Three Dimensional Absolute Nodal Coordinate Formulation for Beam Elements: Theory

2000 ◽  
Vol 123 (4) ◽  
pp. 606-613 ◽  
Author(s):  
Ahmed A. Shabana ◽  
Refaat Y. Yakoub
Keyword(s):  
Coordinate System ◽  
Absolute Nodal Coordinate Formulation ◽  
Mass Matrix ◽  
Local Coordinate System ◽  
Three Dimensional ◽  
Deformation Analysis ◽  
Large Element ◽  
Absolute Nodal Coordinate ◽  
Beam Elements ◽  
Highly Nonlinear

The description of a beam element by only the displacement of its centerline leads to some difficulties in the representation of the torsion and shear effects. For instance such a representation does not capture the rotation of the beam as a rigid body about its own axis. This problem was circumvented in the literature by using a local coordinate system in the incremental finite element method or by using the multibody floating frame of reference formulation. The use of such a local element coordinate system leads to a highly nonlinear expression for the inertia forces as the result of the large element rotation. In this investigation, an absolute nodal coordinate formulation is presented for the large rotation and deformation analysis of three dimensional beam elements. This formulation leads to a constant mass matrix, and as a result, the vectors of the centrifugal and Coriolis forces are identically equal to zero. The formulation presented in this paper takes into account the effect of rotary inertia, torsion and shear, and ensures continuity of the slopes as well as the rotation of the beam cross section at the nodal points. Using the proposed formulation curved beams can be systematically modeled.

Three Dimensional Absolute Nodal Coordinate Formulation for Beam Elements: Implementation and Applications

2000 ◽  
Vol 123 (4) ◽  
pp. 614-621 ◽  
Author(s):  
Refaat Y. Yakoub ◽  
Ahmed A. Shabana
Keyword(s):  
High Speed ◽  
Absolute Nodal Coordinate Formulation ◽  
Mass Matrix ◽  
Three Dimensional ◽  
Large Rotation ◽  
Absolute Nodal Coordinate ◽  
Incremental Methods ◽  
Beam Elements ◽  
High Speed Forming ◽  
The Absolute

This part of these two companion papers demonstrates the computer implementation of the absolute nodal coordinate formulation for three-dimensional beam elements. Two beam elements that relax the assumptions of Euler-Bernoulli and Timoshenko beam theories are developed. These two elements take into account the effect of rotary inertia, shear deformation and torsion, and yet they lead to a constant mass matrix. As a consequence, the Coriolis and centrifugal forces are identically equal to zero. Both beam elements use the same interpolating polynomials and have the same number of nodal coordinates. However, one of the elements has two nodes, while the other has four nodes. The results obtained using the two elements are compared with the results obtained using existing incremental methods. Unlike existing large rotation vector formulations, the results of this paper show that no special numerical integration methods need to be used in order to satisfy the principle of work and energy when the absolute nodal coordinate formulation is used. These results show that this formulation can be used in manufacturing applications such as high speed forming and extrusion problems in which the element cross section dimensions significantly change.

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.

Three-Dimensional Solid Brick Element Using Slopes in the Absolute Nodal Coordinate Formulation

2013 ◽  
Vol 9 (2) ◽  
Author(s):  
Alexander Olshevskiy ◽  
Oleg Dmitrochenko ◽  
Chang-Wan Kim
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Absolute Nodal Coordinate Formulation ◽  
Mass Matrix ◽  
Equations Of Motion ◽  
Test Problems ◽  
Absolute Nodal Coordinate ◽  
Highly Nonlinear ◽  
Brick Element ◽  
The Absolute

The present paper contributes to the field of flexible multibody systems dynamics. Two new solid finite elements employing the absolute nodal coordinate formulation are presented. In this formulation, the equations of motion contain a constant mass matrix and a vector of generalized gravity forces, but the vector of elastic forces is highly nonlinear. The proposed solid eight node brick element with 96 degrees of freedom uses translations of nodes and finite slopes as sets of nodal coordinates. The displacement field is interpolated using incomplete cubic polynomials providing the absence of shear locking effect. The use of finite slopes describes the deformed shape of the finite element more exactly and, therefore, minimizes the number of finite elements required for accurate simulations. Accuracy and convergence of the finite element is demonstrated in nonlinear test problems of statics and dynamics.

