## 2 Theoretical Background for the Nonlinear Analysis of End-Plate Connections
### Structural Behavior of End-Plate Connections  
Steel end-plate connections are commonly used in civil engineering for beam-to-column joints, thanks to their many advantages, such as relatively simple fabrication, high rigidity, and substantial load-bearing capacity. These connections can generally be categorized into two main groups: flush and extended end-plate connections, as presented in Figure 1. As illustrated, the primary distinction between the two configurations lies in the projection of the end-plate beyond the beam flange, which, in the case of extended connections, allows for additional bolt rows above the beam section. This arrangement increases the effective lever arm of the bolts relative to the moment's center of rotation, thereby enhancing the connection's stiffness and load-bearing capacity. Moreover, the extended end-plate configuration provides more space for the inclusion of stiffeners, as shown in Figure 1, which further contributes to structural performance. In contrast, flush end-plate connections exhibit reduced stiffness and resistance, although they may be preferable in situations where fabrication simplicity, cost efficiency, or geometric constraintsare prioritized. The selection of the appropriate connection type depends on the specific design situation and the associated requirements.  
While during the design of steel structures the applied connections are assumed to be either fully rigid or ideally pinned, the connections usually fall between these two borders, thereby being semi-rigid [5]. For such connections, one of the most important parameters is the relationship between the joint rotation \(\theta\) and the corresponding bending moment \(M\). Joint rotation is conventionally defined as the angular change between the beam and the column relative to their original configuration [37].  
To accurately characterize the \(\theta-M\) relationship of the end-plate connection, the joint rotation \(\theta\) can be defined as the relative rotation between the centerlines of the top and bottom beam flanges at the beam end. This rotation can be decomposed into two main components: the shearing rotation \(\theta_{\text{s}}\) and the gap rotation \(\theta_{\text{ep}}\). According to [PERSON] et al. [38], these quantities can be expressed as follows:  
\[\theta=\theta_{s}+\theta_{ep} \tag{1}\]  
\[\theta_{s}=\frac{\Delta}{h_{t}} \tag{2}\]  
\[\theta_{ep}=\frac{\delta}{h_{t}} \tag{3}\]  
where \(\Delta\) denotes the horizontal displacement difference between the centerlines of the top and bottom flanges at the end of the beam, \(\delta\) represents the relative horizontal separation between the end-plate and the column flange at the level of the tensioned beam flange. The term \(h_{t}\) is the vertical distance between the centerlines of the top and bottom beam flanges. The resulting rotational components are illustrated in Figure 2. In this paper, these formulations were used to characterize the \(\theta-M\) relationship. It should be noted that this approach is consistent with practical applications and aligns with the guidelines provided in AISC 360-16 [39] and AISC Design Guide 39 [40].  
As presented in [38], various parameters and structural components--such as end-plate stiffeners and column stiffeners--can significantly influence the shape and characteristics of the \(\theta-M\) curve. Consequently, different failure modes may occur, and relatively small changes in connection details can lead to substantial variations in deformation capacity and moment resistance. Therefore, the precise design and detailing of these connections is essential to ensure both structural performance and safety.  
Figure 1: The two main groups of end-plate connections: (a) flush and (b) extended designs.