## 2 Energy-Based Method  
The total energy is considered constant throughout the dynamic deformation of a structure (Figure 2a) following a sudden column removal scenario, conditional on the assumption that the energy dissipated by other sources, e.g., heat, are neglected [18; 19; 23]. Three components of energy are taken into account when balancing the energy: kinetic energy (KE) originating from the moving sub-structure, strain energy (SE) due to the structural deformation, and release of potential energy (PE) due to the downward movement of the structure. The released potential energy resulting from the unbalanced gravity loads for the remaining structural system leads to dynamic motions and an increase of kinetic energy. In contrast, elastic and inelastic strain energy accumulated by the deformed structure dissipate the kinetic energy, counteracting the motions. If a structure can redistribute the unbalanced loads, the potential energy can be absorbed by the strain energy. Consequently, the kinetic energy can attain a value of zero (i.e., \(\text{KE}=\text{PE}-\text{SE}=0\)). Based on the aforementioned energy balance during a typical column removal scenario, the EBM can be easily interpreted and applied. Generally, the strain energy stored in the deformed system can be calculated through a non-linear static pushdown analysis [18]. In that case, the strain energy is the hatched area under the static capacity curve up to a displacement, namely \(u_{s}\) (Figure 2b). On the other hand, the released potential energy generated by the unbalanced loads following a sudden column removal is equal to the hatched rectangular area for a constant load level, namely \(Q_{d}\). If the kinetic energy is assumed to be zero (at the displacement level, \(u_{s}=u_{d}\)), the potential energy is therefore equal to the strain energy (PE = SE). Note that only the situation in which no collapse occurs is considered. Eventually, the predicted dynamic load \(Q_{d}\) can be calculated as follows [18; 19]:  
\[Q_{d}(u_{d})=\frac{1}{u_{d}}\int_{0}^{u_{s}}Q_{s}(u)du \tag{1}\]  
Figure 1: Layout of this article. where \(Q_{d}\) is the load in the dynamic capacity curve, \(Q_{s}\) is the load in the static capacity curve, \(u_{d}\) is the peak dynamic deflection, and \(u_{s}\) is the static deflection corresponding to the load \(Q_{s}\). The EBM approach is a compromise between accuracy and complexity. The following simplifications are incorporated:  
* The moving sub-structure, subjected to a column failure, is assumed to behave like a single degree of freedom (SDOF) system [17; 18; 23]. The response is controlled by a single deformation mode and the mode keeps constant during the dynamic response. Therefore, the energy of the whole system can be linked to the energy of a point, i.e., every point in the system reaches its maximum displacement response at a same time. However, this never happens in a real structure as the existing infinite number of deformation modes will reach their maximum response at different moments. Consequently, the stored strain energy counted by the EBM is overestimated alongside its calculated deflections [23]. For a structural frame system subjected to an exterior column removal scenario, a non-single deformation response may occur due to complex load redistribution mechanisms (e.g., overturn), such as the results by [4; 13; 46] in relation to the exterior or side column removal scenarios. * All the energy introduced into a system by the loads is switched into pure strain energy. The EBM neglects the energy dissipated by damping or other mechanisms. Therefore, the maximum deflection response will be overestimated. Moreover, it is still a controversial issue regarding how to model the damping mechanism for a sudden column removal scenario [19; 47; 48], e.g., viscous damping or Coulomb damping [23]. For instance, the studies have identified drawbacks associated with the use of Rayleigh damping based on initial stiffness, in which the initial stiffness may introduce in the unwanted artificial viscous forces [14; 49].