## 1 Introduction
### Seismic Isolation Systems  
Seismic isolation systems are currently the most applied passive supplemental damping systems. Seismic isolation is placed on the first floor of the building and disconnects the building from the ground. Seismic isolation equipment, which is quite rigid vertically, has the capacity to make long deformations horizontally. Since the deformations to be caused by earthquake movements in the structure are provided by this deformation in the seismic isolation floor, the damage due to deformations in the superstructure is reduced [6]. There are basically two types of seismic isolation systems: friction and rubber. Friction pendulum bearings (FPBs) consist of a hemispherical slider positioned between two curved sliding surfaces, which provide a horizontal centering force. In order to overcome the challenges of isolating significant displacements using lead rubber bearings (LRBs), FPB base isolation methods are extensively used. It is more appropriate to use FPBs, especially in structures with high displacement demands such as bridges. LRBs are one of the most commonly used base isolators in building-type structures. LRBs consist of laminated rubber layers with reinforced steel plates and a central lead core that provides damping to the building structure due to large shear deformation [7]. In Figure 2, the LRB with  
Figure 1: Advantage of seismic isolation based on the elongation of supplemental damping and fundamental vibration period [6]. 225% horizontal displacement, nonlinear hysteretic behavior, components, and idealized force-displacement curve is presented. Seismic isolation is generally modelled according to the equivalent lateral force method [10; 11]. In this approach, the design process commences with an initially predicted value. If the isolator displacement \(D_{i}\) obtained as a result of the analysis differs from the initially estimated isolator displacement, the cycle is restarted with the calculated value. If the computed and analyzed values approximate each other, the design is finalized. In this method, the nonlinear force-displacement relationship is idealized and the calculations are continued with equivalent linear elastic values. In the idealized force-displacement curve given in Figure 2d, \(F\) is the maximum horizontal force corresponding to the maximum isolator displacement, \(D\) is the maximum isolator displacement, \(F_{y}\) is the yield strength, \(F_{Q}\) is the characteristic strength, \(k_{1}\) is the initial (elastic) stiffness of the isolator before yielding, \(k_{2}\) is the secondary (post-elastic) stiffness after yielding, \(D_{y}\) is the yield displacement, and \(k_{e}\) is the effective stiffness corresponding to the maximum isolator displacement. Using the idealized force-displacement curve, \(F_{Q}\) and \(D_{y}\) should be established first in a seismic isolation design; 0.025 meters is the commonly accepted value for \(D_{y}\)[12]. Under the assumption that the superstructure is linearly modeled, \(k_{2}\) is determined as in Equation (1) using the isolation period [13]:  
\[T_{0}=2\pi\sqrt{\frac{W}{k_{2}\cdot g}} \tag{1}\]  
Figure 2: (**a**) Nonlinear hysteretic behavior, (**b**) 225% shear-strained LRB [8], (**c**) idealized force-displacement curve [9], (**d**) components [8]. where \(W\) is the weight of the structure and \(g\) is the gravitational acceleration. \(F_{y}\) and \(k_{1}\) are calculated using Equations (2) and (3). \[F_{y}=F_{Q}+k_{2}\cdot D_{y} \tag{2}\]  
\[D_{y}=\frac{F_{y}}{k_{1}} \tag{3}\]  
If \(D_{y}\) is assumed as a fixed value, only \(T_{0}\) and \(F_{Q}\) parameters are needed to model the seismic isolation system [14]. The main purpose of this study is to determine these two parameters. When these two parameters are used as optimization variables and structural analyses are performed, \(D\) can be calculated. The effective damping ratio (\(\beta\)) to be used as a constraint in the optimization process is found as in Equation (5), depending on the effective stiffness (\(k_{e}\)).