Basic error estimates for elliptic problems

SUMMARY An assessment of flat triangular plate bending elements with displacement degrees-of-freedom at the three comer nodes only is presented, with the purpose of identifying the most effective for thin plate analysis. Based on a review of currently available elements, specific attention is given to the theoretical and numerical evaluation of three triangular 9 degrees-of-freedom elements; namely, a discrete Kirchhoff theory (DKT) element, a hybrid stress model (HSM) element and a selective reduced integration (SRI) element. New and efficient formulations of these elements are discussed in detail and the results of several example analyses are given. It is concluded that the most efficient and reliable three-node plate bending elements are the DKT and HSM elements.

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A study of three‐node triangular plate bending elements

A simple and efficient finite element is introduced for plate bending applications. Bilinear displacement and rotation functions are employed in conjunction with selective reduced integration. Numerical examples indicate that, despite its simplicity, the element is surprisingly accurate.

A simple and efficient finite element for plate bending

This paper describes a new algorithm to generate interior nodes within any arbitrary multi-connected regions. The boundary nodes and the interior nodes are then linked up to form the best possible triangular elements by a completely revised technique in an efficient and stable manner. Owing to the generality of the central generation program, the global domain is allowed to be divided into as many irregular subdomains as desired, in order to model closely the actual physical situation. Moreover, the boundaries of the sub-domains are updated from time to time when necessary to include the possibilities of progressive refinement around a sharp corner, generating radiating mesh from a prescribed node, generating mesh between two circular arcs, etc. Despite its flexibility and capabilities, data for triangulation have been kept to a minimum by a logical input module; no connectivity information between subregions is needed, and common boundaries are defined once only. All these features have contributed to a powerful method to generate 3-node or 6-node triangular element meshes of great variety within the most irregular heterogeneous regions.

A new mesh generation scheme for arbitrary planar domains

A review of 4-node, 12 degrees-of-freedom quadrilateral elements for thin plates is presented. A new element called DKQ is discussed. The formulation is based on a generalization of the efficient and reliable triangular element DKT presented in References 1 and 2 and on the rectangular element QC presented in Reference 3. These elements are derived using the so-called discrete Krichhoff technique. A detailed numerical evaluation of the behaviour of the DKQ element for the computation of displacements and stresses for thin plate bending problems is presented and discussed. The DKQ element appears to be a simple and reliable 12 degrees-of-freedom thin plate bending element.

Evaluation of a new quadrilateral thin plate bending element

Introduction. The triangular plate bending element with the three nodal values w, wx and wv at the vertices has a particular appeal—its simplicity. But a C1 polynomial interpolation scheme defined over the whole element does not exist. To overcome this Clough and Tocher [1] resorted to subdividing the element into three subelements with the transverse displacement w interpolated individually [2] over each subtriangle so as to maintain a C1 continuity both in the interior of this complex element and on its boundaries. Bazeley et al. [3] interpolated w by some rational functions to obtain the desired variation of w and its derivatives along the sides of the element for assuring a C1 continuity of displacements. They considered also the use of elements which violate the continuity requirements (non-conforming) and for which the variational principle on minimum total potential energy does not hold any more. These elements may, nevertheless, sometimes produce a valid, stable difference scheme and converge to a useful solution. But their excessive flexibility and precarious convergence did not endear them to engineers. Severn, Taylor and Dungar [4, 5] and Allman [6] used a mixed variational principle [7] for generating a compact hybrid [8] finite element. Stricklin et al. [9] (and also others) made a more radical approach to the generation of plate bending elements in general and the nine-degrees-of-freedom triangular element in particular. They started from the basic equations of elasticity rather than from a ready plate theory and did away with the Kirchhoff assumption (the shear) except at the vertices and along the sides of the element (this is the \"discrete Kirchhoff assumption\" [10]). Since the plate is now considered a three-dimensional solid, the in-plane displacements are introduced independently of the transverse displacements and the continuity requirement for them is only C°, as in three-dimensional elasticity. Assuming that the shear energy is at any rate negligible in thin plates, Stricklin neglected it altogether. Removing the Krichhoff assumption from the finite-element analysis of plates and starting with the basic three-dimensional elasticity would seem the most natural approach to the generation of bending elements, particularily since the thin plate is obtained as a limiting case from a three-dimensional solid. However, without the a priori assumptions of Kirchhoff, and with the shear energy retained, the global stiffness matrix becomes violently ill-conditioned [11] as the thickness t of the structure is reduced. The difficulties, then, in constructing thin-plate bending elements directly from three-dimensional elasticity are of a numerical or computational nature. The decline in the conditioning of the matrix may cause, in the computational stage of the solution, grave numerical (round-off) errors, or for a computer with insufficient significant digits the matrix may even be numerically singular.

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Shok Keng Yang

Papers

Triangular, nine-degrees-of-freedom, ⁰ plate bending element of quadratic accuracy

Best Finite Elements Distribution around a Singularity