# Use of Eoofriendly Glass Powder Concrete in Construction of Wind Farms
## 3 Results and Discussion
### Mechanical Properties of Concrete
#### 3.3.1 Mathematical Analysis of the Experimental Results  
A mathematical model has been proposed to further analyze the experimental data. The results obtained fit the following equation:  
\[Cs=Cs_{0}\bigg{(}1-exp\bigg{(}-\bigg{(}\frac{t}{\tau}\bigg{)}^{n}\bigg{)}\bigg{)} \tag{1}\]  
close to the one that defines a first order system in classical mechanics (a similar model has been used by [PERSON] et al. [37] In the above equation, \(Cs\) is the compressive strength, \(Cs_{0}\) is the final stable value of the compressive strength, \(t\) stands for the number of days after setting, \(n\) is a shape constant, and \(\tau\) is a characteristic time that represents the time when the compressive strength reaches a 63% of its final value. The results of the fittings, together with the Root Mean Squared Error, RMSE:  
\[\text{RMSE}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}(\left.Cs\right|_{measured}-\left.Cs \right|_{calculated})^{2}} \tag{2}\]  
are presented in Table 5. The results of the modeling have been plotted in Figure 5 with dashed lines.  
Figure 5: Evolution over time after setting of concrete compression strengths, \(Cs\), for different cases studied (see Tables 1 and 4).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **Control** & **G15** & **G30** & **G45** & **G60** & **G80** \\ \hline \(Cs_{0}\) (MPa) & 41.89 & 33.237 & 31.978 & 26.964 & 16.618 & 11.011 \\ \(\tau\) (d) & 1.5979 & 2.0105 & 2.4732 & 3.3509 & 9.3934 & 59.627 \\ \(n\) & 4.1999 & 3.4191 & 9.4455 & 10.810 & 32.400 & 62.276 \\ RMSE (MPa) & 0.608 & 0.364 & 1.025 & 0.634 & 0.107 & 0.226 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Coefficients of Equation (1), \(Cs_{0}\), \(\tau\), and \(n\), fitted to the measured values of the compressive strength, for each one of the studied cases (see Table 4), together with the Root Mean Squared Error (RMSE) of each fitting.  
The model has proven to simulate quite well the results, as the errors were between 0.65% and 3.21% in relation to the calculated values of \(Cs_{0}\). Results of the model fitting to the \(G_{15}\) sample were slightly different, as the shape of its curve seemed to detach from the general pattern shown by the curves fitted to the data from the other cases. In Figure 6, the coefficients of Equation (1) extracted from the different samples data are plotted in relation to the amount of glass powder used to replace the cement, \(Gp\). The following equations were fitted to the results (see the dashed lines in graphs from Figure 5):  
\[Cs_{0}=41.54-38.06C_{s}\ \left(R^{2}\ =\ 0.967\right) \tag{3}\]  
\[\tau=2.8477+46.875C_{s}^{2}+73.28C_{s}^{4}\ \left(R^{2}\ =\ 0.986\right) \tag{4}\]  
\[n=0.3266+1.0277C_{s}^{2}172\ \left(R^{2}\ =\ 0.857\right) \tag{5}\]  
From Figure 5, it can be seen that the coefficient \(n\) from the \(G_{15}\) sample did not fit Equation (5), making the fitting of Equation (1) to this sample data (see Figure 5) slightly different from the general pattern. This might indicate that the normal errors present in the process of manufacturing the samples has larger effects for lower quantities of glass powder replacing the cement (that is, for lower values of \(Gp\)). Finally, with the coefficients extracted from the above equations for \(Gp\) = 15, the modeling of the sample G15 has been calculated and plotted in Figure 5 with a solid line. The fitting was worse to the experimental data than the one obtained from the direct fitting of Equation (1), with the error being larger (RMSE = 4.32%).