The Three Dimensional Gradient Deficient Beam Element (BEAM9) Using the Absolute Nodal Coordinate Formulation

2014 ◽  
Author(s):  
Abdel-Nasser A. Mohamed ◽  
Jeff Liu
Keyword(s):  
Absolute Nodal Coordinate Formulation ◽  
Three Dimensional ◽  
Beam Element ◽  
Elastic Force ◽  
Nonlinear Material ◽  
Force Model ◽  
Shear Deformations ◽  
Elastic Line ◽  
Absolute Nodal Coordinate ◽  
The Absolute

In this investigation, a three dimensional gradient deficient beam element (BEAM9) using the absolute nodal coordinate formulation (ANCF) is introduced. This element has nine coordinates per node, this includes the position vector and the two gradient vectors rx and ry. Like most of the ANCF elements, this element has constant mass matrix and zero centrifugal and Coriolis inertia forces. The plane strain elastic force model and the elastic line approach are two elastic force models presented in this paper in order to simulate the element internal resistance. Both models support resistance to the general bending and twist moments. The possibilities of employing nonlinear material models will be discussed in future work. Furthermore, the proposed element has the advantage of easy integration over general cross section area that is not easy to perform using the fully parameterized ANCF beam element (BEAM12). Comparing to the ANCF cable element (BEAM6), the proposed element can resist general bending and twist loads. Moreover, shear deformations in the xy plane due to shear force and in the yz plane due to twist moment are considered with the gradient deficient beam element proposed in this work. However, no shear deformations are considered with the ANCF cable element. Comparing to the fully parameterized ANCF beam element, the gradient deficient beam element (BEAM9) avoids some locking issues, shows better computational efficiency and offers better convergency characteristics. Numerical examples are presented in order to validate the proposed gradient deficient beam element and to compare with other ANCF beam elements.

Improvement on Evaluating Axial Elastic Force in Bernoulli-Euler Beam Based on the Absolute Nodal Coordinate Formulation by Accurate Mean Axial Strain Measure

2011 ◽  
Author(s):  
Tsubasa Wago ◽  
Nobuyuki Kobayashi ◽  
Yoshiki Sugawara
Keyword(s):  
Absolute Nodal Coordinate Formulation ◽  
Axial Strain ◽  
Computing Time ◽  
Beam Element ◽  
Elastic Force ◽  
Flexible Beam ◽  
Euler Beam ◽  
Absolute Nodal Coordinate ◽  
The Mean ◽  
The Absolute

This paper presents an improved formulation of axial elastic force in three-dimensional Bernoulli-Euler beam element based on the absolute nodal coordinate formulation. An accurate measure of mean axial strain for evaluating the axial elastic forces characterizes the presented formulation. The presented formulation evaluates the mean axial strain accurately by calculating the length of deformed beam element along its neutral axis. A comparison of the conventional formulations of the axial elastic force and the presented formulation is performed in some numerical examples which contain large bending deformation of flexible beam. As a result, it is verified that the presented formulation can express large deformation accurately with smaller number of elements than the conventional formulation which calculates the mean axial strain with straight-line distance between both element nodes. Moreover, it is also verified that the presented formulation can avoid excessive increase in computing time to simulate the dynamic behavior of flexible beam.

Studies on the Stiffness Properties of the Absolute Nodal Coordinate Formulation for Three-Dimensional Beams

2003 ◽  
Author(s):  
Jussi T. Sopanen ◽  
Aki M. Mikkola
Keyword(s):  
Cross Section ◽  
Absolute Nodal Coordinate Formulation ◽  
Three Dimensional ◽  
Beam Element ◽  
Element Formulation ◽  
Absolute Nodal Coordinate ◽  
Numerical Examples ◽  
Elastic Forces ◽  
Nodal Coordinates ◽  
The Absolute

