# Nonlinear Dynamic Response of a Precast Concrete Building to Sudden Column Removal
## 3 Performance Limit States Related to Column Loss Scenarios  
The following performance limit states (PLS) were assumed related to PCF system:  
1. PLS1: tie or connection yielding. At this stage, the connections start to resist to applied loads in the inelastic range;
2. PLS2: dowel fracture. At this stage, the dowel shear strength is lost, and the resistance to applied loads relies on continuous reinforcements;
3. PLS3: tie or connection fracture. At this stage, the fracture of connection subject to negative moment or the tie fracture elongation is achieved;
4. PLS4: loss of support. At this stage, precast beams experience a lateral displacement which is larger than a threshold value corresponding to a prescribed limit of support width.  
It is worth to note that, although tie or connection fracture could be achieved in some specific positions, the structure could be able to sustain vertical loads if it is verified that the column support is not lost in all joints nearby the column loss location. For this reason, the PLS corresponding to the loss of support is considered the main indicator of catastrophic failure associated with a specific column-loss scenario in the case of PC structures. Referring to Figure 5, the threshold width associated with the support loss is related to the width of the column corbel (\(l_{supp}=250\) mm) reduced by a length equal to \(50\) mm that represents the damage effect in corbel corner. Hence, the support width limit was taken equal to \(200\) mm. This concept will be illustrated in the next section to analyze NLTHA results.  
Referring to MRF system, the following PLSs were assumed:  
1. Achievement of longitudinal rebar yield strain. At this stage, structural members start to resist to applied loads in the inelastic range;
2. Achievement of concrete compressive peak stress;  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{3}{c}{**Negative Moment Connection Strength**} & \multicolumn{3}{c}{**Dowel Shear strength**} \\ \hline \(\theta_{Kxy}\) [rad] & \(\theta_{Kcu}\) [rad] & \(M_{Kxy}\) [\(kN\cdot m\)] & \(M_{Kcu}\) [\(kN\cdot m\)] & \(\Delta_{Rdy}\) [mm] & \(\Delta_{Rd}\) [mm] & \(V_{Rd}\) [\(kN\)] \\ \hline
0.0012 & 0.185 & 383.20 & 690.80 & 0.13 & 4.60 & 51.30 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Calculation example for dowel and negative moment connection strengths.  
Figure 5: Schematic representation of support loss assumption: (**a**) undeformed joint; (**b**) deformed joint with support loss (PLS4) and concrete ejection. (Dimensions in mm).  
3. Achievement of maximum axial load acting on beams above the removed column. This indicator is significant to assess the onset catenary action resisting mechanism;
4. Achievement of maximum tensile longitudinal rebar elongation. At this stage, the tensile rebars achieve maximum strain, which could correspond to the fracture strain.  
Moreover, column and beam shear demands and capacities were also monitored during progressive collapse phenomenon to check brittle failure modes. Exceedance of EC8-conforming shear requirements [53] was notified whether demand exceeded capacity: in that case, shear strength was reduced by 50%.