Stability analysis of buildings

The stability of a building can, and should be, assessed by looking at the stability of its individual elements as well as examining its stability as a whole. National codes have detailed instructions for the first case but the buckling analysis of whole structures is not so well regulated and therefore this chapter intends to address the second case. The designer basically has two possibilities to tackle whole building behaviour in either using finite element packages or relying on analytical methods. The analytical approach is used here. A great number of methods have been developed for the stress analysis of individual frameworks, coupled shear walls and shear walls. Fewer methods are available to deal with a system of these bracing units. The availability of methods for the stability analysis of a system of frameworks, coupled shear walls and shear walls is even more limited. 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, the geometry of the building, or loading. In using an equivalent Timoshenko-beam, Goschy (1970) developed a simple hand-method for the stability analysis of buildings under top-level load. Goldberg (1973) concentrated on plane buckling and presented two simple approximate formulae which can be used in the two extreme cases when the building develops pure shear mode or pure bending mode buckling. The interaction of the two modes is taken into account by applying the Föppl-Papkovich summation formula to the flexural and shear mode critical loads. Using the continuum approach (Gluck and Gellert, 1971; Rosman, 1974), Stafford Smith and Coull (1991) presented a more rigorous analysis for the sway and pure torsional buckling analysis of doubly symmetric multi-storey buildings whose vertical elements develop no or negligible axial deformations. Based on the top translation of the building (obtained from a plane frame analysis) and assuming a straight line deflection shape, MacLeod and Marshall (1983) derived a simple formula for the sway critical load of buildings. In using simple closed-form solutions for the critical loads of the individual bracing frames and coupled shear walls, Southwell’s summation theorem results in a lower bound for the sway critical load of multi-storey buildings (Zalka and Armer, 1992). Even when the critical loads of the individual bracing units are not available, the repeated application of summation formulae leads to conservative estimates of the critical load in a simple manner (Kollár, 1999). In replacing the bracing units of a building with sandwich columns with thick faces, Hegedűs and Kollár (1999)

developed a simple method for calculating the critical load of multi-storey buildings with bracing shear walls and frameworks in an arbitrary arrangement, subjected to concentrated top load. All these methods restrict the scope of analysis in one way or another and none were backed up with a comprehensive accuracy analysis. In taking into consideration all the characteristic stiffnesses of the bracing frameworks and shear walls as well as the interaction among the elements of the bracing structures and among the bracing units themselves (Zalka, 2002), the aim of this chapter is to introduce a simple analytical method for the calculation of the critical load of 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 load of the building is uniformly distributed over the floors and that the location of the shear centre only depends on geometrical characteristics. The critical load of the structures defines the bifurcation point. The procedure for establishing the method for the determination of the critical load of the building will be very similar to the way the method for the calculation of the fundamental frequency was developed in the previous chapter. First, the basic stiffness characteristics will be established for the analysis. The effective shear stiffness will be introduced, which, as in the previous chapter, makes it possible to create an equivalent column by the simple summation of the relevant stiffnesses. Second, based on the equivalent column, the eigenvalue problems characterising the sway buckling and pure torsional buckling problems will be set up and solved. Third, the coupling of the basic (sway and pure torsional) modes will be taken into account. Finally, a comprehensive accuracy analysis will demonstrate the reliability of the method.