8- and 12-node plane hybrid stress-function elements immune to severely distorted mesh containing elements with concave shapes

Abstract

By simply improving the first version of hybrid stress element method proposed by Pian, several 8- and 12-node plane quadrilateral elements, which are immune to severely distorted mesh containing elements with concave shapes, are successfully developed. Firstly, instead of the stresses, the stress function ϕ is regarded as the functional variable and introduced into the complementary energy functional. Then, the fundamental analytical solutions (in global Cartesian coordinates) of ϕ are taken as the trial functions for 2D finite element models, and meanwhile, the corresponding unknown stress-function constants are introduced. Thus, the resulting stress fields must be more reasonable because both the equilibrium and the compatibility relations can be satisfied. Thirdly, by using the principle of minimum complementary energy, these unknown stress-function constants can be expressed in terms of the displacements along element boundaries, which can be interpolated directly by the element nodal displacements. Finally, the complementary energy functional can be rewritten in terms of element nodal displacement vector, and thus, the element stiffness matrix of such hybrid stress-function (HS-F) element is obtained. This technique establishes a universal frame for developing reasonable hybrid stress elements based on the principle of minimum complementary energy. And the first hybrid stress element proposed by Pian is just a special case within this frame. Following above procedure, two 8-node and two 12-node quadrilateral plane elements are constructed by employing different fundamental analytical solutions of Airy stress function. Numerical results show that, the 8-node and 12-node models can produce the exact solutions for pure bending and linear bending problems, respectively, even the element shape degenerates into triangle and concave quadrangle. Furthermore, these elements do not possess any spurious zero energy mode and rotational frame dependence.