Buckling analysis of beam structure with absolute nodal coordinate formulation

2020 ◽  
pp. 095440622094711
Author(s):  
Jia Wang ◽  
Tengfei Wang
Keyword(s):  
Absolute Nodal Coordinate Formulation ◽  
Nonlinear Function ◽  
Shear Stiffness ◽  
Buckling Analysis ◽  
Absolute Nodal Coordinate ◽  
Tangent Stiffness Matrix ◽  
Beam Elements ◽  
Highly Nonlinear ◽  
Analysis Scheme ◽  
Nodal Coordinates

Absolute nodal coordinate formulation (ANCF) was applied to the buckling analysis. A delicate analysis scheme based on dichotomy method was proposed to solve the buckling problem with beam elements whose tangent stiffness matrix is a highly nonlinear function of nodal coordinates. Three existing planar beam elements are employed to show the application. The accuracy and capability of the ANCF beam for buckling analysis was validated with benchmark cases. Additionally, the influence of the shear effect on the buckling load is thoroughly investigated by comparing the solutions associated with different shear stiffness and slenderness.

Dynamic Analysis of Offshore Oil Pipe Installation Using the Absolute Nodal Coordinate Formulation

2013 ◽  
Author(s):  
Jimmy D. Nielsen ◽  
Søren B. Madsen ◽  
Per Hyldahl ◽  
Ole Balling
Keyword(s):  
Dynamic Analysis ◽  
Large Deformation ◽  
Dynamic Behavior ◽  
Absolute Nodal Coordinate Formulation ◽  
Mass Matrix ◽  
Absolute Nodal Coordinate ◽  
Beam Elements ◽  
The Absolute ◽  
External Forces ◽  
Offshore Oil

The Absolute Nodal Coordinate Formulation (ANCF) has shown promising results in dynamic analysis of structures that undergo large deformation. The method relaxes the assumption of infinitesimal rotations. Being based in a fixed inertial reference frame leads to a constant mass matrix and zero centrifugal and Coriolis forces [12]. This makes the method attractive for multibody dynamics implementation. The focus in this paper is the application of ANCF beam elements and their performance on large deformation dynamic analysis. Large dynamic deformation is characteristic for the installation process of offshore submerged oil pipes using oceangoing vessels. In this investigation such an oil pipe is modeled using ANCF beam elements to simulate the dynamic behavior of the pipe during the installation process. Multiple physical effects such as gravity, buoyancy, seabed contact, and fluid damping, are included to mimic the external forces acting on the pipe during installation. The scope of this investigation is to demonstrate the ability using the ANCF to analyze the dynamic behavior of an offshore oil pipe during installation.

Use of the Finite Element Absolute Nodal Coordinate Formulation in Modeling Slope Discontinuity

2003 ◽  
Vol 125 (2) ◽  
pp. 342-350 ◽  
Author(s):  
Ahmed A. Shabana ◽  
Aki M. Mikkola
Keyword(s):  
Finite Element ◽  
Rigid Body ◽  
Coordinate System ◽  
Absolute Nodal Coordinate Formulation ◽  
Mass Matrix ◽  
Global Coordinate System ◽  
Absolute Nodal Coordinate ◽  
Coriolis Forces ◽  
The Absolute ◽  
Slope Discontinuities

A large rigid body rotation of a finite element can be described by rotating the axes of the element coordinate system or by keeping the axes unchanged and change the slopes or the position vector gradients. In the first method, the definition of the local element parameters (spatial coordinates) changes with respect to a body or a global coordinate system. The use of this method will always lead to a nonlinear mass matrix and non-zero centrifugal and Coriolis forces. The second method, in which the axes of the element coordinate system do not rotate with respect to the body or the global coordinate system, leads to a constant mass matrix and zero centrifugal and Coriolis forces when the absolute nodal coordinate formulation is used. This important property remains in effect even in the case of flexible bodies with slope discontinuities. The concept employed to accomplish this goal resembles the concept of the intermediate element coordinate system previously adopted in the finite element floating frame of reference formulation. It is shown in this paper that the absolute nodal coordinate formulation that leads to exact representation of the rigid body dynamics can be effectively used in the analysis of complex structures with slope discontinuities. The analysis presented in this paper also demonstrates that objectivity is not an issue when the absolute nodal coordinate formulation is used due to the fact that this formulation automatically accounts for the proper coordinate transformations.

