Case Study Week1 Essay

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Morley Symposium on Concrete Plasticity and its Application.

University of Cambridge 23rd July, 2007

Mikael W BRAESTRUP Ramboll Oil & Gas, Denmark Keywords: Concrete structures, Mathematical modelling, Slabs & plates.

The mathematical theory of plasticity, or limit analysis, was developed around the middle of the 20th century by Hodge, Drucker, Prager and Greenberg [1-4] in the USA and Hill [5] in the UK. Making use of extremum principles that can be traced back at least to Coulomb [6], the theory aims to predict the collapse load of a structure by means of the three limit analysis theorems: Upper Bound, Lower Bound and Uniqueness. The material properties are described by a yield condition, defining all stress states that can be sustained. The convex frontier of the allowable stress states is the yield surface, which is also the plastic potential for the strain increments that may occur for stress states on the yield surface. Limit analysis can be applied to any material or structure that can reasonably be described as rigid, perfectly plastic. Basically, this requires the absence of stability issues or large deformations, ensuring that elastic strains can be neglected, and adequate ductility, ensuring the necessary redistribution of stresses without excessive damage or rupture. Yield line theory, on the other hand, is an efficient way of determining the collapse load of reinforced concrete slabs, and it pre-dates limit analysis. The term ‘yield line’ (in Danish: ‘brudlinie’, literally meaning ‘line of rupture’) was coined in 1921 by Ingerslev [7] to describe lines in the slab along which the bending moment is constant. In 1931 K W Johansen [8] gave the concept a geometrical meaning as lines of relative rotation of rigid slab parts, and in 1943 [9] published the eponymous theory. Yield line analysis was adopted by the Danish concrete code, and introduced into the curriculum at the Technical University of Denmark. There is anecdotic evidence to the effect that the success of Danish engineers worldwide in the decades immediately following the Second World War owed no small part to their mastery of yield line analysis, allowing them to produce efficient designs of reinforced concrete slabs of any shape and loading, while their foreign colleagues were struggling with tables of elastic moments and influence surfaces, leading to excessive amounts of reinforcement, and applicable to standard cases only.

The fathers of limit analysis immediately recognised yield line theory as an ingenious method of computing upper bound solutions for reinforced concrete slabs (Prager [4]) and in the early 1960s the mathematical theory of plasticity was seized by M P Nielsen, then a research student with Prof Johansen. In the resulting thesis Nielsen [10] derived the yield condition for orthogonally reinforced slabs (in a coordinate system oriented along the directions of principal moment capacity). On this basis he reproduced and supplemented Johansen’s yield line solutions, showing that many were in fact exact, in the sense that coinciding lower bounds exist. At the same time yield line theory was receiving wider attention in the world. Johansen’s pioneering paper [8] had already been published in German in 1932, and after the war the thesis [9] was first translated into Portuguese (for the Brazilian market). The English translation – sponsored by the C & C A - appeared in 1962, followed by the catalogue of solutions (Yield Line Formulae [11]) a decade later. Nielsen’s 1964 thesis [10] was published in English from the start. In the 1960s yield line theory was the subject of considerable interest in the UK, as evidenced by a flurry of papers and monographs, including a special publication issued by Magazine of Concrete Research [12], including contributions by L L Jones, K O Kemp, C T Morley, M P Nielsen and R H Wood. A particular subject under…