## 4 Analytical Study
### Background Theory  
The main aim of this study is to attain a strong calibration of the analytical models to ensure they are consistent with the empirical data obtained from experimental observations. In response to the experimental evaluations, finite element models were systematically developed for the two RC structures subjected to intensive testing. This was accomplished using the sophisticated functionalities of Seismostat V2025 [36]. The software not only provides robust modeling features but is also proficient in incorporating the effects of both material and geometric nonlinearity, which are essential for accurately reflecting the intricate dynamic behaviors of the structures throughout the modeling phase. The fiber-based modeling approach captures the behavior of cross-sections, with each fiber representing a uniaxial stress-strain relationship that facilitates the assessment of the sectional stress-strain state of beam-column elements. This evaluation is derived from the integration of the nonlinear uniaxial stress-strain responses of individual fibers into which the cross-section is logically segmented. A standard discretization method applied to a typical RC cross-section is illustrated in Figure 8.  
Figure 7: Processed ground-motion input used in the shake table tests: (**a**) acceleration time history of the 6 February 2023 Kahramanmaras earthquake and (**b**) normalized spectral acceleration spectra for 2 %, 5 %, and 10 % critical damping. Upon the successful fulfillment of the calibration requirements, conventional BRBs were systematically incorporated into the examined RC frames, thereby facilitating a comparative analysis with the results obtained from the TnT enhanced configurations. This comparison aims to elucidate the performance differences and enhance the understanding of the structural responses under various loading conditions. Ultimately, the findings derived from this comprehensive analysis will contribute significantly to the advancement of knowledge in the field of structural engineering and inform future design practices. Three-dimensional models of the RC structures are illustrated in Figure 9. A third numerical model (hereafter Specimen 3) was assembled by replacing the TnT brace with a conventional X-type BRB of equal axial capacity. No third physical frame was fabricated; the model is used exclusively to benchmark hysteretic energy dissipation and to carry out parametric variations that extend the generality of the experimental findings. As previously stated in Section 3.3, the frame experienced an axial load totaling 60 kN. Therefore, both the traditional BRB and the TnT BRB were engineered to accommodate this specific axial load value. In the context of traditional BRBs, the Stl_BRB model, shown in Figure 10, functions as a uniaxial material framework tailored to assess the characteristics of steel utilized in these structural components. This particular model was initially introduced by [PERSON] et al. [37] The preceding discussion addressed the design methodology for the TnT BRB; thus, the current focus will shift to the analytical calculations pertinent to the conventional BRB design. In this phase, adherence to AISC 341-22 [38] guidelines were maintained. A thorough evaluation of the design parameters led to the formulation of a  
Figure 8: Fiber discretization adopted in the FEM: (**a**) Three-dimensional view of a beam-column element subdivided into Gauss integration sections along its length and (**b**) exploded view of a single section for numerical integration. Figure 9: FEM models: (**a**) Specimen 1, (**b**) Specimen 2, and (**c**) Specimen 3 (numerical benchmark only). core plate engineered to withstand the full axial force without experiencing buckling. This design is crucial for enabling the brace to effectively manage load during seismic activities, thereby contributing significantly to the structural system's overall stability and safety. Subsequently, a restraining mechanism was devised to inhibit global buckling of the core.