# A Proposal of a Method for Ready-Mixed Concrete Quality Assessment Based on Statistical-Fuzzy Approach

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## Abstract

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## 1. Introduction

_{ck}) and constitutes the basis for evaluating the quality of the concrete produced. The proposed procedure for quality assessment of the concrete produced allows for making effective decisions of two types: non-fuzzy (crisp) or fuzzy, which point out to possible solutions and their corresponding preferences.

#### Quality Control of Ready-Mixed Concrete According to EN-206

- (1)
- Individual assessment result criterion f
_{ci}—applied irrespective of the production status (initial or continuous)$${f}_{ci}\ge \left({f}_{ck}-4\right)\mathrm{N}/{\mathrm{mm}}^{2};$$ - (2)
- Mean assessment result criterion f
_{cm}—applied in three methods depending on the production status:- –
- Initial production$${f}_{cm}\ge \left({f}_{ck}+4\right)\mathrm{N}/{\mathrm{mm}}^{2}\u2014\mathrm{method}\mathrm{A};$$
- –
- Continuous production$${f}_{cm}\ge \left({f}_{ck}+1.48\xb7\mathsf{\sigma}\right)\mathrm{N}/{\mathrm{mm}}^{2}\u2014\mathrm{method}\mathrm{B};$$
- –
- The concept of control chart—method C.

- ${k}_{1}=4$—test coefficient value set out by the standard [28],
- $\sigma $—standard deviation for population.

- The concrete acceptance probability is not always a compromise between the producer risk and the customer risk. Applying the standard conformity criteria may lead to an excessive customer risk, especially in the case of an assumption of log-normal distribution of compressive strength.
- Applying the standard conformity criteria may lead the producer to adopting strategies involving higher production costs, as it can unnecessarily require higher mean values of production with higher standard deviations. These criteria are not recommended for production with small deviation and may be a reason for concealing the results for samples of understated compressive strength.
- Applying the standard conformity criteria may produce too high values of the consumer risk.

## 2. Materials and Methods

#### 2.1. Conformity Control of Concrete Compressive Strength in Consideration of Measurement Uncertainty

#### 2.2. Alternative Conformity Criteria for Concrete Compressive Strength

_{c}) of concrete that complies with the conformity criterion can be represented as a fuzzy set (8):

_{c}in interval [0, 1].

_{cm}(Figure 1b). Sporadically, the condition concerning particular test results f

_{ci}is the decisive condition for the fulfilment of the conformity criteria (Figure 1a) [34,35,37,38,41]. Since statistical conformity criteria are found to be insufficient, statistical-fuzzy methods can be applied to define class membership functions, and both standards and expertise can be taken into consideration in the quality control of the concrete produced.

- –
- For method A and sample of size n = 15, (9):$${f}_{cm}\ge {f}_{ck}+4\to T$$
_{cm}is the mean compressive strength of concrete, and f_{ck}is the characteristic compressive strength of concrete. - –
- For method B and sample of size n ≥ 15, (10):$${f}_{cm}\ge {f}_{ck}+1.48\sigma \to T$$
_{cm}is the mean compressive strength of concrete, f_{ck}is the characteristic compressive strength of concrete, σ-estimate for the standard deviation of a population.

_{T}(t) that can be determined for specific concrete classes on the basis of a statistical-fuzzy experiment.

_{x}(t) and p

_{y}(t) of random variables x →N(m

_{x},σ

_{x}) and y → N(m

_{y},σ

_{y}) may be determined. Marginal distribution parameters were determined by means of Monte Carlo simulation methods and the following calculation algorithm [37,44]:

- Generate N groups of random numbers of size n = 3 from normal distribution;
- Randomly select concrete class—Concrete of three adjacent classes C
_{i}_{−}_{1}, C_{i}, C_{i}_{+1}(identical probability of 1/3); - Randomly select standard deviation from 2, 3, 4, 5, 6 MPa with 1/5 probability;
- Repeat (1) and (2) n-times to obtain f
_{ci},…, f_{cn}; - Randomly select defectiveness w from normal distribution;
- Calculate mean compressive strength of adjacent concrete classes from Equation (11):$${f}_{cm({C}_{i-1},{C}_{i})}=\frac{{m}_{{C}_{i-1}}+{m}_{{C}_{i}}}{2}\hspace{1em}\hspace{1em}\mathrm{and}\hspace{1em}\hspace{1em}{f}_{cm}{}_{({C}_{i},{C}_{i+1})}=\frac{{m}_{{C}_{i}}+{m}_{{C}_{i+1}}}{2}$$
- Calculate standard deviation from Equation (12):$${s}_{({C}_{i-1},{C}_{i})}=\frac{1}{n}\sqrt{({s}_{{C}_{i-1}}{}^{2}+{s}_{{C}_{{}_{i}}}{}^{2}}\hspace{1em}\hspace{1em}\mathrm{and}\hspace{1em}\hspace{1em}{s}_{({C}_{i},{C}_{i-1})}=\frac{1}{n}\sqrt{({s}_{{C}_{1}}{}^{2}+{s}_{{C}_{{}_{i-1}}}{}^{2}}$$
- Determine the characteristic compressive strength for the considered and lower concrete classes from Equation (13):$${f}_{ck({C}_{i-1},{C}_{i})}={m}_{({C}_{i-1},{C}_{i})}-t(w){s}_{({C}_{i-1},{C}_{i})}$$$${f}_{ck({C}_{i},{C}_{i+1})}={m}_{({C}_{i},{C}_{i+1})}-t(w){s}_{({C}_{i},{C}_{i+1})}$$
- Calculate mean compressive strength of the considered and lower concrete classes from Equation (15):$${f}_{cm}{}_{({C}_{i-1},{C}_{i})}={f}_{ck({C}_{i-1},{C}_{i})}+4$$$${f}_{cm}{}_{({C}_{i-1},{C}_{i})}={f}_{ck({C}_{i},{C}_{i+1})}+4$$
- Create a table for the probability distribution function of random vector (ξ, η) and determine the histogram of marginal distributions by summing rows and columns. The first marginal distribution is the sum of rows and the classification by the considered and lower concrete classes. The second marginal distribution is the sum of columns and the classification by the considered and higher concrete classes.

_{ξ}(x

_{n}) and p

_{η}(x

_{n}) (marginal distribution parameters) are the basis for determining membership functions of test characteristics for specific concrete classes, i.e., the second stage of calculations.

_{i}for the considered i-class of concrete and higher can be represented by Equation (17):

_{i}for the considered i-class of concrete and higher can be expressed by the following Equation (18):

_{ci}can be calculated from Equation (19) or (20):

_{K}(f

_{cm}) value, the considered concrete batch can be recognized as a specific concrete class. Such recognition might be more or less accurate, depending on the economic requirements and the impact of classification on the quality assessment of the concrete produced.

#### 2.3. Example of Application of the Statistical-Fuzzy Conformity Criteria for Concrete of Class C20/25

_{x},σ

_{x}), i.e., the point of division for concrete of classes C16/20 and C20/25, m

_{x}= 26.5 MPa, and σ

_{x}= 4.48 MPa, respectively. The parameters of marginal distribution of random variable y → N(m

_{y},σ

_{y}), the point of division for concrete of classes C20/25 and C25/30, were estimated as m

_{y}= 39.8 MPa and σ

_{y}= 5.46 MPa, respectively (Figure 4).

## 3. Results and Discussion

## 4. Conclusions

- The proposed concept of quality assessment allows for minimising the risk of wrong classification of a concrete batch, i.e., overstating or understating the concrete class.
- Employing non-standard methods of conformity control of concrete compressive strength may become a useful tool in the investment-related (technology-related) decision-making process.
- The analyses carried out reveal that the statistical-fuzzy conformity control can play an arbitrary role in the quality assessment of the concrete produced.
- Statistical-fuzzy and fuzzy methods allow to take into account the opposing requirements of safety, quality and economy. Taking these requirements into consideration is made possible by determining a degree of membership lower than 1 for the considered concrete class.
- The alternative method of concrete quality assessment is easy to apply; however, it requires a complex calculation procedure, which significantly limits its universal use in the production process. Widespread application of this method would require implementing specialised utility software developed based on specific algorithms.
- The advantages of the statistical-fuzzy approach are particularly observable when employing the concept of concrete families. It allows to minimise the uncertainty connected to the transformation relation between the results for compressive strength of each concrete family member.
- Based on this approach, a risk matrix may be developed for a construction facility in order to verify the assigned reliability class specified in the construction design.
- Statistical-fuzzy methods are fully compatible with the concept of sustainable construction. Accidental understating of the concrete class results in the rejection of a concrete batch by the recipient. An unsuitable concrete mix is then considered as construction waste, which contradicts the principles of rational use of construction materials and mineral resources.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**OC curves for conformity criteria for samples of sizes n = 3 and normal distribution of concrete compressive strength: for a criterion for (

**a**) individual results and (

**b**) mean value.

**Figure 2.**Conformity control of concrete compressive strength according to EN 206 [32], where f

_{cm}is the mean compressive strength of concrete, f

_{ck}is the characteristic compressive strength of concrete, and σ is the estimate for the standard deviation of a population.