The objective of this study is to investigate the accuracy of elastic force models that can be used in the absolute nodal coordinate finite element formulation for the analysis of threedimensional beams. The elastic forces of the absolute nodal coordinate formulation can be derived using a continuum mechanics approach. This study investigates the accuracy and usability of such an approach for the three-dimensional absolute nodal coordinate beam element. This study also presents an improvement proposal for the use of a continuum mechanics approach in deriving the expression of the elastic forces of the beam element. The improvement proposal is verified using several numerical examples. Numerical examples show that the proposed elastic force model of the beam element agrees with analytical results as well as with solutions obtained using existing finite element formulation. The results also imply that the beam element does not suffer from the phenomenon called shear locking. In the beam element under investigation, global displacements and slopes are used as the nodal coordinates, which resulted in a large number of nodal degrees of freedom. This study provides a physical interpretation of the nodal coordinates used in the absolute nodal coordinate beam element. It is shown that the beam element based on the absolute nodal coordinate formulation relaxes the assumption of the rigid cross-section and is capable of representing a distortional deformation of the cross-section.

Performance of Curved Beam Elements Using the Absolute Nodal Coordinate Formulation

2009 ◽  
Author(s):  
Hiroyuki Sugiyama ◽  
Hirohisa Koyama ◽  
Hiroki Yamashita
Keyword(s):  
Absolute Nodal Coordinate Formulation ◽  
Beam Element ◽  
Solution Procedure ◽  
Large Displacement ◽  
Curved Beam ◽  
Absolute Nodal Coordinate ◽  
Curved Beam Element ◽  
Strain Components ◽  
The Absolute ◽  
Longitudinal Stretch

In this investigation, a gradient deficient beam element of the absolute nodal coordinate formulation is generalized to a curved beam for the analysis of multibody systems and the performance of the proposed element is discussed by comparing with the fully parameterized curved beam element and the classical large displacement beam element with incremental solution procedures. Strain components are defined with respect to the initially curved configuration and described by the arc-length coordinate. The Green strain is used for the longitudinal stretch, while the material measure of curvature is used for bending. It is shown that strains of the curved beam can be expressed with respect to those defined in the element coordinate system using the gradient transformation and the effect of strains at the initially curved configuration is eliminated using one-dimensional Almansi strain. This property can be effectively used with non-incremental solution procedure employed for the absolute nodal coordinate formulation. Several numerical examples are presented in order to demonstrate the performance of the gradient deficient curved beam element developed in this investigation. It is shown that the use of the proposed element leads to better element convergence as compared to that of the fully parameterized element and the classical large displacement beam element with incremental solution procedures.

Comparison of Three-Dimensional Flexible Beam Elements for Dynamic Analysis: Classical Finite Element Formulation and Absolute Nodal Coordinate Formulation

2009 ◽  
Vol 5 (1) ◽  
Author(s):  
A. L. Schwab ◽  
J. P. Meijaard
Keyword(s):  
Dynamic Analysis ◽  
Absolute Nodal Coordinate Formulation ◽  
Three Dimensional ◽  
Flexible Beam ◽  
Element Formulation ◽  
Shear Locking ◽  
Elastic Line ◽  
Absolute Nodal Coordinate ◽  
Beam Elements ◽  
Bending Mode

Three formulations for a flexible spatial beam element for dynamic analysis are compared: a Timoshenko beam with large displacements and rotations, a fully parametrized element according to the absolute nodal coordinate formulation (ANCF), and an ANCF element based on an elastic line approach. In the last formulation, the shear locking of the antisymmetric bending mode is avoided by the application of either the two-field Hellinger–Reissner or the three-field Hu–Washizu variational principle. The comparison is made by means of linear static deflection and eigenfrequency analyses on stylized problems. It is shown that the ANCF fully parametrized element yields too large torsional and flexural rigidities, and shear locking effectively suppresses the antisymmetric bending mode. The presented ANCF formulation with the elastic line approach resolves most of these problems.

