## 4 Existing RC Frame
### Numerical Modeling
#### 4.2.2 Nonlinear Static Analysis  
In this section, the main results in terms of capacity curves and distribution of plastic hinges at different time steps of the analysis are presented for both the strong and the weak directions, e.g., the X and Y direction, respectively.  
As for the plastic hinge lengths, according to relation (1), for the existing frame they are in the range 300-330 mm for the deep beams, depending on their length, 290 mm for the flat beam, and 235 mm for the columns.  
The results in terms of base shear vs. top displacement curves are reported for X (strong) direction in Figure 6a and for Y (weak) direction in Figure 6b. In both cases, the two distributions of the lateral loads are applied, according to the first mode and to a uniform pattern proportional to mass. The capacity in terms of base shear is, in the weak Y direction, about half with respect to X direction. The attainment of the LS chord rotation capacity in the first column of the existing frame occurs after about 30 mm from the peak in both pushover analyses (uniform and modal) in the X direction, while for the analyses in the Y direction the rotation capacity of the first column at the ground floor is achieved just after the beginning of the softening branch. Moreover, in Y direction the flat beams achieve their shear capacity close to the peak of the curve.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Material** & **Property** & **Value** \\ \hline \multirow{3}{*}{Concrete: C25} & Mean compressive strength & 33 MPa \\  & Strain at peak stress & 0.002 \\  & Ultimate strain & 0.006 \\ \hline \multirow{3}{*}{Steel: FeB44k} & Mean yield strength & 510 MPa \\  & Hardening ratio 1 & 1.52 \\  & Ultimate strain & 21.7\% \\ \hline \hline \end{tabular} \({}^{1}\) The hardening ratio is defined as ultimate strength over yield strength \(f_{u}/f_{y}\).  
\end{table}
Table 1: Material properties for existing RC frame.  
Figure 5: Three-story RC building. Reinforcement layout of RC element cross-sections: beams in the strong X directions (units in mm).  
The failure mechanisms can be inferred by observing the evolution of the curvatures of the sections at the column and beam extremities during the analyses, reported in Figures 7 and 8 for loading in X and Y direction, respectively. Actually, in the X (strong) direction the first hinges occur at the base of the columns, with some hinges also at the top sections of the columns in the first and second floor, as shown in Figure 7a. Then, a soft-story mechanisms is clearly evidenced with plastic hinges in all of the columns of the ground floor at both base and top sections when the ultimate chord rotation capacity is attained, as depicted in Figure 7b.  
As expected, the behavior in the Y (weak) direction is significantly different, with a much evident deformability (top displacement at maximum base shear almost three times greater than in the strong direction) and the formation of plastic hinges on the flat beams since the first steps of analysis and then a progressive formation of plastic hinges at the base of columns up to the structural failure (Figure 8).  
Figure 6: Capacity curves for RC existing building: (**a**) X direction; (**b**) Y direction. A yellow circle indicates the attainment of the LS chord rotation capacity and a green triangle the attainment of the shear capacity. The type of elements where the limit is reached is marked by the letter C for column and B for beam.  
Figure 7: RC existing building, uniform pushover. Curvature distribution corresponding to a top displacement in X direction of: (**a**) 50 mm; (**b**) 95 mm (attainment of the LS chord rotation capacity). Red line X direction, green line Y direction.