Joint probability density function of the stochastic responses of nonlinear
structures
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Chen Jianbing (³Â½¨±ø) and Li Jie (Àî ½Ü)
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School of Civil Engineering, Tongji University, No. 1239 Siping Road,
Shanghai 200092, China
Abstract: The joint probability density function
(PDF) of different structural responses is a very important topic in the
stochastic response analysis of nonlinear structures. In this paper, the
probability density evolution method, which is successfully developed to capture
the instantaneous PDF of an arbitrary single response of interest, is extended
to evaluate the joint PDF of any two responses. A two-dimensional partial
differential equation in terms of the joint PDF is established. The strategy of
selecting representative points via the number theoretical method and sieved by
a hyper-ellipsoid is outlined. A two-dimensional difference scheme is developed.
The free vibration of an SDOF system is examined to verify the proposed method,
and a frame structure exhibiting hysteresis subjected to stochastic ground
motion is investigated. It is pointed out that the correlation of different
responses results from the fact that randomness of different responses comes
from the same set of basic random parameters involved. In other words, the
essence of the probabilistic correlation is a physical correlation.
Keywords: stochastic response; nonlinearity; joint
probability density function; probability density evolution method; number
theoretical method; finite difference method
