polymer (Fig. 4). In making of these samples percentage of superplastysizer in 0 and 5 % polymer were constantly considered. The cavitation of superplastysizer induces a decrease in flexural strength of these samples. (In making of these samples, the ratio of superplastysizer was kept constant.)

Investigation of Main Effects and Interaction Effects of Factors in Flexural Strength

The main effects of each factor are shown in Fig. 5. It is observed that decreasing of water to binder material from 0.45 to 0.35 increases flexural strength about 13 %. Averagely, 7.5 % of replacement silica fume increases flexural strength about 15 % and also adding 15 % of SBR decreases flexural strength about 9.4 %.

Fig. 5
figure 5

Main effects of diagram for flexural strength

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A single interaction plot of flexural strength is shown in Fig. 6. Parallel lines in an interactions plot indicate there is no interaction effect between factors. The greater the departure of lines from the parallel state indicates the higher the degree of interaction, in this situation, the response at a factor level depends upon the levels of other factors. Therefore, there is no interaction effect between SBR and silica fume. Although, a decrease in flexural strength of samples with 5 % SBR and 7.5–10 % silica fume in water to binder ratio 0.45 are happening. (In making of these samples, the ratio of superplastysizer was kept constant.)

Fig. 6
figure 6

Interaction effects of diagram for flexural strength

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Mathematical Model

Mathematical models based on experimental data are called “free models”, and are generally in regression forms. The regression forms (linear, non-linear, exponential, etc.) should be determined based on experimental data.

The primary factors that affected the flexural strength of concrete are the ratios of water, SBR, silica fume to binder materials and time of curing in water. (In modeling, from effects of superplastysizer on flexural strength is neglected.) Conventional methods of predicting flexural strength of concrete is generally based on either Abrams water-cement ratio or maturity concept of concrete. Since strength development is determined not only by the water to cement ratio but also the influence of other ingredients should be investigated. Therefore, scatter diagrams are prepared and means of values are portrayed in diagrams. A lot of researchers introduce relationship between flexural strength and time of curing in water with a logarithmic equation [15, 16].

The scatter diagrams for all of the specimens exhibited a nonlinear relationship between the flexural strength with the ratios of silica fume and SBR to binder materials. From the scatter diagrams, it was observed that a two degree parabolic might be appropriate for showing the effect of silica fume or SBR in flexural strength of concrete. The relationship of flexural strength with each variable is shown in Figs. 7, 8, 9. The relationship between flexural strength with each factor is determined based on regression.

Fig. 7
figure 7

Relationship between flexural strength and time of curing in water

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Fig. 8
figure 8

Relationship between flexural strength with silica fume to binder materials ratio

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Fig. 9
figure 9

Relationship between flexural strength with polymer to binder materials ratio

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

$${\text{f}}_{\text{l}} = {{{\text{A}} * \left( {0.958 * \log \left( {\text{t}} \right) + 3.896} \right)^{\text{C}} * \left( { - 67.8{\kern 1pt} {\text{s}}^{2} + 12.38{\kern 1pt} {\text{s}} + 4.605} \right)^{\text{D}} * \left( { - 57{\kern 1pt} {\text{p}}^{2} + 6.57{\kern 1pt} {\text{p}} + 5} \right)^{\text{E}} } \mathord{\left/ {\vphantom {{{\text{A}} * \left( {0.958 * \log \left( {\text{t}} \right) + 3.896} \right)^{\text{C}} * \left( { - 67.8{\kern 1pt} {\text{s}}^{2} + 12.38{\kern 1pt} {\text{s}} + 4.605} \right)^{\text{D}} * \left( { - 57{\kern 1pt} {\text{p}}^{2} + 6.57{\kern 1pt} {\text{p}} + 5} \right)^{\text{E}} } {{\text{B}}^{{\left( {{\text{w}}/{\text{b}}} \right)}} }}} \right. \kern-0pt} {{\text{B}}^{{\left( {{\text{w}}/{\text{b}}} \right)}} }}$$
(1)

where fl is the flexural strength (Mpa), w/b is the ratio of water to binder materials, t is the time of curing in water (day), s is the ratio of silica fume to binder materials, 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:

$$\begin{aligned} \log \left( {{\text{f}}_{\text{l}} } \right) = & \log \left( {\text{A}} \right) + \log \left( {\text{B}} \right) * \left( {{\text{w}}/{\text{b}}} \right) + {\text{C}} * \log \left( {0.958\log \left( {\text{t}} \right) + 3.896} \right) \\ & + \,{\text{D}} * \log \left( { - 67.8{\kern 1pt} {\text{s}}^{2} + 12.38{\kern 1pt} {\text{s}} + 4.605} \right) + {\text{E}} * \log \left( { - 57{\kern 1pt} {\text{p}}^{2} + 6.57{\kern 1pt} {\text{p}} + 5} \right) \\ \end{aligned}$$
(2)

That A, B, C, D and E can be determined with multiple linear regressions. The values of these coefficients are shown in the following table. The value of the multiple correlation coefficients (r) has been obtained as 0.9 (Table 1).

