\In this Section, detailed 3D FE analyses are performed on a typical FREI, meshing the device with a refined discretization with 8-noded brick elements, as depicted in Figure \ref{image2}. In Figure \ref{image2}, two different FREI devices are shown, namely FREI-200 and FREI-175. They differ only for the geometry; in particular they have different shape factors, cross areas (the number in the label indicates the edge length in mm of the square cross area) and number of pads used. FREI-200 is used exclusively to benchmark the 3D FE model against some existing experimental data, whereas FREI-175 is the actual low cost isolator used for large scale structural analyses.
Yeoh hyperelastic coefficients are typically estimated through best fitting of uniaxial stretch-strain experimental curves obtained from dog-bone rubber samples loaded up to failure. In particular, we use here existing literature in the field, see \cite{jerrams1998effects,shahzad2015mechanical}, which tunes numerical model coefficients using simple tests of material characterization (like the uniaxial stretch-strain test), where Yeoh coefficients of rubbers are also correlated with rubber hardness. The material considered in the present model is a natural-soft gum rubber with around 40 Shore A hardness \cite{toopchi2008testing}. Instead of steel laminas, glass fibers are assumed as reinforcement to improve the vertical stiffness. As mentioned previously, fiber utilization can reduce the cost of the isolator when compared with steel lamina \cite{calabrese2015shaking}. In this numerical simulation, the fiber is assumed isotropic-elastic with Young modulus E = 40 GPa and Poisson’s ratio=0.20, in accordance with many references \cite{van2017evaluation,toopchi2011bonded,van2014experimental}.
The isolator is used in unbonded type (UFREI), where the upper and lower edges do not exhibit any bond with the supports. Under moderate shear forces, such frictional limited strength allows the isolator to roll-over and facilitates larger deformations. Unlike a bonded model, the peak tensile stress on the rubbers and interfaces significantly decreases. Thus, delamination at high deformation can be avoided.
To realistically reproduce experimental results, a hysteresis behavior of the isolator is required. Two of the authors of this paper proposed a simplified hysteresis behavior of rubber to be used in FE model by mixing elastoplastic concepts and hyperelastic properties \cite{milani2011stretch}. In the present paper, a similar procedure is adopted. First, we generate a nonlinear behavior of the FE model, based only on the hyperelasticity of the rubber pad, i.e., without different loading-unloading paths, see Figure \ref{image4}(b). The dashed line represents the shear behavior obtained by the FE analysis of a UFREI-200. This displacement-force data is then inputted as the properties of a nonlinear spring at a structural level.
Secondly, to generate a hysteresis behavior, a damping coefficient is applied to the spring so that the model can fit the experimental data. In the experiment \cite{toopchi2008testing}, the equivalent damping of the unbonded FREI-200 was reported to be equal to 5.2%. The spring model with damping (Figure \ref{image4}) shows a reasonable fitting with the experiment (Figure \ref{image4}). However, it cannot catch up the sudden increase of damping as seen in the last cycle of the experiment (Figure \ref{image4}). Nevertheless, it is noted that this limitation does not affect the final results.
The nonlinear spring model of UFREI-175 is implemented later at structural level of a two-story isolated masonry building. The implementation is nothing different than a structural identification of the 3D-FE response of the device with a spring-damper element. Such identification is consistent with the simplified model depicted in \ref{image1}. Modeling a nonlinear spring with damping instead of a real 3D isolator reduces significantly the computational time and drastically increases robustness.
Seismic performance of an isolated masonry housing using UFREIs
A FE model of a medium size two-story masonry house with openings and concrete diaphragms at foundation and roof levels is implemented in ABAQUS commercial code. The masonry building taken as reference is an existing one and was constructed in Tawang, India \cite{thuyet2017mitigation} to test the behavior of masonry houses isolated with UFREIs. According to authors’ knowledge, this is probably one of the first examples of masonry housing isolation by means of low cost devices. Dimensions of the building are the following: thickness of the walls 300 mm, each room has 4 x 4 m plan dimensions, height of each floor is 3m, see Figure \ref{image6}. The weight of the roof is assumed equal to 50 kg/m2. To prevent premature damage of the walls, rigid beams on top of windows and doors are inserted, in agreement with the real prototype. The numerical model has thinner walls and significantly smaller isolators compared to the real one built in India \cite{thuyet2017mitigation}.