A two-dimensional shear deformable ANCF consistent rotation-based formulation beam element

Abstract

It has been demonstrated recently that the absolute nodal coordinate formulation (ANCF) can be used to develop lower-order consistent rotation-based formulations (CRBF) that employ finite rotation parameters as nodal coordinates without the need for interpolating the rotation field. The objective of this study is to develop new planar shear-deformable ANCF/CRBF beam elements and demonstrate their use. A cubic ANCF/CRBF shear deformable beam element is first developed starting with the ANCF kinematic description that employs position vector gradients as nodal coordinates. The transverse position vector gradients at the nodes are expressed in terms of finite rotation parameters, leading to a lower-dimensional beam element model that captures the shear deformation, ensures continuity of the stresses and rotations at the nodes, allows for an arbitrary large displacement, and has a kinematic description consistent with geometry methods and suitable for systematically modeling curved structures. The results, obtained using the new planar ANCF/CRBF shear deformable beam element, are compared with the original and more general ANCF shear deformable beam element. Another lower-dimension bilinear CRBF beam element which has three coordinates at each node, two translation coordinates and one rotation parameter, is also developed in this investigation. The formulations of the three finite elements, including the ANCF finite element, considered in this investigation are compared. Numerical results are presented in order to demonstrate the use of the new formulations and test their performances in the analysis of large displacements and deformations. While the ANCF/CRBF assumptions are evaluated numerically, the results obtained show, in general, a good agreement between the elements considered in this study. The results also show that the CRBF finite elements, which have nonlinear mass matrix, can be more efficient for smaller meshes. As the mesh size and the number of finite elements increase, the original higher order ANCF finite elements, which have constant mass matrix and zero Coriolis and centrifugal forces, become more efficient. The ANCF/CRBF approach clearly demonstrates that there is no need for introducing an independent rotation field to capture the shear effect.

Geometrically accurate floating frame of reference finite elements for the small deformation problem

“…The ANCF displacement field can always be written as r f x, t ð Þ ¼ S x ð Þe t ð Þ, where S is the element shape function matrix, e is the vector of element nodal coordinates, and t is time. Using this displacement field and the CRBF coordinate reduction procedure, 30,31 one can write as demonstrated in later sections using a planar example…”

“…The ANCF displacement field can always be written as rf(x,t)=S(x)e(t), where S is the element shape function matrix, e is the vector of element nodal coordinates, and t is time. Using this displacement field and the CRBF coordinate reduction procedure, 30,31 one can write as demonstrated in later sections using a planar example The vector eo defines the initial configuration, while the vector ed defines the displacements. Both terms on the right-hand side of the preceding equation are function of Jo, and therefore, they depend on the initial geometry.…”

Dynamic Analysis of Planar Rigid Multibody Systems modeled using Natural Absolute Coordinates

“…One distinguishing feature that characterizes the motion of a general multibody system is the presence of large displacements, large finite rotations, and possibly small and/or large deformations [4]. Therefore, the correct description of the rotational motion using a coordinate parameterization that is free of kinematic singularities if of fundamental importance for effectively performing dynamic simulations of complex multibody systems [71,100]. On the other hand, the mechanical deformations of the continuum bodies and of the kinematic pairs that form a general multibody system can significantly affect the resulting motion of the system itself.…”

Geometrically accurate infinitesimal-rotation spatial finite elements

“…233 (1) 182-187 preserving the geometry in the reference configuration. 12,13 Geometry and position vector gradients Three configurations are often used to describe the kinematics of continua with complex geometries. These are the straight configuration, the stress-free curved reference configuration, and the current deformed configuration described, respectively, by the parameters or coordinates 10 The position vector r of the material points can be written in the form r ¼ X þ u, where the vector u ¼…”

“…The recently proposed ANCF/CRBF elements can have a number of finite-rotation nodal coordinates equal to those of the conventional elements. 12,13 For the ANCF/CRBF elements, the nodal position gradients can be defined as…”