Table 1 Numeric values of the rate constants from Eqs. (1–2)
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With replacement of one to coefficients that are near one and some simplication, the following equation is obtained.

$${\text{f}}_{\text{l}} = {{5.93 * \left( {0.246\log \left( {\text{t}} \right) + 1} \right) * \left( { - 11.4{\kern 1pt} {\text{p}}^{2} + 1.31{\kern 1pt} {\text{p}} + 1} \right)\;\left( { - 14.72{\kern 1pt} {\text{s}}^{2} + 2.69{\kern 1pt} {\text{s}} + 1} \right)} \mathord{\left/ {\vphantom {{5.93 * \left( {0.246\log \left( {\text{t}} \right) + 1} \right) * \left( { - 11.4{\kern 1pt} {\text{p}}^{2} + 1.31{\kern 1pt} {\text{p}} + 1} \right)\;\left( { - 14.72{\kern 1pt} {\text{s}}^{2} + 2.69{\kern 1pt} {\text{s}} + 1} \right)} {\left[ {3.52^{{\left( {{\text{w}}/{\text{b}}} \right)}} } \right]}}} \right. \kern-0pt} {\left[ {3.52^{{\left( {{\text{w}}/{\text{b}}} \right)}} } \right]}}$$
(3)

From above equation can be determined the flexural strength of samples containing of silica fume and SBR. For investigation of the accuracy of Eq. (3), a diagram of residuals (the difference of observed and fitted values) for flexural strength has been drawn (Fig. 10). The residual diagram shows that the maximum percent error of flexural strength is <10 %.

Fig. 10
figure 10

Diagram of residuals for flexural strength

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The histogram of residuals has been shown in Fig. 11. Although, the distribution residuals doesn’t completely obey the normal distribution but mean and maximum values of histogram of residuals is located near zero. Also, residuals are disturbed randomly.

Fig. 11
figure 11

Histogram of residuals for flexural strength

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In Fig. 12 normal property plot versus residuals is shown. If residuals belong to a normal distribution, points should be collected around of a straight line. Here, normal property plot obey the normal probability distribution approximately, so hypothesis of normal approach is acceptable.

Fig. 12
figure 12

Normal property plot versus residual

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Conclusion

  • In constant water to binder ratio and combined curing system, the flexural strength of concrete in 5 % SBR rises remarkably, but with increasing polymer/binder material ratio, the flexural strength of concrete decreases.

  • Cement replacement up to 7.5 % with silica fume leads to maximize in flexural strength.

  • The decrease of flexural strength is compensated by the reduction of w/b due to the plasticizer effect of SBR. Both phenomena together remain flexural strength approximately constant.

  • The percentage of silica fume that optimizes the flexural strength with adding SBR doesn’t change.

  • In this study, we have shown the practical acceptability of Abram’s law within considered limits of SBR and silica fume and according to main effects of diagrams in each factor is proposed in the following equation:

    $${\text{f}}_{\text{l}} = {{5.93 * \left( {0.246 * \log \left( {\text{t}} \right) + 1} \right) * \left( { - 11.4{\kern 1pt} {\text{p}}^{2} + 1.31{\kern 1pt} {\text{p}} + 1} \right) * \left( { - 14.72{\kern 1pt} {\text{s}}^{2} + 2.69{\kern 1pt} {\text{s}} + 1} \right)} \mathord{\left/ {\vphantom {{5.93 * \left( {0.246 * \log \left( {\text{t}} \right) + 1} \right) * \left( { - 11.4{\kern 1pt} {\text{p}}^{2} + 1.31{\kern 1pt} {\text{p}} + 1} \right) * \left( { - 14.72{\kern 1pt} {\text{s}}^{2} + 2.69{\kern 1pt} {\text{s}} + 1} \right)} {\left[ {3.52^{{\left( {{\text{w}}/{\text{b}}} \right)}} } \right]}}} \right. \kern-0pt} {\left[ {3.52^{{\left( {{\text{w}}/{\text{b}}} \right)}} } \right]}}$$

The proposed model provides the opportunity to predict the flexural strength based on the time of curing in water (t), and water, silica fume and SBR to binder materials ratios that we show with w/b, (s) and (p), briefly.