A triangular element based on an energy-orthogonal free formulation

Abstract

A new Kirchhoff triangular plate-bending element with nine degrees of freedom is derived. The assumed shape functions are nonconforming and consist of a complete quadratic expansion complemented by three cubic modes that are energy-orthogonal to the quadratic expansion. The derivation is based on the free formulation of Bergan and Nygård, which satisfies the conditions of invariance and the patch test. An explicit expression is obtained for the inverse transformation matrix required in this formulation, which makes the stiffness computations extremely efficient. The aspect ratio sensitivity of the element under cylindrical bending modes can be lessened by scaling the higher-order stiffness through coefficients determined by a superlinear patch test technique. Numerical experiments indicate that the new element outperforms previously derived 9-dof triangular elements based on displacement modes.