# A Comparative Analysis of International Standards on Curved Surface Isolators for Buildings
## 3 Design Procedures  
In seismic isolation systems, two primary strategies are employed to reduce the impact of seismic forces on structures: increasing the natural period and enhancing energy dissipation (damping), as illustrated in Figure 4. These two mechanisms are fundamental in ensuring the effectiveness of a seismic isolation system. The natural period of the structure is extended by incorporating an isolation layer between the superstructure and substructure, effectively acting as a harmonic oscillator. This forces the superstructure to oscillate according to the natural frequency of the isolator, thereby reducing the transmission of high-frequency seismic energy. On the other hand, energy dissipation plays a critical role in seismic isolation by transforming the mechanical energy induced by seismic motion into heat. This heat generation results in localized temperature increases, necessitating dynamic testing to assess potential material degradation under cyclic loading. Seismic isolators primarily function as harmonic oscillators, classified into two main categories: Spring-based isolators with a characteristic stiffness constant \(K\); pendulum-based isolators, where the restoring force is determined by the pendulum length \(R\). Among these, FPIs are a widely used category of pendulum-based isolators. FPIs have four key functional characteristics. First, they provide lateral flexibility, enabling controlled horizontal displacement during seismic events. Second, they offer vertical load-bearing capacity, which ensures overall structural stability. Third, they have the self-centering capability, allowing the structure to return to its original position after shaking. Finally, they enable energy dissipation through friction, where the sliding interface converts kinetic energy into heat. This study focuses primarily on single FPIs. However, it is important to note that the kinematic behavior of both single and double friction pendulum systems is essentially equivalent. The single pendulum (Figure 5) consists of a single sliding surface that provides lateral flexibility and energy dissipation through friction, with the restoring force governed by the curvature of the sliding surface. In contrast, the double pendulum (Figure 5) includes an internal articulated slider mechanism that allows for two sequential sliding surfaces, enabling greater displacement capacity and more complex hysteretic behavior. While the two systems differ in complexity and performance characteristics, their kinematic behavior is fundamentally equivalent, particularly under the assumptions adopted in the equivalent linear analysis applied in this research. The key parameters defining FPIs include: equivalent radius (\(R_{\text{eq}}\) or \(R\)); dynamic coefficient of friction (\(\mu\)); maximum design displacement (\(D\)); seismic weight of the structure (\(W\)); effective equivalent stiffness (\(K_{\text{eff}}\)); effective equivalent period (\(T_{\text{eff}}\)) and damping that is equivalent to the total number of seismic and seismic modes. Figure 4: Strategies employed to reduce the impact of seismic forces on structures: increasing the natural period and enhancing energy dissipation (damping). Figure 5: Single pendulum that consists of a single sliding surface that provides lateral flexibility and energy dissipation and double pendulum that includes an internal articulated slider mechanism that allows for two sequential sliding surfaces. is an essential aspect of oscillatory systems, serving to attenuate, constrain, or completely suppress vibrations. In seismic isolation, equivalent viscous damping (\(\xi\)) is commonly used to approximate the behavior of nonlinear single-degree-of-freedom (SDOF) systems. This is completed by equating the energy dissipated in a nonlinear system to that of a linear system with an equivalent viscous damping ratio. For conventional reinforced concrete and steel structures, damping ratios are typically assumed to be 5% and 2%, respectively.