# Estimation of Cyclic Demand in Metallic Yielding Dampers Installed on Frame Structures
## 1 Introduction  
Metallic yielding dampers are a cost-effective technology for passive seismic control of the vibration of structures. In buildings, these types of dampers are generally installed into new or retrofitted framed structures to ensure reliable seismic performance through inelastic energy dissipation. Unlike velocity dependent dampers such as viscous and viscoelastic or other design strategies [1; 2], the force output and amount of dissipated energy does not depend on the loading rate [3], but rather on the magnitude of displacement and on accumulated displacement. [PERSON] [4] observed that pulse-like ground motions have dramatic effects on structural response, such as reduced apparent flexibility of high-rise and base-isolated buildings, increased base shear and inter-storey drifts in high-rise buildings, reduced effectiveness of supplemental damping, and increased ductility demand. Consequently, near-fault ground motions can expose aseismic devices to critical working conditions and a significant reduction in their efficacy. This issue has been studied in base isolated systems [5; 6; 7; 8; 9; 10], raising concerns about the viability of base isolation and leading to specific design provisions in the near-fault to accommodate lateral deformation and optimise supplemental damping. The effects of near-fault seismicity on the seismic performance of aseismic dampers has also been addressed. Although dampers still provide seismic protection compared to bare frames in the near-fault [11], it has been shown that earthquake impulsive effects lead to the need for higher yield strength and to a decrease in the available local ductility in the dampers [12]. This results in lower energy dissipation capacityand requires specific design considerations [13; 14; 15; 16; 17]. The number of equivalent cycles of the hysteretic dampers is also relevant to fatigue issues related to heating due to the rate of energy dissipation. Near-fault records are characterised by shorter durations and tend to produce less inelastic cycles than far-field records. In the case of hysteretic dampers, energy dissipation occurs due to metallic yielding and it is strongly related to cumulative inelastic deformation or cyclic demand in the damper. One approach to tackle cyclic demand is by considering the relationship between dissipated energy at a maximum or target displacement. [PERSON] and [PERSON] [18] investigated the response of elastic-perfectly plastic single-degree-of-freedom (SDOF) systems and noted that structures can absorb more or less energy despite experiencing the same peak displacement. Because energy dissipation must be related to cyclic or cumulative demand, they formulated the concept of equivalent number of yield cycles \(N\). This is defined in Equation (1) and shown in Figure 1, as the ratio of the total energy, \(E_{H}\) that is dissipated by yielding within the structure when subjected to the ground motion of an earthquake to the area under the monotonic load-displacement curve at the same maximum displacement, \(\delta_{m}\), experienced during the vibration. \[N=\frac{E_{H}}{Q_{y}\big{(}\delta_{m}-\delta_{y}\big{)}} \tag{1}\]  
[PERSON] and [PERSON]'s results show that the equivalent number of yield cycles \(N\) is a useful parameter to define the severity or damage potential of the ground motion. It is influenced by (i) the characteristics of the ground motion: \(N\) increases with duration and tends to decrease for near-field records; (ii) the fundamental period: \(N\) increases for structures in the 0.2-2 Hz range; and (iii) ductility: \(N\) greatly increases as the displacement ductility increases. The number of equivalent cycles \(N\) is sometimes termed as \(n_{eq}\) in the literature.