Report - PEER - University of California, Berkeley

instantaneous period of vibration and receives the energy from a set of wavecomponents which correspond to a set of instantaneous period of structure. As aresult, the energy spectrum is obtained from the energy spectrum of the purely elasticsystem through an averaging (smoothing) process. The period which represents a setof instantaneous period is termed to be the effective period. The effective period ofthe damped elastic system is equal to the natural period, since the instantaneousperiod spreads around the natural period. The effective period of the inelastic systembecomes longer than the initial natural period, since the inelastic system is softenedby the plastification. The energy spectrum for design use must reflect the influence ofthe nonlinearity. The nonlinearity which corresponds to the 10% of the fraction of thecritical damping was found to represent the averaging effect met in the practicalstructures.Thus, the energy spectrum of the elastic system with 10% of fraction of criticaldamping is referred to be the energy spectrum for design use. The effective period isinfluenced by the type of restoring force characteristic and the level of nonlinearity.In this paper, the effective period is derived theoretically and the effectiveness ofthe effective period is verified by comparing the result of the direct response analysesand the energy spectra for design use (Akiyama, 1985).2. ENERGY SPECTRUMThe energy input into a single degree of freedom system subjected to a horizontalground motion is written as−∫tE000= Mz && dy(1)where E is the total energy input, M the total mass of the system, & z& 0 the horizontalground acceleration, y the horizontal displacement of the mass relative to the groundand t 0 the duration time of the ground motion. The energy spectrum is defined by thefollowing functions.E(Te)Es (Te) = , (2)M2E(Te)Es (Te) = VE(Te) =(3)Mwhere E s is the energy spectrum and T e the effective period.Eq. (2) is the direct expression of the total energy input per unit mass versus theeffective period. Eq (3) is the equivalent velocity expression of the direct energyspectrum.The energy spectrum of the purely elastic system is denoted by 0 E s . The energyspectra of the damped elastic system and the inelastic system can be described asfollows, based on the energy spectrum of the purely elastic system.For the damped elastic system,386