Large Displacement Analysis of Plates using Hybrid Equilibrium Elements

Small strains can coexist with large displacements, particularly for structures composed of thin plates. This paper is concerned with finite element modelling of such non-linear behaviour when the material retains its linear elastic constitutive relations and Reissner-Mindlin theory can be invoked. This type of behaviour is of particular interest when for example in-plane stiffening becomes significant in a floor slab, or when elastic buckling due to geometric imperfections can occur. The co-rotational concept is invoked whereby the rigid body translations and rotations of elements are accounted for by the associated movements of element local reference axes, and the linear elastic stiffness matrices of elements are maintained unchanged with respect to current local configurations. Such formulations have been described by Izzuddin [

1

] where a more conventional conforming type of quadrilateral element is used. Due to the inherent generality of this approach, it has now been extended to hybrid equilibrium flat shell quadrilateral elements, and this is described in the paper. The formulation of this element is also presented and use is made of local element axes which are defined so as to lead to symmetric tangent stiffness matrices. The hybrid equilibrium model [

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],[

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] enables solutions to be determined which satisfy equilibrium in a strong pointwise sense. This feature is believed to be a novel one in the case of modelling geometrically non-linear behaviour with finite elements, and it allows new meaning to be given to the presentation of equilibrium paths. Thus it may be expected that a plot of a quantity of interest versus a load parameter is more accurate when the quantity is a stressresultant. Numerical examples are presented to compare the performance of the equilibrium models with conventional conforming models, and to investigate the potential advantages of having dual solutions to a geometrically non-linear problem. Theoretical questions regarding the significance of fully equilibrated solutions have not yet been answered for such problems, e.g. can such solutions lead to upper bounds to the energy of the errors in a conforming solution Nevertheless in practice it may be argued that two solutions are always better than one, and useful bounds may be achieved.