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Nonlinear Dyn (2017) 88:1075–1091
DOI 10.1007/s11071-016-3296-x
ORIGINAL PAPER
Higher-order beam elements based on the absolute nodal
coordinate formulation for three-dimensional elasticity
Henrik Ebel ·Marko K. Matikainen ·
Vesa-Ville Hurskainen ·Aki Mikkola
Received: 4 December 2015 / Accepted: 14 December 2016 / Published online: 28 December 2016
© Springer Science+Business Media Dordrecht 2016
Abstract This study thoroughly examines various
higher-order three and four-node beam elements for use
in the absolute nodal coordinate formulation (ANCF).
The paper carefully investigates which potential ben-
efits and drawbacks the utilization of higher-order
ANCF beam elements without in-slope vectors has in
the case of the usage of full three-dimensional elas-
ticity. When the elastic forces for shear-deformable
ANCF beam elements are calculated using full three-
dimensional elasticity—especially in the form of the
St. Venant–Kirchhoff material law—Poisson locking
severely deteriorates the accuracy of the numeric
results. As shown in this paper, an existing approach
to preventing this locking phenomenon for three-node
beam elements can still produce unsatisfying results
in load cases involving bidirectional bending. The
H. Ebel (B)
Institute of Engineering and Computational Mechanics,
University of Stuttgart, Pfaffenwaldring 9, 70569 Stuttgart,
Germany
e-mail: henrik.ebel@itm.uni-stuttgart.de
M. K. Matikainen ·V.-V. Hurskainen ·A. Mikkola
Mechanical Engineering, Lappeenranta University of
Technology, Skinnarilankatu 34, 53850 Lappeenranta,
Finland
e-mail: marko.matikainen@lut.fi
V.-V. Hurskainen
e-mail: vesa-ville.hurskainen@lut.fi
A. Mikkola
e-mail: aki.mikkola@lut.fi
results of this study show that enriching the polyno-
mial basis used to approximate the beam kinematics
provides a natural solution to this issue. As will be
seen, these findings for three-node elements can also
be extended to four-node elements. When using a suffi-
cient approximation order in transverse directions, sat-
isfying accuracy can be achieved both in conventional
one-dimensional bending and in the above-mentioned
bidirectional load case.
Keywords Continuum beam elements ·
Three-dimensional elasticity ·St. Venant–Kirchhoff
material ·Princeton beam experiment ·Numerical
locking
1 Introduction
The computer analysis of multibody system dynam-
ics has become increasingly important in advanced
machine system design. Increased computational power
and enhanced formulations are making it possible
to solve progressively more sophisticated mathemat-
ical models describing the dynamic behavior of com-
plex systems. Nevertheless, computational efficiency
is an important consideration for the multibody system
dynamics analyst, and the level of detail for a mechan-
ical system model should not be any greater than is
needed to yield a sufficiently accurate numerical result.
In multibody system dynamics, an acceptable simpli-
fication for the analysis of motion and forces in many
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