## 3 Numerical Example  
In this section, we examine an existing six-story planar steel frame braced with X-links, as depicted in Figure 4. It is a moment-resisting steel frame braced with a combination of multiple X-braces and single X-braces arranged in a single-stack or staggered configuration. Specifically, there are two X-braces in a single-stack configuration on the first and second floors, with an additional X-brace on the third floor directly above one of the two X-braces on the second floor. On the remaining floors, there are single X-braces arranged in a staggered configuration. The connections between the beams and columns are assumed to be rigid, forming a moment-resisting frame, while the X-bracing is pin-connected to the joints. The frame is assumed to be fixed at its base. This frame is part of a regular in-plan (symmetric) steel building, with reinforced concrete slabs at the floor levels acting as diaphragms. The composite action of the slabs on the behavior of the steel beams is ignored as it is outside the scope of this study. The steel frame features four equal spans of 5 m each. The height of the first floor is 3.5 m, while each of the remaining floors is 3 m high, resulting in a total frame height of 18.5 m. The vertical uniform distributed loads on the beams of the frame are \(g=35\) kN/m, which accounts for the self-weight, additional floor loads, and lightweight partition walls, and \(q=12\) kN/m, which represents the live floor loads. The total vertical uniform distributed load of the seismic combination on the beams of the frame is \(p=g+\psi_{\mathrm{E}}\cdot q\), where \(\psi_{\mathrm{E}}\cdot q\) represents the quasi-live loads with \(\psi_{\mathrm{E}}=0.3\)[38]. This results in a total load of \(p=38.6\) kN/m. These loads result in each floor having an approximate mass of 78 t, contributing to a total frame mass of 468 t. The floor masses and degrees of freedom for the six-story planar steel-braced frame used in the modal analysis are shown in Figure 4b. Figure 4a presents the two elevation patterns of horizontal floor forces (P-1 and P-2) referenced in phase (b) of the methodology and used in the pushover analyses. The steel frame is constructed using steel grade S235, with an average strength of \(f_{\mathrm{sm}}=258.5\) MPa and a maximum strength of \(f_{\mathrm{su}}=396\) MPa. The modulus of elasticity for the steel is \(E_{\mathrm{s}}=210\) GPa. In Figure 5, the constitutive law for the structural steel is depicted along with the performance levels. Figure 4: Six-story, planar, steel-braced frame: (a) static model for the pushover sets (P-1 triangular pattern, P-2 triangular pattern + top force), (b) dynamic model. The existing steel-braced frame was designed according to EN 1998-1 [38] for the high-ductility class (DCH) to withstand a seismic ground acceleration 0.24\(g\) at its base, on soil category D, with a behavior factor of \(q=4\). The columns in the first four floors are constructed with the wide-flange steel profile HEB 450, while the columns in the upper two floors are of the HEB 400 steel profile. The beams on all floors are constructed with the steel profile IPE 400. The braces on all floors are constructed with double-channel steel profile (2 UPN), with varying steel sections across different floors. The steel sections of the structural elements of the steel-braced frame are depicted in Figure 6, which shows the mathematical analysis model. The finite element analysis software SAP2000 [39] is used for all linear and nonlinear analyses within the context of the \"\(M\) and \(P\)\" technique. The columns and beams of the steel frame are modeled as frame elements, while the braces are modeled as link elements to accurately capture significant drops in strength. Rigid zones are defined around the nodes to account for their rigidity. The frame is generally regular in elevation, although there is a minor irregularity due to the first floor being 17% taller than the subsequent floors.