## 2 Modelling Sudden Column Removal on Frame structures
### Dynamic Amplification Factor (DAF)  
The maximum dynamic displacement \(u_{dyn}\) of the SDOF system was computed through the expressions reported in Appendix A assigning an arbitrary value to the repeating variables, for example, \(m=1\) kg, \(k_{e}=1\) N/m and \(F_{y}=1\) N. The solution was implemented into a Matlab script consists in applying, in the order, Equations (A3), (A4), (A9), (A10), (A12), and, finally, Equation (A8). The displacement due to quasi-static loading of the system was determined as  
\[u_{st}=\begin{cases}\frac{f}{k_{e}}&f\leq F_{y}\\ u_{y}+\frac{f-F_{y}}{k_{p}}&f>F_{y}.\end{cases}. \tag{5}\]  
The dimensionless displacement was computed for both the dynamic analysis, \(\Pi_{1}^{dyn}\), and the quasi-static loading \(\Pi_{1}^{st}=u_{st}/u_{y}\). Figure 4 depicts the relationship between normalised displacements, forces and ratios between tangential stiffnesses. Observing the dashed curves, related to the quasi-static loading process, the elastic-plastic law of the connecting device clearly emerges. The curves as \(f/F_{y}=1\) and diverges for larger forces depending on the hardening stiffness. On the contrary, the continuous curves accounting for the dynamics of the system show a different behaviour, diverging for \(f/F_{y}>0.5\).  
The dynamic amplification factor was computed as  
\[\text{DAF}=\frac{\Pi_{1}^{dyn}}{\Pi_{1}^{st}}=\frac{u_{dyn}}{u_{st}}. \tag{6}\]  
Figure 5 reports the DAF for various dimensionless external forces \(\Pi_{2}\) and stiffnesses ratios \(\Pi_{3}\). It is shown that for systems with an elastic-plastic characteristics the \(\text{DAF}\geq 2\). In particular, the well-known value \(\text{DAF}=2\) occurs if \(k_{p}=k_{e}\), that is, the system can be though as pure elastic. The dynamic amplification factor for elastic-plastic with hardening systems increases for \(f>F_{y}/2\) and has a peak for \(f=F_{y}\). This is due to the fact that the mass has its maximum velocity at \(u_{y}\), when the system turns from elastic to plastic, resulting in the maximum discrepancy between the dynamic and the quasi-static maximum displacements. A decreasing trend of the DAF curves is observed for \(f>F_{y}\)  
They asymptotically tend to 2 for large \(f/F_{y}\) ratios since the plastic phase is predominant if \(f\gg F_{y}\), that is, the restoring force in the device depends on the tangent stiffness \(k_{p}\).  
Figure 6 plots the maximum values of the dynamic amplification factor, DAF, for \(f=F_{y}\), that is, \(\Pi_{2}=1\), where the peaks in Figure 5 were observed. The maxima curve has a vertical asymptote for \(\Pi_{3}=0\) corresponding to \(k_{p}=0\), that is, an elastic perfectly plastic system. As observed, the maximum DAF is 2 for \(k_{e}=k_{p}\).  
Figure 4: Loading force - displacement curves for various values of \(\Pi_{3}\). Continuous lines relates to the dynamic analysis, that is, \(\Pi_{1}^{dyn}\), while the dashed lines relate to the quasi-static loading, that is, \(\Pi_{1}^{st}\).  
Figure 5: Dynamic amplification factor for various normalised loading and stiffness ratios. Squares represents the dynamic amplification factor (DAF) obtained through the numerical simulations.