#### Abstract

The paper deals with composition design of self-compacting concrete containing blast furnace granulated slag and polycarboxylate type superplasticizer in a wide range of concrete strength classes. Optimal superplasticizer content, cement, and blast furnace slag consumptions are obtained from the viewpoint of minimum water demand and water-cement ratio, maximum self-compacting concrete mixture workability retention, and concrete strength. A significant effect of joint influence of superplasticizer and blast furnace slag is demonstrated. The possibility of minimizing the cement consumption at optimal superplasticizer content and slag consumption of 100–120 kg/m^{3} is shown. Dependences for calculating SCC strength at 1, 7, and 28 days vs. the water-cement ratio are obtained. It is found that the nature of correlation dependences for SCC splitting and compression strengths when using the blast furnace slag and the superplasticizer is linear. Optimization of SCC compositions using mathematical programming enables to obtain compositions with a lowest possible cost while providing a set of specified parameters.

#### 1. Introduction

Development and implementation of effective superplasticizing additives in technological practice allowed production of self-compacting concrete (SCC) [1]. This high-technological concrete allows energy saving and achieving high physical and mechanical characteristics of structures, especially thin-walled, densely reinforced, and complex ones. Additionally, production technology of SCC has higher environmental parameters [1–3]. However, the need to ensure high SCC mix workability, its durability over time, higher rate of strength growth, as well as high values of the entire set of concrete performance properties makes it difficult to justify its composition, which along with technological criteria should also ensure economic feasibility [3, 4].

Following the recommendations of the European Federation of Specialists Construction Chemicals and Concrete Systems [5], it is more appropriate to use concrete composition design to the original components ratio by volume and not by weight. In the first stage, the ratios between the components are obtained based on typical diapason ranges providing the standardized indicators of the SCC mix.(i)The volume ratio between water and dispersed materials (cement, mineral admixture, and sand fractions smaller than 0.125 mm) from 0.80 to 1.10;(ii)Total content of dispersed materials from 160 to 240 liters (400–600 kg per cubic meter);(iii)Cement content of 350–450 kg/m3 (cement consumption over 500 kg/m3 can increase shrinkage and creep of concrete; consumption less than 350 kg/m3 may be allowed only when using other fine mineral fillers or pozzolanic additives);(iv)Coarse aggregate content from 28 to 35% by the concrete mix volume;(v)The water-cement ratio is assigned based on the requirements of EN 206-1 [6] (usually the water demand does not exceed 200 *l*/m^{3}).

A known method for the SCC design [7, 8] is based on the idea that at the first stage the cement paste and mortar are tested to determine the compatibility of superplasticizer, cement, fine aggregate, and pozzolanic additive, and at the second stage a test SCC mix is tested. An advantage of this method is that it avoids repetition of similar time-consuming tests for the entire concrete mix. However, the disadvantages of the method is, first of all, that not all ready-mixed concrete plants are equipped with necessary equipment to study cement pastes and mortars rheology, in particular by rotary viscometers.

On the other hand, a rather simple method for estimating the rheological characteristics of cement paste was proposed [9]. It is based on dependence of cement paste shear stress on the type and dosage of additives for cement of a certain chemical and mineralogical composition. For example, to obtain self-compacting mixtures of class SF 1, the value of the ultimate shear stress should be, approximately, from 8.9 to 14 Pa, and for mixtures of class SF 2 from 8.7 to 10 Pa.

A simplified method of the SCC composition design is proposed based on the condition of achieving the maximum coarse and fine aggregate packing factor PF [10]. SCC strength is provided by the frame of aggregates glued in the hardened state by cement paste, while the technological properties of mixtures are provided by cement paste in the fresh state. Obviously, the content of fillers in SCC affects the packing factor. A higher PF value results in a higher content of coarse and fine aggregates, reducing the amount of the binder. Accordingly, a high workability of the SCC mixture, its ability to self-compacting at high water content, as well as the compressive strength of concrete will decrease. On the other hand, a low PF value causes increased shrinkage of concrete. Higher content of viscous paste also affects SCC durability and significantly increases its cost. Following [11], the cost of 1 m^{3} of traditional concrete is about $50, while for cast mix and SCC mix, it is $69 and $136, respectively. Table 1 presents data on SCC compositions in different countries [12].

Analyzing the available literature recommendations on selecting of SCC composition, it should be noted that they have mostly general nature and do not offer an algorithm that considers problem formulation features and type of specific source materials [13–15]. Obviously, calculated dependences that should be used in the concrete mix design and should satisfactorily predict concrete properties when changing the main technological factors in a given range [16, 17]. As such, dependencies can be used experimental-statistical models. Such models should be obtained on the basis of well-developed methods for performing so-called active experiments using corresponding methods for their planning, subsequent statistical processing, and obtaining adequate regression equations [18].

