不对称建筑与土壤结构的相互作用英文文献和中文翻译(8)

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Fig. 1. Dynamic model for N-story asymmetric shear building

The 0 and 1 shown in the above-presented equations are the col-umn vectors whose elements are equal to zero and one, respec-tively. On the other hand, hcolumn vector composed of the story heights measured from the ground level to each ﬂoor. In these equations, e and f =eccentricities measured from CR to CM

along the X axis and the Y axis, respectively; m0 =mass of foun-dation; r=radius of gyration of any ﬂoor deck about CM; Ixj and Iyj =moments of inertia of the jth ﬂoor about the axes through the CM and parallel to the X and Y axes respectively; ¨ugx and

¨ugy =ground acceleration records along the X and Y axis, respec-tively; and ﬁnally, U0, V0, 0, 0, 0 =degrees of freedom at the foundation associated with translations, twist and rocking, as ned in Fig. 1. Ujc, Vjc, and

structure deﬁned asdegrees of freedom of the super In Eqs. 3a–3c, Ujc, Vjc, and jc =degrees of freedom of the superstructure about the CM. The physical meaning behind these equations is depicted in Fig. 1. The 3N 3N submatrices on the left upper corner of M*,C*,K*, are the mass, damping, and stiff-ness matrices of the superstructure resting on a rigid base. They are equal to de-

respectively, in which

Kx, Ky, and KR shown in Eqs. 4a and 4b are deﬁned about the CR. The damping matrix shown in Eq. 4a is assumed to be proportional damping. The m shown in the previous equation is an N N diagonal matrix composed of each ﬂoor mass. The in-teraction forces Pxt, Qxt, Pyt, Qyt, and Tt shown in Fig. 1 are expressed by frequency-independent soil springs and dashpotsRichart et al. 1970 as

in which

where MT, IZ, and IR =mass, polar moment of inertia, and moment of inertia for rocking of the rigid body, respectively, which simu-

late the superstructure resting on a massless disc with radius r0. Vs =shear wave velocity of the elastic medium; =Poisson’s ratio, G=shear modulus; and _x0007_=mass density of the half-space. The static impedance functions shown in Eq. 6 as proposed by Ri-chart et al. 1970 were used in this study. The spring-and-dashpot coefﬁcients are made frequency independent for the frequency

range of interest deﬁned as 0 fr0 /Vs 1.5, where f =circular frequency of the applied harmonic excitation.

Multi-Degrees-of-Freedom Modal Equations of Motion

Based on the study by Chopra and Goel 2004, the right-hand side of Eq. 1 can be represented as follows:

here sn =nth modal inertial force distribution equivalent to M*n; n =nth undamped mode shape obtained from K* and M*; and xn and yn =nth modal participation factors deﬁned as

It was assumed that only the nth undamped modal displacement response, Wn, of the whole system will be excited under the load, sn xn ¨ugxt+ yn ¨ugyt. Thus, Eq. 1 can be written as

In Eq. 10, Dn =nth generalized modal coordinate. The nth un-damped modal displacement response, Wn, was redeﬁned as

For proportionally damped elastic systems, the elements of Dn are the same, i.e.,

Hence, Eq. 11 is identical to Eq. 10 for proportionally damped elastic systems. Substituting Eq. 11 into Eq. 9 and premulti-plying both sides of Eq. 9 by TnT, this results in Mn ¨Dn + Cn ˙Dn + KnDn =−Mn xn ¨ugx + yn ¨ugy, n =1–3N +5

where =8 1 column vector with all elements equal to one and

Mn ,Cn ,Kn=8 8 matrices. Eq. 13 is the nth MDOF modal equation of motion. Expansion of Eq. 14 is included in Appen-dix I. It was noted that if the original SSI system was proportion-ally damped, Cn can be represented as

where and =constants. As the original SSI system was non-

proportionally damped, i.e., C* M*+ K*, Eq. 15 becomes

Cn Mn + Kn. This implies that the MDOF modal equations of motion shown in Eq. 13 preserve the characteristics of the non-proportional damping of the original SSI system. The modal re-