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

Abstract

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 moiré fringe and other experimental tests. A simple vibration problem is also discussed. Finally, the problem of a free-edge boundary is considered. In previous studies spurious stresses were calculable on such boundaries. By employing special elements in these regions the required stresses are made indentically zero.