A New Type Component Mode Synthesis Beam Element Based on the Absolute Nodal Coordinate Formulation

2003 ◽  
Author(s):  
Riki Iwai ◽  
Nobuyuki Kobayashi
Keyword(s):  
High Efficiency ◽  
Absolute Nodal Coordinate Formulation ◽  
Mass Matrix ◽  
Synthesis Method ◽  
Beam Element ◽  
Component Mode Synthesis ◽  
Flexible Beam ◽  
Absolute Nodal Coordinate ◽  
New Type ◽  
The Absolute

This paper establishes a new type component mode synthesis method for a flexible beam element based on the absolute nodal coordinate formulation. The deformation of the beam element is defined as the sum of the global shape function and the analytical clamped-clamped beam modes. This formulation leads to a constant and symmetric mass matrix as the conventional absolute nodal coordinate formulation, and makes it possible to reduce the system coordinates of the beam structure which undergoes large rotations and large deformations. Numerical examples show that the excellent agreements are examined between the presented formulation and the conventional absolute nodal coordinate formulation. These results demonstrate that the presented formulation has high accuracy in the sense that the presented solutions are similar to the conventional ones with the less system coordinates and high efficiency in computation.

A Procedure for the Inclusion of Transverse Shear Deformation in a Beam Element Based on the Absolute Nodal Coordinate Formulation

2007 ◽  
Author(s):  
Aki Mikkola ◽  
Oleg Dmitrochenko ◽  
Marko Matikainen
Keyword(s):  
Shear Deformation ◽  
Transverse Shear ◽  
Absolute Nodal Coordinate Formulation ◽  
Beam Element ◽  
Transverse Shear Deformation ◽  
High Frequencies ◽  
Absolute Nodal Coordinate ◽  
Numerical Performance ◽  
The Absolute ◽  
Shear Deformable

In this study, a procedure to account for transverse shear deformation in the absolute nodal coordinate formulation is presented. In the absolute nodal coordinate formulation, shear deformation is usually defined by employing the slope vectors in the element transverse direction. This leads to the description of deformation modes that are, in practical problems, associated with high frequencies. These high frequencies, in turn, complicate the time integration procedure burdening numerical performance. In this study, the description of transverse shear deformation is accounted for in a two-dimensional beam element based on the absolute nodal coordinate formulation without the use of transverse slope vectors. In the introduced shear deformable beam element, slope vectors are replaced by vectors that describe the rotation of the beam cross-section. This procedure represents a simple enhancement that does not decrease the accuracy or numerical performance of elements based on the absolute nodal coordinate formulation. Numerical results are presented in order to demonstrate the accuracy of the introduced element in static and dynamic cases. The numerical results obtained using the introduced element agree with the results obtained using previously proposed shear deformable beam elements.

Shear Correction for Thin Plate Finite Elements Based on the Absolute Nodal Coordinate Formulation

2009 ◽  
Author(s):  
Oleg Dmitrochenko ◽  
Aki Mikkola
Keyword(s):  
Shear Deformation ◽  
Absolute Nodal Coordinate Formulation ◽  
Beam Element ◽  
Transverse Shear Deformation ◽  
High Frequencies ◽  
Absolute Nodal Coordinate ◽  
Plate Elements ◽  
Numerical Performance ◽  
The Absolute ◽  
Shear Deformable

This study is an extension of a newly introduced approach to account transverse shear deformation in absolute nodal coordinate formulation. In the formulation, shear deformation is usually defined by employing slope vectors in the element transverse direction. This leads to the description of deformation modes that, in practical problems, may be associated with high frequencies. These high frequencies, in turn, could complicate the time integration procedure, burdening numerical performance of shear deformable elements. In a recent study of this paper’s authors, the description of transverse shear deformation is accounted for in a two-dimensional beam element, based on the absolute nodal coordinate formulation without the use of transverse slope vectors. In the introduced shear deformable beam element, slope vectors are replaced by vectors that describe the rotation of the beam cross-section. This procedure represents a simple enhancement that does not decrease the accuracy or numerical performance of elements based on the absolute nodal coordinate formulation. In this study, the approach to account for shear deformation without using transverse slopes is implemented for a thin rectangular plate element. In fact, two new plate elements are introduced: one within conventional finite element and another using the absolute nodal coordinates. Numerical results are presented in order to demonstrate the accuracy of the introduced plate element. The numerical results obtained using the introduced element agree with the results obtained using previously proposed shear deformable plate elements.

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