# Effect of Nonlinear Hysteresis Details of Isolation System on In-Structure Response Spectra
## 2 Consideration When Developing ISRS of Fixed and Isolated Structures
### Effect of Initial Stiffness of Isolator on ISRS  
Many factors, such as the mechanical properties of the structure, site amplification, and ground motions, affect the nonlinear behavior of isolated structures, but the factor that has the highest priority is the mechanical characteristics of the isolator. The application of isolators began in the 1970s, and the criteria for displacement and mechanical characteristics of isolators have been established through various analytical and experimental studies [16; 17]. The secondary stiffness and the characteristic strength, which contributed significantly to nonlinear behavior, can be easily derived from the material properties of isolators, so values were suggested for use according to the design purpose. However, because the initial stiffness is difficult to estimate, it is generally expressed as a ratio to the secondary stiffness. There were no clear criteria for this stiffness ratio, so only a recommended range of about 0.05 to 0.11 [16; 17] was proposed, and different values are used in various studies and designs. In the case of lead-rubber bearings, when the weight of the superstructure is very large and a large damping effect is required, an isolator with a large sectional area for lead is used. This causes high initial stiffness and characteristic strength, but it has been confirmed that this amplifies the acceleration response [12; 14]. Figure 3 shows the ISRS with the LRB applied to a five-story beam stick mass-lumped model performed in a previous study [32]. In this previous study, the mass and stiffness of the structure were adjusted so that the natural frequency was the same at 8 Hz, and the mass of the basemat was also set to 30% of the structure mass. As shown in Figure 3, when comparing the ISRS of the fixed and isolated structures with a stiffness ratio of 0.01 and 0.1, the response is completely different depending on the initial stiffness. When the initial stiffness is large (stiffness ratio is 0.01), a large amplification occurs at the frequency of the superstructure itself, and the equipment with a high frequency in the structures may be more vulnerable compared to the fixed structure. An additional point to consider here is that the dominant frequency of the structure shifts. The ISRS peak of the fixed structure occurred at 8 Hz, which is the first natural frequency of the structure, but the peak occurred at about 13 Hz in the isolated structure. Figure 2: In-structure response spectra of the isolated structures in the assumption of equivalent linear and bi-linear isolator models [31]. This was expected to occur due to the isolation system, and the shifted frequency was confirmed with a formula by solving a simple 2-DOF equation of motion. To simplify the formula, the mass and stiffness matrices were constructed using the stiffness ratio of the isolator and structure, and the mass ratio of the basemat and structure. The simple SDOF and 2-DOF systems for solving the equation of motion are shown in Figure 4. The first mode frequency of the fixed structure is defined as \(f_{s,F}\), and the first and second mode frequencies of the isolated structure are defined as \(f_{I}\) and \(f_{s,I}\), respectively. The result of solving the equation of motion is expressed in Equations (1) and (2) using the mass ratio (\(\alpha\)) and the stiffness ratio (\(\beta\)). Through this, the second mode frequency in the isolation system corresponding to the natural frequency of the fixed structure can be expressed as Equation (3). The difference in stiffness between the structure and isolator is very large, and because the stiffness ratio in this range does not have a significant effect, as a result, the natural frequency of the structure in the isolation system can be determined by the mass ratio between the basemat and structure.