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Orthogonal expansion of ground motion and PDEM-based seismic response analysis of nonlinear structures

Li Jie1, Liu Zhangjun^1,2 and Chen Jianbing¹

1. State Key Laboratory of Disaster Reduction in Civil Engineering, Tongji University, Shanghai 200092, China
2. College of Civil & Hydroelectric Engineering, China Three Gorges University, Yichang 443002, China

Abstract: This paper introduces an orthogonal expansion method for general stochastic processes. In the method, a
normalized orthogonal function of time variable t is first introduced to carry out the decomposition of a stochastic process
and then a correlated matrix decomposition technique, which transforms a correlated random vector into a vector of
standard uncorrelated random variables, is used to complete a double orthogonal decomposition of the stochastic processes.
Considering the relationship between the Hartley transform and Fourier transform of a real-valued function, it is suggested
that the first orthogonal expansion in the above process is carried out using the Hartley basis function instead of the
trigonometric basis function in practical applications. The seismic ground motion is investigated using the above method. In
order to capture the main probabilistic characteristics of the seismic ground motion, it is proposed to directly carry out the
orthogonal expansion of the seismic displacements. The case study shows that the proposed method is feasible to represent
the seismic ground motion with only a few random variables. In the second part of the paper, the probability density evolution
method (PDEM) is employed to study the stochastic response of nonlinear structures subjected to earthquake excitations. In
the PDEM, a completely uncoupled one-dimensional partial differential equation, the generalized density evolution equation,
plays a central role in governing the stochastic seismic responses of the nonlinear structure. The solution to this equation
will yield the instantaneous probability density function of the responses. Computational algorithms to solve the probability
density evolution equation are described. An example, which deals with a nonlinear frame structure subjected to stochastic
ground motions, is illustrated to validate the above approach.

Keywords: seismic ground motion; stochastic processes; orthogonal expansion; probability density evolution method;
nonlinear structures; stochastic response analysis