# Seismic Performance Evaluation of Steel Buildings with Oil Dampers Using Capacity Spectrum Method
## 2 Proposal of the Square Root of Sum of Squares (SRSS) Effective Damping Ratio  
The equivalent damping ratio, \(h_{h}\), of a hysteresis damper is defined [26] as,  
\[h_{h}(\mu)=\frac{1}{4\pi}\ \frac{\Delta W_{h}}{W} \tag{1}\]  
where \(\Delta W_{h}\) is the hysteresis energy under one cycle of hysteresis and \(W\) is the area of elastic strain energy. For the bilinear hysteresis, as shown in Figure 1a, they are obtained as,  
\[\Delta W_{h}=4\ K\ \delta_{y}^{2}(\mu-1)(1-p) \tag{2}\]  
\[W=\frac{1}{2}\ K\ \delta_{y}^{2}\ \mu^{2} \tag{3}\]In the Equations (2) and (3), \(\delta_{y}\) is the yield displacement of the system, \(\mu\) is a displacement ductility factor, \(K\) is the initial stiffness, and \(p\) is the ratio of post-yield to pre-yield stiffness. In the same way, the equivalent damping ratio, \(h_{V}\), of an oil damper is defined as  
\[h_{v}(\mu)=\frac{1}{4\pi}\ \frac{\Delta W_{v}}{W} \tag{4}\]  
where \(\Delta W_{V}\) is the area of one cycle of force-velocity relation of the oil damper, given by,  
\[\Delta W_{v}=\pi\ C\omega\ \ \mu^{2}\ \delta_{y}^{2} \tag{5}\]  
where \(C\) is the damping coefficient of the oil damper and \(\omega\) is the equivalent circular frequency of the system. In the system with a hysteresis damper and an oil damper, according to the Building Standard Law of Japan [3], the effective damping ratio, \(h_{eff}\), of a seismically isolated building is defined by the direct sum of three types of damping ratios as given in Equation (6). \[h_{eff,Direct}=h_{0}+h_{h}+\ h_{v} \tag{6}\]  
where \(h_{0}\) is the inherent damping ratio. [PERSON] et al. [27; 28] proposed a formula of the effective damping ratio as the average of the equivalent damping ratio in the range of 0 to \(\mu\) as given in Equation (7). It is reported that the accuracy of this formula is relatively high to estimate the earthquake response of the system. However, it is difficult to obtain an explicit analytical solution for the general hysteresis model. \[h_{eff,Kasai}=\frac{1}{\mu}\int\limits_{0}^{\mu}h_{eq}\big{(}\mu^{\prime} \big{)}\,d\mu^{\prime}=\frac{1}{\mu}\int\limits_{0}^{\mu}\big{(}h_{h}\big{(} \mu^{\prime}\big{)}+h_{v}\big{(}\mu^{\prime}\big{)}\big{)}\,d\mu^{\prime} \tag{7}\]  
The current research study proposed a formula to calculate the effective damping ratio from the sum of the inherent damping, and the SRSS of the equivalent hysteresis and viscous damping ratios as given in Equation (8). \[h_{eff,SRSS}=h_{0}+\sqrt{h_{h}^{2}+h_{v}^{2}} \tag{8}\]  
The comparison of the values of the effective damping ratio for the three procedures (\(h_{eff,Direct}\), \(h_{eff,SRSS}\), and \(h_{eff,Kasai}\)) with respect to the ductility factor is shown in Figure 2, where the post-yield stiffness ratio, \(p\), and the initial damping factor, \(h_{0}\), are varied as \(p=0.01\)-\(0.05\) and \(h_{0}=0.05\)-\(0.3\). It is depicted that the proposed \(h_{eff,SRSS}\) becomes closer to  
Figure 1: (**a**) The force–displacement relation of the bilinear hysteresis model; (**b**) the force-velocity relation of bilinear-type oil damper. the value of \(h_{\textit{eff,Kasai}}\) as \(h_{0}\) increases. On the other hand, \(h_{\textit{eff,Direct}}\) overestimates the effective damping ratio by 30% compared to \(h_{\textit{eff,Kasai}}\). In the case of the bilinear-type oil damper, the equivalent damping coefficient, \(C_{\textit{eq}}\), is proposed in the JSSI manual so that the area under the equivalent linear damping should be equal to the area of bilinear force-velocity relation as shown in Figure 1b [17], and derived as  
\[C_{\textit{eq}}=\frac{C_{1}}{\mu*^{2}}+C_{2}\left(1-\frac{1}{\mu*}\right)^{2} +\frac{2C_{1}}{\mu*}\,\left(1-\frac{1}{\mu*}\right) \tag{9}\]  
where, in Equation (9), \(C_{1}\) and \(C_{2}\) are the pre-relief and post-relief damping coefficients of the bilinear-type oil damper, and \(\mu\) * is the ratio of maximum velocity to relief velocity.