# Estimation of Cyclic Demand in Metallic Yielding Dampers Installed on Frame Structures
## 1 Introduction
Based on the definition of the number of cycles in Equation (6), three cyclic spectral parameters \(N_{A}\), \(N_{V}\), \(N_{D}\) are introduced to characterise cyclic response, depending on whether the structural response is to be governed by acceleration, velocity or displacement histories. The assumption behind this reasoning is that cyclic response depends primarily on the stiffness of the system. It was found that \(N_{A}>N_{V}>N_{D}\) so a stiff structure is likely to experience a larger number of cycles than a flexible structure. Relative stiffness was measured by the ratio of the fundamental period to the earthquake central period. \[N_{cy}=\frac{1}{2}\sum_{i=1}^{2n}\Bigl{(}\frac{u_{i}}{u_{max}}\Bigr{)}^{c} \tag{6}\]  
[PERSON] and [PERSON] [23] claimed that the estimation of cyclic deformation demand resulting from earthquake load is crucial if the damage of the system needs to be evaluated, and they produced a cyclic demand spectra to assess the number of full inelastic cycles, \(N_{f}\). The expression for \(N_{f}\) in Equation (7) includes an energy shape factor \(\alpha_{h}\) that depends on both the ductility and the shape of the hysteresis loops. Structural properties such as \(Q_{y}\) and \(\delta_{y}\) are eliminated from Equation (7) with a focus on seismic design by introducing the relationships between fundamental period \(T\), displacement ductility \(\mu\), and the seismic response modification factor \(R\). As a result, the number of inelastic cycles in a SDOF can be calculated with Equation (8) as a function of the hysteretic energy \(E_{H}\), the mass \(m\), the displacement ductility \((\mu+1)=\delta_{max}/\delta_{y}\), the force reduction factor \(R\), fundamental period \(T\), energy shape factor \(\alpha_{h}\), and spectral acceleration \(S_{a}\). Use of Equation (8) implies prior knowledge of several variables including ground motion parameters, structural parameters, and target ductility. For each specific structural system, a series of empirical expressions can be calibrated to relate \(E_{H}\), \(R\), \(\alpha_{h}\), and \(\mu\), to arrive at simple expressions with only one variable. \[N_{f}=\frac{E_{H}}{4\;\alpha_{h}\;Q_{y}\;\delta_{y}\;(\mu+1)} \tag{7}\]  
\[N_{f}=\Bigl{(}\frac{E_{H}}{m}\Bigr{)}\frac{\pi^{2}}{(\mu+1)\;\alpha_{h}}\Bigl{(} \frac{R}{T\;S_{a}}\Bigr{)}^{2} \tag{8}\]  
[PERSON] [24] underlined that the equivalent number of yield cycles varies largely depending on the characteristics of the ground motion, with values varying from 1 for impulsive records and up to 40 for long-duration records. Statistical regressions of nonlinear time history analyses using two components of 122 European ground motion records revealed that the equivalent number of yield cycles can be predicted with the exponential type expression in Equation (9). It was found that (i) the elastic behaviour is well described by the ratio \(\tau\) of the fundamental period, \(T\), to the corner period within short and medium/long periods in the spectrum, \(T_{G}\), where the spectrum changes from the constant acceleration segment to the constant velocity segment; (ii) the plastic behaviour is described with the strength reduction factor \(R\); and (iii) the seismological parameter with the highest correlation factor is the dimensionless parameter \(I_{d}=2\)g\(\cdot\)\(I_{A}\)/\((\pi\cdot\)\(PGA\cdot\)\(PGV)\), where \(I_{A}\) is the Arias intensity, \(PGA\) is peak ground acceleration, and \(PGV\) is peak ground velocity. Further research by [PERSON] et al. [25] has shown that \(I_{d}\) correlates well to the equivalent number of yield cycles \(N\) both in the vicinity of the source and far from it. Using regression analyses with 128 near-fault and 122 far-field records, sets of parameters in Equation (9) were calibrated in order to assess the number of yield cycles experienced by a SDOF structure in the far-field and in the near-fault. They observed that \(N\) tends to increase with \(I_{d}\) and that \(I_{d}\) tends to increase with the distance to the earthquake source.