## 2 Modelling Sudden Column Removal on Frame structures
### Dimensional Analysis for the Dynamic Phase of the Sudden Damage  
Many quantities determine the behaviour of the previously described SDOF system. With the aim to study the effects of the connector law on the dynamic response, a general approach is required. The main variable of interest is represented by the maximum elongation of the device, that is, the maximum vertical displacement in the frame structure subjected to the sudden column removal. This quantity was previously identified as \(u_{+}\) and can be considered in engineering terms as the dynamic response of the system subjected to a sudden load. For this reason, in the following, this term is recalled as \(u_{dyn}=u_{+}\).  
The value of \(u_{dyn}\) depends on many parameters: the yielding force, \(F_{y}\), the applied force, \(f\), the mass, \(m\), the tangent stiffnesses of the elastic, \(k_{e}\), and the plastic with hardening, \(k_{p}\), rules. In general, \(u_{dyn}\) can be written as  
\[u_{dyn}=\varphi\left(F_{y},f,m,k_{e},k_{p}\right)\,, \tag{2}\]  
where \(\varphi\left(\cdot\right)\) is the solution of the dynamic problem illustrated in the appendix. A dimensional analysis approach was adopted to investigate the influence of the parameters on the maximum  
Figure 3: Elastic-plastic law of the connecting device. Explanations of the letters in the plot are reported in the text.  
Figure 2: Sketch of the single degree of freedom (SDOF) system. The nonlinear device is denoted with letter (\(d\)). On the right-hand side the dynamic forces acting on the mass are reported: \(m\dot{u}\) is the inertia force, \(F_{R}\) is the restoring force of the device, which depends on its elongation \(u\), \(f\) is the applied force.  
displacement. This technique is a mathematical tool that serves for identifying the relationships that describe the physical phenomena [33]. With respect to the previous Equation (2), \(u_{dyn}\) represents the dependent variable, while the remaining are the independent terms. Following the principle of dimensional homogeneity, Equation (2) can be rewritten using [PERSON]'s \(\Pi\)-theorem [34; 35]. Three base dimensions, that is, the length \(\left[L\right]\), the mass \(\left[M\right]\) and the time \(\left[T\right]\) are needed to define the six dependent and independent variables. Hence, it results \(u_{dyn}\doteq\left[L\right]\), \(F_{y}\doteq\left[M\right]\left[L\right]\left[T\right]^{-2}\), \(f\doteq\left[M\right]\left[L\right]\left[T\right]^{-2}\), \(m\doteq\left[M\right]\), \(k_{e}\doteq\left[M\right]\left[T\right]^{-2}\), and \(k_{p}\doteq\left[M\right]\left[T\right]^{-2}\). According to [PERSON]'s theorem, Equation (2) can be contracted to  
\[\Pi_{1}^{dyn}=\Phi\left(\Pi_{2},\Pi_{3}\right), \tag{3}\]  
where \(\Pi_{i}\) (\(i=1,2,3\)) are dimensionless quantities, which number is equal to the number of variables appearing in Equation (2), that is, 6, minus the number of base dimensions, that is, 3. The \(\Pi_{i}\) terms are obtained by identifying among the six variables three repeating terms and using these to turn the remaining into dimensionless quantities. The repeating variables must have independent dimensions, that is, any product between them cannot be dimensionless. In the present approach, the mass \(m\), the elastic stiffness \(k_{e}\) and the yielding force \(F_{y}\) were chosen as repeating variables. Therefore, the resulting three \(\Pi_{i}\) terms were \(\Pi_{1}^{dyn}=u_{dyn}k_{e}/F_{y}\), \(\Pi_{2}=f/F_{y}\), and \(\Pi_{3}=k_{p}/k_{e}\). Considering that the displacement at yielding is \(u_{y}=F_{y}/k_{e}\), for a SDOF system with elastic-plastic with hardening rule, the governing equation is  
\[\frac{u_{dyn}}{u_{y}}=\Phi\left(\frac{f}{F_{y}},\frac{k_{p}}{k_{e}}\right). \tag{4}\]