## 3 Results and Discussion
### Flexural Stiffness
#### 3.7.2 Post-Peak Stiffness  
In order to calculate the post-peak stiffness of the beams, the slope of the curves between the points of the spalling load (80%-85% of peak loading) and the peak load on the descending side of the load-deflection curve was obtained. The loaded beams beyond this section possessed in-elastic characteristics. The stiffness ratios were defined as \(\mathrm{K_{Peak}}=\mathrm{P_{Ppeak}}/\mathrm{\Delta_{Peak}}\) and \(\mathrm{K_{sec}}=(\mathrm{P_{spall}}-\mathrm{P_{Ppeak}})/(\mathrm{\Delta_{ spall}}-\mathrm{\Delta_{Peak}})\), and \(\mathrm{K_{spall}}=\mathrm{P_{spall}}/\mathrm{\Delta_{spall}}\), where \(\mathrm{K_{Peak}}\), secant stiffness (\(\mathrm{K_{sec2}}\)), and spall stiffness (\(\mathrm{K_{spall}}\)) were determined based on the load to the displacement data obtained during the test. The post-peak stiffness values were calculated for all the beams, with the values being presented in Tables 8 and 9 and Figure 14. Obtained for the BCSP-6 beam, it was lower than the service stiffness, indicating a weaker load resisting capability at this loading phase. Generally, the post-peak stiffness for the beams comprised an increase in steel plate thickness, with the exception of the BCSP-15 beam, which recorded a post-peak stiffness of 60.34 kN/mm. With this amount, 11.1% of reduction from the value for the BCSP-10 beam was recorded. Based on the simultaneous evaluation of the toughness and post-peak slope diagrams, the factor of higher flexural toughness values of the BCSP-10 beam was indicated. The lower post-peak slope of the specimens and their higher maximum defections were found to be the main factors of the high flexural toughness. stiffness, ductility, flexural cracking, and mode failure of the beams were compared, followed by the development of several inferences. The single or double helical reinforcement was found to be a satisfactory method to develop the over-reinforced beam section [7]. This finding was proven through the fact that the experimental flexural strengths of all the beams, which are developed with helical reinforcement, are higher than the strength of the calculated ultimate flexural loads. Furthermore, the beams with helical confinements exhibited lesser deformation compared to the control beams without development. Similarly, the BCSP beams investigated in this study exhibited the same characteristic in terms of flexural strength and deflection. Moreover, as the BCSP-system possessed a different curve of load-deflection, the system exhibited more ductile characteristics compared to the conventional system. Besides being dependent on the helical pitch, the helical system's diameters were the main parameters influencing the effectiveness of helical confinement of concrete. The BCSP-system was influenced by the thickness and length of steel plates. The helical reinforcements were found to be more efficient in improving the strength and ductility of confined concrete compared to short stirrups [6]. Comparatively, the bolts-compression steel plate was found to be more effective than helical. It also possessed short ties to enhance the strength, stiffness, and EA of the beams. Therefore, it could be concluded that crushing failure (spalling cover) is present in all methods of the development of over-flexural sections at compression with flexural cracking. Accordingly, the BCSP-system improved the mode failure and transitioned the behavior failure from brittle to ductile. It also avoided sudden failure in the compression zone. Overall, the development of mode failure was observed in this study using failure modes and crack patterns. The details of the flexural behavior are presented in Section 3.1. This failure was due to the strain gradient effect, which occurred in the effective thickness in the BCSP beams subjected to flexural stress. For this reason, the horizontal cracking was reduced to a lower level compared to the cracking in tension.