SELECTING A RATIONAL NUMBER OF CONTROL SAMPLES FOR ASSESSING THE QUALITY OF CONCRETE MIX PREPARATION BY A MIXING-AND-TRANSPORTING MACHINE

V. N. Shchelkonogov, V. G. Lukyanov

(Presented by the scientific seminar of the Department of Geodesy, Mining, and Construction)

Mixing is the most critical operation in the technological cycle of concrete production. As a result of this operation, a uniform mutual distribution of the components of the concrete mix is achieved: binder, aggregates, water, and admixtures. Improving the processes of concrete preparation and quality control is an important task.

In research practice on mixing devices, the quality of the resulting mix is evaluated by various criteria. As a rule, these are indicators of the homogeneity of the mix or concrete being produced.

Until recently, the principal quality indicator for the mixing of concrete mixes having a comparatively complex composition was the uniformity of distribution of the components throughout the volume being mixed [1]. To determine the consistency of individual components in samples, methods are used such as weighing in water and in air, drying, washing and settling, chemical analysis, and others. These methods for evaluating mixing quality give a quantitative characteristic of the uniformity of distribution of the components in concrete mixes by volume and cannot provide a complete picture of the quality of the prepared concrete mix, since they do not take into account the use of the active properties of the binder component—cement.

Recently, for evaluating the efficiency of mixing building mixes, the strength-homogeneity method has found wide application. According to this method, the homogeneity of a mix is characterized by the coefficient of variation of the strength of control cube specimens:

\[ C_v=\frac{\sigma}{\overline{R}}\,100\%, \tag{1} \]

where

\[ \sigma=\sqrt{\frac{\sum_{1}^{N}(R-\overline{R})^2}{N-1}} \]

is the root-mean-square deviation of individual test results from the mean concrete strength, in kg/cm²;

\[ \overline{R}=\frac{\sum_{1}^{N}R}{N} \]

is the mean ultimate strength of concrete, kg/cm²;

\(R\) is the temporary compressive resistance of an individual specimen, kg/cm²;

\(N\) is the number of specimens tested.

The method of assessing the uniformity of a mixture by strength gives an indication of the strength properties of concrete, which are important quality indicators. By comparing the strength characteristics of concretes of the same composition prepared by different mixers, one can judge the effectiveness of the machines used.

In all cases, the quality of mixing is determined on the basis of sample observation, the essence of which is that the characteristic of the entire volume of prepared concrete (the population) is compiled from the results of examining some part of it (a sample) selected at random [2]. It should be noted that in practice deviations are usually observed between the values of attributes obtained as a result of examining a sample of concrete mix and those same attributes for the entire mass of concrete. These deviations are explained by errors of representativeness, which are inherent in the sampling method. As a rule, significant errors arise as a result of an insufficient number of observations. In this case the error is reduced by increasing the sample size.

However, when the number of control cube specimens taken from a concrete sample is increased, an increase in the variability index is observed [3]. This is associated with a number of phenomena and factors that do not depend on the process of preparing the concrete mix itself, but depend on the conditions of sampling, molding, compaction, and the time between the preparation of the first and the last specimens. As a result, there is some deviation from the mean strength value for the last specimens, and the index of strength variability, beginning approximately with the 25th specimen, will increase. A certain increase in $C_v$ does not characterize the operation of the given concrete-mixing machine. In the general case, the “critical” number of specimens after which the conditions of sampling, molding, compaction, etc., begin to have an effect may vary within wide limits depending on the specific conditions of the experiment and its material equipment.

In the course of experimental investigations to determine the quality of concrete preparation by a mixing-and-transporting machine, the task was set of determining the rational number of cube specimens required for a complete strength characterization of a sample of concrete of different grades. Its solution was carried out by an experimental-analytical method [4].

The total number of specimens tested was chosen to be 30 for each of the three grades of concrete produced by the machine. As components of the concrete mix, slag Portland cement with an activity of 390 from the Aktau cement plant and a sand-gravel mixture from the Churubai-Nura quarry were used. Control specimens measuring $15 \times 15 \times 15$ cm were made from the concrete mix immediately after its preparation, with subsequent compaction on a standard laboratory vibrating table. The strength of the concrete specimens was determined in accordance with GOST 10180-67 after 28 days of curing in a metal cabinet with a specified heat-and-moisture regime. The test results were processed by methods of mathematical statistics.

