Report - PEER - University of California, Berkeley

value of the maximum plastic deformation ratios in the positive and negativedirections and k 0 the initial elastic slope.Figure 1. Monotonic load-deformation curve.The natural period, T 0 in expressed asT0M= 2π(11)k0The longest instantaneous period was found to be expressed in terms of theperiod which is associated with the secant modulus as follows (Akiyama, 1999)T = a(12)wherem TT sM 1+µ= 2π = T is the period associated with k s, a T the modificationk qTs0s−factor and k s the secant modulus.The modification factors depends on the restoring force characteristics and aredemonstrated in Table 1 for the typical structural Types. The energy spectra of theHachinohe record in the Tokachi-oki earthquake (1968) are shown in Fig.2. Theenergy spectra are depicted in the equivalent velocity expression shown by Eq.(3).Three structural types are selected. In Fig. 2(a), the abscissa indicates the naturalperiod, T 0 . In Fig.2 (b), the abscissa indicates the effective period, T e . The solid lineindicates the energy spectrum of the damped elastic system with 10% of fraction ofthe critical damping. The broken line indicates the envelope of the energy spectrumdepicted by the solid line. As is indicated by broken line, the energy spectrum isdivided into two range ; the shorter period range and longer period range. Theindividual plot indicates the total energy input which is obtained by the numericalanalysis for the single-mass system under the specified condition of the restoringforce characteristics and the maximum plastic deformation ratios. In shorter periodrange, the energy input increases as the T e increases. As the plastification is deepened.-µ increases and T e is elongated. Therefore, it is natural that the energy inputincreases as - µ increases under the same T 0 as shown in Fig. 2(a). On the other hand,as far as T e is taken as the abscissa, the dependence of the energy input to - µ isdissolved and the energy spectrum of the damped elastic system can be referred toobtain the energy input into highly nonlinear systems (Akiyama, 1999).388