## 3 Collapse Performance of Seismically Isolated Buildings  
### Analysis Models of Isolated Buildings  
A detailed analysis model representing the prototype buildings had been constructed and was then simplified to a lumped-mass-and-spring model that is known to tailor for incremental dynamic analyses requiring a significantly large number of analyses. The responses and dynamic characteristics of a superstructure obtained from both the models will be compared to calibrate the simplified model, which will be discussed in detail later. There are several assumptions implemented into both the analysis models for the prototype buildings. An inherent damping ratio is assumed to be 2% of critical for all modes [24]. Nonlinear analyses are carried out with the Newmark-Beta method that has been typically used to capture nonlinear dynamic responses with reasonable analytical stability. Dynamic equilibrium is calculated every specific time step, which is less than the time-step of earthquake records selected for this study mentioned later. The iterative Newton-Raphson method used for checking the dynamic equilibrium can provide reasonable numerical results with the overshoot tolerance of 1%. A P-delta effect is explicitly considered in the analysis model, and seismic masses in each floor are considered to be lumped at its center with the typical assumption of the rigid diaphragm. This study did not explicitly consider the bounding effects of seismically isolated buildings to the moat wall under a large displacement of isolation systems [26, 27]. The detailed analysis model was constructed using commercial software, PERFORM 3D [28]. Frame Member Compound components were utilized to model structural members in the superstructures and isolators of NRBs and LRBs were modeled with built-in Rubber Type Seismic Isolator elements. According to the prototype test results of isolators presented in [PERSON] and [PERSON] [13] showing that they are almost linear responses under maximum displacements, the NRBs were modeled with linear hysteretic elements. The idealized bi-linear behavior was implemented for the LRBs to model the yielding phenomenon of their embedded lead bars and their hyper-elastic stiffening occurring at large displacements was neglected, since their allowable deformation capacities are assumed to be sufficiently large enough, as considered in their typical designing phase. The FSBs were modeled using built-in Friction Pendulum Seismic Isolator elements with a friction coefficient of 0.0024. The restoring force of the very stiff FSB bearing elements in compression is independent of shear deformations after sliding. Each component consists of rigid end zones located at both ends, and an inelastic element representing its sectional properties between the ends. The lengths of the rigid zones are assumed to be half of the connected column width or beam depth. The deterioration behavior of the elements was modeled, and failure was implicitly accounted for by limiting on a story drift of 2%. The simplified lumped mass model (SLMM) is utilized in the condition that each story in the superstructure above the isolation system is to be modeled with an elasto-plastic spring, and the