## 3 FE Study  
The FE study was performed by considering three phases. The 3D FEM of all BRBs [2], the steel frame with slip-critical bolted connection, and the BRBF were constructed by Abaqus software [39] to simulate their hysteretic behavior under cyclic loading. The 3D hexahedron solid elements with eight integration points are not only suitable for plasticity problems but can also better predict the actual behavior of the test specimens [39]. These elements are therefore usually adopted by researchers [30; 35; 36; 37; 38; 40; 41; 42]. There are three types of 3D hexahedron solid elements with eight integration points: 3D solid element with full integration (C3D8), 3D solid element with reduced integration (C3D8), and 3D solid element with full integration nourished by incompatible modes (C3D8I). The shear locking and hourglass effects determine which type of these elements are used in the FEM. An element bends in reality (see Figure 8a), but it does not bend in the FE analysis. As can be seen in Figure 8b, the angle between the grids decreases to less than 90 degrees. That is, the angle between the grids changes in the FEA, but this is not true,  
Figure 7: Loading protocols [2]. since no shear occurs in the element under bending. The shear locking causes that element to exhibit more rigid behavior in the FE analysis. Due to the computed strain for the C3D8 element being constant, shear locking occurs in these elements. The C3D8R element can overcome the shear locking in the FE analysis because there is only one integration point at its center (see Figure 8c). The grids at this integration point (see Figure 8c) are constant without being affected by changes in the edges of elements. That is, it is not important whether the edge of the element is bent or not, and the angle between grids at the integration point always remains as 90 degrees [39]. Even if the C3D8R element can overcome the shear locking under the bending, singular modes (hourglass modes) can take place due to its insufficient rank of rigidity matrix. These modes cannot generate stress and strain, but they are zero-energy modes that can affect the accuracy of FE analysis. They can also result in unrealistic strains, stresses, bends, and contacts in FE analysis [39]. The 3D hexahedron solid elements with eight integration points that are available can overcome shear locking as well as hourglass modes. The C3D8I element can eliminate the stresses that cause the shear locking through the additional degrees of freedom that are provided with incompatible modes, and the hourglass modes also cannot occur owing to their full integration. For these reasons, the C3D8I element was used in all FE analyses to avoid shear locking and hourglass modes. Furthermore, [PERSON] and [PERSON] [46] suggested that kinetic energy and artificial strain energy should be considered to be lower than 1.0% of internal energy for valid numerical results. The slip-critical bolted connections in both the BRBF and the steel frame were modeled by using the micro-modeling technique. The contact boundary conditions between main plates and splice plates, bolt screws and bolt holes, and bolt nuts and splice plates were determined so as to reconstruct contact, separation, and fixing conditions in the FEA after the slips occurred in the slip-critical connections. While using the surface-to-surface discretization method to prevent surface penetration, the finite sliding approach was used to enable the slips between the contact surfaces [39]. Hard contact and penalty friction were used to define normal and tangential behaviors of contact pairs, respectively. The friction between the contact surfaces was defined by using Coulomb friction [47; 48; 49; 50]. The interactions described above are shown in Figure 9. The bolts are generally modeled as 3D solid or 2D spring elements in the FEM [47; 48; 49; 60; 61].