1. Introduction

2. Materials and Methods

2.2. Methods

2.2.1. Crushability of Volcanic Tuff Aggregate

The crushability of crushed stone is quantified through the measurement of grain destruction during a compression test, in line with the state standard GOST 8269.0-97 [18]. The weight of the crushed stone (gravel) is poured into the cylinder from a height of 50 mm, such that after levelling the upper level of the material by approximately 15 mm, it does not reach the upper edge of the cylinder. Subsequently, the plunger is inserted into the cylinder such that the plunger plate is aligned with the top edge of the cylinder. Should the upper surface of the plunger plate fail to align with the cylinder's edge, it is necessary to either remove or add a small quantity of crushed stone (gravel) until the desired alignment is achieved. Subsequently, the cylinder is positioned on the lower platen of the press. It is recommended that the pressing force of the press be increased by 1-2 kN per second. When testing crushed stone (gravel) in a cylinder with a diameter of 75 mm, the force should be brought up to 50 kN. When testing in a cylinder with a diameter of 150 mm, the force should be increased to 200 kN. The data pertaining to the crushing process is obtained on the hydraulic press IP-1A-500, utilizing a copper cylinder with a plunger.

Calculation of tuff concrete composition was carried out according to the method given by the Departmental Construction Standards (1992) [19].

2.2.2. Preparation of Tuff Concrete Specimens

From the compositions obtained by mathematical planning in laboratory conditions, cube samples with dimensions of 100×100×100 mm were made. The samples were manufactured in accordance with GOST 10180-2012 [20].

2.2.3. Determination of the Average Density of Tuff Concrete Samples.

In accordance with the Government Decree, standard GOST 12730.1-78 [21], the average density of lightweight concrete was determined. In order to calculate the density index of concrete cubes of regular shape, their volume is measured using measuring tools, such as a ruler and a caliper. In this instance, the permissible error is limited to 1 mm. In the case of irregularly shaped samples, the hydrostatic method of determining mass or the use of a volumeter is recommended. Subsequently, the volume is determined by weighing.

2.2.4. Determination of Compressive Strength of Tuff Concrete Samples

The compressive strength of the tufo concrete samples was determined in accordance with the European Standard EN 12390-3:2009 [22]. During the compression test, the cube specimens are placed with one of the selected faces on the lower base plate of the testing machine (press), centred relative to its longitudinal axis, using the marks on the plate of the testing machine (press) or a special centering device.

Once the test specimen has been positioned on the testing machine platens or supplementary steel plates, it is necessary to align the upper platens of the testing machine with the upper support face of the test specimen, ensuring that both surfaces are in complete contact with each other. The specimen is subjected to a constant rate of load increase (0.6 ± 0.2 MPa/s) until failure. Figure 2 presents photographic images of the tuff concrete specimen following fracture.

2.2.5. Determination of Thermal Conductivity of Tuff Concrete

The thermal conductivity of tufobeton was determined by a thermal conductivity meter, the ITP-MG4 (250), in accordance with GOST 7076-99 [23].

The tests were conducted by measuring the temperature of both sides of the samples placed between the heater at 65 °C and laboratory conditions, in comparison to a reference material with known thermal conductivity. To ensure the accuracy of the measurements, the heat flux is determined under steady-state conditions, taking into account both heat conduction within the specimen and heat transfer through the air wall. This typically indicates that measurements were conducted approximately three to four hours after the commencement of the experiment. In this instance, six specimens with dimensions of 250×250×30 mm were prepared for the purpose of determining the thermal conductivity.

2.2.6. Determination of the Fractal Model of Concrete Strength Considering the Scale Effect

The potential for an explanation of the scale effect based on the fractal geometry of B. Mandelbrot is considered. It is demonstrated that the underlying physical essence of the scale effect can be found in the scale invariance of the structure at each scale level of the concrete structure. Consequently, the properties are found to be similar. The strength (in kilonewtons per square centimetre, megapascals) was determined on specimens in the form of cubes with dimensions of 20, 50 and 100 millimetres. Furthermore, the discrete, multilevel character of failure of concrete specimens with dimensions of 20 mm, 50 mm and 100 mm is corroborated by the deformation diagram obtained on a Wille Geotechnik testing machine (model 13-PD/104). The tests were conducted at a loading rate of 0.5 mm/min, with the measurement readings recorded at a rate of 10 readings per second.

2.2.7. Rotatable Method of Experiment Planning

In order to achieve this goal, it is necessary to determine the composition of the concrete mixture. This involves establishing the ratio between the components that form the material's structure and provide a given level of quality indicators.