Theoretically, when the number of tests is $N > 30$, the regularities inherent in any population will be manifested with sufficient accuracy in the sample under study as well; but in practice we deal with a limited number of experimental data. In this connection, while noting a certain error in the data and in the results of processing, it is necessary to establish stable characteristics of each series of strength values of the concrete specimens, identify the errors, and, on the basis of the data obtained, determine the number of specimens that ensure reliable conclusions for quality control of concrete.

For a preliminary assessment of series, in order to identify regularities, we shall apply the method of constructing histograms that graphically express the distribution law of random variables: the density of the distribution, skewness, etc. Histograms characterize a distribution more accurately the larger the number of observations (measurements, tests) considered. But cases are possible when clearly expressed features are also captured with a smaller number of measurements. This may be observed, for example, when constructing histograms of the strength distribution of control specimens for concrete grades 150 and 200 (Fig. 1). In the case of grade-100 concrete, a number of specimens equal to 30 does not give a sufficiently clear idea of the distribution law. In order to draw conclusions about the distribution law, numerical calculations must be carried out.

Fig. 1. Histograms of the strength distribution of control cube specimens for concretes of grades 150 and 200

Analysis of the series by means of histograms shows that the distributions are moderately asymmetric (skewed). As a numerical characteristic of skewness in the case of an asymmetric distribution, we take the coefficient of asymmetry \(S_k\):

\[ S_k=\frac{\mu_3}{\sigma^3}, \tag{2} \]

where \(\mu_3\) is the third central moment.

The standard deviation of the coefficient of asymmetry is calculated by the formula

\[ \sigma_{S_k}=\sqrt{\frac{6}{N}}. \tag{3} \]

When the condition \(|S_k|\leq \sigma_{S_k}\) is satisfied, the empirical distribution may be regarded as close to symmetric with a reliability of 67%.

Another indicator characterizing the distribution law is the excess \(E\), which determines the position of the curve of the given distribution relative to the normal one and is defined as

\[ E=\frac{\mu_4}{\sigma^4}-3, \tag{4} \]

where \(\mu_4\) is the fourth central moment.

The permissible value of excess with a reliability of 67% must not exceed the standard deviation \(|E|\leq \sigma_E\), determined for \(N<50\) as

\[ \sigma_E=\sqrt{\frac{24}{N}}. \tag{5} \]

The absence of outliers in the results of the experiments is expressed by the condition

\[ |R-\overline{R}|\leq 3\sigma, \tag{6} \]

where \(\sigma=\sqrt{D}\) is the root-mean-square deviation.

The results of calculating the central moments

\[ \left(\mu_i=\frac{\sum_{1}^{N}(R-\overline{R})^i}{N}\right), \]

the coefficients of skewness, kurtosis, and the root-mean-square deviations according to formulas (1)—(6) are summarized in Table 1, from which it is seen that the distribution indices of three statistical series of strength values of control specimens for concrete grades 100, 150, and 200 lie within the permissible limits.

Table 1

Parameters of a sample of 30 control cube specimens for different grades of concrete produced by a mixing-and-transporting machine

Consequently, the distribution for these series may be considered close to normal. To test the hypothesis of normality, Pearson’s criterion \((\chi^2)\), Kolmogorov’s criterion, and others may also be used.

The error in calculating \(\overline{R}\) from the sample data, determined by the formula

\[ \sigma(\overline{R})=\sqrt{\frac{\sigma^2}{N}}, \tag{7} \]

for concrete grades 100, 150, and 200, respectively, is equal to:

\[ \sigma(\overline{R})^{100}=\sqrt{\frac{141}{30}}=2.17;\quad \sigma(\overline{R})^{150}=\sqrt{\frac{193}{30}}= \]

\[ =2.54;\quad \sigma(\overline{R})^{200}=\sqrt{\frac{199}{30}}=2.57. \]

The fact that the population mean will not go beyond certain limits can be asserted not with absolute certainty, but only with a certain degree of probability. The probability can be increased by increasing the multiple of the sampling error from 1 to 3. When the sampling error is increased by a factor of 3, the probability, i.e., the reliability of the conclusions, will rise to 0.9975. In practice, such accuracy is not required for determining the strength of concrete; therefore we shall use the quite satisfactory reliability of 0.9.

