Masters Theses

Date of Award

12-1996

Degree Type

Thesis

Degree Name

Master of Science

Major

Engineering Science

Major Professor

Raymond D. Krieg

Committee Members

Jerry Carley, Christopher Pionke

Abstract

In the finite element method as applied to structural mechanics it has been observed that four-node quadrilateral elements (Q4's) tend to perform well when they are aligned with directions of principal stresses. This seemed to suggest that Q4's had poor behavior when placed in shear and that they would therefore perform better when they were aligned with the principal directions. It is also well known that Q4's perform best when they have near right angles at the corners and near one-to-one aspect ratios. It was speculated that the constraint on aspect ratio might be reduced if the shear were eliminated from the element. This thesis explores the possibility of using lines of principal stresses (stress trajectories) to form a finite element mesh with improved performance. No theory was found in the literature to support the claim that alignment of the element would improve performance. It was necessary to know how the accuracy of the Q4 element was affected by its orientation to the principal directions. The procedure was to define a continuum state and observe to what extent an element represented that state as the principal angle was changed. Since the Q4 element can exactly represent any constant strain field, a field with constant strain gradient was used in a theoretical study. A continuum strain field was defined by prescribing constant principal strain gradients for four strain components. Equations were derived which express the displacement field of the element in terms of the principal angle and the four possible constant strain gradients. To carry out the study, the directions of the lines of principal stresses were needed. This might have been done directly by an experimental method using principles of photoelasticity but for this study a preliminary finite element analysis was made from which the stress trajectories were interpolated. A finite element mesh then had to be formed from the stress trajectories by defining nodes and element connectivities. This led to the development of a full mesh generation program. Four different test cases for the method were studied: a simply supported beam with center load, a plate with a hole in tension, a disc with diametral loading, and a cantilevered beam with a distributed load. In each case the problem was initially analyzed with an ordinary mesh and the results used to produce an aligned mesh. Some difficulties were found with creating a mesh which is aligned with stress trajectories. These included the problems of excessive variation in spacing of the lines over the body and the occurance of three- and five-node elements. Also, the effects of nonrectangular element shapes was not fully explored. A convergence study on the ordinary mesh was done by mesh refinement. Three significant values were compared: total strain energy, maximum displacement and maximum stress. The aligned meshes showed no significant improvement in performance in any of the cases.

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