A compatible triangular element including vertex rotations for plane elasticity analysis

An improved two-node timoshenko beam finite element

Publisher Summary This chapter discusses the experience with the patch test for convergence of finite elements. The usual explanation for the astonishing popularity of finite elements is that the technique allows an engineer to specify any feasible problem geometry, and material properties. The program blocks are complex, but they are modular and readily checked. The storage and arithmetic requirements are nicely balanced for modern computers. Most of the innermost loops are such that the cost of the arithmetic can be halved by introducing, as options, only two or three very simple machine-language subroutines. The chapter reviews the origins of the patch test. For if the external nodes of any sub-assembly of a successful assembly of elements are given prescribed values corresponding to an arbitrary state of constant curvature, then the internal nodes must obediently take their correct values. An internal node is defined as one completely surrounded by elements. Conversely, if two overlapping patches can reproduce any given state of constant curvature, they should combine into a larger successful patch, provided that every external node lost is internal to one of the original patches.

Experience with the patch test for convergence of finite elements

On a hierarchy of conforming timoshenko beam elements

A refined four-noded membrane element with rotational degrees of freedom

Shear-deflection terms arise naturally in a finite beam element in bending if the stiffness matrix is obtained on the basis of stress assumption, rather than the more usual displacement assumption.If the displacement assumption is used, an erroneous addition of the shear-deflection terms to the bending terms can be made if an apparently straightforward approach is utilized.

http://journals.sagepub.com/doi/pdf/10.1243/03093247V054239

Inclusion of shear deflection in the stiffness matrix for a beam element

Stiffness and mass matrices are discussed for a plane triangle whose thickness is allowed to vary linearly between the nodes. In-plane and bending actions are considered separately, and the formulation makes use of the ‘hybrid’ approach, in which the from of the stresses is assumed inside, and on the boundary of, the triangle, and displacements additionally assumed on the boundary only. In Appendix 1 the hybrid approach is developed in detail for a simple beam element.The allowance of variable thickness carries with it a greater reliance on efficient use of computers, and an opportunity is taken in the paper to reorganize the hybrid approach to achieve this.The accuracy attainable with this new element is assessed by comparison with results obtained by other methods for a flat plate of variable thickness, simply supported along its edges. A similar plate supported at its corners, and containing holes, is also considered, and the finite-element calculations are compared with moire fringe and other ...

R T Severn

Papers

Inclusion of shear deflection in the stiffness matrix for a beam element

Triangular finite elements of variable thickness and their application to plate and shell problems