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AbstractThis paper presents a global analysis approach to the calculation of the natural frequencies of asymmetric, three-dimen-sional frame structures in which the primary frames run in two orthogonal directions and whose properties may varythrough the height of the structure in a step-wise fashion at one or more storey levels. The governing differential equationsof a substitute system are formulated using a continuum approach and posed in the form of a simple dynamic memberstiffness matrix. Such a formulation allows for the distributed mass and coupled shear–torsion stiffness of the memberand thus necessitates the solution of a transcendental eigenvalue problem. The required natural frequencies are finallydetermined using a stepped cantilever model in conjunction with the Wittrick–Williams algorithm, which ensures thatno natural frequencies are missed. When the structure can be represented realistically by a uniform cantilever, solutionscan be found easily by hand. A parametric study comprising four, three-dimensional, asymmetric frame structures is givento compare the accuracy of the current approach with that of a full finite element analysis. 2006 Elsevier Inc. All rights reserved.Keywords: Asymmetric structures; Torsional vibration; Coupling; Three-dimensional models; Rigid frames; 52599
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Continuum mechanics1. IntroductionThe ability to model complex, three-dimensional, multi-bay, multi-storey structures to a high degree ofaccuracy has become commonplace over the last 20 years due to the widespread availability of powerful desk-top computers and a variety of inexpensive finite element software. The resulting models are often referred toas ‘global’ or ‘holistic’ since they model the whole structure in its entirety. Thus any interaction between struc-tural components such as frames, shear walls, cores, etc. or coupling due to asymmetry of the plan form areautomatically accounted for.Such complex models offer considerable insight into the behaviour of the physical structure and their usefor detailed design and analysis is not in question. However, the development of such a model at an early stage in a design process can be time consuming and unproductive if used during a period of rapid evolution of theconcept. A compelling alternative is to use a simpler model, developed especially for the type of structureunder consideration, which models only the dominant characteristics of the structure. This simplified globalmodel can offer a number of potential benefits. Data preparation and analysis will be quicker; the modellingprocedures are likely to be simpler and more transparent; the accuracy will normally be sufficient for prelimin-ary assessment or checks on solutions obtained elsewhere; it can be used efficiently to improve and develop afeel for structure; it draws directly on the engineer’s experience and judgement and its use is an inclusive pro-cess, where the engineer is at the heart of the solution procedure in a way that can sometimes appear to be lostwhen using fully automated, general software.A considerable amount of research effort has been expended over the years on developing such approximatemethods for the frequency analysis of frame structures. The most widely used methods have utilised a contin-uum approach, in which the building structure is replaced by a cantilever beam with uniformly distributedmass and stiffness.
Research in this area was initially focused on two-dimensional structures, with manyauthors developing a variety of approximate methods for frames (Bolton, 1978; Rafezy and Howson, 2003;Roberts and Wood, 1981; Williams et al., 1983); shear–walls (Rosman, 1974; Rutenberg, 1975); wall–frames(Kollar, 1991; MacLeod, 1970) and three-dimensional symmetric structures comprising frames, coupled walls,wall–frames and braced frames (Delpak et al., 1997; Smith and Crowe, 1986).More recently approximate methods have been developed that can deal with the vibration of asymmetricthree-dimensional structures, in which the translational and torsional modes of vibration are coupled. Kuangand Ng (2000, 2001) considered the problem of doubly asymmetric, proportional structures in which themotion is dominated by shear walls. For the analysis, the structure is replaced by an equivalent uniform can-tilever whose deformation is coupled in flexure and warping torsion. The same authors extended this conceptto the case of wall–frame structures by allowing for bending and shear. In this case however, the wall andframe systems are independently proportional, but result in a non-proportional structural form (Ng andKuang, 2000). Wall–frame structures have also been addressed by Wang et al. (2000), who used an equivalenteccentricity technique that is appropriate for non-proportional structures, but the analysis is limited to findingthe first two coupled natural frequencies of uniform structures with singly asymmetric plan form.Hand methods have also received considerable attention and are particularly suitable for check calcula-tions. In recent papers by Zalka (2001a,b), such a method is presented which can deal with the three-dimen-sional frequency analysis of buildings braced by frameworks, coupled shear–walls and cores. The paper alsoreviews similar related work.The most recent contribution has been made by Kollar, who replaces the original structure by an equivalentsandwich beam that can model both slender and wide structures consisting of frames, trusses and coupledshear walls (Potzta and Kollar, 2003). In a subsequent paper, an alternative approach is adopted in whichthe natural frequencies of the replacement beam are solved approximately.