*3.3. The Strength*

The cubic compressive strengths of the SCCs at a curing age of 28 days were tested in accordance with China code GB/T50081 [37], which is identical to British Standard BS EN 12390-3 [38]. Cubic specimens with dimensions of 150 mm were used. Three specimens were tested as a group. Results are presented in Figure 6.

**Figure 6.** The cubic compressive strength of SCC.

The substitution of manufactured sand with river sand could improve the cubic compressive strength of SCC. Except for R45, other SCCs with ratios of river sand of 27.5% and 35% had higher compressive strength; specifically, their compressive strength was increased by 23.9% and 11.2% compared to R0. Meanwhile, R100 had a higher cubic compressive strength, which was about 6.5% greater than R0. This indicates the good relationship of cubic compressive strength with the density of SCCs with hybrid sand. Manufactured sand is characterized by its rough surface, irregular particle shape, angular edges and the inevitable presence of stone powder, while river sand is characterized by smooth, round and small particles [39]. This different morphology between manufactured sand and river sand could influence the microstructure of SCCs to present their corresponding cubic compressive strengths.

The compressive strength of R35-F30 was only 79.0% that of R35. This is due to the significant dilution effect of fly ash with a lower strength activity index of 84.3% [40–42]. Comparatively, due to the strength activity index of GGBS, which reached 97.6%, the compressive strength of R35-G30 was reduced only by 4.9% compared to R35. Meanwhile, R35-G10F20 and R35-G20F10 had similar compressive strengths to R35. This indicates that the hybrid of fly ash and GGBS had a positive superposition effect on the compressive strength of SCC [11,43].

The maximum particle size of the coarse aggregate had a significant effect on the compressive strength of the SCC. Compared to R35, with a maximum particle size of 20 mm, the compressive strengths of R35-C16 and R35-C10, with maximum respective particle sizes of 16 mm and 10 mm, were reduced by 19.3% and 17.0%. This is similar to the compressive strength observed in conventional vibrated concrete [41]. Under the same mix proportion, the skeleton effect of coarse aggregates in concrete is weakened with decreasing MPS due to the lack of rationally closed packing among coarse aggregates with a similar particle size. At the same time, the decreased compressive strength was also related to the increased crushing index of smaller coarse aggregates, as presented in Table 4.

#### **4. Optimization by Grey Relational Analysis**

Grey relational analysis is a modern mathematics method for factor analysis in a system [26]. It is always used to evaluate the significance of factors in a given system and to judge the factors in order of importance. Its fundamental principle is evaluating the level of similarity for a data sequence by relevancy calculations. A greater relevancy means a higher similarity. In this paper, grey relational analysis is conducted based on the test results as the sample data to select the optimal mixtures from these eleven groups of SCC mixtures.

Assuming that the original data is *D* in the sequence of slump, slump flow, *T*500, density and cubic compressive strength, then,

$$D = \begin{bmatrix} d\_1^1 & d\_2^1 & \dots & d\_n^1 \\ d\_1^2 & d\_2^2 & \dots & d\_n^2 \\ \vdots & \vdots & & \vdots \\ d\_1^m & d\_2^m & \dots & d\_n^m \end{bmatrix} = \begin{bmatrix} 260 & 650 & 11.0 & 2337 & 66.1 \\ 230 & 600 & 11.0 & 2537 & 81.9 \\ 245 & 700 & 3.5 & 2439 & 73.5 \\ 263 & 700 & 7.8 & 2408 & 66.0 \\ 250 & 700 & 10.8 & 2331 & 70.4 \\ 262 & 750 & 3.5 & 2364 & 58.2 \\ 262 & 710 & 4.9 & 2357 & 69.9 \\ 253 & 780 & 5.0 & 2433 & 73.0 \\ 253 & 700 & 7.0 & 2447 & 73.3 \\ 265 & 720 & 8.8 & 2428 & 59.3 \\ 247 & 690 & 11.3 & 2431 & 61.0 \end{bmatrix} \tag{1}$$

where *dk <sup>i</sup>* is the original data of the index *k* at the solution *i*. *m* = 11 and *n* = 5 in this paper.

