# Seismic Response Analysis of Multi-Story Steel Frames Using BRB and SCB Hybrid Bracing System
## 4 Numerical Model  
A total of six CBFs numerical models are created in the earthquake simulation platform OpenSees [44]. Figure 3 displays the numerical model. Figure 3 shows the building information of the prototype frame as well, which presents the cross-sectional shapes of the main members and the building dimension. The designations of \(\mathrm{W14}\times 48\), \(\mathrm{W14}\times 132\) and \(\mathrm{W14}\times 211\) are the steel beams and columns with wide-flange or I-shaped sections. Taking \(\mathrm{W14}\times 48\) for example, it means the steel has a cross-sectional height of 14 in. with a unit weight of 48 lbs/ft. For all stories, Table 1 includes the strength capacity and the initial axial stiffness of the braces. The beams, columns, and braces are modeled using the force-based beam-column elements, considering the displacement interpolation functions of displacement-based elements may deviate from the exact solution [45]. Columns have fixed bases and are continuously modeled throughout building height. The beam-to-column pinned connections are achieved by using two over lapped nodes. The beam and column elements are made of American Society of Testing Materials (ASTM) A992 steel material. The corresponding post-yield stiffness ratio is assumed to be 0.003. The Rayleigh damping ratio of 5% is assumed for the first two modes. Similar to a prior treatment [46], the panel zone is not simulated, with the aim to keep the analysis time to a reasonable level. Further, to simplify the problem, the buckling and fatigue problems are excluded, and thus, the potential degradation of strength and stiffness of steel were not taken into account. In terms of the bracing components, each one is modeled as one single element with four integration points; the corresponding cross sections are an assembly of uniaxial fibers. Due to symmetry, only one bay is numerically modeled. Vertical gravity loads are applied gradually on the model, and then, dynamic analysis under horizontal seismic input is carried out. The structural torsion along with a vertical axis is not triggered.  
Figure 3: Modeling of the prototype frame in OpenSees.  
Figure 3 shows the tributary floor mass is idealized as concentrated mass nodes affiliated to the leaning column. The leaning column is to represent the other spans of frames that have negligible seismic resistance while carry all the building weight. The leaning column is coupled to the braced frame at each floor level in the vertical and lateral displacement directions. This column is vertically rigid and has a large cross-sectional area to carry the applied nodal mass. In the numerical model, the leaning columns are jointly connected at the adjacent levels. In such a situation, the leaning column would not provide stiffness and strength in the seismic input direction for the entire system. The total seismic mass at each floor is equally shared by the braced bays, thus each frame carries 1/6 of the total floor mass. The floor and total mass of each braced frame are \(1.51\times 10^{5}\) kg and \(9.06\times 10^{5}\) kg, respectively. As a result, the leaning column is able to account for the P-\(\Delta\) effect while does not affect the strength or stiffness of the structure.