ABSTRACT
A great number of methods have been developed for the dynamic analysis of individual frameworks, coupled shear walls and shear walls. Fewer methods are available to deal with a system of these bracing elements. This follows from the fact that the interaction among the elements (beams/lintels and columns/walls) of a single framework or coupled shear walls is complex enough but then the bracing units interact with one another not only in planar behaviour but normally also in a three-dimensional fashion. This is why the available analytical methods make one or more simplifying assumptions regarding the characteristic stiffnesses of the bracing units or the geometry of the building. Based on drift calculations and assuming a doubly symmetric structural arrangement, Goldberg (1973) presented several simple methods for the calculation of the fundamental frequency of (uncoupled) lateral vibration and pure torsional vibration. The effect of the axial deformation of the vertical elements was taken into account by a correction factor in his method. The continuous connection method enabled more rigorous analyses (Coull, 1975; Rosman, 1973 and 1981; Kollár, 1992). Using a single-storey torsional analogy, Glück et al. (1979) developed a matrix-based solution for buildings having uncoupled stiffness matrixes. A simple procedure with design tables was made available for asymmetrical buildings developing predominantly bending deformation (Zalka, 2000). Ng and Kuang (2000) presented a simple method for the triply coupled vibration of asymmetric wall-frame structures. However, their method is only applicable to buildings whose vertical bracing elements develop no or negligible axial deformation. In taking into consideration all the characteristic stiffnesses of the bracing frameworks, shear walls and cores, as well as the interaction among the elements of the bracing structures and among the bracing units themselves (Zalka, 2001), the aim of this chapter is to introduce a simple analytical method for the calculation of the natural frequencies of regular multi-storey buildings braced by a system of frameworks, (coupled) shear walls and cores. In addition to the general assumptions made in Chapter 1, it is also assumed for the analysis that the mass of the building is uniformly distributed over the floors of the building and that the location of the shear centre only depends on geometrical characteristics. The equivalent column approach shall be used for the analysis. The equivalent mass and stiffnesses shall be established first, considering deformations due to bending, shear, the lengthening and shortening of the vertical elements and torsion. Closed-form solutions shall then be given for lateral, pure torsional and coupled vibration.