In this study, the slump, slump flow, and slump flow time *T*<sup>500</sup> are the dominant indicators for the workability of fresh SCC. The slump and slump flow are larger for better responses, while the slump flow time *T*<sup>500</sup> is smaller for a better response. The density and the compressive strength are the dominant responses for the hardened SCC, and they are larger for better responses. Thus, the standard matrix *C* can be obtained with the standardizing Equations (2)–(4),

$$\mathbb{C}\_{k}^{l} = \frac{d\_{k}^{l} - \min\left(d\_{k}^{l}, l-1, 2, \dots, m\right)}{\max\left(d\_{k}^{l}, l-1, 2, \dots, m\right) - \min\left(d\_{k}^{l}, l-1, 2, \dots, m\right)} \tag{2} \text{ (for the larger and better response)} \tag{2}$$

*Ci <sup>k</sup>* <sup>=</sup> max(*d<sup>i</sup> <sup>k</sup>*, *<sup>i</sup>*=1,2,...*m*)−*d<sup>i</sup> k* max(*d<sup>i</sup> <sup>k</sup>*, *<sup>i</sup>*=1,2,...*m*)−min(*d<sup>i</sup> <sup>k</sup>*, *<sup>i</sup>*=1,2,...*m*) (for the smaller and better response) (3) *C* = ⎡ ⎢ ⎢ ⎢ ⎣ *C*1 <sup>1</sup> *<sup>C</sup>*<sup>1</sup> <sup>2</sup> ... *<sup>C</sup>*<sup>1</sup> 5 *C*2 <sup>1</sup> *<sup>C</sup>*<sup>2</sup> <sup>2</sup> ... *<sup>C</sup>*<sup>2</sup> 5 . . . . . . . . . *C*<sup>11</sup> <sup>1</sup> *<sup>C</sup>*<sup>11</sup> <sup>2</sup> ... *<sup>C</sup>*<sup>11</sup> 5 ⎤ ⎥ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0.8571 0.2778 0.1667 0.5097 0.3333 0.0000 0.0000 0.0385 1.0000 1.0000 0.4286 0.5556 1.0000 0.5243 0.6456 0.9429 0.5556 0.4487 0.3738 0.3291 0.5714 0.5556 0.0641 0.0000 0.5148 0.9143 0.8333 1.0000 0.1602 0.0000 0.9143 0.6111 0.8205 0.1262 0.4937 0.6571 1.0000 0.8077 0.4951 0.6245 0.6571 0.5556 0.4943 0.5631 0.6371 1.0000 0.6667 0.3205 0.4709 0.0464 0.4857 0.5000 0.0000 0.4854 0.1181 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (4)

where *C*\* is the referenced sequence.

$$\mathbf{C^\*} = [\mathbf{C} \mathbf{1^\*}, \mathbf{C^\*} \mathbf{2^\*}, \dots, \mathbf{C^\*}] = [1.0000 \quad 1.0000 \quad 1.0000 \quad 1.0000 \quad 1.0000] \tag{5}$$

The matrix *C* is the compared sequence, and the correlation coefficient *ξ*j(i) is obtained from Equation (6)

$$\mathcal{J}\_{\hat{\jmath}}(i) = \frac{\min\_{\hat{i}} \min\_{\hat{i}} \left| \mathbf{C}\_{i}^{\*} - \mathbf{C}\_{i}^{j} \right| + \rho \max\_{\hat{j}} \max\_{\hat{i}} \left| \mathbf{C}\_{i}^{\*} - \mathbf{C}\_{i}^{j} \right|}{\left| \mathbf{C}\_{i}^{\*} - \mathbf{C}\_{i}^{j} \right| + \rho \max\_{\hat{j}} \max\_{\hat{i}} \left| \mathbf{C}\_{i}^{\*} - \mathbf{C}\_{i}^{j} \right|} \tag{6}$$

where the value range of *ρ* is [0, 1], and generally takes the form of *ρ* = 0.5. Then, the evaluation matrix of indexes *E* is shown as follows.

