This, together with other simpli-fying assumptions, leads to simple formulae for determining the required natural frequencies (Tarjan andKollar, 2004). A useful tabulated summary of related work by the following authors is also included (Basu,1983; Kopecsiri and Kollar, 1999a,b; Rosman, 1974; Rutenberg, 1975; Skattum, 1971; Smith and Crowe,1986; Smith and Yoon, 1991; Zalka, 2001b).The methods developed in the references above offer solutions of varying accuracy depending on theassumptions employed. Surprisingly, none of them allows for step changes of properties along the height ofthe structure, despite the fact that this is almost inevitably the case in practical building structures of reason-able height. This study therefore seeks to present the simplest model that retains the essential characteristicsnecessary for calculating the natural frequencies of doubly asymmetric, three-dimensional frame structures inwhich the members may be uniform throughout the height of the structure, which leads to a particularly sim-ple hand solution, or may have step changes of properties at one or more storey levels.2. Problem statementThe class of building structure considered herein comprises two sets of orthogonal frames that are not addi-tionally stiffened by bracing or shear walls. The lateral vibration of such frames is well known to be charac-terised by three types of structural action; local bending of the inpidual columns, global bending of the structure as a whole and inter-storey shear, see Fig. 1. The contribution of local bending to the lateral vibra-tion of the frame is normally small and can usually be neglected in most practical cases. On the other hand, theeffect of global bending due to finite axial stiffness of the columns becomes progressively more important as theslenderness of the structure increases.In the theoretical model proposed, the inter-storey shear is fully accounted for, but the effects of local bend-ing are deemed to be insignificant and the columns are assumed to be inextensible and thus do not allow forthe possibility of global bending. The omission of these two latter actions is justified in Sections 7–9, wheretheir combined effects have been assessed through a parametric study that indicates likely limits on the appli-cability of the model.The underlying approach adopted with the model is to dissect the original building structure into segments,by cutting through the structure horizontally at those storey levels corresponding to changes in storey proper-ties.
Thus the storeys contained within a segment between any two adjacent cut planes are identical. A typicalsegment is then considered in isolation. Initially, a primary frame in one direction is replaced by a substituteshear beam. This member has uniformly distributed mass and stiffness and the unusual property that it allowsfor shearing deformation, but not bending deformation. In turn, each frame running in the same direction isreplaced by its own substitute beam and the effect of all these beams is summed to model the effect of the ori-ginal frames. This leads directly to the differential equation governing the sway motion of the segment in thechosen direction. The same procedure is then adopted for those frames running in the orthogonal direction.Once both equations are available it requires little effort to write down the substitute expressions for the coupledtorsional motion. The three equations thus formed are subsequently solved exactly and posed in dynamic stiff-ness form. The resulting coupled shear–torsion beamelement can then be used to reconstitute the original struc-ture by assembling the dynamic stiffness matrices for the inpidual segments in the usual manner.It is clear from the element formulation that the final model has a transcendental dependence upon the fre-quency parameter. The required natural frequencies are therefore determined by solving the model using anexact technique, based on the Wittrick–Williams algorithm, that can be arrested after achieving any desiredaccuracy and which also ensures that no natural frequencies can be missed.3. TheoryConsider the hypothetical layout of a typical floor plan of the asymmetric, three-dimensional frame struc-ture shown in Fig. 2. The plane frames run in two orthogonal directions and it is assumed that the shear cen-tre, S, at each floor level lies on a vertical line that runs through the height of the structure. This condition isautomatically satisfied when the frames running parallel to the x-axis are all proportional, i.e. their stiffnessmatrices can be scaled linearly from that of an arbitrary frame, and likewise for frames running parallel tothe y-axis, although