## 3 Results and Discussion
### Flexural Stiffness  
#### 3.7.1 Service Stiffness  
To calculate the flexural stiffness of every specimen, the slope between the two points corresponded to pre-crack loading (30% of peak loading), spalling load, and yield load (80%-85% of peak loading) on the ascending segment of the load-deflection curve. Normally, the stiffness measured from the slope of the pre-crack loading to the yield load segment of the load-deflection curve was defined as the service stiffness. The service load level for a reinforced concrete beam normally corresponded to 60% until 70% of the ultimate load of the beam based on the load and material factors stipulated in the reinforced concrete design codes [44].  
In this study, the permanent deformation of steel bars was not used to identify the stiffness as the beams were over-reinforced. Furthermore, stiffness was defined based on the response of the beams at the spalling load, yield load, and pre-crack loading. The stiffness ratios were defined as \(\text{K}_{0.3\text{Ppeak}}=\text{P}_{0.3\text{Ppeak}}/\Delta_{0.3\text{Ppeak}}\), \(\text{K}_{\text{sec}}=(\text{P}_{\text{spall}}-\text{P}_{0.3\text{Ppeak}})/( \Delta_{\text{spall}}-\Delta_{0.3\text{Ppeak}})\), and \(\text{K}_{\text{spall}}=\text{P}_{\text{spall}}/\Delta_{\text{spall}}\), where \(\text{K}_{0.3\text{Ppeak}}\), secant stiffness (\(\text{K}_{\text{sec}}\)), and Spall stiffness (\(\text{K}_{\text{spall}}\)) were present. Due to the presence of yield in the BCSP-15 beam, it was dependent on \(\text{K}_{\text{sec}}=(\text{P}_{\text{yield}}-\text{P}_{0.3\text{Ppeak}})/( \Delta_{\text{yield}}-\Delta_{0.3\text{Ppeak}})\). Meanwhile, \(\text{K}_{\text{yield}}=\text{P}_{\text{yield}}/\Delta_{\text{yield}}\) was determined based on the load of the displacement data obtained during the test.  
Of the peak load 30% and 80%-85% were considered as the realistic representations of the service stiffness for the tested beam specimens. Table 8 and Figure 14 illustrate the service stiffness of the CB beam compared to the service stiffness of the development beams. It was observed that the service stiffness (\(\text{K}_{0.3\text{peak}}\)) of the BCSP-6 beam increased up to a ratio of 73.7%. As shown in Table 9, 86% of the increase in stiffness was observed from the BCSP-10 beam, while the stiffness of the BCSP-15 beam amounted to 122.7% in comparison with the CB beam. This increase was possibly the result of the increased thickness of the steel plate, with the absence of buckling steel plate and the slip between the compression steel plate and beam. Furthermore, no bolting slip was present between the steel plate and the beam. These factors contributed to a significant increase in beam stiffness. Therefore, the presence of a steel plate in the compression zone and a higher thickness of the steel plate could lead to higher beam stiffness.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Specimens** & \(\text{P}_{\text{Spall}}\) & \(\text{P}_{\text{Yield}}\) & \(\text{Peak}\) & \(\Delta_{\text{Yield}}\) & \(\Delta_{\text{peak}}\) & \(\Delta_{\text{0.9 spall}}\) & \(\Delta_{\text{0.95 Vield}}\) & \(\mu_{2}=\) & \(\mu_{2}=\) \\
**Designation** & **(kN)** & **(kN)** & **(kN)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & **(mm)** & \(\Delta_{\text{0.9, spall}}/\Delta_{\text{spall}}\) & \(\Delta_{\text{0.9, spall}}/\Delta_{\text{yield}}\) \\ \hline CB & 356 & - & 375 & - & 14.94 & - & - & - & - \\ BCSP-6 & 521 & - & 652.3 & - & 16.4 & 22.1 & - & 2.7 & - \\ BCSP-10 & 635 & - & 837.6 & - & 11.93 & 19.2 & - & 2.9 & - \\ BCSP-15 & - & 935.3 & 935.3 & 10.2 & 15.5 & - & 22.1 & - & 2.2 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Test results of over-reinforced beams for index displacement ductility (\(\mu_{2}\)).