Formulation of the ANCF shear deformable beam element based on the elastic line approach (An efficient calculation strategy with the global and element coordinate systems)

Abstract

The absolute nodal coordinate formulation (ANCF) is one of effective ways for describing the large rotation and deformation in multibody system analysis. A distinctive feature of the ANCF is to use absolute nodal coordinates and global slopes as the element nodal coordinates. Accordingly, it gives a constant and symmetric mass matrix. On the other hand, elastic forces, which are generally expressed by highly nonlinear terms, affects computational performance. Therefore, one of significant topics in the implementation of the ANCF is to derive mathematical descriptions of the elastic forces which can be calculated efficiently. The authors proposed an efficient calculation procedure for the ANCF beam element under the assumption of the Euler-Bernoulli beam theory. Thus, this study is aimed at extending the method to the two dimensional shear deformable ANCF beam element. In particular, we introduce the method called the elastic line approach, which can avoid the shear locking problem. In the present formulation, the strain and the kinetic energies are expressed as functions of the element and the global coordinates, respectively. Then, algebraic constraints regarding the relations between the global and the element coordinate systems are introduced by means of the Lagrange's method of undetermined multipliers. Therefore, this method can be categorized into augmented formulation techniques. The equations of motions of this constrained system are derived by the Lagrange's equation. As the result, equations of motion are given by the differential algebraic equation with index-1. After that, in order to evaluate the proposed method, it is applied to the large deformation problem in the plane case.

Keywords

• flexible multibody dynamics
• differential algebraic equations
• constraint shear deformable beam
• shear locking