On the Calculation of Optimal Fractional Compositions of Concrete Aggregates
Abstract Generated abstract
The paper proposes a computational method for selecting optimal fractionated aggregate compositions for concrete by modeling random aggregate structures on a digital computer and evaluating voidage as the response parameter. A factorial experiment with three aggregate fractions compares physical gravel tests with a probability-geometric digital model, using regression analysis and an F-test to assess whether the model reproduces the relevant response-surface behavior. The results indicate a close linear relation between voidage and aggregate fraction ratios and support using the model to locate extremal compositions. Steepest-ascent and simplex search procedures are then applied, with the simplex plan judged more suitable for machine computation, yielding an approximately stationary composition dominated by the coarse fraction and suggesting reduced reliance on labor-intensive physical experiments.
Full Text
ON THE CALCULATION OF OPTIMAL FRACTIONATED COMPOSITIONS OF CONCRETE AGGREGATES
V. I. SES’
(Presented by the scientific seminar of the Research Institute of EI)
It has been experimentally established that the properties of composite materials are determined to a considerable extent by their internal structure \[1, 3\].
In the present work, the selection of the optimal composition of the aggregate of a composite material, using concrete as an example, is proposed to be carried out from the results of an experiment on a digital computer; a methodology for such an experiment is proposed, and an experimental justification of the methodology is given.
Factorial experiment
It is proposed to obtain an assessment of the optimality of the fractional composition of the aggregate from the results of a factorial experiment on a digital computer. The essence of this method is that a model of a random structure is created on a digital computer, taking into account the influence of the fractional composition of the aggregate. Using methods of active experiment, one finds the aggregate composition for the model corresponding to the extremal value of the evaluated parameter.
Estimates of the geometric characteristics of random structures can be found by the methodology set forth in works \[1, 2\]. To substantiate the possibility of using this methodology, a factorial experiment with gravel was carried out and estimates of the parameters were found on a digital computer by the method of tetrads \[2\]. The design matrix and the results of observations and calculation are given in Table 1. The relative frequencies of granules of the given fraction were taken as factors. The size of the granules of a fraction was taken equal to the arithmetic mean of the diameters of the sieve openings on which the given fraction was separated. The experiment was carried out with a three-fraction aggregate, the granule sizes of which were in the ratio \(1 : 2.12 : 3.66\). Since the factors were taken to be relative frequencies, the number of factors is one less than the number of fractions in the aggregate mixture. The factors \(x_1\) and \(x_2\) are the ratio of the number of granules of the fine and medium fractions to the number of granules of the coarse fraction; \(m^{\mathrm{g}}\) and \(m^{\mathrm{m}}\) are the voidage coefficient (V.C.) for gravel and the digital model, respectively. Observations with gravel were carried out five times under the same conditions to estimate the within-group variance. The corresponding variance of the probability-geometric model on the digital computer \[2\] is equal to zero. The choice of V.C. as the response surface is explained by the linearity of its dependence on the fractional composition. V.C. is found from the expression
\[ m = \Theta(1 - \Theta)^{-1}, \tag{1} \]
Table 1
| Zero level | 18 | 10 |
| Upper level | 28 | 19 |
| Lower level | 8 | 1 |
| Variation interval | 10 | 9 |
| \(x_0\) | \(x_1\) | \(x_2\) | \(m^{\mathrm{g}}\) | \(m^{\mathrm{m}}\) | |
|---|---|---|---|---|---|
| + | + | + | 0,4725 0,4487 0,4675 |
0,4610 0,4663 0,46328 |
0,2534 |
| + | − | + | 0,4942 0,5067 0,4905 |
0,4881 0,4942 0,49474 |
0,2655 |
| + | + | + | 0,4180 0,3961 0,3886 |
0,4192 0,3718 0,39874 |
0,2127 |
| + | − | − | 0,4372 0,4390 0,4338 |
0,4338 0,4392 0,43660 |
0,2264 |
where \(\Theta\) is the ratio of the volume of voids to the total volume of the structure.
The mean value for the group (Table 1, column 2) is underlined. From the results presented in the table, the coefficients for the linear regression equation \(b_1\) and \(b_2\), given below, were found: for the model
\[ m^{\mathrm{m}} = 0,2395 - 0,0064 x_1 + 0,0200 x_2 \]
and for gravel
\[ m^{\mathrm{g}} = 0,4483 - 0,0173 x_1 + 0,0307 x_2. \]
The data for carrying out the regression analysis are given in Table 2.
