# A Comparative Analysis of International Standards on Curved Surface Isolators for Buildings
## 4 Discussion  
The maximum displacement obtained from the ELA is initially estimated at the center of mass of the structure, considering earthquake excitation in a single direction. However, for design purposes, the total displacement must account for the combination of earthquake effects in both orthogonal directions. As per standard practice, this is completed by adding 30% of the displacement in the perpendicular direction to the main directional displacement. For the structural model, the calculated displacements are \(D_{\text{x}}\) = 154.55 mm and \(D_{\text{y}}\) = 139.21 mm. A key advantage of FPIs compared to other isolation systems, such as High-Damping Lead Rubber Bearings (HDRBs), is their inherent ability to ensure that the center of mass and the center of stiffness always coincide. This is because the stiffness of \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Parameter** & **Eurocode 8** & **UBDP** & **LBDP** \\ \hline Displacement \(D_{i}\) [mm] & 136.62 & 113.72 & 163.11 \\ Equivalent stiffness \(K_{\text{eff}}\) [kN/mm] & 323.60 & 449.3 & 278.22 \\ Equivalent period \(T_{\text{eff}}\) [sec] & 3.05 & 2.59 & 3.29 \\ Equivalent viscous damping \(\xi_{\text{eff}}\) [\%] & 26.94 & 37.31 & 20.95 \\ Damping factor \(B_{M}\) & 0.56 & 0.55 & 0.62 \\ Acceleration \(A\) & 0.059 & 0.068 & 0.061 \\ Maximum displacement \(D_{\text{max}}\) [mm] & 136.32 & 113.72 & 163.11 \\ Maximum horizontal force \(F_{\text{max}}\) [kN] & 44,113.84 & 51,094.24 & 45,380.43 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison between Upper Bound Design Properties and Lower Bound Design Properties in the framework of the European standard (EC8) in terms of the key parameters governing isolator design. FPIs is directly proportional to the supported weight, ensuring automatic alignment with the building's mass distribution. In contrast, HDRBs require careful design adjustments to ensure that the gravity center of the building and the stiffness centroid of the isolators are aligned. If they are not, accidental torsional effects may arise, necessitating additional design considerations. To account for such torsional effects, ASCE/SEI 7-22 [58] requires a minimum amplification of 15% on the maximum displacement. Therefore, the maximum design displacement is computed as:  
\[D_{\text{Ed.x,max}}=\sqrt{\left(D_{\text{x}}\right)^{2}+\left(0.3\cdot D_{ \text{y}}\right)^{2}}+D_{\text{x}}\cdot 0.15=183.13\,\text{mm}\]  
\[D_{\text{Ed.y,max}}=\sqrt{\left(D_{\text{y}}\right)^{2}+\left(0.3\cdot D_{ \text{x}}\right)^{2}}+D_{\text{y}}\cdot 0.15=168.51\,\text{mm}\]  
It is important to note that Eurocode standards do not explicitly require displacement amplification to account for accidental torsion. However, in this study, the ASCE/SEI 7-22 approach has been adopted to ensure a more conservative design. According to the above findings, an essential requirement for any seismic isolation system is its self-centering capability, also referred to as the restoring force in U.S. design standards. This characteristic is crucial to prevent significant residual displacements after an earthquake, ensuring that the structure returns to its original position. If excessive permanent deformations remain, the functionality of the structure may be compromised, and the maximum displacement capacity of the isolators could be exceeded, potentially leading to structural failure [67]. All international seismic isolation standards mandate that a structure must return to its initial position (or close to it) after an earthquake. However, the European (EN 15129) [55] and American (ASCE/SEI 7-22) [58] standards adopt fundamentally different approaches to addressing the recentering capability of seismic isolators. These differences are a key aspect of seismic isolation design and require careful consideration when comparing design methodologies across different regulatory frameworks.