## 5 Numerical Comparison of the Existing Mass-Rig Systems
### Numerical Modeling of the Inertial Systems  
To characterize RC column behavior, two modeling methods have been widely used, namely, lumped plasticity and distributed plasticity [17; 18; 19]. In the lumped plasticity approach, the inelastic behavior of the column specimen is specified to occur only at specific regions of the element where the plastic deformation is substantial (plastic hinges). Outside of the plastic hinge regions, the element exhibits linear elastic behavior. The length of the plastic hinge region is indicated by the user and modeled using fibers with two integration points at both ends of the plastic hinge. The properties of the elastic portion of the element can be specified so that the initial stiffness is adequately modeled. In the distributed plasticity approach, the non-linear behavior of the structural component is simulated using non-linear beam-column components discretized using fiber sections, and the total length of the element is divided into several segments or integration points. This method offers a more precise characterization of the inelastic behavior since it allows inelastic deformation to be formed anywhere within the member. It is worth mentioning that the non-linear beam-column element does not consider shear deformation or bond-slip rotation; however, these effects can be added to the RC element through the use of springs. In order to comparatively evaluate the difference in response between the three inertial systems, an RC bridge column model was considered in the modeling. The specimen model was part of an experimental program on large-scale circular cross-section RC bridge columns. Details of the experimental program are presented and discussed in the literature [9]. The cross-section and reinforcement of the column model considered in this numerical study are depicted in Figure 10. The RC column is a scale model of a typical pre-1970 circular RC column part of a multicolumn bridge bent in Oregon, USA. Such columns are commonly seismically deficient according to current seismic provisions, given the poor confinement provided at zones where large inelastic incursions are expected. For comparison purposes, a constant axial load of 0.09\(\cdot f_{\mathrm{c}}^{\prime}\cdot A_{g}=333.6\) kN is considered in the analysis. Non-linear time-history analysis (NLTHA) was performed in OpenSees [10] to compare the performance of the three systems. The input ground motions for the analyses were chosen from the Maule, Chile, M8.8 earthquake, and the Loma Prieta, M6.9 earthquake. For this purpose, the transverse direction of the ground motion recorded at the Curico station (Curico X) was considered from the former earthquake event, and the ground motion recorded at the Capitola station (Capitola X) was considered from the latter event. The Curico X record was amplified using a factor of 1.57, whereas the Capitola X record was kept the same, as it was considered in the experimental program. Details of the scale factors used are discussed in [PERSON] et al. [9] and are not presented here since this study aims to compare the performance of different inertial mass configurations. Figure 11 depicts the original acceleration and integrated displacement histories of the Curico and Capitola records used for the NLTHA. It is worth mentioning that the representation of the earthquake input was the same for all the numerical models and base on the base displacement shown in Figure 11b. Using the displacement histories of the records instead of the acceleration histories was to keep consistency in the input motion applied to the numerical models. While the input time history is applied directly to the column specimen in Model I and Model II, in Model III the ground motion is only applied to the column model. To keep consistency in the modeling approach, OpenSees allows applying the ground motion history to specific nodes in the model by employing the command multi-support excitation pattern, which applies a displacement ground motion.