The solution of such problems is carried out with the help of mathematical methods of the theory of planning the experiment. The use of these methods will allow us to obtain a mathematical model describing the dependence of quality indicators on the level of varying factors.

For the purposes of experimental study, it is recommended that a rotating central composite plan be employed in the matrix F, with n=3.

The effect of the amount of sand and demolished stone, as well as dispersed reinforcement (fiber), on the strength, average density, and thermal conductivity of tuff concrete will be investigated.

The fundamental principle underlying the three-factor planning of experiments and the optimisation of the composition of lightweight structural concrete utilising mathematical statistics is the establishment of a mathematical relationship between the given material properties and the flow rate, the properties of constituent components and the technological factors.

Table 1. Levels of variation of influencing factors.

Table 1. Levels of variation of influencing factors.

No.	Factor	Components	Levels of variation
			-1.682	-1.0	0	+1.0	1.682
1	x1	Tuff sand (TS)	1.318	2	3	4	4.682
2	x2	Tuff coarse aggregate (TCA)	0.682	2	4	6	7.364
3	x3	Basalt fiber (BF)	0.00318	0.01	0.02	0.03	0.03682

The efficacy of an experiment is contingent upon the meaningful formulation of the problem, the appropriate selection of research methodology, the identification of the primary factors and their variations, as well as the comprehensive interpretation of the results in a physical context. The selection of factors and parameters for the optimisation of lightweight structural concrete was based on the criteria of technological and economic feasibility.

A three-factor quadratic dependence experiment was planned, the planning conditions of which are presented in Table 2. For all compositions, the water-cement ratio was assumed to be W/C = 0.4. In the initial stages of the study, a comprehensive factor plan was developed. Subsequently, experiments 1 to 8 were conducted. Subsequently, six parallel experiments were conducted at the centre of the plan, with experiments 15 to 20. The outcomes of these experiments provide compelling evidence that the model exhibits nonlinear characteristics. To provide a more comprehensive description, it is necessary to conduct experiments in star points (experiments 9–14).

A review of the available data suggests that the study employed an experimental design based on a full factorial plan. This involved conducting experiments at various points along the plan, as well as at the centre and at the star points. The results of the centre experiments demonstrated that the model is nonlinear, necessitating the conduct of additional experiments for a more comprehensive description. Consequently, experiments were conducted at the star points with the objective of enhancing the accuracy and completeness of the model. This iterative approach to experiment planning and data analysis is a common practice in scientific research, with the aim of ensuring the reliability and robustness of the results.

The significance of quadratic effects (nonlinearities) is confirmed by experimental data through the process of checking the conditions (Formulas 1 - 3).

$$\frac{1}{8}{\sum }_{\mathrm{i}=1}^{\mathrm{\infty }}{\mathrm{R}}_{\mathrm{i}}=17.5<\frac{1}{6}{\sum }_{\mathrm{i}=15}^{20}{\mathrm{R}}_{\mathrm{i}}=28.0$$

(1)

$$\frac{1}{8}{\sum }_{\mathrm{i}=1}^{\mathrm{\infty }}{\mathsf{\rho }}_{\mathrm{i}}=1835.6<\frac{1}{6}{\sum }_{\mathrm{i}=15}^{20}{\mathsf{\rho }}_{\mathrm{i}}=20.52$$

(2)

$$\frac{1}{8}{\sum }_{\mathrm{i}=1}^{\mathrm{\infty }}{\mathsf{\pi }}_{\mathrm{i}}=0.719<\frac{1}{6}{\sum }_{\mathrm{i}=15}^{20}{\mathsf{\pi }}_{\mathrm{i}}=0.798$$

(3)

Using the experimental data in Table 2, we calculate the values of regression coefficients according to formulas 2-11. As a result, we obtain mathematical models describing the dependence of compressive strength, average density and thermal conductivity on the amount of sand, crushed stone and fiber in the form of Formulas 4-14:

$$y_i=a_0+a_1{x}_1+a_2{x}_2+a_3{x}_3+a_{12}{x}_1{x}_2+a_{13}{x}_1{x}_3+a_{23}{x}_2{x}_3+a_{11}{x}_1^2+a_{22}{x}_2^2+a_{33}{x}_3^2$$

(4)

$$a_0=\frac{A}{N}\begin{array}{cc}[2{\mathsf{\lambda }}^2\left(n+2\right)\left(0y\right)-2\mathsf{\lambda }\mathrm{c}\sum _{i=1}^k\left(iiy\right)& ]\end{array}=28.1$$

