Statistical linear regression models were built for the properties (dependent variables) of binary binder systems, in order to predict these properties for any fly ash (similar to those used in the study) based on the fly ash’s fundamental physical and chemical
characteristics. The fundamental characteristics on which the models were built (independent variables) are listed in Table 5.1. This table also lists the abbre- viations used to label these variables in the models.
The selected independent variables were all known to play a role in the outcome of the dependent variables and the effects are mentioned in Chapter 2.
Separate experimental designs and modeling proce- dures were adopted respectively for the binary and the ternary paste systems. This is because the number of data points (cement + fly ash, binary combinations) available for the binary models was 20 (13 Class C ashes and 7 Class F ashes), whereas the number of data points (cement + fly ash + fly ash, ternary combina- tions) or the number of possible combinations of fly ashes in ternary paste systems were 180 (number of combinations of choosing two fly ashes out of twenty when the proportion of the two chosen ashes is a constant, is 20C2 5 180). Performing the number of
experiments for as many combinations of ternary binder systems is not practically feasible. Hence, a different experimental design (fractional factorial design) was used which allowed to reduce the number of experiments in the ternary systems to nine.
The aim of the modeling process was to use statistical linear regression analysis to identify the best set of inde- pendent variables, which affect a dependent variable (property of the binder) of both binary and ternary paste systems, the most.
The modeling process was not a straightforward linear regression analysis, as it was assumed that the single model to predict the properties for the entire suite of fly ashes might not be feasible. The reasons are as follows.
1. The set of fly ashes used in the study contain two different kinds of ashes, ASTM Class C ashes and ASTM Class F ashes. The ashes were markedly different in their fun- damental physical and chemical compositions and hence, it is likely that their behavior in concrete might be different.
2. The available number of data points for modeling the set of ashes is similar to the number of independent (predictor) variables available to explain the variations in the dependent variables. More so, the number of predictor variables is greater than the number of data points available for Class F ashes.
To counteract the above two challenges, the follow- ing modeling methodology was adopted.
A linear regression analysis was performed on the dependent variables using Statistical Analysis Software (SAS), which included all the twenty data points. The ‘‘best set of variables’’ (which constitute the ‘‘best model’’) found to affect the dependent variable was chosen based on the highest adj-R2of the models. All the data points were in turn predicted using the same models (using the same ‘‘best set of variables’’) built for the dependent variable for the thirteen data points of Class C ashes and seven data points of Class F ashes separately. A plot of the observed and the pre- dicted data values, each for the results obtained for all the data points of Class C and Class F ashes was
plotted. If the prediction of the observed points is accurate, the points on this graph lie close to the 45u line drawn from the origin. The above-mentioned technique is clearly described in the form of a flow chart, Figure 5.1. The trustworthiness of the predictions can be evaluated by using the p-value of the model. Nevertheless, all the regression models were tested by obtaining the dependent variable data for new fly ashes and were validated.
The number and set of variables used to predict the dependent variables (model containing the ‘‘best set of variables,’’ referred to as the ‘‘best model’’) were kept the same for the models of both the classes and at three (with a maximum of four in special cases) for the following reasons.
1. As the number of data points in the models was small (13 for Class C ashes and 7 for Class F ashes), an increase TABLE 5.1
Independent variables used in the modeling process and their abbreviations.
Variables Abbreviations
Physical Properties Mean particle size meansize
Specific surface area measured using Blaine’s apparatus blaines Specific surface area measured using laser particle size analyzer spsurface
Chemical Properties Calcium oxide content cao
Sum of silicon, aluminum and iron oxide contents SAF
Magnesium oxide content mgo
Aluminum oxide content alumina
Sulfate content sulfate
Physico-chemical Properties Loss on ignition carbon
Glass content measured using X-ray diffraction glass
in the number of variables used to describe the variation in the dependent variable would lead to a good fit in the data, but an insignificant model. This would reflect in the ability of the model to predict the dependent variable for a new fly ash, which was not used as a data point in the modeling process.
2. The same set of variables were adopted in the models used to predict the dependent variables in both Class C and Class F ashes because the models which were used to predict the properties of the ternary paste systems are based on a linear relationship between the two binary paste models. In addition, the experimental design for modeling the ternary paste systems (see Chapter 6) involves the use of the variables used in binary paste regression models.
3. An increase in the number of variables used to predict the properties usually leads to
i. A larger number of experiments, which need to be performed for the ternary paste systems based on the experimental design.
ii. The added variable being rendered insignificant compared to the original set of variables.
The statistical modeling of various properties (depen- dent variables) of binary paste systems is explained in the following sections. The analysis of the data includes a table containing the sets of variables of linear regression models, sorted in terms of adjusted R2, and the chosen model with three/four independent variables is highlighted (if present). A table with the predictions of the original data points is included along with a graph showing the deviations of the predictions from the ob- served values.
5.3 Analysis of Results for the Dependent Variables