## 4 Case Study: The Sports Hall
To be on the safe side, these parameters were assumed (and reduced through the safety factor FC = 1.35) as follows: (i) compression strength \(f_{cd}=f_{c,min}/F.C.=5.0 MPa/1.35=3.70 MPa\); (ii) compression elastic modulus \(E_{cd}=4550 MPa\); (iii) shear elastic modulus \(G_{d}=1137 MPa\); and (iv) shear strength without compression \(\tau_{0,d}=0.08 MPa\). Preliminary pushover analyses were carried out in Midas-Gen 2023 v1.1 [51] along both the longitudinal (X) and transversal (Y) horizonal directions on the as-built frame. \begin{table}
\begin{tabular}{l c c c c c c} \hline
**Infill** & \(k_{\text{pr,}nt}\) & \(F_{\text{w,}m}\) & \(d_{\text{w,}m}\) & \(d_{\text{w,}r}\) & \(\theta\) & \(\left(\frac{d_{\text{w,}m}+d_{\text{w,}r}}{2}\right)\cdot\text{cos}\theta\) \\
**Direction/Level/Type** & **(kN/mm)** & **(kN)** & **(mm)** & **(mm)** & **(°)** & **(mm)** \\ \hline DirX-lev1-full & 72.0 & 87.7 & 1.2 & 40.8 & 46.5 & 14.5 \\ \hline DirX-lev1-opening (door) & 22.9 & 27.9 & 1.2 & 40.8 & 46.5 & 14.5 \\ \hline DirY-lev1-full & 70.6 & 190.4 & 2.7 & 90.4 & 25.2 & 42.1 \\ \hline DirY-lev1-opening (door) & 41.9 & 113.1 & 2.7 & 90.4 & 25.2 & 42.1 \\ \hline DirY-lev1-opening (plant) & 50.9 & 137.2 & 2.7 & 90.4 & 25.2 & 42.1 \\ \hline DirY-lev2-full & 86.5 & 171.0 & 2.0 & 66.2 & 19.4 & 32.2 \\ \hline DirY-lev2-opening (plant) & 59.9 & 118.4 & 2.0 & 66.2 & 19.4 & 32.2 \\ \hline \end{tabular}
\end{table} 
Table 1: Mechanical parameters of the masonry infills’ equivalent struts according to the Decanini model [36]. Beam elements with concentrated plastic hinges were enrolled to model the RC column elements in order to simulate, under the average gravity load, \(N_{Ed}=350 kN\), flexural yielding-strengths \(M_{y,Rd}\) equal to \(41.3\,kNm\) and \(59.9\,kNm\) about the two principal axes. The moment-curvature constitutive parameters were calibrated according to FEMA-356 [52] for masonry and EC8:2004 [46] for RC members. In the longitudinal (X) direction, each column was subdivided in two beam elements in order to take into account possible \"soft-storey\" and \"short-column\" effects caused by the non-uniformity of the masonry walls in elevation: a first element from the foundation to the end of the full masonry band, and a second aligned with the window openings. A very stiff elastic law with an associated brittle failure threshold of \(V_{Rd}=22 kN\) calibrated according to EC8:2004 [46] was assigned to the shear hinges located at both end nodes. Nonlinear \"FEMA infill strut-type\" Midas elements [51] were employed to simulate the response of the masonry infills, with the force-displacement parameters of each type of panel (see Table 1) calibrated according to the Decanini model [36; 37]. Nonlinear static analyses revealed the inadequacy of the as-built frame to withstand the seismic loads defined by the Italian NTC-2018 standard. For all analysis cases, the \"seismic safety factor\" \(\zeta_{E}\), calculated as the ratio between the seismic acceleration calculated at the building collapse and the peak ground acceleration (PGA) established by the Italian Building Code [25] for the design new constructions, ranged between 0.29 and 0.65 depending on the adopted loading pattern and direction. The seismic retrofit through dissipative braces required the following propaede utic interventions:  
1. Definition of a seismic joint to separate the main frame from that of adjacent structures housing service rooms and plants (see Figure 7a). 2. Jacketing of the columns to provide anchoring to the dissipative braces through an external layer of C32/40 concrete, including additional \(22\phi 16\) longitudinal B450C rebars and stirrups of \(\phi 10/15\) cm (see Figure 7c). This leads to a significant increase in both flexural strength (\(M_{y,Rd}\), equal to \(374.2 kNm\) and \(497.7 kNm\) about the two principal axes) and shear strength (\(V_{Rd}=177 kN\)). 3.