PDEM-based dimension-reduction of FPK equation for additively excited hysteretic nonlinear systems

Engineering structures usually exhibit hysteretic property when they are subjected to disastrous loadings such as earthquakes, strong wind and huge waves. The solution of FPK equation related to such complex systems are of great significance but is still an extremely challenging problem. In the present paper, the FPK equation for hysteretic systems is studied and the dimension is reduced to render an efficient numerical solution. The equivalence of probability flux in two different treatments, i.e., in the FPK equation which is based on the state space description and in the generalized density evolution equation which is mainly based on the random event description, are first examined. Then the FPK equation for additively excited multi-degree-of-freedom structures exhibiting nonlinear behaviors, including hysteresis together with strength degradation and stiffness degradation, is discussed. By invoking the basic idea of equivalence of probability flux, an equivalent probability flux due to drift could be constructed by the probability density evolution method (PDEM) instead of the direct coupled high-dimensional integral. Numerical implementation procedure is outlined. Two numerical examples are illustrated. Problems to be further studied are discussed.