## 1 Introduction  
The effect of localised damage on frame structures is an open problem in structural engineering. Many studies on progressive collapse presuppose that an initial damage occurs on the structure. Various approaches can be followed to assess the damage tolerance of a structure [1; 2]. Nonlinear analyses accounting for second-order effects and large displacements are usually adopted for assessing the robustness of the structure [3], which usually undergoes plastic deformations. The duration of the damage, that is, the triggering of the progressive collapse, can last from few milliseconds, in case of an explosion, to several years, say, in the case of chemical corrosion.  
Sudden column loss is a dynamic removal phenomenon that can emerge right after explosions or impacts. The response of a structure to sudden column loss can be numerically simulated by adding an extra downward load, accounting for the removed element, to the remaining building [4; 5; 6; 7]. From a dynamical point of view, it was found that column removal time is insignificant for rise time smaller than 10 ms [8; 9; 10], that is, it can be considered instantaneous. Similarly, it was found both that a rise time smaller than \(1/10\) of the vibration period of the structure does not affect its dynamic response [11; 12]. Thus, the response to sudden load application represents an upper bound in terms of displacements [13]. The study of the behaviour of frame structures under such type of events may require sophisticated numerical analysis accounting for material and geometric non-linearities.  
As proved by the large literature available on the topic, many issues are still open and simple rules to design a robust structure are hard to find. This paper deals with a simplified approach to obtain the maximum dynamic displacement of a reinforced concrete frame structure subjected to sudden column removal. The behaviour of such structures is non-trivial since the resisting mechanisms that arise in a damaged frame structure are various and have been recently addressed by experimental [14; 15; 16; 17] and numerical papers [18; 19; 20; 21]. Compressive arch action provides an additional capacity to the flexural mechanism at small deformations, while catenary effects activate for large deformation, preventing the collapse to progress. The proposed methods consists in two tools. The former refers to the possibility to create a synthetic model that simulates the behaviour of the frame subjected to the damage. The structure can be modelled through an equivalent single degree of freedom (SDOF) system [22; 23] since its response to a sudden load mainly depends on its vertical vibration mode. The latter accounts for the fact that the application of a dynamic load produces a response in term of displacement, say \(u_{dyn}\), that is larger than the one that can be obtained applying the same force in a quasi-static loading process, \(u_{st}\)[24]. To study such effect, the dynamic amplification factor (DAF) is adopted as a metrics of the dynamic effect, that is,  
\[\text{DAF}=\frac{u_{dyn}}{u_{st}}. \tag{1}\]  
The value of the parameter is largely dependent on the rise time [25]; in elastic systems subjected to a step load the DAF is equal to 2. On the contrary, the value can be larger in inelastic systems [26; 27; 28].  
In the following, the basics of the modeling of column loss are presented and the synthetic SDOF system is described (Section 2. Thanks to the dimensional analysis approach, the relevant quantities that affect the response of the system to the damage are defined. The simplified approach for the analysis of the damaged frame is then detailed (Section 3). An example on a three storey-four column RC frame is proposed (Section 4) and the accuracy of the results is discussed (Section 5).