## 5 Finite Element (FE) Analysis Models  
The ABAQUS nonlinear FE program [27] was used to predict the behavior of the recentering slit damper devices cyclically tested. The FE models were composed of 3D solid four node elements (_i.e._, C3D4 in the ABAQUS program) associated with nonlinear material properties, geometric nonlinearity, and displacement-controlled load histories. The details of 3D FE models such as structural meshes, displacement loading, and boundary conditions (BC's) are illustrated in Figure 7. The structural meshes were generated by dividing individual parts so that uniform element sizes required for obtaining convergent results should be constructed in the FE model. The SMA bending bars were assumed to be rigidly connected to the flange of the slit damper without clearance between bolt shanks and bolt holes. Therefore, detail modeling for bolt heads, nuts, and contacted bolt threads could be neglected by merging bolt shank and contacted bolt holes in the flange. The displacement-controlled loads were directly imposed on the edge of the flange. The cyclic load histories for refined quasi-static FE analyses were reproduced by using amplitude functions incorporating the static step. The FE analyses were conducted with the same loading history as the experimental slit damper tests (Figure 13a). The nonlinear material properties were aligned to individual parts made up of the solid elements. The isotropic/kinematic hardening material model containing the Bauschinger effect, plastic shakedown, ratcheting and stress relaxation was assigned to FE model parts fabricated with Gr. 50 carbon steel (_i.e._, slit damper and flange). Due to an absence of adequate default material models given to the ABAQUS program, the user-defined material (UMAT) model simulated based on [PERSON]'s subroutine code was employed so as to reproduce the behavior of superelastic SMA materials and the state of 3D stress-strain field contours [27, 28, 29]. This UMAT model completely reflects phase transformation superelasticity at a constant room temperature. The material properties required as input  
Figure 6: Response mechanism of the recentering slit damper system. values to the UMAT subroutine model are acquired from the results of the uniaxial pull-out tests during loading, unloading, and reloading cycles under isothermal conditions. The UMAT subroutine can be implemented on the ABAQUS program associated with the FORTRAN computer language. The stress and strain curves for the superelastic SMA material are illustrated in Figure 8. For this numerical simulation, the required material properties (_i.e._, elastic modulus (40 GPa), Poisson's ratio (0.33), martensite start stress (440 MPa), martensite finish stress (540 MPa), austenite start stress (250 MPa), austenite finish stress (140 MPa), and transformation strain (0.042), phase transformation temperature (22 \({}^{\circ}\)C)) were obtained from uniaxial pull-out tests performed by [PERSON] _et al._[23]. The simulated stress and strain curve was denoted by a series of straight lines whose paths coincide with each phase transformation. During the loading path, the simulated curve shows a good agreement with the experimental curve. The real superelastic SMA materials behave with different elastic properties between martensite and austenite. On the other hand, the UMAT model, produced based on [PERSON]'s subroutine code [28], assumes that the unloading slope during elastic strain recovery is the same as the initial loading slope, in order to easily achieve linear interpolation between phase transformations. Algorithmically, phase transformation displayed at the subroutine proceeded with linearly variable martensite fractions (ranging from 0 to 1.0) in the time-discrete evolution equation. The UMAT model overestimates stiffness upon unloading (reverse phase transformation), and consequently the simulated curve is somewhat fatter when compared to the experimental curve.