# Quantification of Energy-Related Parameters for Near-Fault Pulse-Like Seismic Ground Motions
## 1 Introduction  
A significant number of strong ground motions recorded during recent earthquakes at sites close to the causative seismic fault is characterized by the presence of a dominant pulse-like and often high-amplitude component, which is especially evident in the fault-normal velocity waveform. Such a pulse, which is generated by the constructive interference of the radiated waves due to directivity of the fault rupture, can transmit a large amount of energy to the structures, and can cause prominent variability in the damage potential in the near field [1; 2; 3; 4].  
Within this framework, several studies have been conducted in the past years to analyze the main features of pulse-like seismic ground motions as well as to estimate the corresponding pulse period value [5; 6; 7; 8; 9; 10; 11; 12; 13; 14]. Several studies have been also conducted in order to examine the effects of near-fault pulse-like ground motions on the seismic response of structural systems [15; 16; 17; 18; 19; 20]. In this regard, most studies have focused on the displacement and strength demand for elastic or inelastic structural systems subjected to near-fault pulse-like earthquakes [21; 22; 23; 24; 25; 26]. However, since pulse-like components are usually detected into the velocity time histories of the ground motion, the energy imparted to a structural system likely represents the most relevant factor in determining the damagepotential associated with seismic loads in near-fault areas. In fact, pulse-like ground motion transmits a significant amount of energy in one or two large inelastic excursions accompanied by few reversals of the structural response, thus concentrating the damage in the weakest parts of the structure. This kind of behavior is different from the one that typically occurs under high-frequency seismic motions; in such a case, the energy dissipation takes place over a relatively long time window, and the affected structures undergo to a higher number of reversals in their seismic response without experiencing too much damage if properly designed, since high spectral accelerations usually correspond to a moderate level of the energy transmitted by the earthquake [27].  
In this context, the present study provides a systematic investigation about the seismic demand for inelastic structural systems subjected to the near-fault pulse-like seismic ground motion by focusing on energy-related parameters rather than on those expressed in terms of forces or displacements. In particular, this paper is organized as follows. Section 2 illustrates the different inelastic constitutive models employed to derive relevant spectra of energy-related parameters under the fault-normal horizontal component of near-fault pulse-like seismic ground motions. The considered energy-related parameters are presented in Section 3. The main features of the seismic time histories database are then presented in Section 4. Next, Section 5 illustrates the main results of a large numerical analysis intended to provide a comprehensive assessment of the influence of seismological and structural behavior on the spectra of relevant energy-related parameters. Finally, closed-form approximations for these spectra are proposed in Section 6, where a regression analysis is also carried out to estimate the numerical values of the corresponding parameters (with reference to both mean and standard deviation values) as a function of earthquake magnitude, type of hysteretic behavior (i.e., non-degrading or degrading) and ductility level.