FEM formulation of four and five noded elements using a linearly varying stress assumption

The two-dimensional stress analysis is investigated to develop multi-noded elements that can be used in a finite element mesh. To accomplish this, a hybrid stress element is developed assuming a linear stress variation within the element. The formulation for this type of element is developed for a multi-noded polygon consisting of up to five nodes however simplifies to a constant stress element for a triangle. The formulation differs from the method developed by Pian [AIAA J. 2 (1964) 1333]. In the formulation discussed in this article, the same matrix is used to relate stress to constant stress parameters, and strain to constant strain parameters. This approximation simplifies the formulation, however does not significantly impact the results. The effectiveness of the formulation and code implementation is illustrated through “patch tests” along with other examples to assess this approximation. The linear stress element developed in this article is shown to be robust, yielding reasonable results even when a poorly designed finite element mesh is used in the cantilever beam example. In addition, the article includes a comparison between the linear stress element and the constant strain element to illustrate the advantages of the linear stress element over the constant strain element approach.