Conditional Probabilistic Model for Normalized Hysteretic Energy Given Ductility Ratios

Hysteretic energy, a critical component of seismic input energy, is predominantly dissipated through the hysteretic behavior of structural members in most conventional structures. The motivation is to establish the conditional probabilistic model of normalized hysteretic energy of the structure after determining its displacement, thereby facilitating the estimation of the Park–Ang damage index. This study develops a probabilistic model for normalized hysteretic energy conditional on the ductility ratio. Three macroscopic hysteretic models, representative of the hysteretic behavior of distinct structural types, are employed to quantify the effects of ground motion characteristics (e.g., magnitude, distance, pulse, duration, and site conditions) and structural properties (e.g., post-yield stiffness and damping ratio). The findings reveal that a lognormal distribution effectively characterizes the normalized hysteretic energy. Among the investigated parameters, ground motion duration leads to a significant influence on the distribution of normalized hysteretic energy (maximum difference up to 30%). To facilitate practical applications, a set of predictive expressions is proposed to estimate the mean and standard deviation of normalized hysteretic energy. The resulting conditional distribution reproduces the empirical distribution derived from the original data, with an average error of approximately 5%. Using established expressions, the required ductility capacity under specified performance objectives can be probabilistically determined in seismic design. Moreover, the established distribution can be used to determine the potential hysteretic energy of the structure for assessing its damage state after an earthquake, as demonstrated through a full-scale shaking table test.