#### 2. Research Aim, Scope, and Novelty

From the literature review, SCC is a specific type of concrete, and there are no methods for its composition design. In the frame of this study, a set of experimental-statistical models of concrete mix and hardened concrete properties, depending on the main technological factors, was obtained for normal weight concrete based on Portland cement with addition of the ground slag and the polycarboxylate superplasticizer. The main aim of the present study is to use the obtained models to analyze the individual and combined influence of technological factors and predict the properties of SCC with the granulated blast furnace slag and superplasticizer, as well as optimize SCC compositions. Realization of this aim determines the novelty of this study.

#### 3. Materials and Research Methods

The following materials were used in the frame of the present study: CEM I 42.5 (Dyckerhoff, Ukraine), sand with fineness modulus of *M*_{f} = 2.0, crushed stone fraction 5 … 20 mm, locally available milled blast furnace granulated slag (a specific surface area of 270 m^{2}/kg), polycarboxylate type superplasticizer РСЕ50 (LLC UA-Chemical, Ukraine). Characteristics of raw materials are given in Table 2 and 3.

The research was carried out using the method of experiment mathematical planning [19, 20]. An experimental plan was implemented for three investigated factors (type B_{3} [18] (Table 4),). It allows setting the values of the investigated factors at three levels (lower (-1), average (0), and upper (+1)) and obtain second-order polynomial equations [18].

The following variable factors were selected as follows: cement consumption (C, kg/m^{3} (*X*_{1})), superplasticizer content (SP, % (*X*_{2})), and consumption of the blast furnace granulated slag (GS, kg/m^{3} (*X*_{3})).

The concrete mixture fluidity at experimental points was kept at the level of S5 (Slump (S)) of 260–270 mm, which corresponds to the cone spread of 560–620 mm (F5 according to EN 206–1: 2000 [6]). The workability retention (change in cone slump over time and loss of fluidity after 2 h (ΔS2, %)) were determined for fresh concrete mixtures. Cubic specimens of 10 × 10 × 10 cm were prepared from the concrete mixture. Compressive strength (*f*_{cm}, MPa) and splitting tensile strength (*f*_{ctm}, MPa) were obtained (at 1, 7, and 28 days). The experiment planning matrix and the experimental results are given in Table 4 and 5.

#### 4. Analysis of Experimental Results

Experiments were carried out according to the planning conditions (see Table 5). The concrete composition was changed following the condition that the crushed stone content in SCC should be 0.3–0.35 of the concrete mix volume [5].

Statistical analysis of the experimental results enabled to obtain mathematical models of the output parameters as well as correlation dependences between individual output parameters. The obtained mathematical models have the following general form:where, Y is the output parameter, *X*_{1}…*X*_{3} are variables, *b*_{0}…*b*_{23} are equation coefficients. The coefficients of mathematical models that were obtained are given in Table 6.

The influence of concrete mixture composition factors on the properties of concrete is represented by graphical dependencies (Figures 1–8), calculated using appropriate models (equation (1) (Table 6),).

The parameter of SCC mixtures composition, which determines the required workability, is water demand (W, *l*/m^{3}). Due to the high water-reducing effect of the polycarboxylate superplasticizer, it is possible to reduce the water demand of the SCC mixture by 100–140 *l*/m^{3} that correspondingly yields a lower water-cement ratio or cement consumption (Figure 1).

Significant values of interaction coefficients *b*_{12} and *b*_{23} in the models indicate a significant increase in the water-reducing effect of the additive with increasing cement and blast furnace slag consumptions, i.e., the concrete mixture binder component (Figures 1, 2). Mostly a more noticeable increase in water demand is observed when the cement consumption is more than 400 kg/m^{3}, i.e., outside the constant water demand rule scope [21, 22]. Accordingly, water demand practically does not change at cement consumption from 200 to 400 kg/m^{3} (see Figure 1). Quite high water-reducing effect of the additive significantly eliminates the negative impact of higher dispersed particles content in the concrete mixture. The experimental-statistical model for W/C (see Table 6) shows that this factor varied during the experiment in a fairly wide range from 0.23 to 0.91, which was caused by a significant range of cement consumption (from 200 to 600 kg/m^{3}) (Figure 2).

