FOUR-NODED TRIANGULAR FINITE ELEMENTS

Abstract eng:
The limits and extents of the isoparametric formulation for a four-noded finite element are derived by transforming the parametrized shape function from ξ −η to x−y coordinates. This analytic inversion results in a quadratic equation. The coefficients of this equation dictate whether the form of the shape function is a polynomial, a rational polynomial or an expression containing a square root. Furthermore, in x − y coordinates the C0 shape function for a triangular element with a mid-side node can be derived. Consequently, this study extends the class of elements for which closed form shape functions can be constructed. In 1975 E. L. Wachspress introduced a rational polynomial formulation for finite elements which applies consistently to any convex n-sided polygon; it does not apply to elements with a midside node. Combining these results, convergent shape functions and consistent strain matrices can be formulated for all non-concave quadrilaterals. The associated energy density function can be integrated exactly using the divergence theorem.

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