3.1 Model Generation by DX software
After entering the results into the software, the software suggested models based on the results of completed experimental designs. By default, Design-Expert software automatically shifts to the “Suggested” polynomial model that best fits the criteria presented in the Fit Summary section. Design-Expert calculates the Whitcomb Score (a heuristic scoring system) to choose a default model. In this study, the model suggested by the software for RFc, ∆Fc, and ∆W is a quadratic model, as presented in Tables 9-11.
Table 9 Suggested models for the prediction of RFc
Model
|
Sum of squares
|
df
|
Mean squares
|
R²
|
F-value
|
P-value
|
Linear
|
550.05
|
74
|
7.43
|
0.79
|
123.43
|
< 0.0001
|
2FI
|
406.43
|
61
|
6.66
|
0.84
|
110.64
|
< 0.0001
|
Quadratic
|
173.16
|
59
|
2.93
|
0.93
|
48.73
|
< 0.0001
|
Cubic
|
91.96
|
37
|
2.49
|
0.96
|
41.27
|
< 0.0001
|
Table 10 Suggested models for the prediction of ∆Fc
Model
|
Sum of squares
|
df
|
Mean squares
|
R²
|
F-value
|
P-value
|
Linear
|
1987.71
|
74
|
26.86
|
0.80
|
79.90
|
< 0.0001
|
2FI
|
1279.00
|
61
|
20.97
|
0.87
|
62.37
|
< 0.0001
|
Quadratic
|
680.68
|
59
|
11.54
|
0.93
|
34.32
|
< 0.0001
|
Cubic
|
218.15
|
37
|
5.90
|
0.97
|
17.54
|
< 0.0001
|
Table 11 Suggested models for the prediction of ∆W
Model
|
Sum of squares
|
df
|
Mean squares
|
R²
|
F-value
|
P-value
|
Linear
|
187.86
|
74
|
2.54
|
0.83
|
22.40
|
< 0.0001
|
2FI
|
56.94
|
61
|
0.93
|
0.94
|
8.24
|
< 0.0001
|
Quadratic
|
49.65
|
59
|
0.84
|
0.95
|
7.43
|
< 0.0001
|
Cubic
|
36.34
|
37
|
0.98
|
0.96
|
8.67
|
< 0.0001
|
RFc, ∆Fc, and ∆W are predicted by a second-order model in the form of a quadratic polynomial equation:
Equation (3)
where Y is the predicted response variable, β0, βi, βii, and βij are constant regression coefficients of the model, and xi, xj (i, j= 1, 2, 3, 4; i≠j) represent the coded values of independent variables used in statistical calculations. The relationship between the coded and actual values is as the following equation:
Equation (4)
where Xi is the actual value of the variable, X0 is the actual value of Xi at the center point, and ∆X is the step change of the variable.
Quadratic models for all three response variables, RFc, ∆Fc, and ∆W, are presented in Tables 12-14. For each of the response variables (RFc, ∆Fc, and ∆W), nine different models are presented according to the values of AE and Et.
