## 3 Proposed Method for the Simplified Analysis  
As a result of the previous sections, it results that the dynamic behaviour of the equivalent system is ruled by parameters of the compliance curve, namely, the tangent stiffnesses, the force in the removed element and the yielding force of the system. As emerges in Equation (4), the equivalent mass of the SDOF oscillator is not needed to estimate its maximum dynamic displacement \(u_{dyn}\). Besides, mass information is required if the force is not applied instantaneously. The proposed method consists in computing a compliance law and applying the dimension-less analysis to estimate the dynamic amplification factor, DAF. The analysis is performed with respect to reinforced concrete planar frame structures, for which a quick and preliminary assessment is required. The proposed approach is based on the following assumptions: the frame has regular bays geometry, that is, the widths of the bays are (at least across the considered column) equal; the concrete beams have equal capacity and ductility properties for all the floors (at least across the considered column); following column removal, the damage propagates on the frame structure through a zipper-type progressive collapse mode [36] which can be idealised with the formation of plastic hinges at both ends of the beams spanning from the considered column, as sketched in Figure 7a. The procedure consists in the following steps:  
1. the capacity and the ductility properties of the beams are assessed and a bending moment-curvature (BM-\(\chi\)) relationship is obtained. The bending moment at yielding, \(M_{y}\), and the ultimate bending moment, \(M_{u}\), and the corresponding kinematic quantities, \(\chi_{y}\) and \(\chi_{u}\), are considered as representative for the behaviour of the concrete cross-section;
2. the axial force, N, acting in the removed column is computed considering the loads applied on its tributary area, that is, half of the dead and live loads each beam plus the weight of the upper columns. 3. the mechanism that originates following column removal is studied and the upper bound theorem of the plastic analysis is applied. With reference to Figure 7b, the replacement axial force N previously computed is inserted in the mechanism as an upward force located in A; bending moments m\({}^{*}\) are applied in the plastic hinges. A downward variable force, that is, column removal force, \(\lambda 1\) is inserted in A. The value of \(\lambda\) for which the virtual work principle (VWP) holds defines an upper bound collapse load, \(\lambda\) (\(m^{*}\)) [37];  
Figure 6: Dynamic amplification factor for various stiffness ratios \(\Pi_{3}\) in the range \((0;1]\) for \(\Pi_{2}=1\). The grey line corresponds to DAF=2. Squares represents the DAF obtained through the numerical simulations. the chord rotation of beams ends, \(\theta^{*}\), is computed and an approximate downward displacement of node A is obtained as \(\delta^{*}=\theta^{*}\ell\);
5. the compliance curve is built repeating points 3. and 4. considering the yielding and ultimate capacities of the beams. The yielding force, \(F_{y}\), is computed through the load multiplier \(\lambda\left(M_{y}\right)\) and the corresponding displacement, \(\delta_{y}\), through the chord rotation at yielding, \(\theta_{y}\). A similar approach is required for the ultimate point, that is, for \(F_{u}\) and \(\delta_{u}\). The tangent stiffnesses in the elastic and in the plastic with hardening part of the curve are evaluated as \[k_{e}=\frac{F_{y}}{\delta_{y}}\qquad\qquad k_{p}=\frac{F_{u}-F_{y}}{\delta_{u} -\delta_{y}};\] (7)
6. the displacement due to a quasi-static column removal process is evaluated as \[u_{st}=\frac{\text{N}}{k_{e}}.\] (8) The dimensionless quantities \(\Pi_{1}\), \(\Pi_{2}\) and \(\Pi_{3}\) are computed and the dynamic amplification factor is evaluated following the steps reported in Appendix A;
7. the maximum dynamic displacement is computed as \(u_{dyn}=\text{DAF}\;u_{st}\).