$$E = \begin{bmatrix} \xi\_1(1) & \xi\_1(2) & \dots & \xi\_1(5) \\ \xi\_2(1) & \xi\_2(2) & \dots & \xi\_2(5) \\ \vdots & \vdots & & \vdots \\ \xi\_{11}(1) & \xi\_{11}(2) & \dots & \xi\_{11}(5) \end{bmatrix} = \begin{bmatrix} 0.7777 & 0.4091 & 0.3750 & 0.5049 & 0.4286 \\ 0.3333 & 0.3333 & 0.3421 & 1.0000 & 1.0000 \\ 0.4667 & 0.5294 & 1.0000 & 0.5125 & 0.5852 \\ 0.8975 & 0.5294 & 0.4756 & 0.4440 & 0.4270 \\ 0.3884 & 0.5294 & 0.3482 & 0.3333 & 0.5075 \\ 0.8537 & 0.7500 & 1.0000 & 0.3732 & 0.3333 \\ 0.8537 & 0.5625 & 0.7358 & 0.3640 & 0.4969 \\ 0.9932 & 1.0000 & 0.7222 & 0.4976 & 0.5711 \\ 0.9932 & 0.5294 & 0.4972 & 0.5337 & 0.5794 \\ 1.0000 & 0.6000 & 0.4239 & 0.4859 & 0.3440 \\ 0.4930 & 0.5000 & 0.3333 & 0.4928 & 0.3618 \end{bmatrix} (7)$$

We can then determine the entropy weight for the index *i* by assuming the number of indexes and evaluation objects are *n* = 5 and *m* = 11, respectively:

$$f\_{i\bar{j}} = \frac{\mathfrak{J}\_{\bar{j}}(i)}{\sum\_{j=1}^{m} \mathfrak{J}\_{\bar{j}}(i)}\tag{8}$$

Thus,

$$F = \begin{bmatrix} f\_{11} & f\_{12} & \dots & f\_{1n} \\ f\_{21} & f\_{22} & \dots & f\_{2n} \\ \vdots & \vdots & & \vdots \\ f\_{m1} & f\_{m2} & \dots & f\_{mn} \end{bmatrix} = \begin{bmatrix} 0.105 & 0.065 & 0.066 & 0.091 & 0.076 \\ 0.045 & 0.053 & 0.055 & 0.180 & 0.177 \\ 0.063 & 0.084 & 0.160 & 0.092 & 0.104 \\ 0.121 & 0.084 & 0.076 & 0.080 & 0.076 \\ 0.073 & 0.084 & 0.056 & 0.060 & 0.090 \\ 0.115 & 0.120 & 0.160 & 0.067 & 0.059 \\ 0.115 & 0.090 & 0.118 & 0.066 & 0.088 \\ 0.080 & 0.159 & 0.115 & 0.090 & 0.101 \\ 0.080 & 0.084 & 0.080 & 0.096 & 0.103 \\ 0.135 & 0.096 & 0.068 & 0.088 & 0.061 \\ 0.067 & 0.080 & 0.053 & 0.089 & 0.064 \end{bmatrix} \tag{9}$$

The entropy *h*<sup>i</sup> of the index *i* is shown as

$$h\_i = -k \sum\_{j=1}^{11} f\_{ij} \ln f\_{ij} \tag{10}$$

where *k* = 1/*ln*m = 1/*ln*11 = 0.417.

Thus,

H=[*h*1, *h*2,... , *h*5] = [0.98044, 0.98323, 0.96412, 0.97988, 0.97778] (11)

The entropy weight for the index *i* is

$$
\omega\_{\bar{i}} = \frac{1 - h\_{\bar{i}}}{n - \sum\_{1}^{n} h\_{\bar{i}}} \left( 0 \le \omega\_{\bar{i}} \le 1, \sum\_{i=1}^{n} \omega\_{\bar{i}} = 1 \right) \tag{12}
$$

The entropy weight matrix of various indexes is

$$\begin{array}{c} \mathcal{W} = [\omega\_1, \omega\_2, \dots, \omega\_5] \\ = [0.17076, 0.14638, 0.31319, 0.17567, 0.19400] \end{array} \tag{13}$$

Therefore, the comprehensive evaluation index is

$$R = E \times W^T$$

$$= [0.48197, 0.58253, 0.67393, 0.54055, 0.43551, 0.69897, 0.61890, 0.67207, \dots] \tag{14}$$

$$= 0.54066, 0.54344, 0.41853]$$

This means the sequence is R35-F30, R35, R35-G20F10, R35-G30, R27, R35-C16, R35- G10F20, R45, R0, R100, and R35-C10. The mixtures of R35-F30, R35 and R35-G20F10 are the relatively better groups of the eleven mixtures.