the arbitrary frame does not have to be the same in both directions.The origin of the co-ordinate system is taken to be the shear centre, with the x and y co-ordinates runningparallel to the plane frames. The z-axis then runs vertically from the base of the building and coincides withthe rigidity axis. Point C(xc,yc) denotes the centre of mass at a typical floor level. It is assumed that the floorsystem is rigid in its plane and that the centre of mass at each level lies on a vertical line, the mass axis, that runs through the height of the structure. When the rigidity and mass axes of a structure do not coincide, thelateral and torsional motion of the building will always be coupled in one or more planes.During vibration, the displacement of the mass centre at any time t in the x   y plane can be determined asthe result of a pure translation followed by a pure rotation about the shear centre, see Fig. 3. During the trans-lation phase the shear centre S moves to S0and the mass centre C moves to C0, displacements in each case ofu(z, t) and v(z, t) in the x and y directions, respectively. During rotation, the mass centre moves additionallyfrom C0to C00, an angular rotation of u(z, t) about S0. The resulting translations, uc and vc, of the mass centrein the x and y directions, respectively, areucðz; tÞ¼ uðz; tÞ  ycuðz; tÞð1aÞandvcðz; tÞ¼ vðz; tÞþ xcuðz; tÞð1bÞ More generally, it is clear that the displacements of a typical point (xi,yi) are given by Eqs. (1a) and (1b) whenc = i.The structure is now pided into segments along the z axis by notionally cutting the structure along hor-izontal planes at those storey levels corresponding to changes in storey properties. Fig. 4 shows a typical seg-ment formed by cutting the structure through planes EkFkGkHk and Ek+1Fk+1Gk+1Hk+1 that correspond to thekth and (k + 1)th changes in storey properties. The number of storeys in any one segment can vary from one,to the total number of storeys in the structure if it is uniform throughout its height. However, in any one seg-ment each storey must have the same properties.We now consider a typical segment in isolation and seek to replace each primary frame by a substitute shearbeam that replicates its in-plane motion. We start by considering a typical frame, frame i, that runs parallel tothe y–z plane, see Fig. 2. This whole frame is replaced by the single substitute beam, beam i, shown in Fig. 5.This beam is a two-dimensional shear beam of length L and has uniformly distributed mass and shear stiffness.The mass and elastic axes therefore coincide with the local z-axis and the elastic axis is only permitted sheardeformation vi(z, t) in the y direction, where z and t denote distance from the local origin and time,respectively. The equations of motion for the substitute beam can be developed by considering a typical elemental lengthof the beam, dz. Thus equating the resultant shear force to the mass acceleration givesoQyiðz; tÞoz¼ myio2viðz; tÞot2ð2Þwhere Qyi(z, t) is the shear force on the element and myi is the uniformly distributed mass per unit length.The constitutive relationship for pure shear is given byQyiðz; tÞGAyi¼ oviðz; tÞozð3Þin which GAyi is the effective shear rigidity in the y direction (Smith and Coull, 1991).Substituting the derivative of Eq. (3) into Eq. (2) giveso2viðz; tÞoz2  myiGAyio2viðz; tÞot2¼ 0 ð4Þwhich is the required differential equation of motion for the shear beam element in the y–z plane.If the equivalent procedure is carried out for all of the i frames that run parallel to the y–z plane, thedynamic equilibrium for motion in the y–z plane may be written asoozX nyi¼1GAyioviðz; tÞoz¼X nyi¼1myio2viðz; tÞot2ð5Þwhere ny is the number of frames.Noting that GAyi is constant over the length of the member and substituting for vi(z, t) from Eq. (1b) with creplaced by i givesX nyi¼1GAyio2vðz; tÞoz2þX nyi¼1GAyixio2uðz; tÞoz2 X nyi¼1myio2vðz; tÞot2 X nyi¼1myixio2uðz; tÞot2¼ 0 ð6Þwhere xi is the distance of frame i from the shear centre, S. The second term in Eq. (6) equals zero, since S isthe centre of rigidity of the structure. As C is the centre of mass,Pnyi¼1myixi can be replaced with myxc, wheremy ¼ Pnyi¼1myi , so Eq. (6) can be written as followsGAyo2vðz; tÞoz2  myo2vðz; tÞot2  my xco2uðz; tÞot2¼ 0 ð7Þin whichGAy ¼X nyi¼1GAyi ð8ÞSince the total mass of the segment contributes to its vibration, including the mass of the frames running inthe x direction and the rigid diaphragms, my should be replaced by m, where m is the equivalent distributedmass over the height of the segment. ThereforeGAyo2vðz; tÞoz2  mo2vðz; tÞot2 耦合侧向扭转频率的不对称三维框架结构英文文献和中文翻译(2):http://www.751com.cn/fanyi/lunwen_56549.html