# Energy-Based Prediction of the Displacement of DCFP Bearings
## 2 State-of-the-Art Isolator Displacement Prediction
### Simplified Linear Analysis (Euro Code [7])  
Similar to the equivalent lateral force procedure introduced in ASCE, this method may be applied to isolation systems with equivalent linear damped behavior if they also conform to conditions such as the distance from the site to the nearest potentially active fault, with a magnitude \(M\geq 6.5\), being greater than 15 km and their effective period being smaller than 3 s and larger than three times the fundamental period of the superstructure, assuming a fixed base. It assumes that the superstructure is a rigid solid, and then the displacement of the stiffness center, due to the seismic action, can be calculated in each horizontal direction, from the following expression:  
\[d_{dc}=\frac{M\times S_{c}(T_{eff},\xi_{eff})}{K_{eff,min}} \tag{2}\]  
where \(S_{c}\) (\(T_{eff}\), \(\xi_{eff}\)) is the spectral acceleration defined by an elastic response spectrum that takes into account the appropriate value of the effective damping \(\xi_{eff}\). \(K_{eff}\) is the effective horizontal stiffness of the isolation system, and \(T_{eff}\) is the effective period. Simplified Response Prediction Method for the Maximum Response Using an Equivalent Single-Degree-of-Freedom System (AIJ [8]  
The simple prediction for the maximum response values of an equivalent SDOF system involves the evaluation of the dynamic characteristics of the structure by using viscous damping and hysteretic damping, followed by the prediction of the maximum response using an elastic response spectrum. Two variations are included. In the response prediction method 1, the maximum response is obtained through a convergence calculation that considers the changes in the equivalent period and in the equivalent damping factor due to the yielding of the SDOF system in the spectrum response. For the response prediction method 2, Method 1 is simplified based on the assumption that the response spectrum curve is constant, regardless of the period. The maximum response is also estimated by some response curves. By using these curves, it is possible to grasp visually the tendency of the response reduction in the structures without a convergence calculation. The energy balance method is an analysis method that evaluates seismic resistance based on the balance of the seismic input energy to buildings due to ground motions via energy spectra and the energy absorbed by the building. In this method, the relationship between the maximum shear force coefficient and the maximum displacement (response reduction effect) of the seismic isolation level can be shown in the curves that are related to the seismic isolation period and ground motion level, by assuming that the total hysteresis energy absorbed during the earthquake is equal to the energy dissipated by two repetitions of the steady state loop at the maximum displacement. In the above methods, (1) and (2) are based on equivalent linear analysis with strict application conditions. Meanwhile, (3) and (4) provide the tendency of the response reduction, but the response displacement of the isolation layer cannot be predicted under the specified seismic loads. (II) Simplified Prediction Methods for the Energy Demand of the Isolation System Introduced in Some Design Methods Based on the Energy Concept [9; 10; 11; 12]. Besides the maximum response of the structure, the input energy is also very important for earthquake design methods based on the energy concept [9], in which the energy demand can be taken as the input energy. This value is usually given as the energy spectrum, which can provide the input energy to structures during an earthquake [10]. However, this process is based on response analysis under a certain earthquake record, which makes it difficult to relate the input energy to the characteristics of the group of earthquakes in the construction site.