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SUMMARYA three-noded curved beam element with transverse shear deformation, based on independent isoparametric quadratic interpolations, is designed from field-consistency principles. It is shown that a quadratic element that is field-inconsistent in membrane strain suffers from 'membrane locking'-i.e. an error of the second kind propagates indefinitely as the element length to thickness ratio and/or the element length to radius of curvature ratio increase, in nearly inextensional bending. However, field-inconsistency in shear strain does not lead to 'shear locking' but degrades its performance to exactly that of a field-consistent linear element. It is also seen that field-inconsistency leads to severe axial force and shear force oscillations. Error estimates for locking are derived, wherever possible, and confirmed by numerical experiments. The field-consistent element offered here is the most efficient quadratic curved beam element possible.

Early attempts to derive curved beam and shell elements in a curvilinear system were dramatically unsuccessful. This was wrongly attributed to the failure of these elements to recover strain-free rigid body displacement modes in a curvilinear co-ordinate description. Recent evidence points to a 'membrane locking' phenomenon that arises when constrained strain fields corresponding to inextensional bending are not 'consistently' recovered. This accounts for, more completely and precisely, the failure of such elements.In this paper, a simple linear two-noded Co continuous thick curved beam element based on a curvilinear deep shell theory is derived free from shear and membrane locking. Lack of consistency in the shear and membrane strain-field interpolations in their constrained physical limits (Kirchhoff and inextensional bending limits, respectively) causes very poor convergence due to locking and severe spurious oscillations in stress predictions. Error estimates for these are made and verified. Field-consistent strain interpolations remove these errors and produce the most efficient linear element possible.

SUMMARYThe stiffness matrix for the DKT plate-bending element is formulated explicitly in a global co-ordinate system. This approach avoids transformations of stiffness, and elasticity properties for anisotropic materials, from local to global co-ordinates, which were required in previous formulations. A FORTRAN listing of the algorithm is appended for potential users.

SUMMARYThe fact that finite element models can give rise to violent stress oscillations and that there are optimal locations whcrc strcsscs can bc correctly sampled in spite of the presence of these violent stress Iluctuations has been known for some time. Howcvcr, it is less wcll known that these oscillations arise in a specific class of problems--where there arc multiplc strainfields arising from one or more field-variables and where one or more of these strain-fields must be constrained in particular physical limits. In this paper, we show that unless the interpolations for these constraincd strain-lklds are 'field-consistent', violent oscillations would set in. These osciilations rcprescnt spurious self-equilibrating stress-fields generating spurious energy terms that lcad to 'locking'.The field-consistency interpretation offers a conceptual scheme to delineate these problems and an operational procedure called the functional reconstitution technique allows the errors resulting from fieldinconsistency to be anticipated a priori. We demonstrate the power of this approach through an interesting example of a multi-strain-field problem-t he inextensionallnearly inextensional deformation of a shear flexible curved beam. INTROD UCTlQNThe familiar principles that provide information for the selection of shape functions for a finite element discretization are: thc piecewise approximations: should be compictc polynomials; they should be compatible, i.e. satisfy continuity of these functions or their derivatives, where required, across inter-element boundaries; these functions should be able to represent states of constant strain in the limit; and finally. these functions should be able to recover strain-free rigid-body motion.However, elements derived rigoroiisly from these principles can still behave in dramatically erratic ways when applied to certain important problems even though these elements are quite excellent in other respects. Therefore the need for a new conceptual scheme was apparent. The early work of the authors and their colleague^^-^ focused on some of the strategies, or 'tricks', that were used to salvage elements that suffered from deficiencies such as shear-locking. membrane-locking, parasitic shear-locking at incompressible limits etc. Subsequent modelling of curved beams,6 plane stress modelling of fle~iire,~ plates and shells'.'' and three-dimensional solids with brick led to the formulation of the field-consistency principle'' and the definition of new terminology and operational proccdures' and an error norm" for establishing that errors due to field-inconsistency exist. We shall examine this at greater length in the next section and then use it to carry out an accurate prediction of the nature and magnitude of stress oscillations in a typical constrained multi-strain-field problem.It will be in ordcr here to discuss briefly the study of optimal stress locations with regard to the