David Johnson

School of the Built Environment, Nottingham Trent University,
Burton Street, Nottingham NG1 4BU, UK
E-mail: david. johnson@ntu. ac. uk

The road up and the road down are one and the same

O ye’ll tak’ the high road and I’ll tak’ the low road,
And I’ll be in Scotland afor ye
(Traditional Song)


The comparative merits of hand and automated upper and lower bound techniques for the collapse load estimation of reinforced concrete slabs are examined. Examples, drawn from both theoretical and practical design work, are used to show that both hand and automated upper bound yield line techniques can produce significant, unsafe errors. Automated lower bound solutions, however, are shown to consistently provide safe estimates that are not unduly conservative, provided appropriate formulations are adopted. As long as the engineer is willing to dispense with the crutch of a yield line pattern, it is therefore contended that, whilst Heraclitus may be correct in that both the upper and the lower bound roads can lead to one and the same collapse load, the lower bound road gets you there, certainly more safely, and usually quicker, as the Traditional Song suggests.


A unique solution for the collapse load of a reinforced concrete slab, as predicted by rigid-plastic theory, requires the simultaneous fulfilment of equilibrium, yield and mechanism conditions. This requirement is normally too onerous to be achieved directly and recourse is therefore commonly made to the use of either upper or lower bound solutions (Wood, 1961). Both approaches satisfy the equilibrium condition, but the upper bound approximation ignores the yield requirement, whilst the lower bound technique does not require the specification of a mechanism. The upper bound method can therefore postulate a collapse mechanism which incorporates “unsafe” violations of the slab yield requirement that moments do not exceed the fully plastic moment of the slab section. Lower bound solutions, on the other hand, can postulate moment distributions that are “safe” (in the sense that the plastic moment is nowhere exceeded), but which do not correspond to a collapse mechanism. For both formulations, either hand or automated approaches are possible and the principal features of a range of these will be outlined prior to a comparative study of the merits of the various techniques, as demonstrated by a range of examples.


M. Pandey et al. (eds), Advances in Engineering Structures, Mechanics & Construction, 823-831. © 2006 Springer. Printed in the Netherlands.