A Fundamental Study for Numerical Integration Procedures for the ANCF Consistent Rotation-Based Formulation Beam Elements

Abstract

This study presents a numerical integration technique for a shear deformable beam element based on the absolute nodal coordinate formulation (ANCF). In particular, the consistent rotation-based formulation (CRBF) is used to develop a plane ANCF beam element, namely, the ANCF/CRBF shear deformable element. Even though the ANCF/CRBF beam element introduces rotation parameters as nodal coordinates, it does not need for interpolating the rotation field. Therefore, the ANCF/CRBF formulation defines a unique rotation field, and does not suffer from a coordinate redundancy problem encountered when using the large rotation vector formulation (LRVF). On the other hand, ANCF/CRBF formulation leads to highly nonlinear inertia forces including non-zero Coriolis and centrifugal forces. The objective of this study is to develop a constrained formulation and a numerical integration technique for the ANCF/CRBF shear deformable beam element. In particular, this study focuses on the fact that the global position gradients can be expressed in terms of the finite rotation parameters. In the present method, such algebraic relations are treated as constraints, and the ANCF/CRBF beam element is formulated as the differential algebraic equations (DAEs). As a result, the present formulation ensures the constant mass matrix, however, DAE solvers have to be introduced for solving the resulting equations. In addtion, since the present method can be regarded as a kind of augmented formulation, the number of degrees of freedom is significant from a standpoint of numerical efficiency. In order to cope with increase of the degrees of freedom, this study develops size reduction techniques focusing on the algebraic expressions of constraints. Then, the results obtained from the proposed method are compared with the results obtained using the fully parameterized ANCF shear deformable element.