(5)

$$a_1=\frac{с}{N}\left(iiy\right)=1.3$$

(6)

$$a_2=\frac{с}{N}\left(iiy\right)=6.6$$

(7)

$$a_3=\frac{с}{N}\left(iiy\right)=1.5$$

(8)

$$a_1^2=\frac{A}{N}\begin{array}{cc}[c^2\left[\left(n+2\right)\mathsf{\lambda }-k\right]\left(iiy\right)+c^2\left(1-\mathsf{\lambda }\right)\sum _{i=1}^k\left(iiy\right)-2\mathsf{\lambda }\mathrm{c}(0y)& ]\end{array}=2.6$$

(9)

$$a_2^2=\frac{A}{N}\begin{array}{cc}[c^2\left[\left(n+2\right)\mathsf{\lambda }-k\right]\left(iiy\right)+c^2\left(1-\mathsf{\lambda }\right)\sum _{i=1}^k\left(iiy\right)-2\mathsf{\lambda }\mathrm{c}(0y)& ]=-4.0\end{array}$$

(10)

$$a_3^2=\frac{A}{N}\begin{array}{cc}[c^2\left[\left(n+2\right)\mathsf{\lambda }-k\right]\left(iiy\right)+c^2\left(1-\mathsf{\lambda }\right)\sum _{i=1}^k\left(iiy\right)-2\mathsf{\lambda }\mathrm{c}(0y)& ]-1.9\end{array}$$

(11)

$$a_{12}=\frac{c^2}{N\mathsf{\lambda }}\left(iuy\right)=0.9$$

(12)

$$a_{13}=\frac{c^2}{N\mathsf{\lambda }}\left(iuy\right)=-1.05$$

(13)

$$a_{23}=\frac{c^2}{N\mathsf{\lambda }}\left(iuy\right)=1.05$$

(14)

The regression coefficients of the strength, average density and thermal conductivity equations are given in Table 3.

The coded and natural values of the varied coefficients are given in Table 4.

Statistical analysis of the obtained regression equations by F-criterion confirmed their adequacy.

The analysis of significance of regression coefficients established that the mathematical function of thermal conductivity can be simplified by taking x₃=0, and then it will have the formula 15:

$$\lambda =0.79-0.02x_1-0.02x_2-0.02x_1^2-0.04x_2^2$$

(15)

The results of the experiments shown in Table 5 allow us to determine the optimal values of the coefficients x₁, x₂, x₃. For this purpose, we examine the obtained functions for extrema and obtain a system of equations of the formula 16:

$$\frac{\partial y_i}{\partial x_1}=0;\frac{\partial y_i}{\partial x_2}=0;\frac{\partial y_i}{\partial x_3}=0$$

(16)

Solving these systems sequentially, we obtain that the highest value of strength will have the composition at values (Formula 17):

$$x_1=-0.3;x_2=-1.0;x_3=-0.55;R_b=32MPa$$

(17)

The composition will have the lowest thermal conductivity if you take (Formula 18):

$$x_1+2;x_2=-2;x_3=0;\lambda =0.47W/m·K$$

(18)

For this composition the compressive strength will be 8.7 MPa.

A coefficient is considered to be established if its name and area of definition are specified. In the selected area of definition, it can have several values corresponding to the number of its different states. The quantitative or qualitative states of the factor selected for the experiment are called factor levels [24].

Compressive strength (Formula 19):

$$R_{av}=17.5-1.3x_1-6.6x_2-1.5x_3-2.6x_1^2-4x_2^2-1.9x_3^2+0.9x_1{x}_2-1.05x_1{x}_3+1.05x_2{x}_3$$

(19)

Next, we find the standard deviation ($S_{td}$) by expressing the Formula 20:

$$S_{td}=\sqrt{(\frac{\sum (\mathrm{x}-\mathrm{x}1)2}{\mathrm{n}}})$$

(20)

Which sounds as the root of the sum of squares of differences between the sample elements and the mean value divided by the number of elements in the sample.

Using Formula 20 we obtain the standard deviation for the mean values of experimental and calculated data: S_td1= 8.01; S_td2= 8.3.

The adequacy of the equation is checked by Fisher's criterion. The equation is adequate, because the ratio F thus compiled is less than the theoretical F (Formula 21).

$$F=\frac{S_{ad}^2}{S_{rep}^2}=\frac{2.22}{0.32}=6.9$$

(21)

Since F= 6.9>F_cr = 0.05, the model is considered adequate. Thus, all the coefficients are significant with 99% reliability.

The objective of this investigation is to examine the impact of various factors on the output parameters of lightweight structural concrete, with a particular focus on its average density and strength. The factors under investigation are the quantity of sand, crushed stone and dispersion-reinforcing admixture.

3. Results

4. Discussion

5. Conclusions

References

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