# Seismic Upgrade of Steel Frame Buildings by Using Damped Braces
## 5 Design of the Seismic Rehabilitation
### Step 5: Distribution of the Damped Braces across the Frame  
The distribution of the elastic stiffness and yield force of the dissipative braces at each floor of the real structure was performed according to a proportionality criterion [7; 8; 38] with the goal of matching the modal shape of the retrofitted frame to the first mode shape of the as-built frame. For this reason, the regularity of the frame in plan and in elevation was considered as a fundamental prerequisite for the application of this method. This solution aims at producing the simultaneous yielding of the dissipative devices at all the stories, and thus a global ductility of the damping system coinciding with the ductility of the single dampers. Such condition on one hand legitimates the condensation of the properties of the damping system into an equivalent SDOF device assumed in the design procedure, and in addition maximizes the energy dissipation capacity [33; 34]. At each floor, the properties of the damped braces were evaluated by applying the formulas shown in Figure 9, where \(\phi_{i}\) is the \(i\)-th component of the first mode eigenvector of the main frame, \(N_{yi}^{DB}\) and \(K_{yi}^{DB}\) are the yield force and the elastic stiffness of the single damped brace at the \(i^{\text{th}}\) floor and \(n_{d}\) is the total number of dampers per floor. Eventually the stiffness of the steel braces and the properties of the dampers were calculated based on geometrical considerations depending on the particular brace arrangement. In the present work, steel hysteretic dampers were inserted within a chevron brace in the perimetral bays of the MRFs according to the plan layout shown in Figure 10. In the two-story and four-story structures, the dampers were inserted in X-direction only (Figure 10a), with two dampers per story, while the eight-story frame was upgraded in both horizontal directions (Figure 10b). Each damped brace (DB) was composed of an  
Figure 9: Graphical representation of the procedure for distributing the properties of the damped braces at the stories of the frame. in-series arrangement of a damper (D) and two inclined steel braces (B) according to the typical chevron brace configuration. Therefore, the lateral deformation of each story was accommodated by the deflections of both the damper and the braces in proportion to their stiffnesses. In order to concentrate almost the whole deformation of the story in the damper [90] and increase the amount of energy dissipation, the stiffness of the chevron braces \(K_{i}^{B}\) is set equal to fifteen times the effective stiffness of the damper \(K_{i}^{D}\) in accordance with other studies [31,91]. It must be observed that, while such an assumption is fully realistic for the two-story and the four-story buildings, it provides, for the braces installed at the first floors of the eight-story frame, dimensions that could become economically not practicable though technically feasible. Nevertheless, in this study, it was decided to adopt the same figure of the stiffness ratio also for the eight-story MRF in order to maintain a coherent strategy for all the case studies. In a practical application, obviously the designer will have to verify case by case the convenient value of the stiffness to be adopted, taking into account the resulting dimensions of the braces. The identified properties (elastic stiffness and yield force) of the dampers at each floor are reported in Tables 4-6. The effective stiffness of the single damper \(K_{i}^{D}\) is simply calculated by dividing the elastic stiffness by the relevant damping factor. Some concluding remarks have to be made concerning the current limitations of the procedure. First, it should be noted that the first part of the procedure relevant to the calculation of the lumped properties of the braced system (steps 1 to 4) is valid whether or not the main frame is regular, as it relies on an equivalent SDOF model of the real structure.