## 1 Introduction  
The safeguard of the built heritage is gaining an increasing interest in structural and seismic engineering [1, 2]. The knowledge of the seismic response of the structures plays a fundamental role in the evaluation of the seismic vulnerability of existing structures [3, 4, 5, 6] and in the prediction of the expected losses [7, 8, 9]. The designer, to achieve a correct safety evaluation in terms of the capacity demand ratio, must resort to non\(-\)linear models and non\(-\)linear analyses [10, 11, 12, 13], which, even though they are described in the codes, are very prone to being misinterpreted. The implementation of a non\(-\)linear model needs adequate experience and, first of all, an adequate knowledge of the condition of the building to be modelled in terms of geometry, loads, structural details, materials and degradation [14, 15, 16, 17, 18]. In the perspective of a seismic classification of the built heritage, it is necessary to define a standardizable, unique and user\(-\)friendly methodology [19, 20, 21, 22]; therefore, recently, a simplified assessment approach has been developed concerning steel moment resisting frames (MRFs) [23] based on a trilinear model that needs only the elastic structural analysis and the rigid-plastic analysis and does not require any static or dynamic non\(-\)linear effort. Rigid plastic analysis has generally been used as an assessment tool against vertical and horizontal loads. The novelty point of the procedure is to use the rigid plastic and elastic analysis joined with a performance-based evaluation aimed at easily assessing the seismic performance of existing buildings, also ensuring a rapid mapping of the built heritage in terms of seismic vulnerability.  
This assessment procedure is herein applied to some study cases to support their ease of application. It mainly consists of the definition of a trilinear capacity curve by some characteristic points that define three branches [24; 25]. These points correspond to different limit states. It is important to mention that the code assessment approach does not provide the definition of specific performance points on the pushover curve, thus entrusting the designer to the definition of required target limits which are provided by codes in terms of member chord rotation.  
The method has been set up by performing several parametric analyses, in terms of pushovers, on 420 frames, designed according to three different approaches, i.e., without any seismic detailing (O\(-\)MRFs), by Eurocode 8 (S\(-\)MRFs) and by the theory of plastic mechanism control (G\(-\)MRFs). It is usually observed that structures designed without any seismic provision exhibit a soft storey mechanism while the structures designed according to the modern approach of Eurocode 8 can achieve more robust performances even if the global mechanism can be assured only by the TPMC approach. The results belonging to the pushover analyses performed on the structures have been used to calibrate the assessment relationships, ensuring a wide application of the method. The comparison in terms of capacity and demand can be made according to two alternative approaches: the one proposed by Eurocode 8 [26] and the one proposed by [PERSON] and [PERSON] [27]. The former exploits the concept of ADRS spectrum while the latter has the benefit of being of more easily applicable because it does not single out between low and high periods of vibration. In the following, the main model equations are reported and described.