## 5 Dissipative Bracing Retrofit Solution
### Application of the Design Method to the Case-Study Building
#### 5.2.1 X Direction--Lack of Bending Moment Strength in the Columns  
Step 2. The verification analysis in current conditions highlights that the most critical response parameters in \(X\) direction are the bending moments around \(Y\) (i.e., \(M_{Y}\)) in the first-level columns, with the highest unsafe conditions checked in the four corner columns. By referring to the nomenclature in Section 2.1, the maximum moment \(M_{Y,c}^{a}\) corresponding to the peak response point in Figure 9c and associated with the concurrent axial force \(N_{c}\) = 104 kN mentioned above, is equal to 398.7 kNm. The elastic limit moment \(M_{Y,c}^{e}\) of the corner columns around the \(Y\) axis is equal to 224.8 kNm. Thus, the stress reduction factor \(\alpha_{s}=\alpha_{MY}\) of the critical members in \(X\) direction results as follows:  
\[\alpha_{s}=\alpha_{MY}=\frac{M_{Y,c}^{a}}{M_{Y,c}^{e}}=1.77 \tag{24}\]  
Passing from the member to the frame-structure level (equivalent to the frame structure storey for this case-study building), since all columns have the same cross section, \(\alpha_{F}\) ratio coincides with \(\alpha_{MY}\):  
\[\alpha_{F}=\alpha_{MY}=1.77 \tag{25}\]  
Step 3. On the basis of this value, the equivalent viscous damping ratio of the set of spring-dampers to be installed on the second level is evaluated by means of relation (14), obtaining:  
\[\xi_{eq,\alpha F}=\frac{2}{\pi}\frac{(\alpha_{F}-1)}{\alpha_{F}}=0.277 \tag{26}\]  
Step 4. The \(E_{D}\) energy dissipation capacity of the spring-dampers is calculated by expression (18). The elastic limit values of the level shear \(F_{c}\) (i.e., the sum of the elastic limit shear forces of the columns) and the first interlevel drift \(ILD_{e}\) (replacing \(ID_{e}\) in this case) computed in \(X\) direction, named  
Figure 11: Positions of the dissipative bracing system alignments in plan. \(F_{e,X}\) and \(ILD_{e,1L,X}\), are equal to 969 kN and 22 mm, respectively. Introducing these values, as well as \(\alpha_{F}\) and \(k_{eq,\alpha F}\) values given by (25) and (26), in (18), the following \(E_{D}\) estimate is derived:  
\[E_{D,\alpha F}=2\pi\alpha_{F}F_{e,X}\xi_{eq,\alpha F}ID_{e,1L,X}=65.6\ \rm{kJ} \tag{27}\]  
Dividing \(E_{D}\) by the number of spring-dampers placed in \(X\), the minimum energy dissipation capacity \(E_{D,X,d}\) to be assigned to each of the eight devices in order to reach the target performance at the MCE results as follows: \(E_{D,X,d}\) = 8.2 kJ. The spring-damper type with the nearest nominal energy dissipation capacity, \(E_{n}\), to \(E_{D,X,d}\) has the following mechanical properties, drawn from the manufacturer's catalogue [48]: \(E_{n}\) = 9 kJ; stroke \(s_{max}\) = \(\pm\)30 mm; damping coefficient \(c\) = 9.9 kN(s/mm)\({}^{\gamma}\), with \(\gamma\) = 0.15; \(F_{0}\) = 17 kN; and \(k_{2}\) = 1.74 kN/mm. .2.2 \(Y\) Direction--Lack of Bending Moment Strength in the Columns and Excessive Inter Level Drift  
Step 2. The critical response parameters in \(Y\) direction are represented by the bending moments around \(X\) (\(M_{X}\)) in the second-level columns, with the highest unsafe conditions checked in the four corner columns too, and the second interlevel drifts. The maximum moment \(M_{X,c}^{u}\), corresponding in this case to the peak response point in Figure 9d, is equal to 174.2 kNm, whereas the elastic limit moment \(M_{X,c}^{e}\) is equal to 84.2 kNm. Therefore, the stress reduction factor \(\alpha_{s}=\alpha_{MX}\) of the critical members in \(Y\) direction is:  
\[\alpha_{s}=\alpha_{MX}=\frac{M_{X,c}^{u}}{M_{X,c}^{e}}=2.07 \tag{28}\]  
and thus:  
\[\alpha_{F}=\alpha_{MX}=2.07 \tag{29}\]  
The deformation-related reduction factor \(\alpha_{d}\) given by (19) is calculated for the \(ILD_{max}\) (replacing \(ID_{max}\)) and \(ILD_{e}\) (replacing \(ID_{e}\)) values computed on the second level in \(Y\) direction, named \(ILD_{max,2L,Y}\), \(ILD_{e,2L,Y}\) in the following.