Probabilistic solution for an MDOF hysteretic degrading system to modulated non-stationary excitations

Structures usually exhibit hysteretic and deteriorative behaviors under strong loading, which is generally modeled as a multi-degree-of-freedom nonlinear and hysteretic system under non-stationary random excitations. The resulting governing Fokker-Planck-Kolmogorov (FPK) equation with high dimensions and hysteretic nonlinearity is difficult to solve, especially when the time evolution is involved. It has been rarely analyzed. In this paper, the state-space-split exponential-polynomial closure method, previously proposed for the stationary solution of the high-dimensional FPK equation, is further generalized to consider the non-stationary effect and hysteretic nonlinearity. With the proposed solution procedure, a ten-degree-of-freedom Bouc-Wen hysteretic system under non-stationary excitation is investigated. Different shapes of hysteretic loops with softening and hardening hysteretic behaviors are considered. Accompanied system deterioration is expressed with different levels. Non-stationary responses are analyzed in terms of the expected hysteretic energy and probability density function. Pertinent Monte Carlo simulation method is performed to verify the reliability of approximate solutions. Moreover, the influence of different degradation levels on the response is discussed.