Critical correlation of bi-directional horizontal ground motions

A stochastic model is treated of bi-directional horizontal ground motions (2DGM). It is shown that, in comparison with the Penzien–Watabe model (1975) [3], the cross power spectral density (PSD) function between 2DGM along the building structural axes can be treated in a more general manner by using an extended Penzien–Watabe model introduced in this paper. The auto PSD functions of 2DGM along the building structural axes are assumed to be given and the cross PSD function between these 2DGM is treated as a complex unknown function. A critical excitation problem is then considered for a one-story one-span moment resisting three-dimensional frame. The corner-fiber stress at the column-end is taken as the objective function and the worst cross PSD function of the 2DGM is determined so that the maximum corner-fiber stress at the column-end is maximized. It is shown that the real part (co-spectrum) and the imaginary part (quad-spectrum) of the worst cross PSD function can be obtained by a devised algorithm including the interchange of the double maximization procedure in the time and frequency domains.