## 5 Numerical Comparison of the Existing Mass-Rig Systems
### Numerical Modeling of the Inertial Systems
#### 5.1.1 Model I  
As mentioned previously, the proposed IMS is part of the experimental program presented in [PERSON] et al. [9] and depicted in Figure 5. In general, the test setup includes the IMS, an RC column, an axial load system, and a shake table system. The numerical model of the proposed setup presented in this study is illustrated in Figure 12. The RC column is modeled using the fiber-based distributed plasticity model formulation proposed by [PERSON] et al. [17]. In this strategy, the column is represented using a force-based beam-column element with distributed plasticity, where yielding is allowed at any integration point along the element length [20]. The modeling strategy was calibrated with the experimental results reported in [PERSON] et al. [9]. The chosen modeling strategy incorporates a force-based fiber beam-column element, a zero-length bond segment, and an elastic shear component to model the flexural, bond-slip, and shear components of the total column deflection. A force-based beam-column element with five integration points is used to describe the column, following the strategy used by [PERSON] and [PERSON] [19]. Since most non-linear  
Figure 10: Cross-section of the reinforced concrete (RC) column model. behavior occurs near column base, the Gauss-Lobatto integration scheme is used as the plastic hinge integration method. The column cross-section is divided into three fiber portions: confined concrete core, unconfined concrete cover, and reinforcing steel. A radial discretization scheme is used to discretize the cross-section as follows: 16 radial core divisions, 18 transverse core divisions, 2 radial unconfined cover divisions, and 18 transverse cover divisions. A uniaxial stress  
Figure 11: Time histories for the Curico X (**top**) and Capitola X record (**bottom**): (**a**) Acceleration; (**b**) Displacement. Figure 12: Model I.  
strain model is assigned to each fiber section. The Popovic curve with model parameters proposed by [PERSON] et al. [21] are assigned to both confined and unconfined concrete. The longitudinal reinforcement is modeled using the hysteretic uniaxial material. This material model was chosen because of its capabilities for capturing both the pinching of force and deformation and the damage due to ductility and energy and degraded unloading stiffness caused by cracking and crushing of concrete, bar buckling, and bar fracture. The parameters associated with the pinching behavior, pinchx and pinchy, were both set equal to 1.0 based on the observed pinching behavior in the test specimens. The damage parameter associated with damage due to ductility, i.e., parameter damage1, was set equal to 0.003 to account for the cyclic deterioration observed in the test specimens. The damage parameter associated with damage due to energy, i.e., parameter damage2, was set equal to 0.002 to capture the strength degradation observed in the test specimens. A value of 0.3 was assigned to the degraded unloading stiffness parameter (\(\beta\)). To improve the prediction of column deformation, shear and bond-slip behavior must be included in the model. Shear deformation is ignored with a standard fiber beam-column element approach, which only provides flexural behavior. Additional flexibility from shear behavior is added to the cross-sections at the column-footing interface using a section aggregator. The shear behavior is assumed as an isotropic material with a constant shear modulus, \(G_{\mathit{eff}}\), equal to \(0.2E_{c}\), as recommended by [PERSON] and [PERSON] [22]. Similarly, the conventional fiber beam-column element strategy neglects the added flexibility from the slip of the longitudinal reinforcement at the anchorage because it assumes a complete bond between the concrete and steel. The bond-slip behavior is modeled following the model proposed by [PERSON] [23] and the recommendations of [PERSON] et al. [24]. In this model, the slip behavior is represented using a zero-length fiber segment with the same discretization scheme used for the force-based beam-column element.