Report - PEER - University of California, Berkeley

where M is the total mass of the system, k eq the equivalent spring constant whichgives the fundamental natural period the system, k 1 the spring constant of the firststory, κ 1 =k 1 /k eq , k 1 is expressed as follows.QY1α1Mgk1= =(14)δY1δY1where Q Y1 is the yield shear force of the first story, α 1 : the yield shear forcecoefficient of the first story, δ Y1 : the elastic limit deformation of the first story,g: the acceleration of gravity. Then, T 0 is written asδY1κ1T0= 2π(15)α1gThe shear type frames taken in this paper are conditioned to beδ Y =const., m i =const. (16)where δ Yi is the yield deformation in ith story, m i the mass of ith story, ( ) i : thequantity in the ith story. In such a system, κ 1 is very closely approximated byH 1 = 0.48+0.52N (17)When the type of restoring force characteristics and the maximum deformation ratio−ratio, µ are same in all stories, the secant modulus, kisi is proportional to the initialspring constant, k i . Therefore, T s is expressed as follows.−1+µiTs= T0qGeneralizing Eq. (18),T s for the multi-mass system is obtained as-(18)−1+µ δY1κ1(1+ µ )Ts= T0= 2π(19)qqαg-1−where µ = Σ µ i/N is the mean value of µ and N the number of stories.iiIn the highly nonlinear system, the total energy input is mainly absorbed in aform of cumulative plastic strain energy, W p . In multi-story frames, W p is expressedasNWpΣ= i 1Pi−= W(20)where W pi is the cumulative plastic strain energy in ith story. The damage distributionindex γ 1 is defined as followsWPγ1=(21)WP1W p1 is described as2 2 2 -Mg T α1µ1γ1aWP1= ・(22)24πκ1391