Phase Space Reduction in Stochastic Dynamics

In the field of structural engineering, various analytical procedures have been developed for the analysis of nonlinear structural systems subjected to random dynamic loading. To date, the Monte Carlo simulation (MCS) seems to be the most generally applicable approach for the reliability analysis of large nonlinear multi-degree-of-freedom systems. In this paper, a method that allows for reduction of the computational effort when using the MCS method is presented. First, the method requires the application of digital or analytical techniques (such as equivalent linearization) to obtain the response covariance matrix. Then, by means of the well-known Karhunen-Loéve expansion, the dimension of the system is reduced and MCS is applied. Moreover, in the transformed space the variables with smaller variance are approximated by Gaussian variables. It is shown that this technique allows for a considerable reduction of the computational effort without a significant loss of accuracy. As numerical examples, a 9-degree-offreedom hysteretic structure and a conservative system (i.e., a 10-degree-of-freedom Duffing structure), respectively, are examined.