# Evaluating Collapse Fragility Curves for Existing Buildings Retrofitted Using Seismic Isolation
## 3 Numerical Modeling
### Isolation System  
As discussed in Section 1, important outcomes regarding the collapse performances of existing buildings retrofitted using seismic isolation have been provided by the ReLuis-DPC 2015-2018 RINTC research project [7]. It is worth noting that, in the aforementioned research project, the High Damping Rubber (HDR) element [13], already included in the Opensees library, has been adopted to describe the cyclic behavior of HDRBs. The latter is a two-node element with 12 degrees-of freedom characterized by easy implementation and computational efficiency. The behavior in the axial direction is able to capture cavitation and post-cavitation phenomena in tension. Moreover, the variability of the critical buckling load and the vertical stiffness with horizontal displacement in compression is taken into account. The model of [PERSON] et al. [14] has been implemented to describe the behavior in the horizontal direction. The coupling effect between axial and horizontal directions is partially considered using mechanical properties in the vertical direction dependent on the shear response parameters. More details about the described model can be found in [15]. However, it is worth noting that, in the model proposed within the RINTC project, response parameters in the horizontal direction are independent of those in the axial direction, neither for large displacements nor for large pressures. Therefore, the axial-shear load interaction (at small and large displacements) as well as P-delta effects due to post-buckling behavior are missed. All that considered, in the present paper, the mechanical model of the HDRBs has been improved to overcome those limitations. In particular, the Kikuchi bearing element [16] has been adopted in lieu of the HDR bearing model. The Kikuchi bearing element is a two node link element with multi-spring mechanical model which includes two sets of multiple axial springs (one on the top and the other at the bottom of the bearing height) and a set of mid-height multiple (radial) shear springs, all bound together by rigid links. The number of springs can be selected based on the accuracy/convergence of the numerical analysis. The axial behavior of the multiple axial springs (each one representing an individual fiber of the bearing's cross-sectional area) is described by the non-linear AxialSp uniaxial material, available in Opensees. The multiple shear springs system consists of a series of identical springs arranged radially representing the isotropic behavior of the device in the horizontal plane. The non-linear hysteretic behavior of shear springs is represented through the KikuchiAikenHDR material [17]. The buckling behavior associated with high compressive load is simulated by the tilt of the rigid links and interaction between the axial and shear forces of the multiple shear and axial springs. The main advantage of the Kikuchi bearing element is represented by the possibility of capturing the axial-shear load interaction in particular at large displacements. Moreover, the consequent pre- and post- buckling behavior is also taken into account. Figure 3 compares the cyclic responses exhibited by the same elastomeric device (n. 20 of the SLD case-study), modeled using the twoaforementioned elements, subjected to a certain ground motion record (n. 3). As can be seen, while at lower shear deformations the cyclic responses are broadly aligned, in the large deformation range the Kikuchi bearing element shows a reduction of the lateral stiffness at peak horizontal displacement by increasing the amplitude of the applied deformation. This softening response is mainly associated to the axial-shear load interaction, representing a peculiarity of the Kikuchi model. For the sliders, the same modeling assumptions adopted within the RINTC project have been implemented herein.