Synthesis and Extension of One‐Dimensional Nonlinear Hysteretic Models

Structures under dynamic loading usually exhibit nonlinear hysteretic behavior, especially under strong earthquake excitation. The exact nature of this behavior may vary greatly from structure to structure. Therefore, a lot of one‐dimensional hysteretic models have been developed, including the smooth‐varying and piecewise‐linear (PWL) varieties. In this paper, we present a synthesis and extension of the myriad hysteretic structural models into a general nonlinear hysteretic model, which is mathematically tractable and covers smooth‐varying hysteretic models and various PWL hysteretic models. Because the formulation developed to describe the equation of motion is a nonlinear state equation, we can find the dynamic response explicitly, based on the explicit‐integration approach, and dependably in both deterministic and stochastic ways. Based on the formulation, we also demonstrate the similarity of the responses of the PWL and smooth‐varying hysteretic models. Finally, we add some comments on the extension of stochastic equivalent linearization techniques by smoothing the PWL hysteric models.