## 5 Comparison with Numerical Simulations  
Various numerical simulations were performed to assess the quality of the simplified methodology. SeismoStruct, a fibre-based commercial software capable of a large variety of static and dynamic analyses, was adopted [43]. The geometry of the example structure was inserted in the FE (Finite Elements) model and material and cross-section properties were assigned to the elements. The material nonlinearity derives from the fibre-based approach, for which a uniaxial stress-strain relationship is assigned to each fibre [44]. Concrete behaviour is represented through a uniaxial uniform confinement model [45] while a bilinear stress-strain model with kinematic strain hardening is assumed for the reinforcement steel bars. The sectional state of the elements is obtained through the integration across the fibres that discretize the cross section. Inelastic elements with plastic hinges according to the model proposed by [PERSON] and [PERSON] [46] are adopted in the FE model. The dynamic column removal is simulated with the application of an opposite force on the node onto which the replacement force acts [1,6,30]. In the simulations that are reported in the following, the force application starts at 0.1 s and lasts 0.01 s (column removal time), that is, it can be considered as a sudden column removal. The analysis is performed with 0.001 s time integration steps adopting the integration scheme proposedby [PERSON] et al. [47]. Both middle and side columns removals were simulated. Three different amounts of reinforcement in the beams were adopted: 3+3\(\phi\)16 and 4+4\(\phi\)16 for middle column removal and 3+3\(\phi\)20 for either middle or side column removal. For each configuration, the compliance curve was numerically obtained through a displacement controlled pushdown analysis, the incremental force represented by a downward force acting on the control node. It results that the vertical displacement due to a quasi-static column removal occurs when the incremental force equals in magnitude the axial force in the removed column. Figure 9 reports the compliance curves obtained in the simplified analysis and in the simulations. It is first noted that the range of vertical displacement differs: this is due to the capability of the approximate method to model the true nonlinear behaviour and resisting mechanisms that raise in such phenomenon, such as, arch resisting mechanism and catenary effects [17; 21]. Meanwhile, the yielding forces and the elastic parts of the compliance laws are very close, meaning that the simplified approach approximate the real behaviour of the system well. As expected, the simplified hardening phase is limited due to the ultimate rotational capacity of the element. The previous considerations holds for middle column removal. Referring to side column removal (in violet in Figure 9), it results that the pushdown analysis shows a less rigid elastic part, but with similar yielding force. Figure 10 depicts the results of the FE dynamic sudden column removal. After the removal, the system oscillates around the equilibrium position, which can be different from the one obtained with a quasi-static damage. Side column removal (violet curve) shows oscillations with variable amplitude due to the redistribution of the forces and across the whole frame. The X-axis is limited at 500 ms and depicts the maximum downward displacement of the control node, that is, 63.2 mm that occurs during the second oscillation. The dashed lines indicate the maximum dynamic displacements derived from the simplified analysis. It is shown that the proposed approach overestimates the downward displacement for 3+3\(\phi\)20 (middle column) and 4+4\(\phi\)16 cases, and underestimates it in the remaining cases. Table 1 reports the maximum values and compares the si  
Figure 9: Compliance curves for the four reinforcement amounts in the beams and column removal position. Solid lines (–) correspond to the FE simulations, while dashed lines (- -) to the simplified bilinear compliance law.