#### **5. Discussion**

Based on the test results of eleven SCCs in this study, their workability meets the requirements to be labeled as self-compacting. The SCCs marked as R27, R35, R100, R35- G30, R35-G20F10 and R35-G10F20 meet the target compressive strength of the SCC. Due to the complex influences of the properties and contents of raw materials, including fly ash, GGBS, manufactured sand, river sand and coarse aggregate, different optimal mixtures can be obtained by considering the performance. Considering the slump and the slump flow of fresh SCC, the mixtures R0, R45, R35-F30, R35-G30 and R35-C16 are optimal. Considering the flow time *T*<sup>500</sup> of fresh SCC, the mixtures R35, R35-F30 and R35-G20F10 are optimal. Considering the density of SCC, the mixtures R0, R27, R35 and R35-G10F20 are optimal. Considering the compressive strength of SCC, the mixtures R27, R35, R100, R35-G30, R35-G20F10 and R35-G10F20 are optimal.

Therefore, it is difficult to determine the optimal mixture of SCC only by the test results to satisfy all aspects of performance. In this condition, grey relational analysis is used to obtain the optimal solution among these eleven SCC mixtures after comprehensively considering the workability, density and mechanical properties of the SCCs. As a result, R35-F30, R35 and R35-G20F10 are the better of the eleven mixtures. This agrees well with the experimental results. R35-F30 has a fresh performance with a high slump of 262 mm, a high slump flow of 750 mm and a short flow time *T*<sup>500</sup> of 3.5 s, a density of 2364 kg/m3, and a compressive strength of 58.2 MPa. R35 has a fresh performance with a suitable slump of 245 mm, a high slump flow of 700 mm and a short flow time *T*<sup>500</sup> of 3.5 s, a density of 2439 kg/m3, and a high compressive strength of 73.5 MPa. R35-G20F10 has a fresh performance with a high slump of 253 mm, a high slump flow of 780 mm, a short flow time T500 of 5 s, and a density of 2433 kg/m3 and a high compressive strength of 73.0 MPa.

Given all of the above, the experimental study is suitable for comparing the effects of the raw materials on one aspect of the performance of SCCs, such as workability, density or compressive strength. The grey relational analysis used in this study is applicable to further optimize the mix proportion of SCCs while considering the comprehensive performance of the SCC.

#### **6. Conclusions**

Based on the absolute volume method of mix proportion design of SCC, eleven groups of SCCs were designed with changes in the content of fly ash and GGBS, the content of manufactured sand and river sand, and the maximum particle sizes of coarse aggregates. The design was verified by experimental studies and evaluated by grey relational analysis. Conclusions can be drawn as follows.

The replacement of manufactured sand by a certain amount of river sand could improve the flowability of fresh SCC and increase the compressive strength of hardened SCC. Both the GGBS and fly ash are beneficial for the slump and slump flow of fresh SCC; however, they also increase the flow time. The hybrid effect of GGBS with fly ash would improve the workability of SCC with equal compressive strength. The maximum particle size of the coarse aggregate significantly influenced the performance of the SCC. Compared to an SCC with a maximum coarse aggregate particle size of 20 mm, an SCC with a maximum coarse aggregate particle size of 16mm demonstrated more favorable workability, but the cubic compressive strength of SCC decreased by nearly 20%.

Based on the test results, a grey correlation analysis model was built. Combined with the entropy evaluation method, the optimal mixtures of SCC were selected from these eleven group mixtures. The optimal mixture considering the workability of fresh SCC was found to be R35-F30, which demonstrated fresh performance with a slump of 265 mm, a slump flow of 750 mm and a flow time *T*<sup>500</sup> of 3.5 s. The optimal mixture when comprehensively considering the workability and mechanical properties of SCC was found to be R35, which demonstrated fresh performance with a slump of 245 mm, a slump flow of 700 mm, a flow time *T*<sup>500</sup> of 3.5 s, a density of 2439 kg/m<sup>3</sup> and a compressive strength of 73.5 MPa. This method has improved the objectivity of the evaluation and avoided the loss of information which can be used for the optimization of mix proportions of SCC.

**Author Contributions:** Conceptualization and methodology, X.D. and M.Z.; validation, X.D. and X.Q.; formal analysis, X.Q. and M.Z.; investigation, Y.W. and M.Z.; data curation, Y.R. and M.Z.; writing—original draft preparation, X.D. and Y.W; writing—review and editing, Y.R., M.Z. and X.D; funding acquisition, X.D. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by the Natural Science Foundation of Henan, China, grant number "212300410192"; Key Scientific and Technological Research Project of University in Henan, China, grant number "20A560015".

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available in the submitted article.

**Conflicts of Interest:** The authors declare no conflict of interest.
