Validation of the UEL in a 3D dynamic analysis

For validation purpose, a dynamic 3D analysis on a simple isolated slab is presented. Four UFREIs are used to seismically isolate the square-rigid slab, as seen in Fig. 6. To generate 3D behaviors of the system including torsion and rotation during seismic excitation, four UFREIs with significant different stiffness are used in the isolation system. The rubber properties of the UFREIs are presented in table 1 while the reinforcement is a glass fiber with 40 GPa of Young modulus. Each UFREI has a typical square geometry: 100 x 100 mm2, 5x10mm of rubber pads, and 4x0.55 of fiber lamina. The upper-slab has 4.77 ton of self-weight so that each isolator (with 100x100 mm of surface) experiences 1.17 MPa of vertical pressure. This load corresponds to which experienced by the low cost rubber isolators for masonry housing in the previous work \cite{Habieb_2017} . Two models are presented in this section. The reference model employs detailed 3D UFREIs with refined mesh and the second uses the proposed a UEL as replacement of each UFREI. As aforementioned, the UEL can predict the behavior of UFREI in all 3D DOFs by inputting the shear modulus, damping of rubber and geometry of the corresponding bearing.
To construct an unbonded isolation system, the frictional contact (u coefficient=0.85) is employed on the upper and bottom interface of the bearing. As the analysis of the 3D model with friction events is computationally very expensive, a short accelerogram is selected. The isolated slab is subjected to N-S component (x-axis) of Durzunbey accelerogram with PGA 0.8g (see Figure 17), which is estimated to result a moderate displacement of the isolation system. Three seconds in the end is considered as a recovery time in which the input acceleration is zero to observe the re-centering capability of the UFREI system.
In both models, the 3D movements of the bearing points and point O are monitored during the simulation (see Figure 18). The global torsion magnitude is also monitored through the azimuth rotation of point O. Acceleration at the upper slab is presented in Figure 19. The UEL model gives an excellent prediction of the detailed 3D FE model. Thanks to the damping behavior of the isolation, the peak acceleration at the upper-structure can be reduced. 
Figure 20-22 show the displacement in x, y, and z directions of each isolator, highlighting a good agreement between 3D FE model and UEL model. The figures report a moderate displacement of the isolation system in the major direction (x-axis), with 56 mm of peak displacement at UFREI-2. Figure 20 and 21 indicate that both models have an excellent recovery of shear displacement after seismic excitation. However, a 4 mm of small sliding in x direction is captured in the 3D FE model, but it seems negligible. Meanwhile, no noticeable sliding is reported in minor direction, y-axis (see Figure 21). It is in agreement with the recommendation \cite{Ehsani_2017,Russo_2013} that at least 0.5 MPa of vertical pressure on UFREI can prevent a slip between the unbonded bearing and concrete structures. In the axial deformation (Figure 22), the gravitational load develops in the first two seconds. As predicted, the softest bearing, UFREI 1, experiences the most compressive deformation. However, the 3D FE model gives a more fluctuated axial deformation when the seismic motion is activated. 
The torsional deformation of the isolated slab in Figure 23 presents a reasonable accuracy of the proposed UEL model, which takes into account the torsional stiffness of a single isolator. Although several modes are not captured in the UEL model, some peak torsional angles are well reported by the UEL model.  After post-processing of the simulation results, the shear behavior of each isolator can be reported in the form of a force-displacement loop (see Figure 24 and 25). The UEL model clearly results in an excellent prediction of each UFREI undergoing seismic excitation. The proposed UEL model is therefore validated and ready to be implemented in 3D dynamic analyses of more complex isolated structures.