Report - PEER - University of California, Berkeley

∫ T0+ ∆T0ES(T)dTT0−∆TEs=2∆TFor the inelastic system,∫T0+ 2∆T0ES(T)dTT0Es= (5)2∆TIn the damped elastic system, the instantaneous period of vibration spreads on theboth sides of the natural period, T 0 . In the nonlinear system, the instantaneous periodspreads on the right hand side of T 0 . Let us take up a portion of 0 E s (T) whichcorresponds to T 1 ≦T≦T 2 . The portion of the energy spectrum, 0 E s (T) can beapproximated by the linear relationship which connects 0 E s (T 1 ) and 0 E s (T 2 ).From Eqs. (4) and (5), E s and T e are obtained as follows.0Es(T1)+0Es(T2)Es= , (6)2T1+ T2Te= (7)2Eq. (6) implies that the energy spectrum of the nonlinear system is obtained byaveraging the energy spectrum of the purely elastic system.The effective period, T e is obtained from Eqs. (6) and (7) as follows,For the damped elastic system,Te= T 0(8)For the nonlinear system,Te= T0+ Tm(9)where T m is the longest instantaneous period.3. EFFECTIVE PERIODS OF SINGLE-MASS SYSTEMSAs is shown by Eq.(9), the effective period of the nonlinear system is obtained byknowing the longest instantaneous period, T m . T m can be estimated based on themaximum displacement, δ m , and the shear stress corresponding to δ m . In most cases,the monotonic load-deformation curve under the horizontal loading becomes theenvelope of the hysteretic load-deformation curve under arbitrarily changing lateralloads. Therefore, the secant modulus which corresponds to the maximum deformationresponse can be described as follows, referring to the monotonic load deformationcurve as shown in Fig.1Qmqks= = − ・ k0(10)δm 1+µwhere k s is the secant modulus which corresponds to Q m and δ m , Q m the shear forceunder the development of δ m , q = Qm QY, Q Y the yield shear force, µ =δ m /δ Y -1.0 themaximum plastic deformation ratio, δ Y the elastic limit deformation, - µ the mean(4)387