## 6 Conclusions  
This paper has presented and discussed a simplified approach for the estimation of the maximum dynamic displacement in reinforced concrete frames subjected to sudden column loss. An equivalent load application process has been adopted for exploring the effects of the damage on the structure. Despite simple, the instantaneous load application in a numerical step-by-step analysis provides an upper bound of the true displacement that can be observed if the damage progressively act. Different to the simplified procedure illustrated by [PERSON] et al. [4], following which the compliance curve is obtained applying the gravity loads on an unloaded damaged structure, the present approach considers a damage process occurring on a loaded structure. Awareness about the phenomena that can occur in systems when subjected to forces which magnitude is large enough to engender plastic deformations is necessary for the study the effects of loading scenarios not considered during the ordinary design, such as sudden column removal or explosions on civil engineering structures. It has been proved that, despite simplified, the proposed \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline
**Beams** & **Removed** & \(u_{st,num}\) & \(u_{dyn,num}\) & \(\mathbf{DAF}_{num}\) & \(\begin{array}{c}\mathbf{u}_{dyn,simpl}\end{array}\) & **Error** \\
**Reinforcement** & **Column** & **(mm)** & **(mm)** & **DAF\({}_{num}\)** & **(mm)** & \\ \hline
3 + 3\(\phi\)16 & Middle & 32.4 & 124.4 & 3.84 & 122.5 & \(-1.5\%\) \\
3 + 3\(\phi\)20 & Middle & 23.3 & 47.7 & 2.04 & 49.8 & +4.4\% \\
4 + 4\(\phi\)16 & Middle & 25.8 & 57.6 & 2.23 & 61.3 & +6.4\% \\
3 + 3\(\phi\)20 & Side & 39.2 & 63.2 & 1.61 & 58.7 & \(-7.7\%\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison between numerical and simplified analysis by varying the amount of reinforcement in the beams. Figure 10: Time-histories of the downward displacement of the control point under the sudden column removal process (for both middle and side elements). Column removal starts at 0.1 s and lasts 0.01 s. The dashed lines illustrate the dynamic displacements obtained with the simplified method herein presented. methodology catch the dynamic behaviour of a frame structure subjected to column removal. The comparison between the results obtained with the straightforward approach herein described surprisingly has shown good agreement with the nonlinear dynamic time-history numerical analyses. Membrane actions, which provide a contribution to the capacity of the system under large displacements, have not been considered in the method. For the purposes of the simplified method, the overstrenght provided by membrane effects acts as a safety margin. This approach is suitable in the preliminary design stage, when information about the structure is bare. The Authors implemented it on a spreadsheet into which the basic geometry of the frame, cross-section properties and acting loads are assigned. Solving the procedure, the dynamic displacement is estimated. By varying the variables, say the amount of reinforcement, the size of the elements, a preliminary design for robustness is performed. The validity of the resulting dynamic response lies in the fact that the amplification factor computed with the analytical formula overestimates the true value [13]. Obviously, the simple considerations on the dynamic response of structures to step force can be extended to other loading scenario, such as impacts due to natural phenomena [22]. In addition, this approach could be extended to other types of structures into which it is possible to identify a failure mechanism, say in steel frames or in timber structures. The possibility of including column removal time and strain rate effects into the present simplified formulation is still an open question [30] and need to be addressed in the future. Conceptualisation, V.D.B. and F.K. ; methodology and formal analysis, V.D.B. ; writing, V.D.B. and F.K. ; writing--review and editing, all the authors; funding acquisition, B.C.