Table 12 The prediction of RFc based on the suggested model
AE
|
Et
|
C
|
α
|
γ
|
δ
|
ε
|
ζ
|
Cen
|
7
|
35.91
|
0.13
|
10.73
|
0.032
|
-0.003
|
-3.79
|
28
|
35.42
|
0.09
|
11.44
|
0.032
|
-0.003
|
-3.79
|
56
|
35.52
|
0.08
|
11.43
|
0.032
|
-0.003
|
-3.79
|
PGW
|
7
|
38.36
|
0.14
|
8.74
|
0.032
|
-0.003
|
-3.79
|
28
|
38.12
|
0.10
|
9.46
|
0.032
|
-0.003
|
-3.79
|
56
|
38.31
|
0.09
|
9.45
|
0.032
|
-0.003
|
-3.79
|
Aen
|
7
|
26.94
|
0.13
|
12.34
|
0.032
|
-0.003
|
-3.79
|
28
|
23.59
|
0.09
|
13.05
|
0.032
|
-0.003
|
-3.79
|
56
|
23.75
|
0.09
|
13.04
|
0.032
|
-0.003
|
-3.79
|
Table 13 The prediction of ∆Fc based on the suggested model
AE
|
Et
|
C
|
α
|
γ
|
δ
|
ε
|
ζ
|
Cen
|
7
|
63.87
|
-1.28
|
-24.99
|
0.031
|
0.01
|
5.13
|
28
|
62.98
|
-1.16
|
-26.23
|
0.031
|
0.01
|
5.13
|
56
|
61.96
|
-1.14
|
-26.06
|
0.031
|
0.01
|
5.13
|
PGW
|
7
|
56.06
|
-1.28
|
-19.87
|
0.031
|
0.01
|
5.13
|
28
|
54.66
|
-1.17
|
-21.11
|
0.031
|
0.01
|
5.13
|
56
|
53.42
|
-1.14
|
-20.94
|
0.031
|
0.01
|
5.13
|
Aen
|
7
|
77.24
|
-1.21
|
-27.23
|
0.031
|
0.01
|
5.13
|
28
|
83.12
|
-1.10
|
-28.47
|
0.031
|
0.01
|
5.13
|
56
|
81.94
|
-1.07
|
-28.30
|
0.031
|
0.01
|
5.13
|
Table 14 The prediction of ∆W based on the suggested model
AE
|
Et
|
C
|
α
|
γ
|
δ
|
ε
|
ζ
|
Cen
|
7
|
-7.28
|
0.27
|
1.95
|
-0.02
|
-0.001
|
-0.36
|
28
|
-4.04
|
0.25
|
1.66
|
-0.02
|
-0.001
|
-0.36
|
56
|
-3.38
|
0.25
|
1.98
|
-0.02
|
-0.001
|
-0.36
|
PGW
|
7
|
-12.03
|
0.28
|
4.21
|
-0.02
|
-0.001
|
-0.36
|
28
|
-7.69
|
0.26
|
3.92
|
-0.02
|
-0.001
|
-0.36
|
56
|
-5.42
|
0.26
|
4.24
|
-0.02
|
-0.001
|
-0.36
|
Aen
|
7
|
-0.85
|
0.24
|
0.97
|
-0.02
|
-0.001
|
-0.36
|
28
|
5.65
|
0.22
|
0.68
|
-0.02
|
-0.001
|
-0.36
|
56
|
6.19
|
0.22
|
1.00
|
-0.02
|
-0.001
|
-0.36
|
In Tables 12-14, the terms A and B represent the first (T) and the second (S/M) variables, respectively.
Response variable = C+ α.A + γ.B + δ.AB + ε.A2 + ζ.B2
According to Figure 2, the low distribution of the tested points compared to the predicted line can be observed for the model. One of the main parts of studies of this type is to test the validity of the models presented by the software. For this purpose, an optimal sample with T = 45.86 and S/M = 2.28 was made to be placed in Aen for 7 days. After obtaining the results of the sample made in the laboratory, a very good agreement was observed between the result of the proposed model by the software and the tested sample. According to the model, the results were RFc = 38.74 MPa, ∆W = 5.04%, and ∆Fc = 10.91%, and the results of the tested sample were RFc = 40.01 Mpa, ∆W = 5.24%, and ∆Fc = 11.45%. Therefore, the differences in the results of the tested sample from the modeled sample were 3.17%, 3.96%, and 4.94%, respectively.
After confirming the validity of the models provided by the software, it is possible to evaluate the effect of independent variables on response variables. In this study, the graphs are presented in three categories based on the aggressive environment. Three graphs are presented with different Ets for each response variable, and the effect of two numerical variables is illustrated on that response variable. For convenience in comparing modeled samples with different (S/M) and different (T) samples, it is necessary to introduce the abbreviations of the modeled samples, which are given in Table 15.
