3.1. Phase I: Mix Design Optimization through the Design of Experiments Approach
Cement concrete mixtures are composed of various ingredients, such as cement, aggregates, additives, and water, where all components add up to 100%. The composition of aggregates including coarse, fine, and fillers of concrete mixtures also follows the same principle, i.e., the sum of the components equals 1 (100%). An effective method of studying such blends, where changes in the mixture composition could affect various outcomes (responses), is by applying the mixture design method. Consequently, the effect of various aggregate compositions (gravel, sand, and silt) on the splitting tensile strength (T) of cement concrete paving blocks was studied using the DOE method.
In this study, the input variables were gravel, sand, and silt (%), and the T of the resulting mixtures was selected as the response. To provide a model, the Design of Experiments indicated 12 randomized runs each with a different aggregate gradation. T was obtained based on the EN 1338 standard for each mixture and the average values were included in the DOE (
Table 4).
The highest value for T was obtained for the ninth sample with 0% silt (2.8 MPa), whereas the lowest value (0.4 MPa) was related to the first mixture with 40% silt. Furthermore, the usage of a higher amount of silt increased the water-to-cement ratio of the mixture, limiting the compaction rate. Low T values were also observed for samples lacking coarse aggregates (gravel), such as samples 2, 6, and 12. A good gradation curve is vital to having desirable strength. A very dense or gapped gradation could tamper with the final compaction, resulting in lower mechanical values.
The summary of fit for the produced model is presented in
Table 5, indicating an R
2 value of 90.55. The produced model was then used to predict the corresponding T value for each of the 12 mixtures. The actual versus the predicted T values are depicted in
Figure 4, where the blue line indicates an average T of approximately 1.1 MPa.
Table 6 shows the significance of adding a term to a model given that the other terms are already entered. For instance, sand added the highest significance to the model, followed by gravel. On the other hand, the silt parameter did not have a significant effect on the model (
p < 0.05). In such cases, one could eliminate the insignificant parameter from the model, improving the overall accuracy. However, the elimination of insignificant parameters is not possible for the mixture design approach. A concrete mixture is composed of different ingredients, such as cement, water, and aggregates. By removing any of these components from the design, the produced outcome will no longer be a concrete mixture. Thus, the silt parameter was kept in the model. Moreover, the coefficient of the factors (gravel, sand, and silt) is indicated in the T ratio column of
Table 6. It is observable that silt had a negative value of −0.61, indicating that an increase in the silt content decreased the overall strength of the concrete bricks. This occurred because a high amount of silt increases the need for water in the mixture, which results in lower strength values. Overall, the produced model showed a very high significance value (
p < 0.0002), as indicated by the analysis of variance (
Table 7).
The relationship between aggregate composition and T was studied by producing contour plots (
Figure 5). The final percentages of silt, sand, and gravel directly affected the final strength of the brick samples. In both cases (
Figure 5a, b), the highest value for T was obtained when the percentage of silt was below 5% (total aggregate weight). Moreover, to reach the maximum strength (T > 2.5 MPa), the amount of sand and gravel need to be approximately between 60 and 65% and 33 and 38%, respectively. The silt had a negative interaction with both sand and gravel, i.e., an increase in silt content resulted in lower T results. This is also observable in the contour plots, where the application of more than 30% of silt dramatically decreased the strength.
One of the most important goals when conducting a Design of Experiments through JMP software is to benefit from its powerful optimization tools. The profiler option provides a cross-section of the model, where the curve of each factor shows the relationship with the response based on the levels of other factors. In model optimization, various factors could be considered. One approach is to use the profiler to maximize or minimize the desirability, that is, by simultaneously changing each available variable factor to achieve the target value. However, some design blends may have costly ingredients, or, on the contrary, may include waste/by-products that need to be recycled into certain products. In such scenarios, the variable factor could be fixed at a desired value and the corresponding outcome or response would be optimized accordingly. The current research aimed to maximize the amount of recycled silt in the production of concrete paving blocks. Thus, the profiler tool in the JMP software was used to optimize the concrete mixture accordingly (
Figure 6). The optimum blend had an approximate T of 1.49 MPa, resulting from a mixture consisting of 15, 65, and 20% gravel, sand, and silt, respectively. The amount of silt was fixed at 20% and the remaining parameters were calculated accordingly. In
Figure 6, the values presented in the brackets (1.2112, 1.76643) are the 95% confidence intervals for the tensile splitting strength values. The dotted red lines indicate the selected value for each parameter and the resulting outcome (T) of the model. The slope of the factors represents the coefficient of the parameters, which is also tabulated in
Table 6.
Figure 6 also includes the graphs related to the desirability factor of the mathematical model. The desirability function normalizes the responses from the lowest (0) to the highest obtainable value (1). From the desirability graphs (
Figure 6) it is concluded that by using the 20% silt in the mixture, only 45.5% of the possible T values were obtained.
To further understand the model and study the relationship between different variables and their effect on the final strength, a ternary plot was produced (
Figure 7).
The white section represents the model area. However, the constraints applied to the variable parameters are shown in the grayed-out section of the plot. The splitting tensile strength obtained through the mechanical test for each sample (run) is designated in alphanumerical labels in the ternary plot. The labels are also included in
Table 4. The marker indicates the area in the plot where the optimum mixture was achieved. In addition, the red-dotted line corresponds to a strength equal to 1.489 MPa, as obtained in
Figure 6. The ternary plot also shows that a decrease in silt content and an increase in sand content increased the overall strength of the samples. The plot shows that the same value of 1.489 MPa was achievable using less silt, that is, 8.5% instead of 20%. However, this would lead to recycling less silt into concrete paving blocks.
All in all, according to the profiler optimization, the final values selected for gravel, sand, and silt were calculated as 15, 65, and 20%, respectively. The final mixture was conducted based on the optimized aggregate gradation, keeping the aggregate-to-cement ratio at 4.62. Similar to the production of the first mixtures, enough water was added to achieve slump values close to 0.