## 3 Materials and Methods  
### Formulation of Analytical Models  
In the present study, a design-oriented strain model was proposed based on a large experimental database of 678 confined concrete cylinders (S2 datasheet). The statistical information of the database is shown in Table 2. The whole process was performed using curve fitting technique giving the best \begin{table}
\begin{tabular}{l l l} \hline \hline \multicolumn{1}{c}{**Model**} & \multicolumn{2}{c}{**Expression for Strain of Confined Concrete**} \\ \hline [PERSON] and [PERSON] [15] model & \(\varepsilon_{\alpha}\) = \(0.002+0.001\frac{F_{\alpha}t}{D_{\alpha}^{2}}\) & (7) \\ [PERSON] et al. [16] model & \(\frac{\varepsilon_{\alpha}}{\varepsilon_{\alpha}}\) = \(1+5(\frac{F_{\alpha}}{f_{\alpha}}-1)\) & (8) \\ [PERSON] and [PERSON] [17] model & \(\varepsilon_{\alpha}\) = \(\varepsilon_{\alpha}+0.01\frac{F_{\alpha}t}{f_{\alpha}}\) & (9) \\ [PERSON] et al. [18] model & \(\varepsilon_{\alpha}\) = \(\frac{f_{\alpha}-f_{\alpha}}{f_{\alpha}}\) & (10) \\  & where \(f_{\alpha}\) = \(0.872f_{\alpha}^{\prime}+0.371f_{l}+6.258\) & \\  & and \(E_{2}\) = \(245.61f_{\alpha}^{\prime\alpha\,0.2}+1.3456\frac{F_{\alpha}t}{D}\) & \\ [PERSON] [19] model & \(\frac{\varepsilon_{\alpha}}{\varepsilon_{\alpha}}\) = \(1+(310.57\varepsilon_{h,\,\mathit{rup}}+1.90)(\frac{F_{\alpha}t}{f_{\alpha}}-1)\) & (11) \\ [PERSON] and [PERSON] [20] model & \(\frac{\varepsilon_{\alpha}}{\varepsilon_{\alpha}}\) = \(1.75+12\varepsilon_{h}p_{\alpha}^{1.45}\) & (12) \\ [PERSON] et al. [21] model & \(\frac{\varepsilon_{\alpha}}{\varepsilon_{\alpha}}\) = \(1.75+6.5\varepsilon_{k}^{0.8}\rho_{\varepsilon}^{1.45}\) & (13) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Previous strain models for confined concrete. Figure 1: Confinement mechanics of fiber-reinforced polymer (FRP) sheets. fit after minimizing some statistical parameters such as R\({}^{2}\), RMSE, and the sum of squared errors (SSE) between the predictions of previous models and experimental values. The evaluations of strain models given by [PERSON] and [PERSON] (1958), [PERSON] et al. (2016), [PERSON] and [PERSON] (2017), [PERSON] et al. (2018), [PERSON] (2019), [PERSON] and [PERSON] (2020) and [PERSON] et al. (2021) were performed using the collected database. The models selected for evaluation have wider use in literature. The statistical parameters R\({}^{2}\) and RMSE are represented by Equations (12) and (13), respectively. \[R^{2}\ =\ (\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{\big{[}n\sum x^{2}-(\sum x )^{2}\big{]}[n\sum y^{2}-(\sum y)^{2}}\big{]}^{2}} \tag{5}\]  
\[RMSE\ =\ \sqrt{\frac{\sum{(x-y)}^{2}}{n}} \tag{6}\]  
Where n is number of data points; x is the experimental value and y is the predicted value of confined concrete strain. The value of parameter RMSE varies from zero to one giving a perfect prediction at zero. Similarly, the value of parameter R\({}^{2}\) also varies from zero to one; one shows a perfect correlation and zero represents that there is no correlation between experimental and predicted values of strains. The difference between experimental and predicted values was taken as error and SSE was minimized to achieve the best fit. A smaller value of SSE shows a tight fit of the empirical model to the experimental data. The performances of the previously proposed confined concrete strain models with R\({}^{2}\) and RMSE were represented in Figure 2. \begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline  & \(D\) (mm) & \(H\) (mm) & \(nt\) (mm) & \(Es\) (GPa) & \(\mathbf{f^{\prime}_{co}}\) (MPa) & \(\mathbf{\varepsilon_{co}}\) (\%) & \(\mathbf{\varepsilon_{cc}}\) (\%) \\ \hline Min & 51.00 & 102.00 & 0.09 & 10.00 & 12.41 & 0.17 & 0.33 \\ Max & 406.00 & 812.00 & 5.90 & 663.00 & 188.20 & 1.53 & 4.62 \\ Mean & 153.70 & 307.57 & 0.92 & 174.59 & 43.52 & 0.27 & 1.60 \\ + S.D & 43.91 & 87.83 & 1.06 & 120.06 & 22.31 & 0.14 & 0.82 \\ \({}^{*}\) COV & 0.29 & 0.29 & 1.16 & 0.69 & 0.52 & 0.52 & 0.52 \\ \hline \hline \end{tabular} + Standard Deviation. \({}^{*}\) Coefficient of Variance.  
\end{table}