**Figure 4.**Marginal distribution and membership functions for C20/25 and every adjacent concrete class: C16/20 and C25/30.

**Figure 5.**Marginal distribution and membership functions for C20/25 and every second adjacent concrete class: C12/15 and C30/37.

**Figure 6.**Real results of compressive strength assessment and values for particular classes transformed in relation to the referential concrete in the concrete family.

**Table 1.**λ’ values for correlated results of mixed size samples [34].

Number (n) of Results | Value λ′ |
---|---|

3 | 2.67 |

15 | 1.48 |

**Table 2.**Fragment of the table presenting the conformity assessment of the population of results analysed.

Number Sample | Compressive Strength | Criterion 1 | Assessment | Criterion 2 | Assessment | Compressive Strength + Uncertainty | Criterion 2 + Uncertainty | Assessment |
---|---|---|---|---|---|---|---|---|

[-] | f_{ci}[MPa] | f_{ci}[MPa] | [-] | f_{cm}[MPa] | [-] | f_{ci}[MPa] | f_{ci min}[MPa] | [-] |

1 | 23.1 | 23.1 | met | - | 24.2 | - | ||

2 | 29.7 | 29.7 | met | - | 30.8 | - | ||

3 | 29.1 | 29.1 | met | 27.3 | unmet | 30.2 | 28.4 | unmet |

4 | 29.5 | 29.5 | met | 29.4 | met | 30.6 | 30.5 | met |

5 | 27.5 | 27.5 | met | 28.7 | unmet | 28.6 | 29.8 | met |

6 | 34.3 | 34.3 | met | 30.4 | met | 35.4 | 31.5 | met |

7 | 28.1 | 28.1 | met | 30.0 | met | 29.2 | 31.1 | met |

8 | 31.9 | 31.9 | met | 31.4 | met | 33.0 | 32.5 | met |

9 | 26.1 | 26.1 | met | 28.7 | unmet | 27.2 | 29.8 | met |

10 | 28.0 | 28.0 | met | 28.7 | unmet | 29.1 | 29.8 | met |

11 | 29.4 | 29.4 | met | 27.8 | unmet | 30.5 | 28.9 | unmet |

12 | 32.6 | 32.6 | met | 30.0 | met | 33.7 | 31.1 | met |

13 | 33.8 | 33.8 | met | 31.9 | met | 34.9 | 33.0 | met |

14 | 33.0 | 33.0 | met | 33.1 | met | 34.1 | 34.2 | met |

15 | 32.8 | 32.8 | met | 33.2 | met | 33.9 | 34.3 | met |

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**MDPI and ACS Style**

Skrzypczak, I.; Kokoszka, W.; Zięba, J.; Leśniak, A.; Bajno, D.; Bednarz, L.
A Proposal of a Method for Ready-Mixed Concrete Quality Assessment Based on Statistical-Fuzzy Approach. *Materials* **2020**, *13*, 5674.
https://doi.org/10.3390/ma13245674

**AMA Style**

Skrzypczak I, Kokoszka W, Zięba J, Leśniak A, Bajno D, Bednarz L.
A Proposal of a Method for Ready-Mixed Concrete Quality Assessment Based on Statistical-Fuzzy Approach. *Materials*. 2020; 13(24):5674.
https://doi.org/10.3390/ma13245674

**Chicago/Turabian Style**

Skrzypczak, Izabela, Wanda Kokoszka, Joanna Zięba, Agnieszka Leśniak, Dariusz Bajno, and Lukasz Bednarz.
2020. "A Proposal of a Method for Ready-Mixed Concrete Quality Assessment Based on Statistical-Fuzzy Approach" *Materials* 13, no. 24: 5674.
https://doi.org/10.3390/ma13245674