Table 2
| Cause of deviation | Degrees of freedom | Formula | Gravel | Model |
|---|---|---|---|---|
| Total sum | 3 | \(S_{00} = \Sigma m_1^{\mathrm{g}} - n m^{\mathrm{g}}\) | 0,05 | 0,0018 |
| 1 | \(\displaystyle \sum_{1}^{n} m_i x_{1i}\) | −0,6932 | −0,0258 | |
| 1 | \(\displaystyle \sum_{1}^{n} m_i x_{2i}\) | 0,12268 | 0,0798 | |
| Regression | 2 | \(\displaystyle \sum_{1}^{n} b_i S_{0i}\) | 0,0049 | 0,0017 |
Multiple correlation, which serves as a measure of the relationship between the dependent and independent variables, is evaluated by the quantity
\[ R^2 = \Sigma b_i S_{0i}/S_{00} \]
and is equal to 0.98 and 0.94 for the gravel and the model, respectively. The large value of the multiple correlation indicates a close linear relationship between K.P. and the factors \(x_1\) and \(x_2\). The residual sum of squares \(R_0^2 = S_{00} - \sum b_i S_{0i}\) for the spheres and the gravel is \(10^{-4}\).
To substantiate the possibility of using the model, it is necessary to prove the validity of the hypothesis \(H_0\): equality of the coefficients \(b_1^{\mathrm{g}} = b_1^{\mathrm{m}},\ b_2^{\mathrm{g}} = b_2^{\mathrm{m}}\) in the regression equations for the gravel and the model, which indicates coincidence of the equations up to the constant term. The criterion is specified by the variance ratio
\[ \frac{R_2^2 - R_0^2}{2}\ \bigg/\ \frac{R_0^2}{n+n' - 6}, \]
with 2 and \((n+n' - 6)\) degrees of freedom, where \(n\) and \(n'\) are the sample sizes, \(R_2^2\) is the residual sum of squares of the combined sample with \((n+n' - 6)\) degrees of freedom.
\[ R_2^2 = S_{00}^{\mathrm{g}} + S_{00}^{\mathrm{m}} - \sum b_i'' S_{0i}'', \]
where \(S_{0i}'' = S_{0i}^{\mathrm{g}} + S_{0i}^{\mathrm{m}}\) are the “corrected” sums of products of the combined sample, and \(S_{0i}^{\mathrm{g}}\) and \(S_{0i}^{\mathrm{m}}\) are the corresponding sums from Table 2.
The constants \(b_i''\) are found from the expression \(S_{0i} = b_1'' S_{1i}'' + b_2'' S_{2i}''\) and are equal to
\[ \begin{aligned} b_1'' &= -0.012, & b_2'' &= 0.025,\\ S_{01}'' &= -0.096, & S_{02}'' &= 0.203. \end{aligned} \]
Then \(R_2^2\) is equal to 0.0004 and the variance ratio is equal to 3. The tabular value of the \(F\)-ratio with \((2,2)\) degrees of freedom and significance 0.95 is equal to 19; therefore, the deviation may be considered insignificant and the hypothesis \(H_0\) valid. Consequently, the extreme points of the response surface for the gravel can be determined from the proposed model.
Methodology of the Extremal Experiment on a Digital Computer
The use of the model makes it possible to replace a labor-intensive and not always possible physical experiment by a machine experiment. One of the methods of an extremal factorial experiment is implemented in an algorithm for a digital computer, by which the conditions for carrying out the calculation and the direction of movement into the extremal region are determined from the results of preliminary calculations.
Table 3 gives the design matrix of a 2-factor extremal experiment and the results of the search for the extremal value of K.P. by the steepest-ascent method implemented on the BESM-4. For comparison and substantiation of the methodology, the results of a physical experiment for the same compositions of the aggregate mixture are also given here.
The direction of movement was determined by the coefficients \(b_i^{\mathrm{m}}\) and \(b_i^{\mathrm{g}}\), found from the results of four experiments. Experiments 5, 6, and 7 were calculated with changes in the factors \(x_1\) and \(x_2\) by steps of 0.1 and 0.0026, respectively. Up to experiment 6, K.P. decreases, and in experiment 7 it increases. Therefore, a new design is composed for compositions close to the composition of the sixth experiment, and a new set of four calculations is carried out. This procedure is repeated until an almost stationary region is reached.
To compare the methods, a program of the machine experiment was compiled according to a simplex design; the calculation of the parameters is performed using the given model. The results are presented in Table 4.
The experiment was planned as a 2-factor one; the percentage content of the 3rd fraction is determined by the content of the first two. The smallest value of K.P. was obtained in experiment 7. In parentheses are the numbers of the experiments whose conditions are replaced by the given ones.
