# Novel Predictive Model of the Debonding Strength for Masonry Members Retrofitted with FRP
## 4 Results and Discussion  
The performance of GEP in training and testing sets is evaluated in terms of two usual statistical measures: correlation coefficient (\(R\)) and root mean square error (RMSE), which are given as follows:  
\[\text{RMSE}=\sqrt{\frac{1}{N}{\sum\limits_{i=1}^{N}{(t_{i}-o_{i})^{2}}}} \tag{2}\]  
\[R=\sqrt{1-\frac{\sum\limits_{i=1}^{N}{(t_{i}-o_{i})^{2}}}{\sum\limits_{i=1}^{N }{(o_{i}-\overline{o_{i}})^{2}}}} \tag{3}\]  
in which \(t_{i}\) denotes the target values of the maximum debonding force; while \(o_{i}\) and \(\overline{o}_{i}\) denote the observed and averaged observed values of the maximum debonding force, respectively; and \(N\) is the number of samples. A new formulation of debonding strength for masonry elements retrofitted with FRP was derived using an experimental database from [PERSON] et al. [44]. Table 2 lists 134 data points that were considered in this study. In the GEP model, \(b_{p}\), \(t_{p}\), \(E_{p}\),\(f_{ut}\), \(b_{m}\), and \(l_{b}\) were considered as inputs and the maximum debonding force (\(P_{\text{max}}\)) as the independent output. Data were randomly grouped into two subsets: a training set of 100 data points and a testing set of 34 data points--approximately 75% and 25% of the 134 data points, respectively. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Model** & **Equation (Units: N, mm)** & **References** \\ \hline [PERSON] & \(P_{\text{max}}=110.2\times 10^{-6}\times\left(f^{\prime}_{cm}/42\right)^{2/3}E_{p}\,t _{p}l_{e}\,b_{p}\) & [45] \\  & \(l_{e}=e^{6.134-0.58\times\ln(E_{p}\,t_{p})}\) & (unit of \(E_{p}\) is GPa) & \\ \hline [PERSON] & \(P_{\text{max}}=b_{p}\sqrt{2E_{p}\,t_{p}G_{c}}\) & [46] \\  & \(G_{c}=1.43\,\text{Nmm}/\,\text{mm}^{2}\) & [42] \\ \hline [PERSON] & \(P_{\text{max}}=0.5l_{b}\,b_{p}f_{ut}\) & [47] \\ \hline [PERSON] & \(P_{\text{max}}=(b_{p}+7.4)\sqrt{2E_{p}\,t_{p}G_{c}}\) & [47] \\  & \(G_{c}=0.514f^{\prime}_{cm}\)/\({}^{236}\) & [22] \\ \hline Accardi & \(P_{\text{max}}=b_{p}\sqrt{12E_{p}t_{p}\sqrt{f^{\prime}_{cm}}}\) & [22] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**References** & **No. ** & \(t_{p}\) **(mm)** & \(E_{p}\) **(GPa)** & \(b_{p}\) **(mm)** & \(l_{b}\) **(mm)** & \(b_{m}\) **(mm)** & \(f_{\pi t}\) **(MPa)** & \(P_{\text{max}}\) **(kN)** \\ \hline
[51] & 1 & 6.35 & 40.8 & 6.35  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline
**References** & **No. ** & \(t_{p}\) **(mm)** & \(E_{p}\) **(GPa)** & \(b_{p}\) **(mm)** & \(l_{b}\) **(mm)** & \(b_{m}\) **(mm)** & \(f_{\tt eff}\) **(MPa)** & \(P_{\tt max}\) **(kN)** \\ \hline
[43] & 62 & 1.2 & 162 & 15 & 241 & 230 & 3.55 & 46.8 \\  & 63 & 1.2 & 162 & 15 & 328 & 2  
The statistical analyses of the data are summarized in Table 3. As can be observed in Table 3, the statistics for both the training and testing sets are in good agreement, meaning that both of them represent almost similar populations. Table 4 shows the functional set and operational parameters used in GEP modeling. The GEP-based explicit formulation of maximum load is given in the following equation:  
\[P_{\max}=[f_{ut}^{1.5}L_{b}-L_{b}t_{p}c_{1}]+[f_{ut}^{3}b_{m}-b_{p}^{2}t_{p}]+[ L_{b}^{1.5}+c_{2}+\frac{t_{p}}{c_{0}}(L_{b}+E_{p})] \tag{4}\]  
in which \(c_{1}\) = 9.941346, \(c_{2}\) = 9.58728, and \(c_{0}\) = 9.351684. The variation of observed and estimated values by GEP model are illustrated in Figure 2 for the training and test periods. It is clear from the figure that the GEP estimates closely follow the corresponding observed experimental data in the both periods. The scatterplots of the observed and estimated debonding forces are shown in Figure 3. From the fit line and \(R^{2}\) values, it is evident that the estimates of the GEP model are very close to the ideal line. Table 5 shows the values of two performance indicators (\(R^{2}\) and RMSE) for some existing models and the proposed model in this study, regarding both training and testing phases.