# A Comparative Analysis of International Standards on Curved Surface Isolators for Buildings
## 3 Design Procedures
### European Standard Approach  
The European standard (EN 1998-1--Eurocode 8) [57] follows the ELA method for the design and assessment of seismically isolated structures. This approach relies on an iterative procedure, where an initial displacement assumption is made (\(D_{i}\)), and calculations are refined until the final displacement converges to the initial assumption. To achieve this convergence, it is necessary to determine the key mechanical parameters that characterize FPIs, including:  
* Initial displacement \(D_{i}\): \[D_{i}=400\,\mathrm{nm}\] (2)
* Equivalent stiffness (\(K_{\mathrm{eff}}\)), which governs the dynamic response of the isolation system (where \(W\) is the seismic weight of the structure, equivalent radius (\(R_{\mathrm{eq}}\) or \(R\)), \(D_{i}\) the initial displacement in Equation (2), and \(\mu\) the dynamic coefficient of friction): \[K_{\mathrm{eff}}=W\bigg{(}\frac{1}{R}+\frac{\mu}{D_{i}}\bigg{)}=233.34\,\mathrm{ kN}/\,\mathrm{mm}\] (3)
* Equivalent stiffness (\(T_{\mathrm{eff}}\)), representing the effective oscillation period of the isolated structure: \[T_{\mathrm{eff}}=2\pi\sqrt{\frac{W}{gK_{\mathrm{eff}}}}=3.59\,\,\mathrm{sec}\] (4)* Equivalent viscous damping (\(\xi_{\text{eff}}\)), which accounts for energy dissipation in the system: \[\xi_{\text{eff}}=\text{min}\left(\frac{2\mu}{\pi\left(\mu+\frac{D_{i}}{K}\right) }\right)=12.73\%\] (5)
* Reduced spectral acceleration, adjusted by the damping factor (\(\eta\)), to incorporate the effects of increased damping in the isolation system: \[\eta=\text{max}\left(\sqrt{\frac{10}{5+\xi}},0.55\right)=0.75\] (6)  
This expression is derived by equating the energy dissipated per cycle by frictional sliding to the energy dissipated in a linear viscous damper. The formula effectively captures the hysteretic energy dissipation of the isolator in terms of an equivalent damping ratio that can be used with linear response spectra. The radius \(R\) controls the restoring stiffness, while the coefficient of friction \(\mu\) directly governs the level of energy dissipation, with higher values of \(\mu\) resulting in greater damping. Since the CSCR-10 response spectrum does not provide a direct mathematical equation to estimate spectral acceleration (\(S_{a}\)) for a given period, unlike the EC8 [57] and ASCE/SEI 7-22 [58] standards, an approximation method is required for regions of interest. To address this limitation, a curve-fitting tools has been developed to derive an approximate equation for spectral acceleration, ensuring accurate interpolation at the equivalent period of interest. This approach allows for a practical estimation of \(S_{a}(T)\) in regions where the response spectrum lacks explicit analytical formulation. Figure 6 presents the CSCR-10 [64] response spectrum for Zone III, considering soil type S3, ductility factor \(\mu=2\), and over-strength factor \(SR=1\). To enhance clarity, Figure 6a provides a zoomed-in view of the portion of the spectrum where the natural period is higher, along with the approximate equations (Figure 6b) corresponding to the constant velocity and constant displacement branches of the spectrum. This approach ensures that the spectral acceleration estimation remains consistent with engineering standards, providing reliable input for further structural analysis. Therefore, the seismic coefficient (C), which corresponds to the spectral acceleration, can be determined based on the required period in the response spectrum, considering the previously defined seismic parameters. The spectral acceleration is estimated as a function of the selected period, ensuring consistency with the response spectrum and seismic design criteria:  
\[0.5\text{s}\leq T_{\text{eff}}\leq 3.78\text{s}\ \rightarrow\ C=0.3219 \text{x}^{-1} \tag{7}\]  
\[3.78\text{s}\leq T_{\text{eff}}\leq 10\text{s}\ \rightarrow\ C=1.2274 \text{x}^{-2.007} \tag{8}\]  
where \(x\) is the natural period variable.