Report - PEER - University of California, Berkeley

⎧1⎫⎧0⎫⎪ ⎪ ⎪ ⎪ιx= ⎨0⎬ι y = ⎨1⎬⎪ ⎪ ⎪ ⎪⎩0⎭⎩0⎭(2)respectively, where each element of 1 is equal to unity and of 0 is equal to zero. Thus,the effective earthquake forces are⎧m1⎫ ⎧ 0 ⎫⎪ ⎪ ⎪ ⎪peff =− s g =−⎨ 0 ⎬ gx ⎨m1⎬gy⎪ ⎪ ⎪ ⎪⎩ 0 ⎭ ⎩ 0 ⎭() t u&& () t u&& () t or u&&() t(3)The spatial distribution of the effective forces (Eq. (3)) over the building is defined bythe vector s and the time variation by u&& g () t = u&& gx() t or u&& gy () t . This forcedistribution can be expanded as a summation of modal inertia force distributions s n(Chopra 2001, Section 13.3):3N3Ns = ∑ s n = ∑Γn Mφ nn= 1 n=1(4)where φ n is the nth natural vibration mode of the structure consisting of threesubvectors, φ xn, φ yn,andφ θ n, and() t()ΤL⎧nΤ⎪φxnm1for u&&gxΓ n = Mn = φn Mφn Ln = ⎨M Τn⎪⎩ φynm1for u&&gy tThus() t =− u () tpeff,ns n&&g(5)(6)is the nth-mode component of effective earthquake forces.In the MPA procedure, the peak response r n of the building to p eff,n () t — orthe peak “modal” demand—is determined by a nonlinear static or pushover analysisusing the modal force distribution⎧mφxn⎫* ⎪⎪sn= ⎨mφyn⎬⎪ ⎪⎪⎩Ioφθn⎪⎭(7)336