# A Comparative Analysis of International Standards on Curved Surface Isolators for Buildings
## 4 Discussion
The isolation system must be designed to generate a lateral restoring force such that the period corresponding to its tangent stiffness, at any displacement up to the total design displacement, remains below 6 s: \[T=2\pi\sqrt{\frac{R}{g}}<6\ \sec\ \cong R<8900\ \mathrm{mm}\]
2. The restoring force must be greater than the restoring force at 50% of the design displacement by at least \(W/80\), where \(W\) represents the seismic weight. For a sliding pendulum isolator, the variation in force is given by: \[\Delta F=K\cdot D\] where \(D\) is the design displacement and \(K\) the lateral stiffness of the frictionless pendulum base isolator. Therefore, by the previous requirement it may be satisfied that: \[\Delta F=K\cdot D\cdot 0.5\ \geq\frac{W}{80}\Delta F=\frac{W\cdot D}{R\cdot 2}\ \geq\frac{W}{80}\]  
An equivalent requirement can be derived based on the equivalent radius (\(R\)) from the previous relation:  
\[\frac{W\cdot D}{R\cdot 2}\ \geq\frac{W}{80}\ \equiv\ \frac{80\cdot W\cdot D}{W \cdot 2}\ \geq R40\cdot D\ \geq R\]  
In the case of ASCE/SEI 7-22, the restoring force at the design displacement must be greater than the restoring force at 50% of the design displacement by at least W/40. This requirement is determined using the same procedure:  
\[20\cdot D\ \geq R\]The EN 15129 [55] approach differs significantly as it is based on an energy-based methodology. This approach was originally proposed in the theoretical studies in [68], where over 100 time-history analyses of base-isolated structures utilizing various types of isolators were conducted. It is also important to note that energy dissipation and self-centering capability are inherently opposing functions. Their relative significance primarily depends on the specific case under examination, as highlighted in [68]. The energy balance approach in seismic isolation design is based on the following fundamental expression:  
\[E_{i}=E_{S}+E_{H}+E_{V} \tag{16}\]  
where \(E_{i}\) represents the mechanical energy transferred to the structure due to seismic excitation; \(E_{S}\) is the reversibly stored energy (elastic strain energy and potential energy) within the isolation system and any structural elements influencing its response; \(E_{H}\) accounts for the energy dissipated through hysteretic deformation or friction within the isolators; and \(E_{V}\) denotes the energy dissipated through viscous damping mechanisms. According to EN 15129 [55], the re-centering capability requirement is applicable only when using the ELA. The standard mandates that the stored elastic energy (\(E_{S}\)) must satisfy the following condition relative to the energy dissipated by hysteretic mechanisms (\(E_{H}\)):  
\[E_{S}\geq 0.25E_{H} \tag{17}\]  
This requirement ensures that the isolation system retains sufficient restoring force, preventing excessive residual displacements after an earthquake. By maintaining an adequate balance between stored and dissipated energy, the structure remains functional and returns to its original position, preserving the effectiveness of the isolation system. The elastically stored energy (\(E_{S}\)) and the energy (\(E_{H}\)) dissipated through hysteretic deformation or friction for the sliding pendulum isolator can be expressed, respectively, as follows:  
\[D_{\text{Ed.x,max}}=\sqrt{\left(D_{\text{x}}\right)^{2}+\left(0.3\cdot D_{ \text{y}}\right)^{2}}+D_{\text{x}}\cdot 0.15=183.13\,\text{mm}\]0  
\[D_{\text{Ed.x,max}}=\sqrt{\left(D_{\text{x}}\right)^{2}+\left(0.3\cdot D_{ \text{y}}\right)^{2}}+D_{\text{x}}\cdot 0.15=183.13\,\text{mm}\]1  
and by proper substitution into Equation (17):  
\[D_{\text{Ed.x,max}}=\sqrt{\left(D_{\text{x}}\right)^{2}+\left(0.3\cdot D_{ \text{y}}\right)^{2}}+D_{\text{x}}\cdot 0.15=183.13\,\text{mm}\]2  
This implies that the energetic approach outlined in EN 15129 [55] imposes a limitation on the friction coefficient of sliding pendulum isolators, whereas US standards do not address this aspect, as they follow a different approach.