Finite element analysis of stress concentrations and singularities

Abstract

Stress concentrations or singularities occur in a structure whenever there is a sharp change in geometry, or discontinuities in loading or material properties. In the analysis of problems with significant concentrations, the effectiveness of the finite element method can be increased by using (1) elements which can conform closely to the peripheral geometries, and (2) special shape functions in the elements in which stress gradients are large.

For plates with circular holes, truncated sector elements are the obvious choice for simulating the geometry. In this thesis, we develop three sector elements, two for plate extension and one for plate bending. Direct and natural stiffness methods were used to derive element stiffness matrices for extensional elements, and a comparative study is included. The effectiveness of sector elements for study of stresses around circular holes is brought out through a series of annular plate problems.

The ability of the finite element method to determine the stress field around a stress concentration or singularity is enhanced by proper choice of the element and its shape functions. Recognizing this fact, we have developed special “primary elements” whose kinematic description includes terms relevant to the nature of the local concentration. The selection of such shape functions is facilitated by a local continuum solution. In any given problem, a large region enclosing the source of stress concentration or singularity is treated as a primary element. The rest of the region is covered with simple conventional elements, which we designate as “secondary elements.” In effect, the primary element is equivalent to a large stack of conventional finite elements in the region of stress concentration or singularity so that the orders of matrices are significantly reduced and it is possible to achieve high orders of accuracy with small computational effort.

In this thesis, primary elements are developed for the analysis of stresses around circular holes and stress singularities at cracks in homogeneous and compound plates in extension. A series of examples demonstrate the effectiveness of this method, in which special primary elements are hybridized with conventional secondary elements.