# Nonlinear Dynamic Response of a Precast Concrete Building to Sudden Column Removal
## 2 Case Study and Modelling Technique
### FE Modelling and Analysis Technique  
Nonlinear time history analyses (NLTHA) were carried out using Seismonstruct FE code [46] to investigate threat-independent, sudden column-loss scenarios of the selected case study. The force-based (FB) fiber modelling approach [47] was used to predict the nonlinear response of the case-study building. Beam-column fiber elements with distributed plasticity approach were used to model the frame members. Mechanical and geometrical nonlinearities were included in the model to account for the actual structural behavior under large displacements and rotations. Each member nearby the column loss location was modelled using two FB elements, each of them with 5 integration points and 300 fibers, which were considered enough to describe the deformed shape during progressive collapse analysis. Sectional stresses and strains were computed at individual fibers through direct integration of the uniaxial material response. Reinforcing steel behavior was modelled though a bilinear stress-strain relation, whereas the Mander curve [48] was used for concrete, which is able to account for confinement effect provided by stirrups. Strain rate effects [49] were conservatively neglected in this study. In Figure 2a,b, FE models for MRF and PCF systems under edge column loss scenario are shown. Gravity loads were applied to the structure through masses per unit length over beams. The column base nodes were fixed to the ground. The resisting contribution of the floor diaphragms was conservatively neglected and only their self-weight was considered. Moreover, the diaphragm was removed at the column loss level, while keeping an in-plane rigid floor at the 2 nd and 3 rd floor levels. Column loss scenarios were simulated by deactivating designated elements after a time equal to 0.01 s, whereas the total time of NLTHA was set to 3 s and the convergence tolerances were based on displacement/rotation criteria. A tangent stiffness-proportional Rayleigh damping and the Newton-Raphson algorithm were used. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Floor** & **Edge-Q\({}_{b,e}\)** & **Interior-Q\({}_{b,i}\)** & **Corner-Q\({}_{b,c}\)** & **Unit** \\ \hline
1 st, 2 nd & 47.50 & 69.70 & 47.50 & [kN/m] \\
3 rd & 19.00 & 38.00 & 19.00 & [kN/m] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Line loads at different floor levels and column removal locations. In the case of PCF system, the shear and rotational behaviors of the connections between precast beams and columns were modelled in Seismostat using \"link\" elements between two overlapping nodes, as shown in Figure 2c. Furthermore, referring to Figure 2d,e:  
1. An isotropic hardening quadrilinear asymmetric relation was assigned to the connection moment-rotation behavior;
2. An isotropic hardening quadrilinear symmetric relation was assigned to the connection shear-displacement behavior. It is noted that link deformations are related to relative displacement and rotation between nodes 1_1 and 1_2 (see Figure 2c). Link degrees of freedom (DOFs) are uncoupled, so it is not possible to account for the interaction between shear and moment. The remaining DOFs were considered fixed. For the sake of simplicity, links were inserted in the only bays  
Figure 2: FE modelling: (**a**) FE model for MRF system; (**b**) FE model for PCF system; (**c**) links, ties and frame elements; (**d**) moment–rotation spring behavior; (**e**) shear-displacement spring behavior. directly affected by column removal. The beam-to-column eccentricity was modelled by adding a rigid element between the precast column and beam nodes, see Figure 2c, which corresponds to 450 mm (refer also to Figure 1c). The following two variants of PCF system were considered:  
1. PC1: in addition to the connections modelled as springs, the tying reinforcement was considered by adding truss elements (see Figure 2c), which simulates the continuity provided by ties along the beam members, as provided in the fib Bulletin 63 [34];
2.