# Seismic Upgrade of Steel Frame Buildings by Using Damped Braces
## 1 Introduction
Even if it has some analogies with other approaches based on the response spectrum and utilizing either the initial stiffness [7; 9] or the secant stiffness to the maximum response level [6; 10], the strength of the proposed method relies on its simplicity: the procedure requires to perform initially a pushover analysis to evaluate the capacity curve of the as-built structure, while in the subsequent steps the capacity curve of the retrofitted frame is calculated via simple analytical equations where the unknown parameter is the yield force of the equivalent damped brace. Analytical equations are also used to distribute the stiffness and strength properties of the dissipation braces at every floor of the real building with the target of matching the lateral deformation of the retrofitted frame to the first mode deformation of the as-built frame. This condition, which ensures the simultaneous engagement of the dampers at the different floors, is usually sought in all the design methods [10; 34], because it maximizes the dissipation capacities of the system. The iterative procedure can be automated within a spreadsheet, resulting very convenient for professionals. Another attractive feature is that the procedure is code-compliant, since it has been formulated by taking into explicit account the provisions of the Italian Building Code (IBC) [51; 52] and the Eurocode [53]. It is worth noting that the above method is very similar to the one developed independently for steel MRFs by [PERSON] and [PERSON] [33], as both methods consist of a recursive procedure where the properties of an equivalent SDOF dissipation brace are defined by the comparison between the inelastic response spectrum demand and the capacity of the retrofitted frame in order to match a specified performance displacement. The main difference lies in the fact that the procedure by [PERSON] and [PERSON] [11], which has been formulated in compliance with IBC [51], represents the capacity curves of the equivalent SDOF frame which are used in the ADRS space for the application of the Capacity Spectrum Method by means of equivalent bilinear curves with post-yield hardening, in accordance with the recommendations of IBC's Commentary [52]. The method [11] was applied in previous works to mid-rise residential RC buildings characterized by a fundamental period of about 1 s [11; 12]. In comparison to RC buildings, steel structures are characterized by a substantially more flexible structural response, with a non-negligible influence of P-delta effects, and a non-negligible column deformability especially in case of tall buildings, which can affect the vibration mode. The present study aims at assessing the viability of the retrofit procedure for steel MRFs and highlighting the possible limitations of the method. Three archetype configurations of MRFs with increasing number of floors (2, 4 and 8) affected by evident deficiencies such as non-equal bidirectional strength and stiffness in the main horizontal directions are considered [54; 55]. The retrofit aims at upgrading the seismic performance of the buildings in order to comply with larger seismic demands, such as e.g., in cases of an update to the hazard maps and/or code regulations. For this purpose, the seismic hazard characteristics associated with the municipality of Lamezia Terme (seismic class 1 according to IBC [51], and soil type C) are considered. The structure of the paper will follow this scheme. Chapter 2 provides fundamentals to the modeling choices adopted in the OpenSees framework for the non-linear behavior of steel members, with the further purpose of providing some useful indications to researchers who use this software program. The OpenSees software program (Open System for Earthquake Engineering Simulation) [56; 57] has been indeed used since it facilitates the analysis of multi-story buildings in the non-linear regime with high accuracy and computational efficiency [58].