The development of damage-based energy factor under far-fault ground motions for the seismic demand evaluation of structural systems in energy-balance theorem

The objective of this investigation is to develop the damage-based energy factor (gamma) for seismic demand evaluation of Reinforced Concrete (RC) and steel structures under probable future ground motions regarding the energy-balance theorem and use in the Performance-Based Plastic Design (PBPD) procedure. Before this, the energy factor was determined considering different ductility levels, where the procedure could be extended at constant specific damage levels in damage-based design theory to consider the influence of the hysteresis energy, frequency content, ground motions amplitude, and duration for design purposes. Hence, the popular Park-Ang damage index was employed as a damage model to obtain energy factor under 400 far-fault ground motion records by investigating the influence of structural and earthquake properties on it, including period of vibration, damage level, ultimate ductility ratio, stiffness hardening, structural deterioration, beta factor, soil class type, magnitude (Mw), and source-to-site distance. Due to the influence of ground motion characteristics by using statistical analysis, the Ap/Vp ratio is employed to determine the energy factor, which depends on soil class type, magnitude, fault mechanism, and source-to-site distance. Also, a simple equation based on nonlinear regression analysis among data is suggested to estimate the energy factor based on influential structural and earthquake characteristics, and its error is demonstrated by the two concepts of bias and standard deviation. Finally, three empirical structures validated by numerical modeling, consisting of a full-scale RC bridge pier, a full-scale fourstory RC building, and the three-story steel frame, are considered to validate the accuracy of the proposed method and equation. Statistical results illustrate that the difference between estimated displacements and obtained ones from direct time history analysis is not higher than 20 %, especially compared to the existing damage-based coefficient method.