# Proposal of a Design Procedure for Steel Frames with Viscoelastic Dampers
## 1 Introduction  
Buildings designed according to conventional capacity design techniques [1] experience only minor structural damage in the case of low-to-moderate-intensity earthquakes and ensure an adequate safety level to their occupants in the case of major events. However, after severe earthquakes the immediate functionality of the building or even the possibility and the cost-effectiveness of repairing are not guaranteed since both the dissipative members and the non-structural elements are expected to be strongly damaged. To limit or even avoid damage in structural and non-structural components, passive control techniques [2] have been developed and successfully applied in the last decades to both new and existing buildings in order to control the seismic response [3, 4, 5]. The present paper focuses on a passive control system based on viscoelastic solid dampers and applied to new moment-resisting frame (MRF) steel structures. A viscoelastic solid damper (VED) exploits the viscous-hysteretic behavior of elastomers to dissipate strain energy in the form of heat when subjected to cyclic deformation in dynamic load conditions [6, 7]. When subjected to dynamic cyclic loading, the response of the elastomer shows a lag between the applied strain and the stress developed in the material. In the stress-strain plane, the response to harmonic loading is characterized byinclined elliptical hysteretic cycles. Mechanical properties of elastomers are affected by strain rate and ambient temperature, as widely reported in the literature [8; 9; 10]. In the case of prolonged cycling, elastomers are prone to softening due to self-heating [11; 12], and this effect increases with the imposed deformation [13; 14]. A VED consists of layers of elastomer and steel plates bound together through a vulcanization process. Steel plates are linked to the structure so as to allow the elastomeric layers to deform in shear. VEDs are usually mounted in series with a bracing system, typically consisting of diagonal braces (e.g., [15; 16]) or chevron braces (e.g., [17; 18]), but other configurations can be found in the literature [19; 20; 21; 22]. The key features of a VED, as a passive control device, are the capability of providing the structure with additional stiffness and damping [23] and the capability of recovering its initial undeformed shape at the end of seismic events [15; 24]. To the best knowledge of the authors, the first applications of VEDs in buildings date back to the 1960s for the control of wind-induced vibrations [21]. The first experimental and theoretical studies on the use of VE dampers as seismic devices date back to the early 1990s [8; 9; 25; 26; 27; 28]. Recently, design guidelines for new buildings equipped with viscoelastic dampers, based on nonlinear response spectrum analysis, have been included in the draft of the new Eurocode 8 [29]. However, various design approaches for buildings equipped with VEDs already exist in the literature, i.e., procedures based on the construction of the elastic or elastoplastic reduction curve [30; 31], energy-based procedures [32], procedures based on the direct displacement-based design method [33; 34], and procedures based on the Modal Strain Energy (MSE) method [35; 36; 37; 38]. The MSE method is an analytical tool initially formulated by [PERSON] and [PERSON] [39] and developed for engineering purposes by [PERSON] and [PERSON] [40]. The MSE method allows the prediction of the equivalent viscous damping ratio of a structure equipped with viscoelastic dampers provided that the characteristics of the viscoelastic material and the modal shape of the undamped structure are known. Also, capacity spectrum method procedures have been proposed to assess the performances of buildings with high added damping [41; 42; 43]. Other studies focused on the effects of earthquakes with different characteristics on structures equipped with seismic protection devices, with specific reference to near-fault earthquakes [44; 45; 46].