Table 15 Abbreviations for all model-based samples
T (°C)
S/M
|
40
|
60
|
80
|
1
|
MGPC1-40
|
MGPC1-60
|
MGPC1-80
|
2
|
MGPC2-40
|
MGPC2-60
|
MGPC2-80
|
3
|
MGPC3-40
|
MGPC3-60
|
MGPC3-80
|
3.2 Residual compressive strength of control samples
As mentioned before, it is essential to make control samples (OPC) to have criteria for evaluating the performance of MGPC samples. The RFc values of OPC in intervals of 0, 7, 28, and 56 are presented in Figure 3 to represent the criteria for MGPC and OPC comparisons. The compressive strength in time 0 is the 28-day compressive strength of mortar sample that hasn’t been exposed to aggressive environment yet.
According to the output model of DX software, all graphs are categorized based on the aggressive environment in which samples are placed.
3.3 Model-based MGPC samples exposed to the chloride environment
Figure 4 shows the effect of numerical variables on the RFc value of MGPC samples in the chloride environment.
Figure 4 Values of RFc for MGPC samples exposed to the chloride environment at Et = 7(a), 28 (b), and 56 (c) based on the proposed model
According to Figure 4, the best RFc for all three values of Et belongs to the MGPC2-40 sample. The RFc value for the MGPC2-40 sample reached 44.4 MPa after 56 days of exposure to Cen. RFc of OPC in Et = 56 is equal to 47Mpa, and the superiority of OPC over MGPC is valid for all Et’s. Following the RFc values, the values of ∆Fc and ∆W for model-based MGPC samples and actual control samples are represented in Figures 5 and 6.
The lowest decrease in compressive strength after 56 days of exposure to Cen is related to MGPC2-60 sample. Figure 5 shows MGPC2-40, MGPC2-60, and MGPC3-60 samples with less compressive strength reduction than OPC samples, which is related to the low permeability of these samples. At Et =7, the value of ∆Fc is -1/91 and this indicates an increase in compressive strength. According to a study by Albitar, all types of concrete may experience an initial increase in compressive strength when exposed to an aggressive environment [7]. Albitar attributes this initial increase to the hydration process of calcium silicate and other pozzolanic reactions. Following these reactions and the pressure caused by the expansion of the elements, internal blockage occurs in the concrete structure, which has a positive effect on the increase of compressive strength.
Among all MGPC’s, in 56 days, the lowest ∆W value is related to MGPC3-40. In terms of weight change, MGPC shows more weight reduction and thus poorer performance than OPC concrete after 28 and 56 days exposure to Cen.
3.4 Model-based MGPC samples exposed to the acidic environment
According to Figure 7, the highest RFc is related to MGPC2-40 for all Et values. MGPCs at S/M = 2 show the highest value. The RFc value for the MGPC2-40 sample decreased to 36.8 MPa after 56 days of exposure to Aen, while RFc decreased to 39.17 MPa for OPC. In Aen, the superiority of OPC over MGPC about the RFc value is valid for all Et values.
In Figure 8, after 56 days of exposure, the lowest decrease in compressive strength among MGPC’s is related to the MGPC3-40 sample. In these graphs, all MGPC samples show a greater reduction in compressive strength than OPC samples. Among MGPC samples, MGPC3-40 shows a very close compressive strength reduction to OPC samples.
The lowest ∆W among all MGPCs is related to MGPC3-80 after 56 days of exposure. Figure9 (a) shows the better performance of MGPC concretes featuring S/M = 3. By comparing these graphs with those related to ∆Fc, it can be concluded that the better performance of MGPC in ∆W graphs can be due to weight gain resulting from the entry of the solution material into the concrete, although the result of weight change shows weight loss. In fact, the more compressive strength reduction of MGPC than OPC at Aen and Et =7 indicates that the low weight loss of MGPC specimens with S/M = 3 compared to OPC occurred only due to weight gain.
3.5 Model-based MGPC samples exposed to PGW
As shown in Figure 10, the maximum RFc value belongs to MGPC1-40. The value of RFc is maximum at S/M = 1 or 2 depending on Et and T. the RFc value for the MGPC1-40 sample decreased to 44.26 MPa after 56 days of exposure to PGW while the RFc for OPC is 45.17 Mpa after 56 days of exposure. Regarding RFc, the superiority of OPC over MGPC in PGW is valid for all Et values.
