## 2 Analytical Methods for Calculating the Failure Load  
The load capacity of a dapped-end beam is defined by its flexural strength in the mid-section and the strength of the supporting section. As mentioned, dapped-end beams mostly fail due to shear forces acting on the dapped end. If the element is reinforced with straight prestressed strands in the tension zone, the flexural strength's impact is further reduced. The complex stress distribution in the notched end prevents accurate calculation of the failure load through conventional methods of determining shear strength. To calculate the load capacity of dapped-end beams, specialized methods are chosen. These methods typically involve assessing the potential failure modes such as those shown in Figure 3. According to the PCI [6], these failure modes can be described as one of the following: flexure (cantilever bending) and axial tension in the extended end; direct shear at the junction of the dap and the main body of the component; diagonal tension emanating from the reentrant corner; diagonal tension in the extended end; diagonal tension in the undapped portion. To determine the failure load, the following specialized methods were chosen:  
Figure 3: Five potential failure modes of a dapped-end beam [6]. 1 Flexure (cantilever bending) and axial tension in the extended end; 2 direct shear at the junction of the dap and the main body of the component; 3 diagonal tension emanating from the reentrant corner; 4 diagonal tension in the extended end; 5 diagonal tension in the undapped portion. Figure 2: Forces acting on the point A at the re-entrant corner of the dapped end [14]. * PCI dapped-end beam design method, which is defined in Section 5.6.3[6];
* EC2 strut-and-tie (STM) model, which is defined in Section 6.5[7];
* The semi-empirical truss model of [PERSON] et al. in 2005[19]. It is crucial to note that the specialized methods employed do not account for the effects of prestressing. These methods are designed and optimized for a specific configuration of the dapped end and cannot evaluate the impact of prestressed reinforcement if such reinforcement is not used in the methods. The decision to employ these methods was taken to achieve two objectives. The first objective was to compare the bearing capacity of the prestressed beam in the numerical model with analytical methods and to determine if the impact of prestressing was significant enough to cause an overestimation of the bearing capacity and pose a risk of premature failure. The second objective was to identify which methods most accurately reflect the behavior of a non-prestressed dapped-end beam in the numerical model so they can be utilized as a baseline for future research to modify these methods and evaluate the impact of prestressing. The primary mode of failure observed in dapped-end beams is attributed to shear forces. Therefore, the use of the shear calculation method outlined in Eurocode 2 (EC2) Section 6.2[7] is considered to be an alternative approach for analytically determining the failure load of the dapped-end beams. Employing analytical calculation methods, a parametric analysis of the failure load was conducted based on the height of the dap \(h_{d}\), the distance of the support from the re-entrant corner \(l_{s}\), the concrete strength \(f_{cm}\), and the area of shear reinforcement \(A_{eh}\). The varying parameters of the dapped end are highlighted in blue in Figure 4.  
The impact of reinforcement prestressing was examined utilizing the proposed model presented in Section 4 of this study. The proposed analytical model was founded on the shear computation methodology described in EC2, which is explained in Section 6.2, and was derived from the parametric analysis conducted within the scope of this research. In the EC2 methodology, prestressing affects the load-bearing capacity of compressive concrete struts.