# Estimation of Cyclic Demand in Metallic Yielding Dampers Installed on Frame Structures
## 1 Introduction
[PERSON] [19; 20; 21] addressed the correspondence between the cumulative inelastic deformation ratio, \(\eta\) defined in Equation (2) (i.e., dissipated hysteretic energy normalised to the yield force \(Q_{y}\) and displacement \(\delta_{y}\)) and the inelastic deformation ratio \(\mu\) defined in Equation (3) (i.e., plastic deformation normalised to \(\delta_{y}\)) through numerical analyses with elastic-plastic systems with 5 degrees of freedom. Equation (4) shows that \(N\) and the ratio \(\eta/\mu\) take the same value. The ratio \(\eta/\mu\) is also referred to as the \"efficacy\" of the structural element. \[\eta=\frac{E_{H}}{Q_{y}\,\delta_{y}} \tag{2}\]  
Figure 1: Energy dissipation in an elastoplastic system: (**a**) Cyclic force-displacement Q-\(\delta\) curve; (**b**) Energy dissipated by yielding \(E_{H}\); (**c**) Energy dissipated in a quarter-cycle at maximum displacement. \[\mu=\frac{\delta_{m}-\delta_{y}}{\delta_{y}} \tag{3}\]  
\[N=\frac{E_{H}}{Q_{y}\!\left(\delta_{m}-\delta_{y}\right)}=\frac{\eta}{\mu} \tag{4}\]  
[PERSON] highlighted that, if known in advance, the ratio \(\eta/\mu\) allows us to estimate the maximum expected displacement for a certain amount of dissipated energy demand. Regardless of any other parameters, he proposed an approximate design value of \(\eta/\mu=2\) for bare elastic-perfectly plastic frames with no strength deterioration. This expression suggests that the efficacy of the structure (in terms of the relationship between dissipated energy and maximum displacement) can be predicted for a certain lateral carrying system under the design earthquake. [PERSON] also proposed design values [19; 20; 21] of the ratio \(\eta/\mu\) in Equation (5) for flexible-stiff mixed frame multi-storey systems, consisting of (i) a flexible part or frame element able to restrain the lateral deformations within the elastic range for the maximum drift expected under the seismic loading (Figure 2a), and (ii) a stiff part or energy absorbing elements (namely, dampers) with high elastic rigidity and large deformation capacity (Figure 2b). Figure 2c represents the behaviour of the mixed system. Because of the added complexity of the mixed multi-storey system, a parametric study was used to focus on structures with predetermined lateral stiffness and strength distributions, and changing the parameter \(r_{q}=/Q_{m}/\delta_{y}\), that is, the ratio of the maximum force in the flexible part, \(/Q_{m}\), by the yield strength of the stiff part, \({}_{s}Q_{y}\). From the results of these studies, it can be concluded from Equation (5) that as \(r_{q}\) increases, the number of equivalent yield cycles in the dampers increases, leading to the following relationship:  
\[N=\frac{\eta}{\mu}=\left\{ \begin{array}{ll}4+r_{q}&r_{q}\leq 1\\ 8\sim 12&r_{q}>1\end{array} \right. \tag{5}\]  
[PERSON] [22] proposed a cyclic demand spectra that specifies the number of cycles for which the seismic load must be resisted by a structure, and noted that both the amplitude and the number of cycles should be considered in assessing the damage potential of ground motions. [PERSON]'s alternative definition of the number of cycles \(N_{xy}\) is based on dividing cumulative damage by the damage caused by a full cycle of largest amplitude \(u_{max}\). The Coffin-Manson law was used for damage and damage is accumulated by the Miner's rule. The number of equivalent cycles of amplitude \(u_{max}\) that cause  
Figure 2: Load-deformation curves in flexible-stiff mixed systems: (a) flexible part; (b) stiff part; (c) mixed system. the same damage to the structure as the entire deformation history of \(n\) half-cycles of amplitude \(u_{i}\) is given by Equation (6), where \(c\) is a structural damage parameter that requires experimental calibration. Cyclic demand spectra are site-specific, so they contain all uncertainties related to the ground motion and yield level, so, the only variable is the fundamental period. Exponent \(c\) must account for the rest of the structural characteristics.