Energy Spectra of Constant Ductility Factors for Orthogonal Bidirectional Earthquake Excitations

Abstract

Current research findings about energy spectra are almost entirely based on the single-degree of freedom system that is only fit for two dimensional analyses. Therefore, it is necessary to establish energy spectra of multidimensional systems subjected to multidirectional earthquake excitations. The bidirectional energy balance equation is established based on the single-mass bi-degrees of freedom system. The equation of normalized energy spectra is presented with earthquake motions, periods, and strength reduction factors as variables. The approximate relation of strength reduction factors between the x and y directions are derived from the seismic influence coefficient spectra. The 89 pairs of bidirectional earthquake motion records are selected as earthquake excitations to establish mean normalized input energy spectra and hysteretic energy spectra. The spectral values of normalized energy are compared between the single-mass bi-degrees of freedom system and single degree of freedom system, which shows that in some cases traditional energy spectra may underestimate the energy responses. The influence factors of mean spectra such as period, ductility factor, period ratio, and soil site are analyzed. Analytic results show that the coupling influences of nonlinear responses of structural two lateral components are considered in normalized energy spectra and are theoretically more rational than traditional spectra. In addition, two kinds of energy spectra have the typical spectral pattern features of the soil site, while the pattern and values of the two kinds of spectra are affected by the ductility factor, period ratio and the soil site.