Without bonding on the upper and bottom surfaces, the U-FREI bearing is then subjected to a cyclic horizontal displacement with 1 Hz of frequency and a constant vertical pressure 1.17 MPa. The maximum magnitude of this shear test is 120% of total thickness of rubber pads. The result of FE simulation is presented in Figure 2 and Figure 3, showing a remarkable softening of the horizontal stiffness due to rolling-over deformation. A softening is also captured on the compressive behavior of the U-FREI, see Figure 4, during horizontal deformation. In Figure 4, only a last cycle is presented to simplify the analysis of the vertical behavior.
The softening is affected by the combined action of the vertical load and the horizontal deformation. Therefore, a suitable variable is introduced into the classic horizontal stiffness equation to generate the softening effect  (Figure ??). The shear stiffness kH depends on the bearing area Ar, which varies due to the shear displacement u. Consistently, a similar method is applied for the vertical stiffness evaluation (see Fig.2b). The hardening is generated by estimating the magnitude of shear displacement when the full-contact mode takes place. At this stage, the stiffness is multiplied by a hardening factor (see Fig. 2a).  The rotational and torsional stiffness of the UFREI are adjusted by introducing a reduction factor. A study recently presented shows that the rotational stiffness of UFREIs is 25% lower than that of bonded isolators \cite{Al_Anany_2015}. In the present model, the rotational and torsional stiffness are considered constant, as they do not affect the global behavior of an isolated structure significantly.
The modified equations are then implemented into a UEL beam element with two nodes (each node has 6 DOFs), connecting the structure to the ground, as seen in Fig. 3. To produce a hysteresis behavior, a Bouc-Wen model is implemented in the UEL code, taking into account a coupling mechanism between damping force in x and y direction (eq. 3 and 4), as proposed in \cite{Kumar_2014} . The hysteretic behavior of the beam is then evaluated by means of a standard Newton- Raphson procedure. UEL is implemented as Fortran Abaqus subroutine.