Base force element method (BFEM) on potential energy principle for elasticity problems
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Base force element method (BFEM) on potential energy principle for elasticity problems Yijiang Peng • Zhanlong Dong • Bo Peng Yinghua Liu
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Received: 16 January 2011 / Accepted: 22 June 2011 / Published online: 6 July 2011 Ó Springer Science+Business Media, B.V. 2011
Abstract Using the concept of base forces, the problem of the explicit tensor expression of the element stiffness matrix for a new finite element method (FEM)—the base force element method (BFEM) was studied. The mathematic model of the BFEM on potential energy principle was established by using an explicit expression. A code for the BFEM was developed using MATLAB computer language. The approach was used to solve several problems of elasticity theory, and various shape elements were compared and analyzed. The numerical results are consistent with those of the theoretical solutions, and the applicability of this approach is verified. Keywords Base force element method Finite element method Stiffness matrix Potential energy principle Elasticity problem
1 Introduction An important problem in the use of the finite element method (FEM) is to define a reasonable and applicable Y. Peng (&) Z. Dong B. Peng Key Lab of Urban Security & Disaster Engineering, Beijing University of Technology, Beijing 100124, China e-mail: [email protected] Y. Liu Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China
stiffness matrix. However, even for triangular elements, the expression of stiffness matrix is rather complicated. Even more difficult are the expressions for the stiffness matrices for arbitrary polygonal or polyhedral elements. In general, the expression for the stiffness matrix of the traditional FEM is written as Bathe (1982, 1996), Cook (1981), Zienkiewcz (1977): K¼
Z
BT DBdV
ð1Þ
V
where B is the strain–displacement matrix, D is the elasticity matrix, and V is the volume of an element. The shortcoming of Eq. 1 is that, due to the complexity of the integrand in the above integration, the integration is usually carried out numerically. This will bring some loss of the precision to the calculation. Moreover, due to the difficulty of determining the shape function for arbitrary polygonal or polyhedral elements, the quadrilateral isoparametric element is often used. Gao (1999, 2003) presented the base forces concept for describing stress state at a point. By means of the base forces, the equilibrium equation, boundary condition and elastic law are written in simple form. As a result, many typical large strain problems of elasticity have been solved in recent years (Gao and Gao 2000; Gao and Chen 2001; Gao 2002, etc.). The base forces concept also showed advantages in linear elastic problems. The explicit expression for the stiffness matrix of the FEM was
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given using the base forces concept by Gao (2003), and the expression can be written as: E 2m KIJ ¼ mI mJ þ mIJ U þ mJ mI 2V ð1 þ mÞ 1 2m
The three forces acting on the front surfaces of the hexahedron are denoted by dTi , let
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