# A Comparative Analysis of International Standards on Curved Surface Isolators for Buildings
## 4 Discussion
The re-centering capability of seismic isolation systems is a fundamental performance requirement that ensures the structure returns to its original position after an earthquake, minimizing residual displacements and preserving post-event functionality. In essence, re-centering reflects the system's ability to restore equilibrium without requiring external intervention. This property is particularly critical for essential facilities, such as hospitals, where operability after a seismic event is vital. In base isolation systems, re-centering is achieved through the restoring force generated by the geometry of the isolator (e.g., the curvature in sliding systems) or the elasticity of the materials (e.g., in elastomeric bearings). The primary parameters influencing re-centering are the radius of curvature in pendulum-type isolators, the friction coefficient, which opposes re-centering, and the displacement demand imposed by seismic loading. The ASCE/SEI 7-22 standard defines a minimum requirement for restoring force by stipulating that, at the design displacement, the restoring force must exceed the restoring force at 50% displacement by at least W/40, where W is the seismic weight. This criterion, while simple and enforceable, does not explicitly account for the effects of friction, which can significantly hinder the re-centering behavior in systems with high frictional resistance. On the other hand, the European standard EN 15129 introduces a more refined approach based on the energy balance principle, specifically requiring that the reversibly stored elastic energy be at least 25% of the hysteretically dissipated energy. This condition ensures that even in systems with substantial energy dissipation, a sufficient elastic restoring force remains available to bring the structure back toward its original position. In friction-based isolators, re-centering capability depends on the balance between the geometrically induced restoring force and the opposing frictional resistance. While low-friction systems tend to ex hibit excellent re-centering performance, high-friction systems may experience permanent offsets unless carefully controlled. Therefore, it is crucial to assess this performance aspect not only through force-based checks but also via energy-based evaluations, particularly in cases involving high friction coefficients or complex loading conditions. In this study, re-centering capability has been carefully evaluated under both code frameworks. The results suggest that while both standards aim to prevent large residual displacements, the energy-based approach adopted by EN 15129 offers a more transparent and physically consistent methodology, especially for isolators with significant nonlinearity or frictional behavior. Conversely, the ASCE/SEI 7-22's restoring force criterion may lead to underestimation of residual drift, particularly in systems where friction dominates the response. In U.S. seismic design standards, the restoring force of an isolation system is assumed to depend on three key parameters: the isolated structure's fundamental period; the yield friction level of the isolator; the yield displacement of the isolation system. The ASCE/SEI 7-22 specifies that the restoring force at the design displacement must be at least W/40 greater than the restoring force at 50% of the design displacement, where \(W\) represents the total seismic weight of the structure. This ensures that the isolation system possesses adequate self-centering capability, minimizing residual displacements after an earthquake. Similarly, the AASHTO Guide Specifications for Seismic Isolation Design (AASHTO-GSID-14) [59] imposes comparable criteria, reinforcing the necessity of a sufficient restoring force to prevent excessive permanent deformations. One of the primary requirements of this standard is to guarantee that the isolation system can return the structure to its near-original position post-earthquake, thereby preserving the functionality and integrity of the building:  
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