Bp Naganarayana

Field-consistency analysis of the isoparametric eight-noded plate bending element

The eight-node isoparametric plate bending element based on the serendipity shape functions behaves very poorly even after reduced integration of the shear strain energy. It has therefore been the subject of considerable study using various devices to improve it—mixed methods, enforcing of constraints, tensorial transformations, etc. In this paper, we shall proceed from the field-consistency paradigm to understand why the original element and even the element modified by the 2 ∗ 2 Gaussian rule cannot achieve, consistently, the true shear strain constraints in the penalty limit of thin plate behaviour. We then derive the optimal shear strain definitions that leave the element free of all problems in the rectangular form, for most sets of practical boundary suppressions. From this, we next determine the optimum manner of co-ordinate transformation that preserves the true constraints even in the form of a general quadrilateral. This is achieved within the context of iso-P Jacobean transformations and without having to bring in tensorial or base vector definitions and transformations. This should be the simplest displacement type version of this element.

Consistent thermal stress evaluation in finite elements

Abstract The computation of thermal stresses in displacement finite elements through a formal theoretical basis using the minimum potential principle leads to oscillating stress predictions. Many finite element packages have therefore used an average temperature in simple elements; thermal stresses were computed only at the centroids of such elements to avoid these problems. In this paper we trace this difficulty to a consistency requirement—the requirement that stress fields derived from temperature fields (or initial strains) must be consistent with the total strain field interpolations used in the finite element formulation. The principle governing the problem is developed from the minimum total potential theorem and the Hu-Washizu theorem. This gives it a formal rational basis and leads to an orthogonality condition that provides the procedure for determining consistent, thermal stresses in a variationally correct manner. The princple is demonstrated using some simple problems. A four-noded laminated plane-shell element is also considered to prove the rigour and generality of the approach presented here.

Consistent and variationally correct finite elements for higher-order laminated plate theory

Abstract Higher-order plate theories and the corresponding finite element formulations have become popular recently, owing to the increasing demand for accuracy in transverse stress predictions in thick and laminated structures. In this paper, several displacement-type finite elements based on the Lo-Christensen-Wu higher-order description of transverse structural deformation are examined. The errors of locking, delayed convergence and stress oscillations are critically studied. The accuracy of the elements is then demonstrated using some well-established bench-mark tests. It is also observed with reference to a high-precision three-dimensional finite element solution that the higher-order transverse deformation theories and the corresponding finite element formulations, in general, cannot capture the transverse stress distribution correctly near the structural boundary of thick beams and plates.

ACCURATE THERMAL STRESS PREDICTIONS USING C0-CONTINUOUS HIGHER-ORDER SHEAR DEFORMABLE ELEMENTS

The behavior of C 0 -continuous higher-order shear deformable finite element formulations when applied to thermal stress conditions is analyzed critically. It is shown that, unlike in the first-order formulations, the inconsistent initial strain field can also disturb the displacement recovery in the higher-order shear deformable elements. In this paper, the possible errors due to such inconsistent terms are a priori predicted in an analytical sense and validated by using linear and quadratic beam element formulations. The paper suggests how the total and the thermal strain components have to be consistently reconstituted to get optimal accuracy from higher-order shear-deformable finite elements for thermal stress analysis. A conflict between the consistency and the continuity requirements is noticed in such formulations. The extraneous errors observed in the variationally correct and consistent formulations are attributed to this fact.

Beam elements based on a higher order theory—II. Boundary layer sensitivity and stress oscillations

The flexure of deep beams and thick plates and shear flexible (e.g. laminated composite) beams and plates is often approached through a finite element formulation based on the Lo-Christensen-Wu (LCW) theory. A systematic analytical evaluation of beam elements based on the LCW higher order theory was carried out recently. It turns out that the availability of a large number of degrees of freedom to prescribe end/boundary conditions leads to discontinuity effects that trigger off wiggles (sharp oscillations) in some of the higher order displacement terms. These wiggles propagate outward from the point of excitation and disturb the transverse normal stress predictions. This paper examines the origin of these oscillations and how these boundary layer effects can be contained by refined modeling within the boundary layer zone or region when beam elements based on this higher order theory are used. A similar difficulty should be present in plate elements based on the same theory.

DISPLACEMENT AND STRESS PREDICTIONS FROM FIELD- AND LINE-CONSISTENT VERSIONS OF THE EIGHT-NODE MINDLIN PLATE ELEMENT

Abstract The eight-node isoparametric Mindlin plate bending element based on the serendipity shape functions has a long history of investigation behind it, and has seen various devices to improve it—mixed methods, enforcing of constraints, tensorial transformations, etc. Only very recently have successful versions free of locking in general quadrilateral form and without kinematic modes emerged. In this paper, we shall examine two of the most successful displacement method procedures (a field-consistency approach and a line-consistency approach) and proceeding from these, design three very accurate versions—one based on a variationally correct field-consistency paradigm alone, and two versions derived from the need to ensure consistency of tangential shear strains along principal reference lines so that the usual patch tests are exactly passed. The latter two have shear strain definitions that leave the element free of all problems (locking and kinematic modes) for all boundary suppressions and element distortions whereas the former has two kinematic modes. These line-consistent elements, however, introduce spurious quadratic shear stress oscillations as they have not been derived in a variationally correct sense. The recovery of accurate transverse shear stress resultants must therefore be performed very carefully, and a filtering technique is implemented for this.

Force and moment corrections for the warped four-node quadrilateral plane shell element

Abstract A four-node quadrilateral plate element can be used as a facet shell element only if provision is made to allow for non-coplanarity of the four nodes. The element stiffnesses are generated for a ‘mean plane’ equidistant from the four nodes, and are corrected by introducing equilibrated forces and moments as the element is ‘moved’ to the original nodes. In this paper we introduce a rational procedure for determining the ‘kick-off forces’ and show that the procedure used in NASTRAN violates a virtual work condition and leads to difficulties in certain warped configurations.

Field-consistency analysis of 27-noded hexahedral elements for constrained media elasticity

Most 3-dimensional stress analysis is currently performed using the 8-noded linear and the 20-noded quadratic (serendipity) hexahedral elements. Recent experience with the quadratic plate/shell elements show that the 9-noded Lagrangian elements are superior to the 8-noded serendipity elements in applications to thin plate/shell flexure. Extension of this logic suggests that a 27-noded Lagrangian hexahedral element formulation with field-consistency corrections is needed for robust and reliable 3-D modelling of constrained media problems. In this paper we examine the field-consistency aspects involved in the formulation of such an element.

Expert systems and finite element structural analysis — a review

Finite element analysis of many engineering systems is practised more as an art than as a science. It involves high level expertise (analytical as well as heuristic) regarding problem modelling (e.g. problem specification, choosing the appropriate type of elements etc.), optical mesh design for achieving the specified accuracy (e.g. initial mesh selection, adaptive mesh refinement), selection of the appropriate type of analysis and solution routines and, finally, diagnosis of the finite element solutions. Very often such expertise is highly dispersed and is not available at a single place with a single expert. The design of an expert system, such that the necessary expertise is available to a novice to perform the same job even in the absence of trained experts, becomes an attractive proposition.In this paper, the areas of finite element structural analysis which require experience and decision-making capabilities are explored. A simple expert system, with a feasible knowledge base for problem modelling, optimal mesh design, type of analysis and solution routines, and diagnosis, is outlined. Several efforts in these directions, reported in the open literature, are also reviewed in this paper.