# 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  
Reaction forces, energy dissipation, stiffness properties, and overall performance under cyclic loading conditions are computed from the hysteresis curve (Figure 6a). This curve compares the reaction forces at the lower bolts against the applied displacements. The bilinear behavior assigned to the material reveals a nonlinear model prediction when the loop is examined. This is attributed to the inhomogeneity of local stress behavior;  
Figure 5: Fields of (**a**) equivalent von Mises stress, (**b**) equivalent cumulative plastic strain (AEPS), (**c**) equivalent plastic strain, and (**d**) strain energy on the ultimate applied displacement. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Equivalent Stress (MPa)** & **AEPS** & **Equivalent Strain Energy (mJ)** & **Energy per Unit Volume (mJ/mm3)** & **Dissipated Energy (mJ)** \\ \hline
475.2 & 0.925 & 0.055 & 33,070 & 22.7 & 51,166 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Summary of simulation results of damper behavior at the ultimate applied displacement. while some parts are under elasticity, localized zones reach plasticity, converting the overall performance to nonlinear. The symmetrical skeleton curve from the force peaks at the end of each load step is attributed to isotropic hardening. The predicted increasing peak force at the end of each cycle is due to the hardening behavior that alters the yield limit of the material with each load. To validate the numerical model, the hysteresis curve is compared in Figure 6a with experimental data from [PERSON] et al. [19]. Their experiments involved testing UD-40 damper specimens with similar dimensions, fabricated from JIS SM490 steel, under horizontal cyclic loading. The specimens were fixed at their upper and lower ends to a horizontal axial machine using bolted connections. A 500-kN actuator applied sinusoidal displacement-controlled loading at an average speed of 10 mm/s and amplitudes of 10, 30, 50, and 100 mm. Displacement measurements were acquired using wire displacement transducers located in the upper supporting beam, while load measurements were obtained using a built-in load cell located below the lower plate. The computational model predicts higher reaction force values within the investigated displacement range, which is a direct result of the bilinear approximation of the hardening behavior. Figure 1 shows clearly that the higher slope of the bilinear model compared to the true stress-strain curve affects the estimated stress levels in the damper. This overestimation of strain hardening is most evident in the prediction of yield strength from initial yielding up to 3% strain. The discrepancy in hardening is identified as the primary source of error, given that the damper primarily experiences local strains ranging from 0 to 5.5%. However, the average relative error of the model in the stress values is only 7.8%, which falls within an acceptable range for validating the effectiveness of the computational model in predicting the performance of the UD-40 damper. A key feature of this hysteresis curve is that the area within the loop directly represents the energy dissipated by the damper as plastic work. The stiffness levels are calculated via Equation (7). The stiffness is divided into two types: the first stiffness or elastic stiffness produced in the elastic zone, and the second stiffness or postelastic stiffness produced in the plastic zone. Both are calculated via Equation (7), as proposed by [PERSON] and [PERSON] [39]. \[k_{e}=\frac{1}{\frac{2 LR^{2}}{E_{y1}}+\frac{\pi R^{3}}{2E_{ly2}}} \tag{7}\]  
\[I_{y1}=\frac{w_{1}t^{3}}{12} \tag{8}\]  
\[I_{y2}=\frac{w_{2}t^{3}}{12} \tag{9}\]  
Figure 6: (a) Hysteresis curve and (b) plastic strain energy vs. displacement. In Equation (7), \(k_{e}\) is the first stiffness. \(L\) is the arm's length of the damper; \(R\) is the radius of the curve; and \(E\) is the Young's modulus.