Quite a significant interaction between the factors of the superplasticizer and slag contents shows that while simultaneous increase in the SP and blast furnace slag contents yields noticeable decrease in concrete W/C, i.e., a positive effect of organic-mineral complex on SCC strength is expected (Figure 2).

SCC mix workability retention was studied by the changes in SCC workability in time (Figure 3). To qualitatively assess workability retention, the indicator of SCC fresh mix workability change after 2 hours (ΔS2, %) was used. All the investigated factors in transition from the lower level to the upper level contribute to reduction of workability loss at 2 hours, i.e., increase of SCC mix workability retention.

The highest impact on preservation of workability loss over time is caused by SP content (*X*_{2}) in transition from the lower level to the upper level (0–1%) is observed and a change in workability retention from 20 to 40%. The obtained results demonstrated that increasing of blast furnace slag content reduces the superplasticizer effectiveness as SCC mixture workability stabilizer (Figure 3). In turn, in SCC mixtures that do not contain SP, increasing slag content prevents a significant workability loss.Figure 4.

To maintain workability, Portland cement and the blast furnace slag should be considered as a common binder. Increase in binder content has a positive effect on SCC workability [23]. This becomes especially noticeable when the amount of cement in concrete is low (Figure 7).

As expected, factors *X*_{1} (cement consumption) and *X*_{2} (SP content), cause a significant decrease in W/C and yield a significant increase in strength. Following the mathematical models (see Table 6) there is a significant interaction coefficient between these factors, due to which there is a valuable increase in strength while increasing the content of these components. Thus, the increase in cement consumption in SCC causes an increase in one-day strength f_{cm1} from 1.5 to 2 times (without using SP) and from 5 to 7 times (with a maximum content of SP).

The influence of the *X*_{1} factor on SCC strength is almost linear, *X*_{2} has an extreme effect. The existence of an optimal content of the superplasticizer is felt, causing the maximum increase in strength. Optimum SP content is significantly related to cement consumption, at minimum content of C to maximize the SCC strength 0.5% of SP is sufficient, at the maximum cement consumption, optimal SP content is from 0.8 to 1.1% (Figure 5). There is also a noticeable interaction between the content of SP and blast furnace slag consumption. In this case, with increasing slag content, optimal SP content and its effectiveness is somewhat reduced (Figure 5).

The extreme effect of blast furnace slag content at cement consumption of 600 kg/m^{3} is more noticeable for SCC strength at 28 days [24]. Optimal slag content that provides maximum strength in this case is 40–50 kg/m^{3} and at cement consumption of 200 kg/m^{3} it becomes 120–150 kg/m^{3} (Figure 6).

In order to study the effectiveness of the investigated factors on SCC strength at 28 days, the specific consumption of cement per unit strength was calculated.

The mathematical model for C_{f} (see Table 6 and Figure 7) shows that the minimum cement consumption per unit strength (4–4.8 kg/MPa), which indicates the effective concrete composition is achieved at maximum polycarboxylate type SP content (0.75–1%) at ground blast furnace slag consumption corresponding to the average level (100–120 kg/m^{3}).

SCC splitting tensile strength was determined at 1 and 28 days. The values of f_{ctm1} ranged from 0.4 to 5.44 MPa and those of f_{ctm28} from 1.35 to 8.9 MPa. The mathematical models show that the influence of variable factors on splitting tensile strength is almost the same as on compressive one (Figure 8). A more noticeable positive effect of blast furnace slag content on f_{ctm} at 28 days at low cement consumption is notable. It is known [25] that tensile strength is mostly determined by the amount of cement stone in concrete and its strength. With long-term hardening, low cement consumption, and a large amount of the slag, this effect is quite expected [26].

For specific source materials correlation dependences of concrete strength on the water-cement ratio (W/C), obtained by approximating experimental results (Tables 4 and 5), can be used for determining water demand of the SCC mixture at the compositions design stage. For the investigated materials, such dependences are given in Table 7. For splitting tensile strength correlation dependences that relate it to compressive strength have been also obtained (Table 7).

It should be mentioned that calculations according to the equations given in Table 8 yield the value of C/W (for cement compressive strength at 28 days *R*_{C} = 50 MPa) by known Bolomey’s formula [27–29].

#### 5. Optimization of SCC Compositions

The set of obtained experimental-statistical SCC models (Table 6) enabled to perform mathematical optimization of SCC compositions to provide the desired parameters (*P*_{1}*…P*_{m}) at minimum cost of concrete mix.By providing the necessary quality indicators, we havewhere *C*_{C}, C_{GS}, and C_{SP} are cost of cement, granulated slag, and superplasticizer, respectively; C, GS, and SP are cement, slag, and superplasticizer consumptions, respectively, kg/m^{3}; *P*_{1}*…P*_{m} are the required concrete quality parameters; *x*_{1}*… x*_{3} are composition factors; *a* and *b* are the borders of the factors' values.