Table 3
| No. | Factor levels | Ratio of the weight of the fine fraction to the coarse fraction, \(x_1\) | Ratio of the weight of the middle fraction to the coarse fraction, \(x_2\) | Void coefficient: gravel \(m^{\mathrm{g}}\) | Void coefficient: model \(m^{\mathrm{m}}\) |
|---|---|---|---|---|---|
| 1 | Zero level (0) | 0.07 | 0.07 | ||
| 2 | Upper level (+1) | 0.1 | 0.1 | ||
| 3 | Lower level (−1) | 0.04 | 0.04 | ||
| 4 | Variation interval | 0.03 | 0.04 | ||
| 5 | Experiment 1 | −1 | −1 | 0.6437 | 0.2541 |
| 6 | Experiment 2 | +1 | −1 | 0.5918 | 0.2352 |
| 7 | Experiment 3 | −1 | +1 | 0.6294 | 0.2538 |
| 8 | Experiment 4 | +1 | +1 | 0.5797 | 0.2330 |
| 9 | \(b_1^{\mathrm{g}}\) | −0.0254 | −0.0066 | ||
| 10 | \(b_i^{\mathrm{m}}\) | −0.0099 | −0.0006 | ||
| 11 | Step | 0.1 | 0.026 | ||
| 12 | Experiment 5 | 0.17 | 0.096 | 0.5316 | 0.2270 |
| 13 | Experiment 6 | 0.27 | 0.122 | 0.5044 | 0.2174 |
| 14 | Experiment 7 | 0.37 | 0.148 | 0.5104 | 0.2236 |
| 15 | Zero level (0) | 0.27 | 0.122 | ||
| 16 | Upper level (+1) | 0.34 | 0.202 | ||
| 17 | Lower level (−1) | 0.2 | 0.042 | ||
| 18 | Variation interval | 0.07 | 0.08 | ||
| 19 | Experiment 8 | −1 | −1 | 0.4900 | 0.2176 |
| 20 | Experiment 9 | +1 | +1 | 0.4794 | 0.2121 |
| 21 | Experiment 10 | −1 | +1 | 0.5142 | 0.2249 |
| 22 | Experiment 11 | +1 | +1 | 0.5081 | 0.2185 |
| 23 | \(b_1^{\mathrm{g}}\) | −0.042 | 0.0132 | ||
| 24 | \(b_i^{\mathrm{m}}\) | −0.030 | 0.0034 |
Table 4
| Factors | \(x_1\) | \(x_2\) | \(x_3\) | \(m^{\mathrm{m}}\) |
|---|---|---|---|---|
| Zero level, % | 33 | 26 | ||
| Variation interval, % | 10 | 10 | ||
| Experiment 1 | 38 | 29 | 33 | 0.275 |
| » 2 | 28 | 29 | 43 | 0.263 |
| » 3 | 33 | 20 | 47 | 0.255 |
| » 4(1) | 23 | 20 | 57 | 0.245 |
| » 5(2) | 28 | 11 | 61 | 0.233 |
| » 6(3) | 18 | 11 | 71 | 0.223 |
| » 7(4) | 32 | 02 | 75 | 0.211 |
| » 8(5) | 13 | 02 | 85 | 0.243 |
As a result of the experiments, an almost stationary region was found with the fractional composition in % by weight given in Table 5 and with the value \(m^{\mathrm{m}} = 0.2110 \div 0.2121\).
Table 5
| Relative sizes | 1 | 2.12 | 3.66 |
|---|---|---|---|
| Relative weight of fraction, % | 23–27 | 2–4 | 69–75 |
However, formalizing the computation according to the simplex plan is considerably simpler, since there are strict rules for the sequence of operations when conducting the experiment, whereas the steepest-ascent method provides for the experimenter’s intervention. In view of the foregoing, the simplex plan is the most acceptable for a machine experiment.
The results of the analysis confirm that the method proposed in the present work can be successfully used to find the optimal granulometric composition of concrete aggregate.
Application of the present method will make it possible to improve the properties of concrete while reducing its cost.
In addition, a labor-intensive physical experiment is replaced by computation on a digital computer, taking into account in the optimality criterion factors that are difficult, and sometimes impossible, to evaluate in a physical experiment.
REFERENCES
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V. I. Soroker, V. I. Galaktionov. Selection of optimal mixtures of fractionated aggregates for concrete at reinforced-concrete-products plants. Izvestiya vuzov, “Construction and Architecture,” 5, Novosibirsk, 1966.