# Seismic Performance Evaluation of Steel Buildings with Oil Dampers Using Capacity Spectrum Method
## 3 Proposed CSM Procedure for Buildings Equipped with Oil Dampers  
This section demonstrates the procedure of the proposed CSM using the equivalent single degree of freedom (ESDOF) system defined by [PERSON] et al., 2000 [29]. The ESDOF consists of a mass, \(m\), a frame damping coefficient, \(C_{p}\), and a stiffness, \(K_{f}\). This study assumed that the natural period of the ESDOF was 0.5 s and the base shear coefficient (i.e.,  
Figure 2: Comparison of different procedures to estimate the effective damping ratio. the ratio of yielding force to the total weight) was 0.2. The ESDOF was equipped with the bilinear-type oil damper which had a damping coefficient, \(C_{D}\), and a stiffness, \(K_{D}\), as shown in Figure 3a. The ESDOF parameters are summarized in Tables 1 and 2. The capacity curve of the system is shown in Figure 3b. The demand curve is presented in the form of an acceleration-displacement spectrum of the selected earthquake ground motion. In this study, the Level-2 earthquake defined in the Building Standard Law of Japan [3] was used to generate the design acceleration-displacement response spectrum with a 5% damping factor. The latter was adjusted according to the value of the effective damping using the reduction factor, \(F_{h}\), in Equation (10), adopted by the JSSI manual [17] from the study conducted by [PERSON] et al. [28,30]. \[F_{h}=\frac{\sqrt{1+25h_{5\%}}}{\sqrt{1+25h_{eff}}} \tag{10}\]  
The proposed CSM was performed as following:  
* The multi degree of freedom (MDOF) was converted to the ESDOF [29];
* For selected ductility factor values (e.g., \(\mu\) = 1, 2, 3... ), the respective effective damping ratios (e.g., \(h_{1}\), \(h_{2}\), \(h_{3}\)... ), not including the contribution of the oil damper \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Natural Period** & **Weight** & **Damping** & **Stiffness** & **Yielding** \\
**(T)** & **(W)** & **(h\({}_{0}\))** & **(K\({}_{\mu}\))** & **Force (F\({}_{y}\))** & \(p=\frac{\text{K}_{h}}{\text{K}_{i}}^{*}\) \\ \hline
**s** & **kN** & **\%** & **kN/cm** & **kN** \\ \hline
0.5 & 5000.0 & 5.0 & 805.68 & 1000.0 & 0.05 \\ \hline \hline \end{tabular} \({}^{*}\)\(K_{1}\) and \(K_{2}\) represent the pre-yield and post-yield stiffness of the ESDOF system, respectively.  
\end{table} 
Table 1: Technical parameters of the equivalent single degree of freedom (ESDOF) system. Figure 3: **(a)** The equivalent oil damper configuration in the ESDOF system; **(b)** the initial demand curve (5% damping) and capacity curve of the ESDOF system. devices, were computed using Equation (11) [3], and the demand spectra were adjusted by Equation (10), and plotted together with the capacity curve of the ESDOF as shown in Figure 4a; \[h_{eq}=0.8\frac{2}{\pi}\left(1-\frac{1}{\mu}\right)\frac{1-p}{1+p(\mu-1)}+0.05\] (11)
* The spectral ordinates (spectral acceleration, \(S_{A}\), and displacement, \(S_{D}\)) of the demand spectrum at the intersection points with equivalent stiffness lines were determined and named as initial performance points;
* For each initial performance point, the equivalent circular frequency of the ESDOF was estimated by Equation (12); \[\omega_{eq}(\mu)=\sqrt{\frac{S_{A}}{S_{D}}}\] (12)
* The corresponding story displacements (\(\delta_{1}\), \(\delta_{2}\), \(\delta_{3}\)... ) were computed from the ESDOF.