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ð47Þwhich is satisfied whenxðkÞj ¼ k 12 pbjj ¼ 1; 2; 3 k ¼ 1; 2; 3; ... ð48ÞHence the required natural frequencies can be determined easily using Eq. (48).7. ExamplesThe work of this section consolidates the foregoing theory by performing a parametric study on four framesof varying slenderness (height devided by least plan dimension) and comparing the lower natural frequencieswith those obtained from a full finite element analysis. The frames, which have 5, 20, 40 and 60 storeys, respec-tively, all have the same doubly asymmetric floor plan and equal storey height of 3 m. Each structure consistsof five plane frames in the y direction (F1–F5) and four plane frames in the x direction (F6–F9) which areconnected to each other by typical rigid diaphragms at each floor level with the arrangement shown inFig. 7. In the 5 and 20 storey buildings, the properties of the structural elements do not change along theheight of the structure, so each structure can be modelled using a single substitute beam element and the nat-ural frequencies can be determined from the theory of Section 6. In the 40 and 60 storey buildings, the prop-erties of the structural elements change in a step-wise fashion every 20 storeys. Tables 1a and 1b show thecross-sectional area of all columns and the second moment of area of the beams and columns of all the build-ings. The member properties have been carefully selected to ensure that the original structures are fully rep-resentative of practical buildings. For simplicity in determining member masses, half the mass of the columns framing into and emanatingfrom a floor diaphragm, together with the mass of the diaphragm and any associated beams, is stated asan equivalent uniformly distributed floor mass at that storey level.
Thus the centre of mass is at the geometriccentre of the floor plan. This corresponds precisely to the automatic idealisation process in ETABS (Wilson et al., 1995) and additionally only requires the total mass of the floor to be converted into the equivalent uni-formly distributed mass of the member in the substitute beam approach. Arbitrarily the mass is assumed tohave a constant value of 360 kg/m2at each floor level, even where the stiffness properties of the memberchange. Young’s modulus for all members is taken to be E =2 · 1010N/m2.All the plane frames in this example are proprtional, so that the shear centre at each floor level lies in avertical line through the building. The eccentricities in the x and y directions can then be calculated as follows(Cheung and Tso, 1986)xc ¼ 5:454 m; yc ¼ 5:00 mThe distributed mass of the shear–torsion beam (smeared from the diaphragm) and the polar mass radius ofgyration of the diaphragms about the shear centre can be calculated as follows 8. Numerical resultsColumn 2 of Tables 2–5 show the coupled natural frequencies (Hz) of the 5, 20, 40 and 60 storey framesobtained from the proposed three-dimensional shear–torsion beam theory, respectively. The third and fourthcolumns in each table show the results of a full finite element analysis of the original frames for the two casesof extensible columns, where A took the values given in Tables 1a and 1b and inextensible columns, whereA = 1. These results were obtained using the vibration programme ETABS in which the automatic idealisa-tion process was utilised that assumes uniformly distributed mass on rigid floor diaphragms. Relevant com-parisons are made in columns five to seven.9. DiscussionIt was mentioned in Section 2 that the vibration of three-dimensional frame structures is characterised bythree types of structural action; local bending, global bending and inter-storey shear. The proposed shear–tor-sion model deals accurately with inter-storey shear, but lacks any stiffness contribution stemming from localbending. The proposed model is therefore more flexible than the original structure in those areas where localbending is important and hence the natural frequencies will be underestimated. On the other hand, the col-umns are assumed to be inextensible and the model will therefore overestimate those frequencies that are sig-nificantly influenced by global bending. Since local bending is most prominent in buildings with lowslenderness and global bending is most prominent in buildings with relatively high slenderness, there willbe a useful range of buildings for which the model yields perfectly acceptable results.These arguments are borne out by the results of Tables 2–5.In Table 2 the results for the five storey frameare least accurate due to significant local bending of the inpidual columns. However, the difference betweenthe model results and those of the finite element analysis still lie below 9%. As the slenderness increases and the源-自/优尔+文,论`文'网]www.youerw.com