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...

[1]  Cen Song,et al.  Development of eight-node quadrilateral membrane elements using the area coordinates method , 2000 .

[2]  J. Z. Zhu,et al.  The finite element method , 1977 .

[3]  J. H. Yeo,et al.  Remedies to rotational frame dependence and interpolation failure of US‐QUAD8 element , 2007 .

[4]  Song Cen,et al.  Area co-ordinates used in quadrilateral elements , 1999 .

[5]  J. A. Stricklin,et al.  On isoparametricvs linear strain triangular elements , 1977 .

[6]  A. Stahel.  Calculus of Variations and Finite Elements , 2012 .

[7]  R. Cook,et al.  Concepts and Applications of Finite Element Analysis , 1974 .

[8]  Min-zhong Wang,et al.  General Representations of Polynomial Elastic Fields , 2012 .

[9]  Hai-chʿang Hu,et al.  Variational Principles of Theory of Elasticity with Applications , 1984 .

[10]  O. C. Zienkiewicz,et al.  The Finite Element Method: Basic Formulation and Linear Problems , 1987 .

[11]  Song Cen,et al.  Quadrilateral membrane element family formulated by the quadrilateral area coordinate method , 2007 .

[12]  K. M. Liew,et al.  A novel unsymmetric 8‐node plane element immune to mesh distortion under a quadratic displacement field , 2003 .

[13]  Ren-Hong Wang,et al.  A new 8-node quadrilateral spline finite element , 2006 .

[14]  Yuqiu Long,et al.  Generalized conforming triangular membrane element with vertex rigid rotational freedoms , 1994 .

[15]  Fumio Kikuchi,et al.  Modification of the 8-node serendipity element , 1999 .

[16]  K. Washizu,et al.  Variational Methods in Elasticity and Plasticity, 3rd Ed. , 1982 .

[17]  Polynomial stress functions of anisotropic plane problems and their applications in hybrid finite elements , 2012 .

[18]  Chenfeng Li,et al.  Shape-free finite element method: The plane hybrid stress-function (HS-F) element method for anisotropic materials , 2011 .

[19]  S. Kunimatsu,et al.  The analysis of interpolation precision of quadrilateral elements , 2004 .

[20]  Song Cen,et al.  Some basic formulae for area co-ordinates in quadrilateral elements , 1999 .

[21]  Song Cen,et al.  8- and 12-node plane hybrid stress-function elements immune to severely distorted mesh containing elements with concave shapes , 2011 .

[22]  Song Cen,et al.  A 4-node hybrid stress-function (HS-F) plane element with drilling degrees of freedom less sensitive to severe mesh distortions , 2011 .

[23]  K. Bathe,et al.  Effects of element distortions on the performance of isoparametric elements , 1993 .

[24]  R. L. Harder,et al.  A proposed standard set of problems to test finite element accuracy , 1985 .

[25]  K. Washizu.  Variational Methods in Elasticity and Plasticity , 1982 .