ABSTRACT
Stability problems concerning multi-storey buildings can be investigated on two levels. An element-based “local” analysis can be carried out, step by step, aimed at certain key structural members. Codes of practice normally follow this avenue and have detailed instructions for the analysis. This approach makes it possible to carry out the analysis in a relatively simple way but has disadvantages. It leaves the designer with the task of identifying all the key members and it cannot address the full-height, three-dimensional global behaviour of the multi-storey building. The “local” approach may also lead to uneconomic solutions as the elements of the whole structure tend to work together and, with the local approach, the possibility to take into account the effects of interaction is normally limited to the neighbouring members only. The other approach is the global approach. The concept of the global critical load ratio has been around for some time. Around but not in use. Or at least not in use to such an extent as it should have been. As the results of an illustrative example given in this chapter will demonstrate, the global critical load ratio is far more than a stability parameter: It is a generic characteristic with which the designer can monitor the overall performance of the whole bracing system. It also links the three important areas of analysis: the stress, stability and dynamic analyses. The way the structure responds to the loads-in two or three-dimensional manner-is automatically taken into account and made clear to the designer. Around the middle of the last century Chwalla (1959) emphasized the importance of the global approach and recommended the introduction of a “global factor”. Halldorsson and Wang (1968) suggested that a “general safety factor” should be used for building structures, and its importance was comparable to that of the “overturning factor” used in the design of dams. Dowrick (1976) drew attention to the importance of the overall stability of structures. Dealing with plane structures Stevens (1983) linked theory and practice and underlined the importance of the critical load in the design of frameworks. The idea of a global safety factor also surfaced in connection with the structural design of large structures (Zalka and Armer, 1992). MacLeod and Zalka (1996) and MacLeod (2005) advocated the use of the critical load ratio emphasizing its ability to handle torsional behaviour in a relatively simple way. The importance of torsional behaviour cannot be overemphasized, especially considering the fact that up to the emergence of the personal computer, relatively little attention had been paid to the three-dimensional behaviour of complex structures in university textbooks and in national and international codes of practice. However, the situation seems to be changing. More and more powerful computers, sophisticated software packages and advanced guidelines on modelling
complex structures (MacLeod, 1990 and 2005) make it easier to carry out true three-dimensional analyses. The global approach and methods developed with the global aspect in mind have also been emerging in structural designer handbooks and in codes of practice themselves (EN 1992, 2004; EN 1993, 2004; Martin and Purkiss, 2008). Applying the global approach, the structural engineer can rely on two types of technique: full-blown, albeit time consuming, analyses can be performed using advanced computer modelling, or quick, less accurate, but more descriptive investigations may be carried out that use specialized but simpler models (Howson, 2006) like those presented in this book. Depending on which direction the situation is looked at from, the global critical load ratio can be defined in two ways. First, it can be defined as
(6.1)
where
is the total vertical load of the regular multi-storey building with Ncr global elastic critical load for buildings subjected to uniformly distributed floor load L, B plan length and breadth of the building Q intensity of the uniformly distributed floor load n number of storeys Practicing structural engineers may prefer the reciprocal definition when the global critical load ratio is the ratio of the global elastic critical load and the total vertical load:
=λ (6.3)
as it carries a practical meaning that is easy to relate to the safety of the structure. Somewhat confusingly, codes of practice use both definitions. In this book, from now on, the reciprocal definition [Equation (6.3)] will be used. When there is significant extra load at top floor level (e.g., a swimming pool), its detrimental effect cannot be ignored. For such cases Equation (6.3) can be amended and the global critical load ratio can be obtained using
(6.4)
where
F is the extra concentrated load at top floor level Fcr is the critical load for the concentrated top load case The global critical load can be determined carrying out a full-blown second order analysis using a computer program or by approximate analytical solutions e.g., the ones presented in Chapter 5. The global critical load ratio can be used in different ways. Codes of practice normally concentrate on its use as an indicator whether or not second order analysis is needed. If the condition
10≥λ (6.5)
is satisfied, then the suitability of the bracing system is proved and the vertical load bearing elements can be considered as braced (by the bracing system) and neglecting the second-order effects (due to sway and torsion) may result in a maximum 10% error. If condition (6.5) is not satisfied, the stability of the building may still be acceptable but it must be demonstrated using a second-order analysis. However, there is a warning here: it is widely accepted in practical structural engineering that the absolute minimum for a critical load ratio is four. Another simple use of the global critical load ratio may be as a global safety factor: the greater the value of the global critical load ratio, the greater the safety of the multi-storey building against buckling. The global critical load ratio can also be used as a performance indicator. As its value is calculated using the basic (sway and pure torsional) critical loads and taking into account the coupling of the basic modes, any weakness either in the bending/shear and torsional stiffnesses or in the geometrical arrangement of the bracing units (on which the detrimental coupling depends) is picked up automatically. As it happens, any weakness detected during the course of the stability analysis leads to unfavourable behaviour when the fundamental frequency and the maximum deflection of the building are calculated. This is demonstrated below when the structural performance of a building is monitored using the global critical load ratio. The case study concentrates on a 10-storey building whose detailed global analysis is presented in Section 12.1 and only the main results are summarised here. The plan length of the building is 15 metres and the breadth is 9 metres, resulting in a plan area of 135 m2 (Figure 6.1).
