Keywords: equilibrium models, limit analyses, hyperstatic fields, concrete slabs.

This paper is concerned with lower bound limit analysis of plates, with particular application to the design or assessment of reinforced concrete slabs. Three important aspects of such an analysis are the formation of statically admissible fields of stress-resultants, the definition of appropriate yield criteria, and the use of mathematical programming for optimization.

The main focus of the paper is on finite element models composed of triangular equilibrium elements having moment fields of degree less than or equal to two, that are suitable when (a) only moments are included in yield criteria (the Kirchhoff model), or (b) both moments and shear stress-resultants are included (the Reissner-Mindlin model). In either case the direct formation of independent hyperstatic fields is presented with the aim of providing an efficient and highly sparse set of constraints in terms of static variables for use in a linear or a second order cone programme.

Methods of formulating particular fields of stress-resultants that equilibrate with applied loads are proposed, from (a) a yield line analysis based on a similar mesh of rigid elements, or (b) a linear elastic analysis based on a similar mesh of hybrid elements. It is emphasized that the former enables both upper and lower bounds to be obtained, useful in the context of limit analyses, whilst the latter is useful in complying with building codes when limited ductility can be accounted for by limiting the extent of moment redistribution from a linear elastic solution.

Numerical results of these formulations are presented based on Kirchhoff models for the benchmark problem of a square homogeneous isotropic slab with fixed sides and supporting a uniformly distributed load [1] and a case study originally presented in Reference [2].

The main conclusions based on the application of Kirchhoff models are (i) that the ability to formulate directly the particular and the hyperstatic fields represented by the static variables in an hierarchic manner as regards degree can lead to very sparse systems and an efficient solution from linear programmes based on interior point methods; and (ii) reasonably close lower and upper bounds can be achieved from similar finite element models. Further work is required to extend the numerical investigations to applications such as flat slabs where Reissner-Mindlin models are warranted so as to incorporate shear into the yield criteria.

1

E.N. Fox, "Limit analysis for plates: the exact solution for a clamped square plate of isotropic homogeneous material obeying the square yield criterion and loaded by uniform pressure", Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 277(1265), 121-155, 1974. doi:10.1098/rsta.1972.0057

2

G. Kennedy, C.H. Goodchild, "Practical yield line design", The Concrete Centre, 2004.

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