# Seismic Upgrade of Steel Frame Buildings by Using Damped Braces
## 5 Design of the Seismic Rehabilitation
### Step 1: Definition of the Main Frame Capacity Curve  
The capacity curve of the as-built structure (\(V_{F}-d_{F}\)) was determined via NLSAs as recommended in IBC [51]. The NLSAs were performed in the positive and in the negative Xand Z- directions of the structure (Figure 1), by applying two horizontal force distributions proportional to either mass (UNIFORM distribution) or modal (MODAL distribution) properties (Figure 3), and considering 5% accidental eccentricity of the center of mass of each story. The \(V_{F}-d_{F}\) capacity curve of the MDOF structure was then converted to the \(V_{F}^{*}-d_{F}^{*}\) capacity curve of the equivalent SDOF system through the modal participation factor \(\Gamma\), as shown in Figure 3.  
In each of the two main horizontal directions, the lowest capacity curve was used to evaluate the properties of the equivalent SDOF system. As shown in Figure 4, for each MRF, there is a substantial difference between the capacity curves in X- and Z-directions, due to the preferred orientation of the cross-sections of the columns, which have their strong direction aligned to the Z-axis.  
Step 2: Definition of the Target Displacement and Construction of the Equivalent Bilinear Capacity Curve of the Main Frame  
The target displacement \(d_{p}\) was defined based on the assumed building performance criteria, such as the story drift limits for the non-structural components or the plastic deformation limits for the structural members [39; 86; 87]. In agreement with reference [88], the immediate occupancy structural performance level, i.e., the condition in which the structure is immediately accessible as it retains its original strength and stiffness, was defined by the ending point of the elastic branch on the pushover curve. In the present case, the target displacement \(d_{p}\) was defined by multiplying the yield displacement \(d_{y}\), identified as the limit of the elastic branch on the capacity curve [88] (Figure 4), by a ductility factor \(\mu_{F}=1.5\), in accordance with reference [55]. The choice of a ductility factor of 1.5, which involves a partial engagement of the main frame in the dissipation of the seismic energy, corresponded to the design strategy of relatively small dissipation braces, requiring minor  
Figure 4: MODAL capacity curves of the as-built structures (MDOF) with indications of the target displacement \(d_{p}\) (green dots) and the yield displacement \(d_{y}\) (red dots).  
Figure 3: Determination of main frame capacity curves according to IBC.  
Figure 6 shows, as an example, the comparison between the equivalent bilinear capacity curves of the four-story structure in the X- and Z- directions and the relevant demand response spectra. As evident, in the Z-direction the main frame meets the target performance; instead, in the X-direction, retrofit is necessary to control the lateral deformation. Same results were obtained for the two-story structure; on the contrary, for the eight-story, MRF retrofit is required in both the X- and Z-directions. For sake of brevity, only the results relevant to the four-story frame are shown.