# Effect of Dynamic Loading Conditions on Maximizing Energy Dissipation of Metallic Dampers
## 3 Dynamic Analysis of Static Optimization
### Dynamic Analysis of the Statically Optimized Damper  
The optimized shape of the presented damper is derived from the optimization process with a static analysis using an approach that is similar to previous studies [13; 14; 22; 29]. However, this approach can have some limitations. As stated in Section 3.3, it is difficult to induce specialized shapes for the different capacities required. The control of the stroke size or cycles scarcely influences the optimized shape. Moreover, since the shape-optimized damper becomes stronger and stiffer than the initial model, the plastic deformation of the damper can decrease with a small amount of ground acceleration. Because of this reduction of deformation, the optimized damper might dissipate less seismic energy during small earthquakes.  
To consider the dynamic characteristics of a metallic damper, a dynamic analysis was conducted through ABAQUS to examine the capability of the previously optimized damper under dynamic loading. In order to evaluate the tendency as simply as possible, a prototype bridge model was constructed (Figure 9). This structural model was composed of two prototype piers and one prototype superstructure in a \"simply supported\" condition. The structure was assumed to be excited through the ground, only in x-direction (in-plane). The lengths of the piers and the superstructure were 10 m and 30 m, respectively. The same material, i.e., steel, was used, which was applied in the modeling of the damper, with both the prototype column and superstructure having the same circular cross section with a diameter of 0.5 m. Thus, the mass of the super structure was 49.45 ton, and the first natural frequency of the structure was 1.23 Hz. Here, the total lateral stiffness of the bridge was mainly due to the stiffness of the prototype columns. The intrinsic damping ratio of the material was presumed to be zero, in order to ensure that the dissipation of seismic energy occurred only through the plastic deformation of the damper. Because this prototype bridge model did not consider the structural design aspect, the self-weight of the structure was disregarded.  
To analyze the tendency of the optimized damper's behavior with respect to the scales of the ground motion, two simple ground accelerations were applied to the model. The accelerations were presumed on the basis of the sine functions of one cycle with two maximum accelerations: 0.1 g and 0.2 g (where g indicates the gravitational acceleration) having a frequency of 0.5 Hz. The sine function with a maximum of 0.1 g represents a small earthquake, and the sine function with a maximum of 0.2 g represents a larger earthquake. The FE analysis was conducted using dynamic-implicit time integration in ABAQUS with default settings. This time integration scheme uses a Hilber-Hughes-Taylor operator, which is an extension of the Newmark \(\beta\)-method [32; 33]. In ABAQUS, the operator utilizes a very slight numerical damping as a default value to enhance the numerical stability, but the damping effect of the integration scheme was almost negligible in this study.  
The results of the analysis are provided in Table 2. The initial shape and one optimized case are compared. Case S1 represents the results of the static optimization without the thickness variable, due to the similarities between the complete results. As in the previous static, the optimized damper dissipates more energy than the initial shape damper in the case of the larger 0.2 g acceleration. However, the plastic energy dissipations of S1 decrease at a smaller acceleration of 0.1 g. This result confirms that the optimized damper will dissipate less energy for small ground acceleration with high probability. Therefore, further optimization-aimed dynamic modeling was constructed to overcome this limitation of the static optimization approach.