## 5 Design and Verification of Vertical Steel Damper
### Evaluations of Energy Dissipation Performance  
To evaluate the damping performance of the considered steel dampers in Table 9, force-displacement analysis was performed for the cyclic displacement load. Figure 12 shows the detailed finite element analysis model for the steel damper. As shown in the figure, a three-dimensional solid element (SOLID181) was used for the steel damper model, and it was modeled to have a sufficient number of elements and aspect ratio to enable more accurate plastic deformation analysis. As a boundary condition, a virtual node that can transmit the load from the superstructure was set and modeled so that a cyclic displacement load could be applied to the end of the steel damper using the connecting element (MPC184). As shown in Figure 12, all displacements at two pin holes are assumed to be constrained as fixed conditions, and the displacement-controlled cyclic load is applied at the end of the pinned joint. Figure 11: Configurations and dimensional parameters of steel damper. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Steel Damper** & \(\mathbf{L1}\) & \(\mathbf{L2}\) & \(\mathbf{L3}\) & \(\mathbf{H1}\) & \(\mathbf{H2}\) & \(\mathbf{H3}\) & \(\mathbf{W}\) \\ \hline SD1 & 40 & 40 & 80 & 20 & 8 & 5 & 5 \\ SD2 & 40 & 35 & 75 & 15 & 10 & 5 & 4 \\ SD3 & 40 & 10 & 70 & 20 & 14 & 6 & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Dimensions for steel dampers (mm). In order to accurately describe the plastic behavior of Type 316 SS, the material of the steel damper used in this study, for the finite element analysis, the following [PERSON]'s inelastic constitutive equations [25, 26] were used for the kinematic hardening model:  
\[\dot{\alpha}_{ij}=\sum_{k=1}^{3}\left[\frac{2}{3}C_{k}\dot{e}_{ij}^{p}-\gamma_{k }\left(\alpha_{ij}\right)_{k}\dot{p}\right] \tag{5}\]  
where \(\dot{\alpha}_{ij}\) and \(\dot{p}\) indicate the revolution of back stress and an accumulated plastic strain, respectively, and \(C_{k}\) and \(\gamma_{k}\) (\(k\) = 1-3) are material constants to be used in the ANSYS program. For the isotropic hardening model, the inelastic Vece model [27] is used as follows:  
\[\dot{R}=b[Q-R]\dot{p} \tag{6}\]  
where \(\dot{R}\) indicates the revolution of drag stress, and \(b\) and \(Q\) are material constants. Table 10 is the material constants for Type 316 stainless steel used in the above inelastic material constitutive equations [28]. To evaluate the hysteretic damping performance of the steel damper, the vertical cyclic displacement range was set to \(\pm\)30 mm and inelastic finite element analysis was performed on the triangular waveform input. Figure 13 shows the distribution of an equivalent plastic deformation for the steel damper of SD2 during five cycles. As shown in the figure, the maximum equivalent plastic strain occurs at the upper and lower surfaces of the middle part of the steel damper by about 2.5%, and after the final fifth cyclic load, the maximum residual plastic strain remains about 0.56% at the transition region. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Material**} & \(\sigma_{\mathbf{yo}}\) & \(E\) & \(C_{1}\) & \(C_{2}\) & \(C_{3}\) & \(\gamma_{1}\) & \(\gamma_{2}\) & \multirow{2}{*}{\(\gamma_{3}\)} & \multirow{2}{*}{\(b\)} & \(Q\) \\  & **(MPa)** & **(GPa)** & \(\times\)**10\({}^{9}\)** & \(\times\)**10\({}^{9}\)** & \(\times\)**10\({}^{9}\)** & \(\times\)**10\({}^{3}\)** & \(\times\)**10\({}^{3}\)** & \(\times\)**10\({}^{3}\)** & \(\times\)**10\({}^{6}\)** \\ \hline Type 316 SS & 135 & 190 & 120 & 20.2 & 10. Figure 14 presents the analysis results of an equivalent plastic strain time history at the node where the maximum plastic strain occurs for SD2. As shown in the figure, the maximum equivalent plastic strain in the middle of the steel damper is about 2.6% at the time of the maximum and minimum cyclic displacement loads.