Modeling of Slope Discontinuities in Flexible Body Dynamics Using the Finite Element Method

2002 ◽  
Author(s):  
Ahmed A. Shabana ◽  
Aki M. Mikkola
Keyword(s):  
Finite Element ◽  
Coordinate System ◽  
Absolute Nodal Coordinate Formulation ◽  
Mass Matrix ◽  
Global Coordinate System ◽  
Absolute Nodal Coordinate ◽  
Coriolis Forces ◽  
Body Dynamics ◽  
Definition Of ◽  
Slope Discontinuities

A large rigid body rotation of a finite element can be described by changing the definition of the axes of the element coordinate system or by keeping the axes unchanged and change the slopes or the position vector gradients. In the first method, the definition of the local element parameters (spatial coordinates) changes with respect to a body or a global coordinate system. The use of this method will always lead to a nonlinear mass matrix and non-zero centrifugal and Coriolis forces. The second method, in which the axes of the element coordinate system do not rotate with respect to the body or the global coordinate system, leads to a constant mass matrix and zero centrifugal and Coriolis forces when the absolute nodal coordinate formulation is used. This important property remains in effect even in the case of flexible bodies with slope discontinuities. The concept employed to accomplish this goal resembles the concept of the intermediate element coordinate system previously adopted in the finite element floating frame of reference formulation. It is shown in this paper that the absolute nodal coordinate formulation that leads to exact representation of the rigid body dynamics can be effectively used in the analysis of complex structures with slope discontinuities.

A New Plate Element Based on the Absolute Nodal Coordinate Formulation

2001 ◽  
Author(s):  
Aki M. Mikkola ◽  
Ahmed A. Shabana
Keyword(s):  
Absolute Nodal Coordinate Formulation ◽  
Mass Matrix ◽  
Body Motion ◽  
Deformation Analysis ◽  
Plate Element ◽  
Large Rotation ◽  
Shell Elements ◽  
Absolute Nodal Coordinate ◽  
Plates And Shells ◽  
The Absolute

Abstract In this investigation, a method for the finite rotation and large deformation analysis of plates is presented. The method, which is based on the absolute nodal coordinate formulation, leads to a plate element capable of representing exact rigid body motion. In this method, continuity conditions on all the displacement gradients are imposed. Therefore, non-smoothness of the plate mid-surface at the nodal points is avoided. By developing such a plate element, a constant mass matrix is obtained, and as a consequence, the centrifugal and Coriolis forces are equal to zero. Generalization of the formulation to the case of shell elements is discussed. Numerical results are presented in order to demonstrate the use of the proposed method in the large rotation and deformation analysis of plates and shells.

A Numerical Approach to Solving Flexible Multibody Systems Using the Absolute Nodal Coordinate Formulation

1999 ◽  
Author(s):  
R. Y. Yakoub ◽  
A. A. Shabana
Keyword(s):  
Absolute Nodal Coordinate Formulation ◽  
Sparse Matrix ◽  
Mass Matrix ◽  
Numerical Approach ◽  
Absolute Nodal Coordinates ◽  
Absolute Nodal Coordinate ◽  
Velocity Transformation ◽  
Flexible Multibody ◽  
Nodal Coordinates ◽  
The Absolute

Abstract By utilizing the fact that the absolute nodal coordinate formulation leads to a constant mass matrix, a Cholesky decomposition of the mass matrix can be used to obtain a constant velocity transformation matrix. This velocity transformation can be used to express the absolute nodal coordinates in terms of the generalized Cholesky coordinates. In this case, the inertia matrix associated with the Cholesky coordinates is the identity matrix, and therefore, an optimum sparse matrix structure can be obtained for the augmented multibody equations of motions. The implementation of a computer procedure based on the absolute nodal coordinate formulation and Cholesky coordinates is discussed in this paper. A flexible four-bar linkage is presented in this paper in order to demonstrate the use of Cholesky coordinates in the simulation of the small and large deformations in flexible multibody applications. The results obtained from the absolute nodal coordinate formulation are compared to those obtained from the floating frame of reference formulation.

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