Field‐consistency and violent stress oscillations in the finite element method

Abstract

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

A new shear flexible cubic spline plate element for vibration analysis

“…This study, in general, suggests that for the plate thickness parameter a=h of the order 10, the original spline element does not seem to su er from locking problems whereas it is susceptible to shear locking for the fairly thin as well as extreme thin cases. Furthermore, one can conclude that unlike element employing the reduced integration scheme, which may not be suitable for the problems wherein a higher order of integration is needed (for instance tapered plates), consistent element frees the order of integration and thus, is applicable for both thick and extreme thin situations, as expected [19][20][21]. Finally, it can be viewed that the consistency approach with constant level (CS-3) alone yields accurate results for fairly thick to extremely thin plates.…”

“…At the simplest level of ÿeld-consistency, we consider the use of ÿeld redistributed substitute interpolation functions which include only those speciÿc terms that must be made ÿeld-consistent, as outlined in References [19][20][21]. This is achieved here by smoothing the original interpolation function in a least-squares accurate fashion to the desired form i.e.…”

“…The occurrence of shear locking phenomenon with respect to problems concerning with shear deformation theory in conjunction with B-spline functions has been eliminated by choosing di erent order of spline functions for the constrained ÿeld variables [13], by constructing B-spline displacement ÿeld based on the inspection of the Timoshenko beam mode functions [15], which is the function of thickness parameter, and by introducing modiÿed shear modulii [11; 12]. The shear locking behaviour in the ÿnite element analysis of beams, plates and shells has been studied by many authors [16][17][18][19][20] in the context of elements employing conventional polynomial functions for the ÿeld variables. The more versatile approach, among the available techniques for alleviating locking such as reduced=selective integration scheme and assumed strain ÿelds is the ÿeld consistent formulation [19; 20].…”

Shear flexible field-consistent curved spline beam element for vibration analysis

“…This study, in general, suggests that for the beam thickness parameter¸/t of the order 100, the original spline element does not seem to su!er much from locking problems whereas it is susceptible to membrane and shear locking for the extreme thin case. Furthermore, one can conclude that unlike elements employing the reduced integration scheme, which may not be suitable for the problems wherein a higher order of integration is needed (for instance tapered beams), consistent element frees the order of integration and thus, is applicable for both thick and extreme thin situations, as expected [22,23]. Finally, it can be opined that the consistency approach with constant level (CMCS-3) alone produces good results for fairly thick to extremely thin beams.…”

The Exact Two-Node Timoshenko Beam Finite Element Using Analytical Bending and Shear Rotation Interdependent Shape Functions

“…Carpenter et al [4] credited the coupling between bending and shear rotation in the Timoshenko beam equations as causing shear locking and suggested that these be decoupled through the use of appropriate strain fields. Using the concept of field consistency, Prathap [5,6] attributed shear locking to the choice of interpolation functions used for displacement fields, which impose spurious constraints that occur because the strains developed from the displacement approximation are inconsistent. Discussions of different interpolation schemes for the transverse displacement and sectional rotation which determine various Timoshenko beam finite element models have been given by Reddy [7].…”