Analytical trial function method for development of new 8‐node plane element based on the variational principle containing Airy stress function

Purpose − The purpose of this paper is to propose a novel and simple strategy for construction of hybrid‐“stress function” plane element. Design/methodology/approach − First, a complementary energy functional, in which the Airy stress function is taken as the functional variable, is established within an element for analysis of plane problems. Second, 15 basic analytical solutions (in global Cartesian coordinates) of the stress function are taken as the trial functions for an 8‐node element, and meanwhile, 15 unknown constants are then introduced. Third, according to the principle of minimum complementary energy, the unknown constants can be expressed in terms of the displacements along element edges, which are interpolated by element nodal displacements. Finally, the whole system can be rewritten in terms of element nodal displacement vector. Findings − A new hybrid element stiffness matrix is obtained. The resulting 8‐node plane element, denoted as analytical trial function (ATF‐Q8), possesses excellent performance in numerical examples. Furthermore, some numerical defects, such as direction dependence and interpolation failure, are not found in present model. Originality/value − This paper presents a new strategy for developing finite element models exhibits advantages of both analytical and discrete method.