## 1 Introduction  
In order to prevent significant structural damage by guaranteeing adequate ductility and energy dissipation capacity, [PERSON] and [PERSON] [1] proposed the well-known design methodology called \"Theory of Plastic Mechanism Control (TPMC)\" in 1997. The original version of this method is based on the extension of the kinematic theorem of plastic collapse to the concept of mechanism equilibrium curve, and, recently, a closed-form solution has been developed and applied to different steel structural typologies [2, 3]. This approach has been applied only to new buildings in order to avoid partial failure mechanisms, such as the well-known soft-storey mechanism, by ensuring the achievement of the global mechanism type under severe seismic actions. However, recently, a new performance design method, also based on the TPMC approach, has been developed to evaluate the behaviour of the existing buildings. In fact, the assessment of the seismic vulnerability of existing structures has become, in recent years, a topic of increasing importance among those developed by structural engineering [4, 5, 6, 7, 8, 9, 10, 11]. Infrastructures, but also simple buildings, need to be checked and verified against seismic loading to design any intervention in seismic enhancement or retrofitting. The planning of seismic risk mitigation has shown that it is of fundamental importance to carry out a rapid classification of the existing building heritage in terms of seismic performance [12, 13, 14, 15, 16, 17, 18]. The performance of a building is evaluated by comparing its ability to dissipate incoming seismic energy due to structural ductility and seismic demand. If the demand for ductility is lower than the capacity, the structure, even if damaged, does not collapse, ensuring, in this case, the primary requirement of performance from the point of view of the Ultimate Limit States (ULS), namely the protection of human lives. Otherwise, i.e., when the demand for ductility is higher than the capacity, the structure collapses because it is not able to dissipate the incoming seismic energy satisfactorily and, therefore, to develop an adequate collapse mechanism [19; 20; 21]. The main difficulties in creating a satisfactory model for the prediction of seismic damage concern the quantitative definition of the degree of damage corresponding to each expected performance level. The complete knowledge of the seismic performance of a structure is the result of sophisticated numerical procedures such as static or nonlinear dynamic analysis [22; 23; 24; 25; 26]. Such analyses require the structural model to be adequately accurate to capture nonlinear behaviour. However, especially for large-scale assessments and the planning of seismic risk mitigation interventions, a simplified methodology able to describe the capacity curve analytically and quickly without resorting to more complex analyses could be extremely useful, given a code implementation. As a result, a simplified methodology for different structural types has been developed to evaluate the capacity of an existing building without resorting to static or dynamic nonlinear analysis [27; 28; 29; 30]. To calibrate this model, a wide parametric analysis was carried out on 420 frames designed according to three different approaches [27]. The non-dimensional pushover curve has been simplified by defining three branches: an elastic response branch, a plateau branch corresponding to the maximum bearing capacity, and a descending branch corresponding to the mechanism equilibrium curve. The result is a simplified trilinear model. In addition, this model allows the definition of four characteristic points (A, B, C, D) corresponding to specific performance levels, associated with the limit states provided by the existing codes. (\"Fully Operational\" (FO), \"Operational\" (O), \"Life Safety\" (LS), \"Near Collapse\" (NC)). This paper validates the proposed method by applying it to 420 steel MRFs designed using three distinct approaches.