To sum up, the softening is affected by the combined action of the vertical load (Pl) and the horizontal deformation (u). Therefore, a suitable variable is introduced into the classic horizontal stiffness equation to generate the softening effect.  The governing equation is briefly discussed in this report. In equation 1-3, is the length of the isolator (perpendicular to the figure), l0 and lt are the initial and current length of rubber in shear direction (see Fig.\ref{661589}). The shear stiffness kH depends on the bearing area Ar, which varies due to the shear displacement u, while ml is a modifying factor to determine the contribution of the free-stress zone.  Consistently, a similar method is applied for the vertical stiffness evaluation (see Fig.2b). The fictitious hardening line 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.  The rotational and torsional stiffness of the UFREI is 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}. Meanwhile, through a 3D FE simulation, we did not find any difference in torsional stiffness between bonded and unbonded type, under 1 MPa of vertical pre-loading. 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 \ref{994117}. 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, as proposed in \cite{Kumar_2014}. The hysteretic behavior of the beam is then evaluated by means of a standard Newton-Raphson procedure. The UEL is implemented as Fortran Abaqus subroutine.