The enlarged limits of deviations for a reliability of 0.9 will be determined by the formula

\[ \Delta=t\sigma(\overline{R}), \tag{8} \]

where \(t=1.67\) (Table 9.1 [2]) is the error-multiplicity coefficient, or confidence coefficient.

\[ \Delta^{100}=1.67\cdot 2.17=3.63;\quad \Delta^{150}=1.67\cdot 2.54=4.23;\quad \Delta^{200}=1.67\cdot 2.57=4.29. \]

The size of the nonrepeated sample required to ensure an accuracy \(\Delta\) with reliability 0.9 is calculated by the formula

\[ n=\frac{t^{2}\sigma^{2}N}{N\Delta^{2}+t\sigma^{2}}: \tag{9} \]

for concrete grade 100

\[ n^{100}=\frac{2.78\cdot141\cdot30}{30\cdot13.2+2.78\cdot141}=14.9\simeq15; \]

for concrete grade 150

\[ n^{150}=\frac{2.78\cdot193\cdot30}{30\cdot17.9+2.78\cdot193}=15.02\simeq15; \]

for concrete grade 200

\[ n^{200}=\frac{2.78\cdot199\cdot30}{30\cdot18.4+2.78\cdot199}=15.03\simeq15. \]

Thus, the rational number of cube specimens in a concrete sample should be taken as 15.

The calculated value of the sample size agrees satisfactorily with the experimental data for determining mixing uniformity, assessed by the variability index according to formula (1). The results of determining \(C_v\) for different numbers of control specimens and for the grades of concrete produced—100, 150, and 200—are presented graphically in Fig. 2.

Fig. 2. Dependence of the variability index \(C_v\), %, on the number of control specimens: 1, 2, 3—respectively for concrete grades 100, 150, 200

It is evident from the graph that the reduction in strength variation occurs as the number of specimens is increased to 15–20 for all concrete grades. A further increase in the number of specimens leads to an increase in the variability index.

On the basis of the data obtained, recommendations were developed for carrying out continuous quality control of the concrete produced during the period of industrial testing of mixing-and-transporting machines. The variability index, determined from 15 specimens for concretes of grades 150 and 200, was within the range 5.7–8.1%. For grade-100 concrete, used in shotcreting work, \(C_v\) varied within the range 6.7–11%. Such indices for concretes obtained under production conditions should be considered satisfactory.

Conclusions

In the practice of research on concrete mixers, a method is widely used for assessing the uniformity of a mixture by the strength of control cube specimens, based on sample observation. This method gives pre-

idea of the strength properties of the concrete produced, which are important indicators of quality.

As the number of control specimens in the sample increases, a regular increase is observed in the variability index $C_v$, which characterizes the homogeneity of the concrete mix. The increase in the variability index is explained by the duration of the processes of sampling, molding, and compacting a large number of specimens.

Using methods of mathematical statistics, it has been established that the strength of the control specimens of a concrete sample follows the normal distribution law. The rational number of specimens for a non-replicated sample should be taken as 15.

On the basis of the data obtained, a methodology for monitoring concrete quality has been developed and used in conducting industrial tests of mixing-and-transporting machines. The proposed method for determining sample size can also be applied to other conditions of preparing construction mixes.

REFERENCES

1. M. V. Bunin. On criteria for the quality of mixes. Izvestiya vuzov. Stroitel’stvo i arkhitektura, 1958, No. 10.

2. N. N. Ryauzov. General Theory of Statistics. Moscow, Statistika Publishing House, 1971.

3. V. T. Bogatyrev, V. N. Shmigalsky. Testing of Mixing Machines. Novosibirsk, Nauka, 1966.

4. V. N. Shchelkonogov, V. G. Lukyanov. Mechanization of the construction of monolithic concrete lining of horizontal mine workings using mixing-and-transporting machines. In the present collection.