ICE - Institution of Civil Engineers - Virtual Library

This paper reviews the relationship between yield line theory, developed for reinforced concrete slabs by Johansen in the 1930s and 1940s, and limit analysis, formulated by the Hodge–Hill–Prager school around 1950. Recalling work from the 1970s, it is shown that not only is yield line theory fully compatible with limit analysis, but the Johansen ‘stepped’ yield criterion is in fact the only yield condition for the slab that allows coinciding upper and lower bounds in the sense of limit analysis to be derived by yield line theory. The yield condition for an arbitrarily reinforced slab is reproduced in closed form, obviating the need for coordinate transformations, and allowing the permissibility of stress states to be checked for anisotropic slabs with non-coinciding principal directions of top and bottom reinforcement.

Mass concrete is a brittle material. If a structure is cast in unreinforced concrete—a dome, say, or a Roman cross vault—then that structure will inevitably crack in response to the movements imposed by the environment (settlements of foundations, spread of abutments). Such cracks are in themselves not signs of danger; they allow the structure as a whole to deform in a ductile way, and its behaviour may be interpreted by the use of plastic theory.

The paper considers the broad question of when and to what extent the theory of plasticity is useful in the analysis and design of reinforced concrete structures at the ultimate limit state of collapse. It covers early lower- and upper-bound approaches to design, and discusses various objections to the application of plasticity theory to concrete, concentrating on the need for adequate ductility – nowadays perhaps provided by fibre rather than bar reinforcement – and the crucial matter of size effect. It concludes that plasticity theory can be useful for the design of ordinary concrete structures known to have ductile failure modes, but that great care and wariness are needed when attempting to extrapolate its use to unusual structures and particularly to structures of large size without adequate ductility.

Twentieth-century researchers responded to the need to produce a ‘plastic’ building material based on concrete. In the twenty-first century the search should be for potentially superior building technologies that are less burdensome on the environment. The destruction of the twin towers of the World Trade Center in New York, USA in 2001 calls into question some current construction practices. Integration of construction teaching, alongside theory and design, is overdue. This leads to the question, ‘where to from here?’.

An automated yield-line analysis technique for reinforced concrete slab structures has been developed at the University of Cambridge and implemented in a computer program called COBRAS for use in assessing the ultimate flexural strength of concrete bridges. The program employs recent developments in solid modelling theory to create a three-dimensional model or ‘picture’ of a bridge and all its components within the computer. This model is built up from a number of separate layers of polygonal regions, each of which represents the plan location within the bridge of a particular structural feature. In combination, these polygons provide all the information necessary to describe the structural parameters of the material components and dimensions of the bridge, the external loading acting on the bridge and the geometric information required for the collapse mechanism analysis. This enables realistic bridge and load configurations to be analysed rigorously and very rapidly. An extensive theoretical and experimental validation programme was undertaken in collaboration with the Transport Research Laboratory and the program has been adopted widely by bridge engineers in the UK and also abroad, primarily for use in assessing the load capacity of short-span reinforced-concrete slab bridges.

In the construction industry, the need for flat slabs supported on columns in an irregular layout is rising. Design methods validated by experimental evidence for flat slabs supported on non-rectangular grid are, however, scarce. To fill this vacuum, two irregular flat slabs were tested at Nanyang Technological University. Those slabs, however, failed at almost twice the design ultimate load, showing the need for more work. In this paper, existing design methods based on lower bound approach are reviewed. Gurley's bimoment concepts are used to check the validity of zero twisting assumption in some of these methods. Also, experimental details of the nine flat slab panels, supported on non-rectangular grid, designed using the structural membrane approach and tested at Cambridge University Engineering Department are provided. Based on the observations, some suggestions are made for practical application of the structural membrane approach to design flat slabs on non-rectangular column grid.

Yield-line analysis is conventionally undertaken using collapse mechanisms that comprise rigid regions that intersect at yield lines, which behave in uniaxial bending only. It is well established that the flexural capacity of reinforced concrete (RC) slabs subjected to biaxial bending can be less than that when they are subjected to uniaxial bending. Concerns must therefore result regarding whether the inability of conventional yield-line mechanisms to model regions of biaxial bending could lead to significantly unsafe (upper bound) results. A simple method is presented to enable zones of biaxial bending to be taken into account in yield-line analysis. In this method ‘wide’ yield-line mechanisms are used which comprise rigid regions that are connected by ‘yield lines’ of finite and constant width and with uniform finite curvature over their area. Two examples of the method are presented. From the results, it appears unlikely that the reduced flexural capacity of RC slabs in biaxial bending will have a highly significant effect on their overall capacity in many cases. Perhaps of greater significance is the variation in ductility observed for different bending cases.

A method is developed that allows automated lower bound analyses of concrete slabs to be carried out, with the specific aim of producing a technique that can be applied to existing structures which are being assessed after many years in use and for which the original calculations, and hence the assumed methods of load distribution, have been lost. It is shown that any load distribution within the slab can be expressed in terms of an equilibrium state and a set of states of self-stress. Optimisation techniques are developed that allow the highest lower bound to be found. It is shown that an analysis based on the original Hillerborg two-strip model fails to find a reasonable lower bound when the slab is subjected to point loads and in the corners of slabs but a refinement using four sets of strips at 45° to each other can give very good agreement with both exact analytical solutions and upper bound methods.

This paper presents a view of the state of concrete plasticity as seen by a retired consulting engineer/structural designer. The author discovered the ‘bimoment’ method for the design of torsion-free Hillerborg slabs using upper bound collapse mechanism methods. He has also provided collapse mechanisms for plane-stress problems including shear in ordinary beams where longitudinal and transverse reinforcement may both be yielding.