After 56 days, the lowest ∆Fc in PGW is related to the MGPC2-60 sample. In Figure11, the MGPC2-60 and MGPC2-40 samples show a lower reduction in compressive strength than that of OPC samples.
The value of ∆W = -0.33% for MGPC1-40 indicates an increase in sample weight after 7 days, and this finding is in accordance with Albitar's research in which a slight increase occurred in weight followed by a decrease [7]. This initial weight gain may be due to the entry of chemical particles existing in the aggressive solution into the MGPC pores.
After observing the graphs, it can be generally argued that with an increase in T a decrease happens in RFc in all AEs and for all values of Et. An increase in T affects both the initial compressive strength (28 days) and the change in compressive strength. According to studies by Rovnanik, large porosities are formed in MGPC samples and the volume of cumulative porosity is increased by increasing the T value above 60 °C, which has a negative effect on the mechanical properties of MGPC at the age of 28 days [16]. Okoye concluded that this increase in porosity affects the permeability of the specimens, and an increase in porosity is causes an increase in permeability [14]. In this study, increasing the temperature reduced the initial compressive strength(28 days) and increased the porosity. thus, the increased porosity of concrete increased its permeability in exposure to aggressive environment, and the increase in the permeability of concrete causes more decreases its compressive strength in exposure to aggressive environments. According to Albitar's research, the reduction of GPC resistance can be due to the leakage of alkaline-activating substances, such as NaOH, in the corrosive solution [7].
3.6 Optimization
The optimization module of DX software searches for a combination of factor levels that simultaneously satisfy the criteria defined for each response and factor. To include a response in the optimization criteria, it must have a model fit through analysis or be supplied via an equation-only simulation. Moreover, factors are automatically included “in range. Numerical optimization is a hill-climbing technique. In addition to the design points, a set of random points are examined to see if there is a more desirable solution. Finding an initial feasible region can be difficult. We start with a small value of a penalty function in a downhill simplex (Nelder-Mead) multi-dimensional pattern search, which converges at either a fixed point or a design space boundary. In this study, it is desirable that RFc would be the maximum, and ∆Fc and ∆W would be the minimum. According to the suggested models and the defined desirability for each response variable, the optimal values for the independent variables in each of AE and Et are represented in Table 16.
Table 16 Suggested values for producing optimal samples and predicted values for the response variables of each sample
Categorical variable
|
Numerical variable
|
Response variable
|
AE
|
Et
|
T
|
S/M
|
RFc
|
∆W
|
∆Fc
|
Cen
|
7
|
45.86
|
1.95
|
45.04
|
2.34
|
0.18
|
28
|
45.86
|
2.11
|
43.70
|
4.08
|
1.51
|
56
|
45.86
|
2.07
|
43.60
|
5.38
|
2.12
|
PGW
|
7
|
45.97
|
1.42
|
45.42
|
1.39
|
2.80
|
28
|
45.86
|
1.39
|
44.35
|
4.47
|
5.09
|
56
|
45.86
|
1.29
|
44.15
|
6.90
|
5.78
|
Aen
|
7
|
45.86
|
2.28
|
38.74
|
5.04
|
10.91
|
28
|
45.86
|
2.49
|
34.40
|
9.65
|
18.68
|
56
|
45.86
|
2.52
|
34.12
|
10.88
|
19.03
|
As shown in Figure13, there is a slight difference between the RFc values of OPC control samples and model-based optimized MGPC samples, and no difference of more than 5 Mpa is observable. Since all samples conform to the criteria accepted by the world civil society, such as ACI318 and ASHTO, it is feasible to use optimized MGPC samples as an alternative to OPC samples. Following the RFc graphs, ∆Fc graphs are presented as the criteria to evaluate the durability performance of samples. As shown in Figure 11 (c), the model-based optimized MGPCs have better performance over OPC control samples. According to Figure 11 (b), ∆W values for MGPC samples are higher than those for OPC samples, As explained before, the lower ∆W of OPC compared to MGPC samples is attributed to the more porous microstructure and, consequently, more pore solution content of OPC control samples.