Keywords: locking, finite elements, shell element, large displacement analysis.

Ever since the emergence of the displacement-based finite element method, a most serious problem that has influenced its application in linear and nonlinear structural analysis is related to the locking phenomenon. There are many aspects to this phenomenon, depending on structural form, element shape, and type of analysis. Early forms of locking were observed in the modelling of plate bending problems using the Reissner-Mindlin hypothesis [1], where the inability of a mesh of conforming elements to bend without inducing transverse shear strains leads to deteriorating performance as the plate thickness is reduced, a phenomenon referred to as 'shear locking'. Other forms of locking can arise with conforming elements, such as 'membrane locking' when modelling curved shells, and 'distortion locking' when employing isoparametric mapping with irregular element shapes.

Whilst locking phenomena may be viewed from several different perspectives, depending on the context of element application, a common feature is the degradation in the approximation of various strains over the element domain, principally due to polluting higher-order strains.

This paper presents a general approach for optimising the performance of isoparametric elements with the objective of minimising locking. The proposed approach can be used for planar, plate and shell elements, and is applicable to linear as well as nonlinear analysis. Optimisation is performed by correcting the conforming strain field, towards the highest-order strain distribution afforded by the conforming element, using suitable hierarchic modes. In this respect, it is shown that the application of the new approach leads naturally to assumed strain fields, which are most effectively obtained as a transformation of the conforming strain fields at sampling Gauss points. Furthermore, two alternative transformations are presented, corrective and objective, relating to whether the assumed strain field is taken as a correction of the conforming strain field or as the objective strain field, respectively.

The proposed optimisation approach is applied to relieve membrane, shear and distortion locking for a previously developed nine-node conforming shell element [2], which is employed within a co-rotational framework for large displacement analysis. Several examples are finally provided, which demonstrate the accuracy of the optimised element and, consequently, the effectiveness of the proposed optimisation approach towards lock-free finite elements. In this respect, it is shown the most effective application of the new approach is based on the first level of hierarchic optimisation with an objective transformation for the assumed strains.

1

O.C. Zienkiewicz, R.L. Taylor, The Finite Element Method, Vols. 1 & 2, 4th edition, McGraw-Hill, 1991.

2

B.A. Izzuddin, Z.X. Li, "A Co-rotational Formulation for Large Displacement Analysis of Curved Shells", in "Developments in Mechanics of Structures and Materials", A.J. Deeks and H. Hao (Editors), Vol. 2, Taylor & Francis Group, London, 1247-1253, 2004.

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