# Energy-Based Prediction of the Displacement of DCFP Bearings
## 2 State-of-the-Art Isolator Displacement Prediction
### Simplified Linear Analysis (Euro Code [7])
Thus, some methods that can predict the total input energy based on earthquake characteristics were proposed in some earthquake design methods based on energy [11]:  
(1) Approximate Inelastic Input Energy Spectra  
Based on [PERSON]'s study [10], it was found that an inelastic input energy spectrum could be approximated by an elastic spectrum that corresponds to an equivalent viscous damping and an equivalent natural period. However, [PERSON] [12] has shown that this method is not suitable for ground motions with a highly harmonic nature. (2) Another Method Proposed by [PERSON] [11]  
This method can predict the maximum input energy of a structure with a specified ductility ratio if the strong motion duration at a given site is known. However, this maximum value is the maximum input energy that is evaluated throughout the whole period range and means that the maximum input energy of a structure within a specific period cannot be predicted. The energy balance equation used in this study is shown in Equation (3) [11]. When an isolation system consisting of DCFP bearings is subjected to earthquake ground motions, the input energy \(E\) is accumulated in the elastic strain energy of the bearing, \(E_{s}\), and the kinetic energy of the upper structure, \(E_{k}\), and is mainly consumed by the frictional resistance of the bearing in \(E_{h}\) and by the damping of the system in \(E_{\xi}\):  
\[E=E_{s}+E_{k}+E_{h}+E_{\xi} \tag{3}\]  
where,  
\[E=-\int m\times ag\times dy \tag{4}\]  
\[E_{k}=\frac{m\times v^{2}}{2} \tag{5}\]  
In these formulas, \(E\) is the total input energy (the relative input energy in [11]), \(E_{s}\) is the elastic strain energy, \(E_{k}\) is the kinetic energy (the relative kinetic energy in [11]), \(E_{h}\) is the hysteresis energy, \(E_{\xi}\) is the damping energy, m is the mass of the upper structure, \(ag\) is the ground acceleration, \(dy\) is the a segment of the relative displacement between the upper structure and the ground and \(\
u\) is the relative velocity between the upper structure and the ground. [PERSON] has shown that the total input energy, \(E\), based on the SDOF system, can provide a very good estimate of the input energy for multi-story buildings [9]. Therefore, in this study, an SDOF system with a damping ratio of 0% was applied in order to simplify the computation and to make the influence of the earthquake inputs on the response of the DCFP bearings clearer. As a result, the input energy can be expressed as:  
\[E=E_{s}+E_{k}+E_{h} \tag{6}\]  
To sum up, in the early stages of the structural design, simple response prediction methods are of vital importance. Two categories of prediction methods are usually applied, as introduced above: methods based on equivalent linear analysis and methods based on energy demand and supply. The first category mainly focuses on the response displacement of the isolation layer, and the second category mainly concerns the input energy of the isolation system. Furthermore, most methods in both categories function according to the response spectrum or the energy spectrum from the response analysis. However, this study proposed a method that is directly based on the characteristics of earthquake records and can relate both the response displacement and the input energy to the ground velocity history with sufficient accuracy. Therefore, a deeper understanding can be gained in order to provide another perspective for the design methods for isolation systems.