The optimization problem can be solved using the MS Excel facilities. The following examples demonstrate recommended methods for solving such optimization problems. Task 1 SCC optimal compositions are designed with *S* = 250–270 cm and strength classes from C12 / 15 to C60/75. The following raw materials should be used for concrete production: Portland cement CEM I 42.5, sand with *M*_{f} *=* 2.0, crushed stone fraction 5–20 mm, ground blast furnace slag, and polycarboxylate type superplasticizer. The part of crushed stone in the aggregate mixture is 0.35 of the concrete mixture volume (crushed stone density *ρ*_{CS} *=* 2.8 kg/m^{3}). The following values were used for SCC cost calculations: cement – 88 EUR per 1 ton, blast furnace granulated slag – 27 EUR per 1 ton; polycarboxylate type superplasticizer – 2.85 EUR per kg. The optimization results are shown in Table 8. Task 2SCC compositions are designed with workability *S* = 250–270 mm and strength classes from C12 / 15 to C60/75. The concrete mix should have workability loss at 2 hours, less than 5%. Raw materials for SCC production meet Task 1. The optimization results are shown in Table 9. Task 3Compositions of high-strength SCC are designed with workability *S* = 250–270 cm and strength class at 1 day С20/25. Concrete mix should have workability loss at 2 hours, less than 5%. Raw materials for SCC production meet Task 1. The optimization results are shown in Table 10.

Analyzing the optimization results, it should be noted that the investigated organic-mineral complex, including the ground granular blast furnace slag and the polycarboxylate type superplasticizer, is an effective and allows achieving resource-saving SCC with required strength in a wide concrete classes range. Thus, SCC strength in the range of classes from C12/15 to C30/35 (Table 8) is achieved just by increasing the contents of the blast furnace slag and the superplasticizer at the lowest possible cement consumption. Further increase of SCC strength to class C60/75 requires increase of cement consumption to 457 kg/m^{3} and reduction of slag content in the binder (cement + slag) from 42 to 16%, with a gradual increase in superplasticizer consumption. As expected, the main factor for increasing SCC strength over the entire range of classes is water-binder ratio reduction.

Following the obtained results of SCC composition optimization according to Task 2 (Table 9), to ensure stable SCC mix workability, the concrete should have a high amount of organic-mineral complex, especially for the low strength classes. Thus, to provide the required SCC mix workability at two hours for classes C 12/15–C 25/30 for the investigated technology, the contents of the blast furnace slag and the superplasticizer should be increased up to 53% in the binder and up to 0.56%, respectively.

For concrete classes C30/35 and C32/40, workability is provided by increasing consumptions of the slag and the superplasticizer. SCC compositions providing strength values corresponding to classes C 35/45 and higher simultaneously ensure the required fresh mix workability retention. It is obvious that the main criterion of SCC mix workability retention is the high amount of the binder (at least 340 kg/m^{3}) at SP consumption of at least 0.5-0.6%.

According to the obtained results (Table 10), ensuring specified SCC strength at 1 day of hardening is possible by increasing the clinker component of the binder and, accordingly, reducing the consumption of the blast furnace slag.

#### 6. Conclusion

Experimental-statistical models of the water-cement ratio, water demand, and self-compacting concrete mixture workability retention, as well as SCC strength at 1, 7, and 28 days, vs. consumptions of cement, the blast furnace granulated slag, and superplasticizer content were obtained. The models enabled to quantify the influence of the investigated factors and their interaction effects.

The obtained quantitative dependencies allowed to find optimal superplasticizer content, consumptions of cement and the blast furnace slag from the viewpoint of minimum water demand and the W/C ratio, maximum self-compacting concrete mixture workability retention, and concrete strength, as well as to establish a significant combined effect of the superplasticizer and the blast furnace slag in SCC. At the same time, possibility of minimizing cement consumption at optimal content of the superplasticizer and slag consumption of 100–120 kg/m^{3} was demonstrated.

Correlation dependences of SCC strength at 1, 7, and 28 days vs. the water-cement ratio were obtained. Linear nature of correlation dependences for splitting and compression strengths of SCC with the blast furnace slag and the superplasticizer was demonstrated.

Using mathematical programming, optimization of SCC compositions with a set of specified properties at minimum cost was performed. The proposed methodology allows obtaining rational values of the main technological parameters depending on the specified properties of concrete mix and SCC.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.