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Concrete In Australia : December 2013
54 Concrete in Australia Vol 39 No 4 DISCUSSION PAPER change every 5-10 years? Or do we want to show students basic alternatives that lie almost in sight or just around the corner – especially solutions that seem to properly respect basic statics? Structural design and drafting should never be just a matter of relying on software to implement the current code. People at all levels should have an intelligent understanding for what is going on. The paper is deliberately repetitive in that the basic message is repeated three times: in the figures, in the text and in the appendices. One hopes this is helpful, and Braestrup thinks that it is. But the paper could otherwise be about half the length. Should the author produce an alternative minimum length paper? The author thinks that the statics are correct, but one would be really delighted if someone suggested a simpler, perhaps more graphic approach to the statics involved. Reference Beletich, Hymas, Reid and Uno “Reinforced Concrete The Designers Handbook”. Cement and Concrete Services <www. cementandconcrete.com> Some History of Structural Plasticity Research into plastic theory for structural steel design began at Cambridge University before World War II and spread to Lehigh, US, and Sydney, Australia. The book, The Steel Skeleton (2 enormous volumes, Cambridge University Press), by Baker, Horne & Heyman, all of Cambridge, was published in 1954. Jacques Heyman later wrote a book, The Stone Skeleton, which explored the plastic analysis of monumental stone buildings including the great gothic cathedrals. Jacques was alive and well when one last visited Cambridge in 2007. The Steel Skeleton reported many strain measurements on large buildings. It concluded that there were usually large strains and stresses locked in by the construction process. The usual assumption of elastic analysis is that the only significant stresses are those generated by the gravity and horizontal loads, and The Steel Skeleton makes it clear that this is just not so. From a plastic point of view, the main significance of elastic analysis is that it is a valid lower-bound analysis, but there are others. More recently, one’s Danish friends reported that redistribution in real structures begins below service loads, and that a plastic analysis provides moment diagrams that are quite appropriate for calculating deflections. Jack Roderick came from Cambridge to the University of Sydney as head of the school of civil engineering about 1950, when it was still the only school of civil engineering in New South Wales. He arranged for Dr Max Lay to write the Australia Standard for Steel Structures as the first Australian plastic code, published about 1967 – a truly excellent code. Rigid-plastic theory is quite clear that there are just three types of solution: • upper-bounds, which provide a kinematically-correct collapse mechanism but do not necessarily attempt to satisfy equilibrium everywhere • lower-bounds, which satisfy equilibrium everywhere but do not necessarily supply a collapse-mechanism • exact solutions, which both supply a collapse mechanism and satisfy equilibrium everywhere. Exact solutions are not necessarily unique, but the collapse load is unique. These bound theorems apply to rigid-plastic steel structures and also to rigid-plastic concrete structures including yield-line theories for both: • slabs loaded perpendicular to their own plane and • concrete in plane-stress as in beams in shear. There may be a hope that these will someday merge to provide a 3D yield-line theory dealing, for example, with transfer of moments and shears from flat slabs to supporting columns. K.W. Johansen, M.P. Nielsen and other Danes first discovered the solely upper-bound analysis of floor slabs from the mid- 1930s. Prof Arne Hillerborg 1964 of Lund, Sweden suggested the lower-bound analysis of slabs as torsion-free grillages and wrote a textbook about it. This author discovered the bimoment method for the upper- and sometimes lower-bound (hence sometimes exact) yield-line analysis of floor slabs, regarded as torsion-free grillages in Auckland 1979. This has been published extensively with ICE London Magazine of Concrete Research. Hillerborg wrote in 1979: “A very noticeable property of the bimoment method is the simplicity of the numerical calculations for many cases met with in everyday design work. Therefore it can be recommended as a useful tool in design offices.” Fox, of Cambridge, published a 42 page exact solution of a full-torsion slab for one particular case. A torsion-free analysis of that same case takes a line or two with a pocket calculator and turns out to be 75% of the Fox solution. Denmark, in dealing with plane-stress yield-line problems in concrete, generally addresses both the collapse-mechanism and the complete equilibrium solution. Germany and Switzerland, where the S&T method originated, tend to address the equilibrium solution but not the collapse mechanism. The author does suggest that his paper addresses both upper- and lower-bound criteria following the Danish precedents. The late Roderick agreed with me. Designers of concrete structures would normally want to take over the analysis methods originally developed for steel structures. AS 3600 c 6.7 PLASTIC METHODS OF ANALYSIS says that this can be done subject, of course, to the requirement for ductile ‘N’ rebar. This will be true for members that are designed to fail at Whitney (plane, perpendicular) hinges but not for members failing in shear until appropriate mechanisms are discovered and published. Limitations and speculations This note is intended as an appendix to recent and, perhaps, future papers on the subject of shear in reinforced beams. The need for this note has arisen from discussion by others. The first limitations of this author’s work on shear in reinforced beams are that: • the tensile strength of concrete is everywhere assumed zero but • the target is not just beams in shear but the wider field of structural concrete in plane-stress. The implication of the first limitation is that the present approach cannot address shear in slabs and slab-like band- beams up to say 2400 wide (the width of a piece of plywood) 49-56 - Discussion.indd 54 49-56 - Discussion.indd 54 25/11/13 4:15 PM 25/11/13 4:15 PM