Error Analysis Of Spatially-varying Seismic Ground Motion Simulation By Spectral Representation Method

Strong earthquakes will cause damage to and even collapse of long-span bridges. The accurate prediction of bridge’s structural responses to seismic ground motions is therefore a crucial necessary process in the design of long-span bridges. Consequently, the simulation for generating sample time histories of spatially varying ground motions is often a prerequisite for predicting seismic responses of a long-span bridge. The spectral representation method(SRM) is often used for generating sample of spatially-varying seismic ground motions via evolutionary power spectral density(EPSD) model. Nevertheless, error analysis is scarce for the simulation of spatially-varying seismic ground motions.As a result, this paper concerns the theory of evolutionary power spectral density and methods for its estimation, together with the error analyze of spectral representation method in spatially-varying ground motion simulation, which contains the following two main parts:First, this paper studies methods for evolutionary power spectral density estimation. Based on windowed filtering method, this paper presented a new method for evolutionary power spectral density estimation, which is gained by the Fourier transformation of the time varying correlation function that calculated from samples of time histories and is fully verified by numerical example and comparison between existed method.Second, this paper concerns statistical errors in the simulation of spatially-varying seismic ground motions modeled by evolutionary Gaussian vector processes with zero mean and simulated by the spectral representation method. The bias and random error formulas are derived for the simulated evolutionary power spectral density, time-varying correlation function and standard deviation. The closed-form error formulas for the evolutionary power spectral density are further simplified under specified practical conditions. It is shown that the simulated non-stationary characteristics are all unbiased, the closed-form random error formulas for the evolutionary power spectral density can reduce to those for stationary simulations, and in the numerical example, the predicted random errors match those given by ensemble average, which proofs that the proposed closed-form error formulas are valid. By using the random error formulas, the factors influencing the random errors are investigated. The results show that the spectral representation method implementation scheme involving both random amplitudes and phase angles would cause large random errors and that increasing the number of samples and frequency intervals can reduce random errors. The closed-form error formulas are finally used to estimate the errors in the simulation of spatially-varying seismic ground motions for the Tsing Ma suspension bridge in order to illustrate the practical value of the proposed formulas.