## 4 Results
### Impact of Prestressing Intensity  
Numerical analysis revealed premature cracking at the re-entrant corner of the dapped end. To assess the severity of this effect, another parameter was analyzed: the load causing crack formation, based on the intensity of the prestress. The beam was loaded until cracking occurred at the re-entrant corner. Figure 20 depicts the relationship between the cracking load at the re-entrant corner and the compressive stress ratio (\(\sigma_{cp}/f_{cm}\)), the ratio between the compressive stress caused by prestressed reinforcement and the average compressive concrete strength. The first crack was observed to have occurred at a load of 91.20 kN, which corresponds to 31.62% of the maximum load. This is in accordance with the studies of [PERSON] et al. [25], who specified a range of 27-42% of the maximum load, and [PERSON] [29], who indicated a range of 20-33% of the maximum load for the formation of the first cracks at the re-entrant corner. The results of this prestressing intensity analysis suggest that the tensile stresses induced by the prestressed reinforcement significantly reduced the resistance to cracking at the re-entrant corner. A reduction of approximately 50% of the cracking load was observed  
Figure 19: Comparison of varying cross-section areas of shear reinforcement on the failure load when prestressed reinforcement is used. when the compressive stress ratio was in the 0.20-0.25 range. Furthermore, at around a 0.30-0.35 ratio, the formation of cracks at the re-entrant corner of the numerical model was observed during the prestressing release stage before the beam was subjected to an external load. The final numerical analysis was performed to determine the significance of the force P\({}_{2}\) shown in Figure 2 for the bearing capacity of the dapped-end beam. The drawings of the model are shown in Figure 6 and the parameters are depicted in Figure 4. The distance \(l_{s}\) = 80 mm and the shear reinforcement \(A_{vh}\) = 144.66 mm\({}^{2}\). The beam model is illustrated in Figure 21.  
The analysis was performed with the application of prestress at varying intensities. Shear failure mode was observed in the results of the nonlinear numerical analysis. This is evidenced in Figure 22. The results obtained are presented in Figure 23, which depicts the relationship between the bearing capacity and the ratio of compressive stress caused by prestressing (\(\sigma_{cp}/f_{cm}\)). Figure 21: General view of the numerical model for the analysis of prestressing’s impact on bearing capacity. Figure 20: Relationship between the cracking load and the compressive stress ratio. The results depicted in Figure 22 show that the impact of prestressed reinforcement on bearing capacity is significant. The non-prestressed dapped-end beam failed at a load of 264.56 kN, whereas the prestressed beam with the highest analyzed prestressing intensity failed at a load of 241.25 kN. The decrease in bearing capacity was found to be linear approximately up to a 0.265 \(\sigma_{cp}/f_{cm}\) ratio. Once that ratio was reached, the influence of prestressing declined, and looking further, the bearing capacity decreased only slightly. The results demonstrate a significant reduction in bearing capacity due to prestressing, which must be considered during the design of dapped-end beams. The results of the parametric analysis and the prestressing impact analysis were used to propose equations for assessing the effects of prestressing when the prestressed tendons induce force P\({}_{2}\), as depicted in Figure 2. The proposed analytical model was modelled after the EC2 design for shear forces described in Section 2 of this study. The EC2 methodology was modified by replacing the \(V_{Rd,max}\) in Equation (7) with the calculated maximum shear capacity of concrete of a prestressed dapped end \(V_{u,c}\). It is estimated with the following equation:  
\[V_{u,c}=V_{c,max}\cdot\alpha_{\mathrm{d}}\cdot\alpha_{\mathrm{dp}}, \tag{15}\]  
Figure 23: Relationship between the bearing capacity and compressive stress ratio.