# Computational Modeling of U-Shaped Seismic Dampers for Structural Damage Mitigation
## 4 Simulation Results of Damper Behavior
### Overall Performance: Stiffness, Reaction Forces, and Energy Dissipation Capacity
\(I_{y1}\) and \(I_{y2}\) represent the stiffness to bending for the arms and curve of the damper, respectively. Specifically, in Equation (8), \(w_{1}\) is the average width of the arms, and for Equation (9), \(w_{2}\) is the average width of the curve section of the damper. For both, \(t\) is the thickness. Although the second stiffness exhibited during the nonlinear behavior of the damper cannot be deduced from a simple analytical expression, experimental observations suggest that it may approximate 0.1 to 0.5 times the initial stiffness [39]. For the purposes of this paper, the second stiffness is assumed to be 0.1 times the initial stiffness. Additionally, an analytical comparison between the stiffnesses of each damper is performed using the basics of springs from Equations (10) and (11) [40]. These equations do not consider the geometrical data of the damper, but only the reaction forces (\(F\)) and the displacement (\(\delta\)). The analysis will be performed in the first step, using a maximum displacement of 10 mm. \[k_{e}=\frac{F}{\delta} \tag{10}\]  
\[k_{p}=\frac{\Delta F}{\Delta\delta} \tag{11}\]  
where \(k_{p}\) is the second stiffness, \(\Delta F\) is the difference in the forces applied in the plastic zone of the curve, and \(\Delta\delta\) is the difference between the displacements in the same zone. The load protocol for the analysis considers 20 steps from 0 to \(\pm\)100 mm applied to the upper bores. Table 4 shows the comparison between the stiffness given by the geometrical parameters (GS) and that by the reaction/displacement parameters (RDS). The experimental first stiffness reported in [PERSON] et al. [19] is included, while the second stiffness is computed from the reported experimental hysteresis curve in similar displacement intervals. Table 4 demonstrates a notable difference between the two methods, with the RDS method yielding higher values for the first stiffness compared to the GS method. This difference can be attributed to the conditions of the study, where the RDS method considers both the geometrical and boundary conditions, against the GS, which only depends on the geometry of the damper. The second stiffness with RDS methods provides values closer to experimental data than GS. This is related to the nonlinearity of plastic behavior in the damper, with local values reaching plasticity while others behave elastically, a phenomenon that cannot be described by the GS method. Compared to experimental results reported in ref. [19], GS yields accurate prediction of stiffness with a 16% relative error compared to the RDS method with a 46% global relative error. These discrepancies could be affected by the accuracy of experimental data, error bars, intervals of considered displacements, and the non-reported method selected by the authors to compute the experimental stiffness [19]. Figure 6b shows the increase in energy dissipated by the damper with the applied displacement. The vertical distance between the peaks at each displacement is increasing with each cycle, indicating that the energy dissipated grows exponentially. In the model damper, under ten cycles, the maximum dissipated energy reaches 50 kJ. The number of cycles and the energy that the damper can dissipate are ongoing research topics that involve calibrating appropriate damage-fatigue models. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Stiffness (N/mm)**} & \multicolumn{3}{c}{**Relative Error (\%)**} \\  & **1 st** & **2 nd** & **1 st** & **2 nd** & **Global** \\ \hline Experimental [19] & 2000 & 100 & & & \\ GS [39] & 1920 & 192 & 4 & 96 & 46 \\ RDS & 2308 & 82 & 15 & 16 & 16 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Stiffness obtained via the GS and RDS methods compared to experimental data.