Introduction

The book deals with the structural analysis of the bracing systems of multi-storey building structures and intends to offer useful tools to both researchers and practicing structural engineers. As a consequence, the material is divided into two parts: Part I presents the theoretical background and Part II gives worked examples. A couple of decades ago approximate methods played a very important and normally dominant role in the structural design of large structures as often, because of the lack of computer power, it was not feasible, or practical, or sometimes possible, to carry out an “exact” analysis of big and complex structures. Then more and more powerful computers with more and more sophisticated programs started to become available to wider and wider structural engineering communities. Soon the debate started with questions like “Do we need old-fashioned approximate methods?” and “Should we rely on brainless number-crunching machines that cannot think?” and “Shall we just input all the data, press <Enter> and by tomorrow the structural analysis is done?” and “Computers in the design office: boon or bane” (Smart, 1997). This debate will perhaps go on for a long time. But one thing seems to be certain: simple analytical methods and closed-form approximate solutions do and will play an important role in practical structural engineering and theoretical research (Howson, 2006). Not only because they offer important independent checking possibilities to help to avoid CAD (Computer Aided Disaster) (Brohn, 1996) but also because the development and use of such methods help to understand the complex behaviour of large structures such as multi-storey buildings. They are also useful tools in developing structural engineering common sense and a feel for the behaviour of structures. When multi-storey buildings are investigated, two main avenues are available for the structural engineer: sophisticated and powerful computer packages can be used or “conventional” calculations can be made. Perhaps the best way to tackle the task is to employ both approaches: at the preliminary design stage simple hand methods can quickly help to establish the main structural dimensions and to point to efficient bracing system arrangements. More detailed computer-based analyses can follow. Before the final decision is made, it is essential to check the results of the computer analysis and recheck the adequacy of the key elements of the bracing system. Here, again, suitable analytical methods can play a very useful part. The fact that the methods in the book are all based on continuous models has another advantage. When the results of a finite element analysis (based on discrete models) are checked, it is advantageous to use a technique that is based on a different approach, i.e., on continuous media. Structural analysis is normally carried out at two levels. The structural

engineer has to ensure that a) the individual elements (beams, columns, floor slabs, etc.) are of adequate size and material to carry their load and b) the structure as a whole has adequate stiffness and the bracing system fulfils its main role to provide sufficient stability to the building. The book does not deal with individual structural elements. Its aim is to present simple analytical methods for the complex global analysis of whole structural systems in the three main structural engineering areas. Closed-form solutions will be given for the maximum rotation and deflection, the fundamental frequency and the critical load of the building, assuming three-dimensional behaviour. The continuum method will be used which is based on an equivalent medium that replaces the whole building. The discreet load and stiffnesses of the building will be modelled by continuous load and stiffnesses that make it possible to use analytical tools to produce relatively simple, closed-form solutions to the resulting differential equations and eigenvalue problems. It will be assumed that the structures are • at least four storeys high with identical storey heights • regular in the sense that their characteristics do not vary over the height • sway structures with built-in lower end at ground floor level and free upper

end and that • the floor slabs have great in-plane and small out-of-plane stiffness • the deformations are small and the material of the structures is linearly elastic • P-delta effects are negligible. Structural engineering research and practice often see researchers/structural designers who have specialized in one area with limited knowledge elsewhere. Designers are often reluctant to deal with theoretical matters; researchers often have little practical knowledge (or attitude); those dealing with stress analyses are sometimes ignorant of stability matters; people engaged in earthquake engineering may not be very good at the optimisation of bracing systems, etc. This book offers a unified treatment for the different structures (frameworks, coupled shear walls, shear walls and cores) and also for the different types of investigation (deflection, rotation, frequency, stability). The same terminology will be used throughout, and it will be shown that these seemingly independent areas (deformations, frequencies, critical loads-or stress, dynamic and stability analyses) are in fact very closely related. In addition, the global critical load ratio links them to the performance of the bracing system in a rather spectacular manner. Numerous approximate methods have been published for structural analyses. However, it is surprising how few, if any, have been backed up with comprehensive accuracy analysis. Here, in this book, dozens/hundreds of bracing units/systems are used to demonstrate the applicability and accuracy of the methods presented. Although real multi-storey buildings seldom develop planer deformation only, Chapter 2 (dealing with the planar analysis of individual bracing units) is probably the key chapter of the book in the sense that it introduces all the characteristic stiffnesses that will be used for the three-dimensional investigations

of whole systems later on. It is also shown here how the complex behaviour can be traced back to the local bending, global bending and shear deformation (and their torsional equivalents) of the bracing system. All the characteristic types of bracing unit are covered here: sway-and infilled frameworks, frameworks with crossbracing, coupled shear walls, shear walls and cores. Deflections and rotations are the subject of Chapter 3 where the main aim is to present simple, closed-form solutions for the maximum deflection and rotation of the building. The investigations spectacularly show the contribution of the two key (bending and shear) stiffnesses as well as the interaction between them. Chapter 4 deals with the frequency analysis of buildings. Closed form formulae and tables make it possible to calculate the lateral and torsional frequencies of the building. The coupling of the lateral and torsional modes can be taken into account by a simple summation formula or, if a more accurate result is needed, by calculating the smallest root of a cubic equation. The often neglected but very important area of stability is covered in Chapter 5. In using critical load factors, simple (Euler-like) formulae are presented for the lateral and torsional critical loads. The combined sway-torsional critical load is obtained as the smallest root of a cubic equation. Chapters 2, 3, 4 and 5 end with a demonstration of the accuracy of the method(s) presented in the chapter. Chapter 6 introduces the global critical load ratio which is a useful tool for monitoring the “health” of the bracing system and indicates if the bracing system is adequate or more rigorous (second-order) analysis is needed. The global critical load ratio can also be used to assess different bracing system arrangements in minutes in order to chose the most economic one. The results of a comprehensive example illustrate the practical use of the global critical load ratio. Part II presents sixteen examples worked out to the smallest details, with step-by-step instructions. The examples range from the deflection or frequency or stability analysis of individual bracing units to the complex deflection and frequency and stability analyses of bracing systems, considering both planar and spatial behaviour. Although most of the formulae in the book are of the back-ofthe-envelope type, due to the complexity of global three-dimensional analyses, some of the calculations may still seem to be rather cumbersome to carry out by hand. It is very rare, however, that a structural engineer today would wish to do actual hand-calculations, however simple they may be. Convenient spreadsheets and calculation worksheets make it possible to do the structural analysis and document its result at the same time in minutes. All the methods presented in the book are suitable for this type of application; in fact the worksheet version of all the sixteen worked examples has been prepared and made available for download. These one-to-eight page long worksheets cover a very wide range of practical application and can also be used as templates for other similar structural engineering situations.