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Research Papers

Structural and Continuum Mechanics Approaches for a 3D Shear Deformable ANCF Beam Finite Element: Application to Buckling and Nonlinear Dynamic Examples

[+] Author and Article Information
Karin Nachbagauer

Johannes Kepler University Linz,
Altenbergerstraße 69,
Linz 4040, Austria
e-mail: karin.nachbagauer@jku.at;
karin.nachbagauer@fh-wels.at

Johannes Gerstmayr

Austrian Center of Competence in Mechatronics
Altenbergerstraße 69,
Linz 4040, Austria
e-mail: johannes.gerstmayr@lcm.at

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the Journal of Computational and Nonlinear Dynamics. Manuscript received December 20, 2012; final manuscript received April 11, 2013; published online October 15, 2013. Assoc. Editor: Hiroyuki Sugiyama.

J. Comput. Nonlinear Dynam 9(1), 011013 (Oct 15, 2013) (8 pages) Paper No: CND-12-1226; doi: 10.1115/1.4025282 History: Received December 20, 2012; Revised April 11, 2013

For the modeling of large deformations in multibody dynamics problems, the absolute nodal coordinate formulation (ANCF) is advantageous since in general, the ANCF leads to a constant mass matrix. The proposed ANCF beam finite elements in this approach use the transverse slope vectors for the parameterization of the orientation of the cross section and do not employ an axial nodal slope vector. The geometric description, the degrees of freedom, and a continuum-mechanics-based and a structural-mechanics-based formulation for the elastic forces of the beam finite elements, as well as their usage in several static problems, have been presented in a previous work. A comparison to results provided in the literature to analytical solution and to the solution found by commercial finite element software shows accuracy and high order convergence in statics. The main subject of the present paper is to show the usability of the beam finite elements in dynamic and buckling applications.

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References

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Nachbagauer, K., Gruber, P., and Gerstmayr, J., 2013, “Structural and Continuum Mechanics Approaches for a 3D Shear Deformable ANCF Beam Finite Element: Application to Static and Linearized Dynamic Examples,” Computat. Nonlin. Dyn., 8(2), p. 021004. [CrossRef]
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Figures

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Fig. 1

Overview of a selection of structural beam and continuum mechanics/solid beam finite elements

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Fig. 2

The geometric description of the elements is based on a position vector r(i) and two slope vectors r,η(i) and r,ζ(i) in the i-th node. These vectors are defined on a scaled and straight reference element, given in coordinates (ξ,η,ζ).

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Fig. 3

Perspective and front view of the cross section frame vectors

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Fig. 4

Geometry of the right-angle frame under linearly increasing in-plane end load F

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Fig. 5

A jump in the tip displacement in z-direction at an applied load F≅1.087 shows that the computed results match the given results in the literature quite well since the critical load in Argyris et al. [21] is given as Fcr = 1.088 N

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Fig. 6

Geometry of the right-angle frame under end moments Mz and -Mz

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Fig. 7

A jump in the tip displacement in z-direction at applied moments of magnitude M≅0.62 Nm shows that the computed results are in good accordance with given results in the literature since the critical moment in Argyris et al. [21] is given as Mcr = 0.62477 Nm

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Fig. 8

Comparison of the load-displacement curves computed with four proposed quadratic ANCF beam finite elements and the results by Argyris et al. [21, Fig. 24.6, p. 53]. Due to good convergence behavior of the proposed elements, the computations with only four elements lead to the same curves as compared to the curve in Ref. [21], which has been computed with ten beam elements.

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Fig. 10

Example 3 (model A): global position of the tip of the flexible beam discretized by four elements using the continuum-mechanics-based formulation, CMF (locking-free), compared to the results in Sugiyama and Yamashita [23, Fig. 11], which have been found by a continuum-mechanics-based formulation using a locking-compensation based on Hellinger–Reissner's principle and a discretization with four elements

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Fig. 11

Example 3 (model A): This convergence plot shows the errors of the tip positions |x-x*| listed in Table 1. As the reference value x* for computing the error, the converged solution computed with 256 quadratic CMF (locking-free) elements is taken.

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Fig. 9

Visualization of the rigid-flexible double pendulum as in Ref. [23]

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Fig. 12

Visualization of the rigid-flexible double pendulum (model B) as in Ref. [22]

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Fig. 13

Example 4 (model B): comparison of linear and quadratic ANCF beam elements with continuum-mechanics-based formulation, CMF (locking-free), compared to the results from a continuum-mechanics-based formulation in Sugiyama et al. [22, Fig. 15]

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