## 5 Retrofitted Building
### Numerical Modeling
#### 5.2.2 Nonlinear Static Analysis  
In this section, the main results in terms of capacity curves and distributions of plastic hinges at different time steps of the analysis are presented, for seismic action in X and Y directions.  
The plastic hinge lengths, for the RC-framed skin elements according to relation (1), are about 280 mm for the beams and 285 mm for the columns.  
The results in terms of base shear vs. top displacement curves are reported for X (strong) direction in Figure 12a and for Y (weak) direction in Figure 12b, for both the lateral forces distribution, i.e., uniform and modal. Due to the effect of the reinforcing RC-skin frame, the capacity in terms of base shear is significantly higher than the capacity of the existing building and the values for actions in X and Y directions are now comparable, leading to a structure without a weak direction. Actually, the capacity is slightly higher in the Y direction respect to that in the X direction, since there are more reinforcing columns along the sides parallel to Y direction. Moreover, it is possible to observe that, in both directions and for both pushover analyses (uniform and modal force distributions), the LS chord rotation capacity of the first column of the existing building is achieved just after the beginning of the softening branch.  
Figure 13 shows the curvature of the different sections at two steps of the pushover in X direction with uniform distribution of forces: Figure 13a refers to yielding corresponding to a top displacement of about 100 mm, while Figure 13b refers to the attainment of ultimate chord rotation capacity in the columns of the existing building. It is possible to observe the mechanism with the formation of plastic hinges at the base and the top section of all the existing columns of the first floor and some of the existing columns of the second floor,  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Material** & **Property** & **Value** \\ \hline \multirow{3}{*}{Concrete: C30} & Compressive strength & 38 MPa \\  & Strain at peak stress & 0.002 \\  & Ultimate strain & 0.006 \\ \hline \multirow{3}{*}{Steel: B450C} & Yield strength & 514 MPa \\  & Hardening ratio 1 & 1.19 \\  & Ultimate strain & 25.3\% \\ \hline \hline \end{tabular} \({}^{\dagger}\) The hardening ratio is defined as ultimate strength over yield strength \(f_{a}/f_{y}\).  
\end{table}
Table 2: Material properties for RC-framed skin.  
Figure 12: Capacity curves for the retrofitted building: (**a**) X direction; (**b**) Y direction. A yellow circle indicates the attainment of the LS chord rotation capacity and the type of elements where the limit is reached is marked by the letter C for column.  
together with the formation of plastic hinges in the columns of the reinforcement RC-skin frame in the first and second floor.  
Concerning the pushover in Y direction, Figure 14a depicts the curvature of the different sections for a top node displacement of about 100 mm: it is possible to observe that the plastic hinges occur at the bottom and top sections in the reinforcing columns at the first and second floor while, in the existing frame, the hinges are again at the base of the columns and on the flat beams. Figure 14b shows the curvature corresponding to the attainment of LS chord rotation capacity of the columns of the existing building. It is interesting to note that no shear failure was detected on the existing flat beams, as opposed to the pushover in Y direction of the existing building without reinforcement (see Section 4.2.2).