3.1. Development and Analysis of the Regression Model
The experimental results for the actual mix proportions of the composite cementitious material are presented in
Table 8. The corresponding data from
Table 8 were input into Design-Expert software, which processed and analyzed the experimental data using the least squares method. The regression analysis produced the response surface equation, as shown in Equation (2).
An analysis of variance (ANOVA) was conducted to examine the sources of error in the regression model and to evaluate its overall significance. The results are presented in
Table 9. As shown in the table, the overall F-value of the constructed response surface regression model was 8.48, and the corresponding
p-value was 0.0175, which is well below the significance level (
p < 0.05). These results indicate that the model is highly statistically significant, with only a 1.75% probability that the observed effects are due to random noise. This indicates that the model demonstrates a strong fitting ability for the response values within the range of the experimental data and effectively captures the regular variation in the dependent variables with respect to the independent variables. Meanwhile, the model’s lack of fit F-value was 0.2626, and the corresponding
p-value was 0.8498, which is considerably higher than 0.1. This suggests that the lack of fit term is not statistically significant, indicating that the deviation from pure error can be attributed to random variation rather than model underfitting. In summary, the regression equation developed in this study demonstrated strong statistical significance, and its lack-of-fit term was insignificant. These results indicate that the model exhibits high fitting accuracy and reliability within the current design space and can be effectively used for subsequent response prediction and parameter optimization.
The error analysis results of the fitted model are shown in
Table 10. A correlation coefficient (R
2) closer to 1 indicates a stronger correlation and higher model reliability. When the adjusted R
2 is ≥0.85, the fitted equation can be regarded as effectively representing the relationship between factors and variables. To ensure experimental precision, the coefficient of variation (C.V.) should be less than 10%, and the signal-to-noise ratio should exceed 4. As shown in
Table 10, the R
2, adjusted R
2, and predicted R
2 are 0.9521, 0.9042, and 0.8153, respectively. The C.V. is 1.89%, and the signal-to-noise ratio is 11.5515, indicating that the regression model possesses high accuracy and strong reliability. By integrating the analysis of variance results in
Table 9 with the error analysis in
Table 10, it can be concluded that the response surface regression model demonstrates strong statistical significance, excellent fitting performance, and robust predictive capability within the current range of factor levels. Therefore, this model can serve as a theoretical foundation and analytical tool for optimizing the mix design of composite cementitious materials, providing reliable support for subsequent strength regulation, parameter optimization, and performance prediction.
By generating three-dimensional response surface plots and two-dimensional contour maps, the effects of various factors on the 28-day compressive strength of the composite cementitious material, as well as their interactions, can be visually illustrated. As shown in
Figure 10, the interaction effects among the different components on the 28-day compressive strength exhibit the following trend: In the region with low SS content, the compressive strength decreases slightly as the CFS content increases (and the DG content decreases), but the overall increasing trend is not pronounced. This indicates that the interaction between CFS and DG has a relatively weak effect on strength in this region, mainly because CFS possesses low hydration activity and the calcium ion concentration in the system is limited. Therefore, although the CFS dosage increases, the improvement in specimen strength remains negligible. In the region with high SS content, the compressive strength of the specimens increases markedly with increasing CFS content, and the response surface displays a steep slope. This suggests that as the CFS content rises, the specimen strength progressively improves. This behavior is primarily attributed to the high SS dosage, which supplies sufficient calcium ions to promote the reaction between the sulfate ions in DG and the active silica and alumina in CFS, leading to the formation of ettringite. The formation of ettringite activates the cementitious properties of CFS, thereby effectively enhancing the specimen strength. In the region with low DG content, as the SS content decreases (and the CFS content increases), the response surface exhibits slight fluctuations, but the overall trend in specimen strength is downward. This suggests that the reduction in high-calcium SS content is the primary reason the cementitious activity of CFS cannot be effectively activated. In the region with high DG content, the response surface displays an arch-shaped pattern, with specimen strength initially increasing and then decreasing as SS content rises. This behavior is primarily attributed to the excess sulfate ions supplied by high DG content, which, under conditions of elevated SS content, promote the formation of excessive ettringite, thereby reducing the specimen strength. Across the full range of CFS content, the response surface exhibits minor fluctuations in some regions but shows an overall downward trend. In summary, SS plays a dominant role in the activity of the composite cementitious system, while SS and CFS demonstrate a strong synergistic contribution to strength development. However, the dosage of DG should be carefully controlled to avoid excessive use.
Using Design-Expert software and regression equation predictions, the optimal mix proportion of the composite cementitious material was determined as 46.5% SS, 41.5% CFS, and 12% DG. When converted to the practical mix proportion, this corresponds to 37.2% SS, 33.2% CFS, 9.6% DG, 20% cement, 4% sodium silicate, and 0.1% superplasticizer. At this mix proportion, the predicted 28-day compressive strength reaches 41.2 MPa, indicating that the composite cementitious material possesses excellent mechanical properties.
To further validate the predictive accuracy and applicability of the regression model, composite cementitious mortar specimens were prepared according to the optimal mix proportion obtained from the optimization design (37.2% SS, 33.2% CFS, 9.6% DG, 20% cement, 4% sodium silicate, and 0.1% superplasticizer). After 28 days of curing in accordance with standard procedures, compressive strength tests were conducted. To minimize the influence of random errors on the results, three parallel tests were carried out. The test results are presented in
Table 11. The compressive strengths of the three specimens were 40.3 MPa, 39.6 MPa, and 42.6 MPa, yielding an average value of 40.8 MPa. The measured strength of the composite cementitious material exhibits only a slight deviation from the predicted value of the response surface regression model (41.2 MPa), demonstrating that the model developed in this study has strong predictive accuracy and practical engineering applicability.