## 1 Introduction  
Prestressed concrete beams are widely used for constructing large load-bearing structures and bridging long spans. The prestressing force on the concrete is transferred via shear stresses over a certain length referred to as the transmission length. The Hoyer effect, due to the dilation of the strands after release, contributes to the transfer of the prestressing force. Splitting forces due to the local introduction of prestressing forces in the end zones occur, resulting in a complex three dimensional stress field. Splitting, spalling, and bursting cracks can occur in the end zones, which are undesirable. Modelling the end zones of a pretensioned girder creates challenges when applying analytical formulae [1; 2; 3] or strut-and-tie models [4; 5; 6; 7; 8] in particular when the 3D behavior is considered. In this situation, finite element modelling (FEM) becomes beneficial, because it can manage these complexities more effectively. FEM has also been used by other researchers to study the stress distribution in end zones. [PERSON] et al. [9] model a pretensioned beam considering the strands as truss elements. In this application, the radial expansion due to the Poisson effect is not taken into account. As a result, the model is developed as a linear elastic model, such that the actual nonlinear behavior of the interface between the strands and the surrounding concrete is not accurately simulated. [PERSON] and [PERSON] [10] define the pretensioning process in two stages: (1) the tendon tensioning at time t\({}_{0}\) and (2) the transfer of prestress to concrete at time t\({}_{1}\). As a result, the accuracy of the model increases. In their model, local friction between the prestress element and the concrete is introduced. The disadvantage of this approach is that it neglects the concrete tensile strength at the interface between the two materials. The elastic loss in prestressing force is also neglected. [PERSON] et al. [11] examine the stresses along a strand using a simple model with a seven-wire strand in the center of the beam. Two models are developed. In the first one, the strand is modelled as a linear element embedded in concrete. In the second one, the strand consists of a solid cylinder with an eight seed meshing. The friction is defined as a surface-to-surface contact, and the behavior in the normal direction is defined as hard contact. To increase the accuracy of the calculation, the augmented Lagrange multiplier and finite sliding are used. For the friction coefficient, values of 0.70 to 1.40 are taken into account. [PERSON] and [PERSON] [12] and [PERSON] et al. [13] designed a 3D non-linear FEM model to study end zones. The Concrete Damaged Plasticity (CDP) model is applied to account for the non-linear concrete properties. The compression behavior is defined according to the guidelines of both the fib Model Code 2010 [14] and AASHTO LRFD Bridge Design Specifications [15] while the tensile behavior is solely defined by the fib Model Code 2010. The model does not contain strands. Instead, the prestress is modelled by leaving gaps at the strand locations and applying tangential forces on the gap's inner surface. Although the Hoyer effect is not included in this model the results are in strong agreement with experimental data. [PERSON] et al. [16] built further on this approach, focusing on BT-78 girders, and proposed recommendations for strand debonding and strand draping. Finally, [PERSON] et al. [17] also use this approach to study the effects of debonding in 72 W Wisconsin bulb-tee girders, and propose a simplified approach for Y-crack tensile strains although the correlation is limited. [PERSON] et al. [18] study the prestress force combining three adaptations to the previously mentioned works. (1) The strands are modelled such that they exhibit radial deformations due to the Hoyer effect. Since these deformations produce large radial stresses, it is beneficial to implement them into the model.